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rst t t rtt r

  1. ✐✈ ❘✐♥❣r❛③✐❛♠❡♥t✐ ❉❡s✐❞❡r♦ r✐♥❣r❛③✐❛r❡ ✐❧ Pr♦❢❡ss♦r ❆❧❡ss❛♥❞r♦ ❇♦tt❛r♦ ♣❡r ❛✈❡r♠✐ ❞❛t♦ ❧✬♦❝❝❛s✐♦♥❡ ❞✐ ❧❛✈♦r❛r❡ ❛ q✉❡st❛ t❡s✐ ❡ ♣❡r ❛✈❡r♠✐ ♦✛❡rt♦ ♠♦❧t❡ ♦♣♣♦rt✉♥✐tà ♣❡r ✐❧ ❢✉t✉r♦✳ ❘✐♥❣r❛③✐♦ ✐❧ ♠✐♦ ❝♦rr❡❧❛t♦r❡✱ ❧✬■♥❣❡❣♥❡r ❊❞♦❛r❞♦ ❆❧✐♥♦✈✐✳ ❊✬ s❡♠♣r❡ st❛t♦ ❞✐s♣♦♥✐❜✐❧❡ ❡ ♠✐ ❤❛ ❞❛t♦ ❝♦♥s✐❣❧✐ ✉t✐❧✐ ❞✐ ❝✉✐ ❢❛rò t❡s♦r♦✱ ❣r❛③✐❡✳ ❘✐♥❣r❛③✐♦ t✉tt✐ ✐ ♠✐❡✐ ❛♠✐❝✐✱ ❝❤❡ ♠✐ s♦♥♦ st❛t✐ ✈✐❝✐♥✐ ❡ s♦♥♦ r✐✉s❝✐t✐ ❛ r❛❧❧❡❣r❛r❡ ❧❡ ❣✐♦r♥❛t❡ ♣❡❣❣✐♦r✐✳ ❘✐♥❣r❛③✐♦ s♦♣r❛tt✉ttt♦ ❈❧❡❧✐❛✱ ❝❤❡ è s❡♠♣r❡ st❛t❛ ❛❧ ♠✐♦ ✜❛♥❝♦✱ ♠✐ ❤❛ ❝♦♥❢♦rt❛t♦ ♥❡✐ ♣❡r✐♦❞✐ ❞✐✣❝✐❧✐✱ ❡❞ è r✐✉s❝✐t❛ ❛ ❢❛r♠✐ ✈❡❞❡r❡ ✐❧ ❧❛t♦ ♣♦s✐t✐✈♦ ❞❡❧❧❡ ❝♦s❡ ❛♥❝❤❡ q✉❛♥❞♦ ✐♦ ♥♦♥ r✐✉s❝✐✈♦ ❛ tr♦✈❛r❧♦✳ ■♥✜♥❡✱ ✉♥ r✐♥❣r❛③✐❛♠❡♥t♦ s♣❡❝✐❛❧❡ ✈❛ ❛✐ ♠✐❡✐ ❣❡♥✐t♦r✐ ❙❛✉r❛ ❡ ❙❡❜❛✱ ❝❤❡ ♠✐ s♦♥♦ st❛t✐ ✈✐❝✐♥✐ ♥❡✐ ♠♦♠❡♥t✐ ❞✐✣❝✐❧✐ ❞✐ q✉❡st✐ ❛♥♥✐✳

  2. ❈♦♥t❡♥ts ✶ ■♥tr♦❞✉❝t✐♦♥ ✶ ✶✳✶ ❉r❛❣ r❡❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❚❤❡ r✐❜❧❡ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❚❤❡ ❢r❛❝t❛❧ ❣❡♦♠❡tr✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❆✐♠ ♦❢ t❤❡ ✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹✳✶ ❈❛s❡ ♦❢ st✉❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹✳✷ ❉❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷ ▼✐❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠ ✶✷ ✷✳✶ ❱✐s❝♦✉s s✉❜❧❛②❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ▼❛t❤❡♠❡t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷✳✶ ❙t♦❦❡s ❡q✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷✳✷ ❉❡❝♦✉♣❧✐♥❣ ♦❢ t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ Pr♦tr✉s✐♦♥ ❤❡✐❣❤ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✹ ❇♦✉♥❞❛r② ✐♥t❡❣r❛❧ ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹✳✶ ❇■❊ ❢♦r t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹✳✷ ❇■❊ ❢♦r t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺ ❇♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✺✳✶ ❇♦✉♥❞❛r② ❞✐s❝r❡t✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✺✳✷ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② q✉❛♥t✐t✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✺✳✸ ❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✺✳✹ ▼❛t❧❛❜ ❛❧❣♦r✐t❤♠ ❞❡s❝r✐♣t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✻ ❱❛❧✐❞❛t✐♦♥ ♦❢ t❤❡ ❝♦❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✼ ❘❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✈

  3. ❈❖◆❚❊◆❚❙ ✈✐ ✷✳✼✳✶ ❈♦♥str✉❝t❡❞ ❣❡♦♠❡tr✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ 120 ◦ tr✐❛♥❣❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼✳✷ ✹✹ 90 ◦ tr✐❛♥❣❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼✳✸ ✺✺ ✸ ▼❛❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠ ✻✹ ✸✳✶ ◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✶✳✶ ❋✐♥✐t❡ ✈♦❧✉♠❡ ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✶✳✷ ❋r❛❝t✐♦♥❛❧ st❡♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✸✳✷ ❈♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥ ❛♥❞ ❣r✐❞ s♣❛❝✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✸✳✸ ◆❛✈✐❡r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✹ ❱❛❧✐❞❛t✐♦♥ ♦❢ t❤❡ ❝♦❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✸✳✺ ❘❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹ ❈♦♥❝❧✉s✐♦♥ ✽✽ ✹✳✶ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✹✳✷ ❋✉t✉r❡ ❞❡✈❡❧♦♣❡♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶

  4. ❈❤❛♣t❡r ✶ ■♥tr♦❞✉❝t✐♦♥ ✶

  5. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✷ ✶✳✶ ❉r❛❣ r❡❞✉❝t✐♦♥ ❚r❛♥s♣♦rt ✐s t❤❡ ♠♦✈❡♠❡♥t ♦❢ ♣❡♦♣❧❡✱ ❛♥✐♠❛❧s ❛♥❞ ❣♦♦❞s ❢r♦♠ ♦♥❡ ❧♦❝❛t✐♦♥ t♦ ❛♥✲ ♦t❤❡r✱ ❛♥❞ ✐t ✐s ✐♠♣♦rt❛♥t ❜❡❝❛✉s❡ ✐t ❡♥❛❜❧❡s tr❛❞❡ ❜❡t✇❡❡♥ ♣❡♦♣❧❡✱ ✇❤✐❝❤ ✐s ❡ss❡♥t✐❛❧ ❢♦r t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❝✐✈✐❧✐③❛t✐♦♥s✳ ❯♥❢♦rt✉♥❛t❡❧② ♣❡r❢♦r♠✐♥❣ ✐ts ❢✉♥❞❛♠❡♥t❛❧ r♦❧❡✱ tr❛♥s♣♦rt❛t✐♦♥ ♣r♦❞✉❝❡s ❛ ❧♦t ♦❢ ❛✐r ♣♦❧❧✉t✐♦♥✱ ✐t ✐♠♣♦✈❡r✐s❤❡s ❧♦❝❛❧ ❛✐r q✉❛❧✐t②✱ ❝❛✉s❡s ❛❝✐❞✐✜❝❛t✐♦♥ ❛♥❞ ✐t ✐s ❛ ♠❛❥♦r ❡♠✐tt❡r ♦❢ CO 2 ✳ ❚❤❡ ❧❛st r❡♣♦rt ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❊♥✲ ❡r❣② ❆❣❡♥❝② ❛♥❞ t❤❡ ■♥t❡r♥❛t✐♦♥❧ ❯♥✐♦♥ ♦❢ ❘❛✐❧✇❛②s ♣♦✐♥t ♦✉t t❤❛t t❤❡ 23 , 1% ♦❢ t❤❡ CO 2 ✇♦r❧❞ ❡♠✐ss✐♦♥ ✐♥ ✷✵✶✷ ✇❛s ❞✉❡ t♦ t❤❡ ✇❤♦❧❡ s❡❝t♦r ♦❢ tr❛♥s♣♦rt❛t✐♦♥ ✭❩❛♠❜✐♥✐ ❬✷✽❪✮✳ ❆❧t❤♦✉❣❤ t❤❡ t❡❝❤♥♦❧♦❣✐❡s ❡♠♣❧♦②❡❞ ✐♥ t❤✐s s❡❝t♦r ❣r♦✇ ✉♣ ❡①♣♦♥❡♥t✐❛❧❧②✱ t❤❡ ♥❡❣❛t✐✈❡ ❡✛❡❝ts ♦♥ t❤❡ ❛♠❜✐❡♥t st✐❧❧ ✐♥❝r❡❛s❡❀ ❢♦r ❡①❛♠♣❧❡✱ ✐♥ ✷✵✶✹ t❤❡ ✇❤♦❧❡ ❡♠✐ss✐♦♥ ♦❢ t❤❡ tr❛♥s♣♦rt❛t✐♦♥ s❡❝t♦r ✐♥ t❤❡ ❊✉r♦♣❡❛♥ ❯♥✐♦♥ ✐♥❝r❡❛s❡ ♦❢ ✷✵ ✪ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✶✾✾✵✳ ❋♦r t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❛♥❞ ❡✈❡♥ ❡❝♦♥♦♠✐❝❛❧ r❡❛s♦♥s✱ t❤❡ ❢✉❡❧ ❝♦♥s✉♠♣t✐♦♥ r❡❞✉❝t✐♦♥s ✐s ♣✉rs✉✐t❡❞ ❢r♦♠ ②❡❛rs ❜② ❛❧❧ t❤❡ ❝♦♠♣❛♥✐❡s ✐♥✈♦❧✈❡❞ ✐♥ t❤✐s s❡❝t♦r✳ ❚❤❡ t❤r✉st ✐♥ t❤✐s ✇❛② ❤❛s ❜r♦✉❣❤t ❛ ❧♦t ♦❢ ✐♥♥♦✈❛t✐✈❡ s♦❧✉t✐♦♥s✱ ✉♣♦♥ t♦ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❡❧❡❝tr✐❝ ❝❛r✱ t❤❛t ❝❡rt❛✐♥❧② ✇✐❧❧ ❜❡ t❤❡ ❢✉t✉r❡ ♦❢ tr❛♥s♣♦rt❛t✐♦♥✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ❛ ❧♦t ♦❢ ♦t❤❡r ♠❡❝❤❛♥✐s♠s t❤❛t ❛❧❧♦✇ t♦ ♦❜t❛✐♥ ❛ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❢✉❡❧ ❝♦♥s✉♠♣t✐♦♥✱ ❛♥❞ ❛♠♦♥❣ t❤❡s❡ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✜♥❞ t❤❡ ✈✐s❝♦✉s ❞r❛❣ r❡❞✉❝t✐♦♥✳ ❙❦✐♥ ❢r✐❝t✐♦♥ ♦r ✈✐s❝♦✉s ❞r❛❣ ✐s ❝✉rr❡♥t❧② ❝♦♥s✐❞❡r❡❞ t❤❡ ♠❛❥♦r ❜❛rr✐❡r t♦ t❤❡ ❢✉r✲ t❤❡r ♦♣t✐♠✐③❛t✐♦♥ ♦❢ ♠♦st ❛❡r♦❞②♥❛♠✐❝ ❛♥❞ ❤②❞r♦❞②♥❛♠✐❝ ❜♦❞✐❡s✳ ❋♦r ✈❛r✐♦✉s r❡❛s♦♥s✱ s❡✈❡r❛❧ ❝❧❛ss❡s ♦❢ tr❛♥s♣♦rt✱ ♥♦t❛❜❧② ❛✉t♦♠♦❜✐❧❡s✱ tr✉❝❦s✱ ❛♥❞ ❤❡❧✐❝♦♣t❡rs✱ ❛r❡ st✐❧❧ ✐♠✲ ♣❛❝t❡❞ ❜② ♣r❡ss✉r❡ ❞r❛❣ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡s❡ ❞❡✈✐❝❡s ✇♦✉❧❞ ♥♦t ❜❡♥❡✜t ♠❛t❡r✐❛❧❧② ❢r♦♠ ✈✐s❝♦✉s ❞r❛❣ r❡❞✉❝t✐♦♥ ❛t t❤❡ ♣r❡s❡♥t t✐♠❡✱ s✐♥❝❡ ✈✐s❝♦✉s ❞r❛❣ ✐s ❛ s♠❛❧❧ ♣♦rt✐♦♥ ♦❢ t❤❡✐r ♦✈❡r❛❧❧ ❞r❛❣ ❜✉❞❣❡t✳ ■♥st❡❛❞✱ t❤❡ s❦✐♥ ❢r✐❝t✐♦♥ r❡❞✉❝t✐♦♥ ✐s ♦❢ ✐♥t❡r❡st ❢♦r s✉❝❤ ❛♣♣❧✐✲ ❝❛t✐♦♥s ❛s✿ ❤✐❣❤✲s♣❡❡❞ ❛✐r❝r❛❢t✱ ❛❡r♦s♣❛❝❡ ♣❧❛♥❡s ✭♦r❞❡r ♦❢ ✸✵✲✹✵ ✪ s❦✐♥✲❢r✐❝t✐♦♥ ❞r❛❣✮✱ tr❛♥s♣♦rt ❛✐r❝r❛❢t✱ s❤✐♣s ♦♣❡r❛t❡❞ ❛t ❧♦✇ ❋r♦✉❞❡ ♥✉♠❜❡r ✭♦r❞❡r ♦❢ ✺✵ ✪ s❦✐♥✲❢r✐❝t✐♦♥ ❞r❛❣✮✱ ♠♦st ✉♥❞❡r✇❛t❡r ❜♦❞✐❡s ✭✼✵ ✪ ♦r ❣r❡❛t❡r s❦✐♥✲❢r✐❝t✐♦♥ ❞r❛❣✮ ✭❇✉s❤♥❡❧❧ ❬✽❪✮✳ ■♥ t❤❡ q✉❡st ❢♦r ❞r❛❣ r❡❞✉❝t✐♦♥ t❤r❡❡ ♠❡t❤♦❞s s❡❡♠ t♦ ❜❡ t❤❡ ♠❛❥♦r ❝♦♥t❡♥❞❡rs t♦❞❛② ✭▲✉❝❤✐♥✐ ❡t ❛❧✳❬✷✵❪✮✿ • ❉❡❧❛②✐♥❣ t❤❡ s❡♣❛r❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ❜② tr✐❣❣❡r✐♥❣ ❛♥ ❡❛r❧② tr❛♥s✐t✐♦♥ t♦ t✉r❜✉❧❡♥t ✢♦✇✳

  6. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✸ • ❆✛❡❝t✐♥❣ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ❜② ✐♥❥❡❝t✐♦♥ ♦r s✉❝t✐♦♥ ♦❢ ✢✉✐❞✳ • ▼♦❞✐❢②✐♥❣ t❤❡ ✈✐s❝♦s✐t② ♦❢ t❤❡ ✢✉✐❞ ✐♥ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ❜② ✐♥❥❡❝t✐♦♥ ♦❢ ❛ s✉✐t❛❜❧❡ ❞✐✛❡r❡♥t ✢✉✐❞ ♦r ❜② ❝❤❛♥❣✐♥❣ ✐ts t❡♠♣❡r❛t✉r❡✳ • ❚❡①t✉r✐♥❣ t❤❡ ✇❛❧❧ ✇✐t❤ ♠✐❝r♦✲❣r♦♦✈❡s ✭✇✐t❤ ❛ ❞✐♠❡♥s✐♦♥ ♦❢ ❛❜♦✉t 100 µm ✮ ❛❧✐❣♥❡❞ ❛❧♦♥❣ t❤❡ ♠❛✐♥ ✢♦✇ ❞✐r❡❝t✐♦♥✳ ❙✉❝❤ ❛ ❣r♦♦✈❡s t❛❦❡ t❤❡ ♥❛♠❡ ♦❢ r✐❜❧❡ts✳ ❚❤❡ ✜rst ♦❢ t❤❡s❡ ♠❡t❤♦❞s ✐s t❡❝❤♥♦❧♦❣✐❝❛❧❧② t❤❡ ♠♦st ❛❞✈❛♥❝❡❞✱ ❛♥❞ ✉s❡❞ ❢♦r ❛ ❧♦♥❣ t✐♠❡ ✐♥ ❛✐r♣❧❛♥❡ ✇✐♥❣s ✐♥ ❛t ❧❡❛st s♦♠❡ ❢♦r♠❀ t❤❡ s❡❝♦♥❞ ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ♠❛✐♥❧② ❢♦r ✐♥t❡r♥❛❧ ✢♦✇s✱ s✉❝❤ ❛s t❤❡ tr❛♥s♣♦rt❛t✐♦♥ ♦❢ ✈❡r② ✈✐s❝♦✉s ✢✉✐❞s ✐♥ ❧♦♥❣ ♣✐♣❡❧✐♥❡s ✭Pr❡③✐♦s✐ ❡t ❛❧✳ ❬✶✻❪✮✳ ■ts ❛♣♣❧✐❝❛t✐♦♥ t♦ ❡①t❡r♥❛❧ ✢♦✇s ❜❡✐♥❣ ❝♦♥s✐❞❡r❡❞ t♦♦ ❡①♣❛♥s✐✈❡✳ ❚❤❡ ❢♦✉rt❤ ♠❡t❤♦❞✱ ✐s ✈❡r② ❛♣♣❡❛❧✐♥❣ ❜❡❝❛✉s❡ ♦❢ ✐ts ❝♦♠♣❧❡t❡❧② ♣❛ss✐✈❡ ♥❛t✉r❡✱ ❜✉t ✐ts ♠❡❝❤❛♥✐s♠ ♦❢ ♦♣❡r❛t✐♦♥ ✐s ♥♦t ✇❡❧❧ ✉♥❞❡rst♦♦❞✱ ❛♥❞ ✐♥ ❢❛❝t ♦♥❡ ♠❛② ❡✈❡♥ ✇♦♥❞❡r✱ ❛ ♣r✐♦r✐✱ ✇❤② ✐t s❤♦✉❧❞ r❡❞✉❝❡ ❞r❛❣ ❛t ❛❧❧✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ♦❜s❡r✈❛t✐♦♥ t❤❛t s✉❝❤ ❝♦rr✉❣❛t✐♦♥s ♦❝❝✉r ♥❛t✉r❛❧❧② ✐♥ s❤❛r❦ s❦✐♥ ✭❇✉r❞❛❦ ❬✸❪❀ ❈❤❡r♥②s❤♦✈ ❡t ❛❧✳ ❬✹❪✮ tr✐❣❣❡r❡❞ t❤❡ ✐♥t❡r❡st ♦❢ r❡s❡❛r❝❤❡r ♦♥ t❤❡ r✐❜❧❡ts✳ ❉✉r✐♥❣ t❤❡ ✽✵s ✐t ✇❛s ❡①♣❡r✐♠❡♥t❛❧❧② s❤♦✇♥ ✭❇❡❝❤❡rt ❡t ❛❧✳ ❬✷✻❪❀ ❙❛✇②❡r ❬✷✸❪❀ ❈❤♦✐ ❬✺❪❀ ▼❝▲❡❛♥ ❡t ❛❧✳ ❬✶✼❪✮ t❤❛t ❛ r❡❞✉❝t✐♦♥ ♦❢ ✹✲✼ ✪ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❞r❛❣ ♦❢ ❛ s♠♦♦t❤ s✉r❢❛❝❡ ❝❛♥ ✐♥❞❡❡❞ ❜❡ ❛❝❤✐❡✈❡❞ ✐♥ t✉r❜✉❧❡♥t ✢♦✇s✳ ❚❤❡ ♣❡r❝❡♥t ♦❢ r❡❞✉❝t✐♦♥ s❡❡♠s t♦ ❜❡ s♠❛❧❧✱ ❜✉t✱ ❢♦r ❡①❛♠♣❧❡✱ ❡✈❡♥ ✈✐s❝♦✉s ❞r❛❣ r❡❞✉❝t✐♦♥s ❛s s♠❛❧❧ ❛s ✶✵ ✪ ♦♥ t❤❡ ❢✉s❡❧❛❣❡ ❛❧♦♥❡ ❛r❡ s✐❣♥✐✜❝❛♥t ✇❤❡♥ ✐t ✐s r❡❛❧✐③❡❞ t❤❛t t❤❡s❡ s♠❛❧❧ r❡❞✉❝t✐♦♥s tr❛♥s❧❛t❡ ✐♥t♦ ❛ ✸✺✵ ♠✐❧❧✐♦♥ ♣❡r ②❡❛r ♦❢ ❢✉❡❧ s❛✈✐♥❣s ❢♦r t❤❡ ❛✐r❧✐♥❡s ❝♦♠♣❛♥✐❡s ✭❲❛❧s❤ ❡t ❛❧✳ ❬✷✺❪✮✳ ❚❤❡ ♣r❡s❡♥t ✇♦r❦ ✐s ❢♦❝✉s❡❞ ♦♥ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ t❤r♦✉❣❤ ❛ ♣❛rt✐❝✉❧❛r s❤❛♣❡ ♦❢ r✐❜❧❡ts✳

  7. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✹ ✶✳✷ ❚❤❡ r✐❜❧❡ts ■♥ ✶✾✼✷ t❤❡ ◆❆❙❆ ▲❛♥❣❧❡② ❘❡s❡❛r❝❤ ❈❡♥t❡r ❜❡❣✉♥ ❛ ♥❡✇ s❦✐♥✲❢r✐❝t✐♦♥ ♣r♦❣r❛♠ ✐♥ ♦r❞❡r t♦ r❡❞✉❝❡ t❤❡ ❛✐r❝r❛❢t ❢✉❡❧ ❝♦♥s✉♠♣t✐♦♥✳ ❆❢t❡r ❛ ❧♦t ♦❢ st✉❞✐❡s✱ ✐♥ ✶✾✼✻✱ ❉r✳ ▲✳ ❘✳ ❆s❤ ❤❛❞ t❤❡ ✐❞❡❛ t❤❛t s♠❛❧❧ ✢♦✇✲❛❧✐❣♥❡❞ ❢❡♥❝❡s ♠✐❣❤t ♠♦❞✐❢② t❤❡ ♥❡❛r✲✇❛❧❧ str✉❝t✉r❡ ♦❢ t❤❡ t✉r❜✉❧❡♥t ❜♦✉♥❞❛r② ❧❛②❡r✱ t❤✉s r❡❞✉❝✐♥❣ t❤❡ s❦✐♥✲❢r✐❝t✐♦♥✳ ❚❤❡ s♠❛❧❧ ❢❡♥❝❡s ✇❡r❡ ❝❛❧❧❡❞ ✧r✐❜❧❡ts✧✱ ❛♥❞ ❛ s♠❛❧❧ r❡s❡❛r❝❤ st✉❞② ✇❛s ✐♥✐t✐❛t❡❞ ❛t ◆❆❙❆ ▲❛♥❣❧❡② ❘❡s❡❝❤ ❈❡♥t❡r ✭❇✉s❤♥❡❧❧ ❬✽❪✮✳ k ❋✐❣✉r❡ ✶✳✶✿ ❆ ♠♦❞❡❧ ♦❢ ❛ tr✐❛♥❣✉❧❛r r✐❜❧❡t s✉r❢❛❝❡❀ t❤❡ ②❡❧❧♦✇ ❛rr♦✇ ✐♥❞✐❝❛t❡s t❤❡ ✢♦✇ ❞✐r❡❝t✐♦♥✳ ❚❤❡ ✐❞❡❛ t❤❛t ❛ ❧♦♥❣✐t✉❞✐♥❛❧❧② ❣r♦♦✈❡❞ s✉r❢❛❝❡ ❝♦✉❧❞ r❡❞✉❝❡ t❤❡ t✉r❜✉❧❡♥t s❦✐♥ ❢r✐❝t✐♦♥ ❞r❛❣ ❛r✐s❡s ❢r♦♠ ❛ ❧♦t ♦❢ ❡①♣❡r✐♠❡♥t❛❧ st✉❞✐❡s✳ ❆♠♦♥❣ t❤❡s❡✱ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ ❑❧❡❜❛♥♦✛ ❬✶✺❪ ✇❤✐❝❤ ✐♥❞✐❝❛t❡ t❤❛t ♦✈❡r ✺✵ ✪ ♦❢ t❤❡ t✉r❜✉❧❡♥❝❡ ❡♥❡r❣② ✐s ❣❡♥❡r❛t❡❞ ✇✐t❤✐♥ t❤❡ ✺ ✪ ♦❢ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r✱ ❛♥❞ t❤❡ st✉❞② ♦❢ ❑❡♥♥❡❞② ❡t ❛❧✳ ❬✶✹❪✱ t❤❛t s❤♦✇s t❤❛t t✉r❜✉❧❡♥t s❤❡❛r str❡ss ✐s r❡❞✉❝❡❞ ❜② ✹✵✲✹✺ ✪ ✐♥ t❤❡ ❝♦r♥❡rs ♦❢ ❛ sq✉❛r❡ ❞✉❝t✳ ❆♥♦t❤❡r ✐♠♣♦rt❛♥t ✜❡❧❞ ♦❢ r❡s❡❛r❝❤ t❤❛t ✐♥❝r❡❛s❡ t❤❡ ✐♥t❡r❡st ✐♥ r✐❜❧❡ts s✉r❢❛❝❡s✱ ✇❛s t❤❡ st✉❞② ♦❢ t❤❡ s❤❛r❦ s❦✐♥✳ ❚❤❡ ✐♥t❡r❡st ♦❢ ❡♥❣✐♥❡❡rs ✐♥ s✉❝❤ ❛ ♠❛tt❡r ❛♣♣❛r❡❞ ❜❡❝❛✉s❡ t❤❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r ♦❢ ❢❛st s❤❛r❦s ✐s ❝♦♠♣❛r❛t✐✈❡❧② ❤✐❣❤ ✭ Re ∼ 10 6 − 10 7 ✱ ❝❛❧❝✉❧❛t❡❞ ✇✐t❤ t❤❡ ❜♦❞② ❧❡♥❣t❤✮✱ ❛♥❞ ♣♦t❡♥t✐❛❧ ❞r❛❣ r❡❞✉❝✐♥❣ ♠❡❝❤❛♥✐s♠s ❞❡r✐✈❡❞ ❢r♦♠ s❤❛r❦ s❦✐♥ ❛r❡ ♦❢ t❡❝❤♥♦❧♦❣✐❝❛❧ ✐♥t❡r❡st ✭❇❡❝❤❡rt ❬✷✻❪✮✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ❛r❡ s♦♠❡ ♣❡❝✉❧✐❛r✐t✐❡s ✇✐t❤ s❤❛r❦s✳ ❚❤❡② ❛r❡ t❤❡ ♦❧❞❡st ✜s❤ ✐♥ t❡r♠s ♦❢ t❤❡✐r ❡✈♦❧✉t✐♦♥❛r② ❤✐st♦r②❀ t❤❡ ❛♥✲ ❝❡st♦rs ♦❢ ♣r❡s❡♥t ❞❛② s❤❛r❦s ❛❧r❡❛❞② ❡①✐st❡❞ ✸✺✵ ♠✐❧❧✐♦♥ ②❡❛rs ❛❣♦ ❛♥❞ s♦♠❡ ❢❛♠✐❧✐❡s ♦❢ ❧✐✈✐♥❣ s❤❛r❦s ❤❛✈❡ ❛❧r❡❛❞② ❡①✐st❡❞ s✐♥❝❡ ✶✾✵ ♠✐❧❧✐♦♥ ②❡❛rs✳ ❚❤✉s✱ t❤❡r❡ ❤❛s ❜❡❡♥ ❛

  8. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✺ ❋✐❣✉r❡ ✶✳✷✿ ❚❤❡ ■s✉r✉s ❖①②r✐♥❝❤✉s✱ ❡✈❡♥ ❝❛❧❧❡❞ ▼❛❦♦✱ ✐s ♦♥❡ ♦❢ t❤❡ ❢❛st❡st s❤❛r❦ ♦♥ t❤❡ ♣❧❛♥❡t✳ ■t ❝❛♥ r❡❛❝❤ ❛ ♠❛①✐♠✉♠ s♣❡❡❞ ♦❢ 70 km/h ✳ ❝♦♠♣❛r❛t✐✈❡❧② ❧♦♥❣ t✐♠❡ ❢♦r ♦♣t✐♠❛❧ ❛❞❛♣t✐♦♥ ❞✉r✐♥❣ t❤❡✐r ❡✈♦❧✉t✐♦♥✳ ❚❤✐s ♦♣t✐♠✐③❛✲ t✐♦♥ ✐s ❛❧s♦ s❤♦✇♥ ❜② ♦t❤❡r ❢❡❛t✉r❡s r❡❧❛t❡❞ t♦ s✇✐♠♠✐♥❣✱ ❧✐❦❡✱ ❡✳❣✳✱ t❤❡ ♦♣❡r❛t✐♦♥ ❛♥❞ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ✜♥s✳ ■♥ ❛❞❞✐t✐♦♥✱ ❜❡❝❛✉s❡ ♦❢ t❤❡✐r ♣❛rt✐❝✉❧❛r ♠♦❞❡ ♦❢ ✐♥t❛❦❡ ♦❢ ♦①②❣❡♥ t❤r♦✉❣❤ t❤❡✐r ❣✐❧❧s✱ ✇❤✐❝❤ ✇♦r❦s ♦♥❧② ✐♥ ❢♦r✇❛r❞ ♠♦t✐♦♥✱ t❤❡② ♠♦✈❡ ❝♦♥st❛♥t❧②✱ ❞❛② ❛♥❞ ♥✐❣❤t✱ ❢✉rt❤❡r♠♦r❡✱ t❤❡ ❜✉rst s♣❡❡❞ ♦❢ ♦✛✲s❤♦r❡ ♣r❡❞❛t♦r② s❤❛r❦s ✭✜❣✉r❡ ✶✳✷✮ ✐s ❝♦♠♣❛r❛t✐✈❡❧② ❤✐❣❤✱ ✐t ✐s ❜❡❧✐❡✈❡❞ t♦ ❜❡ ❛❜♦✉t 10 ✲ 20 ♠✴s✳ ❚❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r✐❞❣❡ str✉❝t✉r❡ ♦❢ s❤❛r❦ s❝❛❧❡s ♦❢ ❢❛st s❤❛r❦s ❛♥❞ t❤❡ ❞r❛❣ r❡❞✉❝✐♥❣ ♠❡❝❤❛♥✐s♠ ♦❢ r✐❜❧❡ts ❤❛❞ ❜❡❡♥ r❡❝♦❣♥✐③❡❞ ❜② ❘❡✐❢ ❛♥❞ ❉✐♥❦❡❧❛❝❦❡r ❬✷✶❪✳ ❘❡✐❢ ❝♦♠♣✐❧❡❞ ✐♥❢♦r♠❛t✐♦♥ ♦♥ s❤❛r❦ s❝❛❧❡s ❢♦r ♠♦r❡ t❤❛♥ ✹✵ s♣❡❝✐❡s ❛♥❞ ❢r♦♠ ❞✐✛❡r❡♥t ❣r♦✇t❤ st❛❣❡s✱ ❛♥❞ t❤❡ s❝❛❧❡s ✇❡r❡ t❛❦❡♥ ❢r♦♠ ❞✐✛❡r❡♥t ♣❛rts ♦❢ t❤❡ s❤❛r❦ ❜♦❞✐❡s✳ ❲✐t❤ t❤❛t ♠❛t❡r✐❛❧ ❤❡ ❝♦✉❧❞ ❡st❛❜❧✐s❤ ❛ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ r✐❞❣❡ str✉❝t✉r❡ ❛♥❞ ✈❡❧♦❝✐t② ♦❢ ❞✐✛❡r❡♥t s♣❡❝✐❡s✳ ❚❤❡ r❡s✉❧ts ♦❢ t❤✐s ❝❧❛ss✐✜❝❛t✐♦♥ r❡✈❡❛❧ t❤❛t ❛❧❧ ❢❛st ♦✛✲s❤♦r❡ s❤❛r❦s ❤❛✈❡ s❝❛❧❡s ✇✐t❤ ✜♥❡ r✐❞❣❡s ✭s♣❛❝✐♥❣ 35 ✲ 105 µm ✮ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✶✳✸✱ ✇❤❡r❡❛s r❡❡❢ s❤❛r❦s ❤❛✈❡ ❛ s✐♠✐❧❛r s❝❛❧❡ str✉❝t✉r❡ ❜✉t s❧✐❣❤t❧② ✇✐❞❡r r✐❞❣❡ s♣❛❝✐♥❣✳ ■♥ ❝♦♥tr❛st t♦ t❤❛t✱ s❧♦✇ s❤❛r❦s ❤❛✈❡ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t ❛♥❞ ❧❛r❣❡❧② ✈❛r②✐♥❣ s❝❛❧❡ ♣❛tt❡r♥s ✇❤✐❝❤ ♠❛② ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❞❡✈✐❝❡s ❢♦r ♣r♦t❡❝t✐♦♥ ❛❣❛✐♥st ❛❜r❛s✐♦♥✱ ♣❛r❛s✐t❡s ❛♥❞ ♣r❡❞❛t♦rs✳ ❚❤❡r❡❢♦r❡✱ ✐♥ t❤❡ s♣✐r✐t ♦❢ ❜✐♦♠✐♠❡t✐❝s✱ ❛ ❧♦t ♦❢ ✧❛rt✐✜❝✐❛❧ s❤❛r❦✲s❦✐♥✧ ❤❛✈❡ ❜❡❡♥ ❜✉✐❧t ❛♥❞ t❡st❡❞✳ ❆❧❧ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ♠❡❛s✉r❡♠❡♥t ❝♦♥✜r♠ t❤❛t s✉❝❤ ❛ s✉r❢❛❝❡ ❝❛♥ r❡❞✉❝❡ t❤❡ s❦✐♥✲❢r✐❝t✐♦♥ ❢r♦♠ ✹ ✪ ✉♣ t♦ ✶✵ ✪ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s✉r❢❛❝❡✳ ❖t❤❡r t❡❝❤♥✐q✉❡ ❝❛♥ ❜❡ ✉s❡❞ t♦ r❡❞✉❝❡ t❤❡ s❦✐♥ ❢r✐❝t✐♦♥✱ ❤♦✇❡✈❡r t❤❡ ❣r♦♦✈❡❞ s✉r❢❛❝❡s ❛r❡ t❤❡ ♠♦st ❛♣♣❡❛❧✐♥❣ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❡❛t✉r❡s✿

  9. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✻ ❋✐❣✉r❡ ✶✳✸✿ ❉✐✛❡r❡♥t s❤❛r❦s ❤❛✈❡ ❞✐✛❡r❡♥t ❦✐♥❞ ♦❢ s❦✐♥✱ ❜✉t t❤❡ ❢❛st❡r s❤❛r❦s ❤❛✈❡ ❛❧❧ t❤❡ s❛♠❡ ♠✐❝r♦s❝♦♣✐❝ r✐❞❣❡s✳ • t❤❡② ❛r❡ ❛ ♣❛ss✐✈❡ ❞r❛❣✲r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡ ✭r❡q✉✐r❡s ♥♦ ❛❞❞✐t✐♦♥❛❧ ❡♥❡r❣②✮✳ • ❚❤❡✐r ❡✛❡❝t✐✈❡♥❡ss ✐s r❡❧❛t✐✈❡❧② ✉♥❛✛❡❝t❡❞ ❜② ♠✐s❛❧✐❣♥♠❡♥t ✇✐t❤ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♦✇ ✭❲❛❧s❤ ❡t ❛❧✳ ❬✷✺❪❀ ❈❤♦✐ ❡t ❛❧✳ ❬✺❪✮ ♦r ❜② ❝♦♠♣r❡ss✐❜✐❧✐t② ❡✛❡❝ts ✭▼❝▲❡❛♥ ❡t ❛❧✳ ❬✶✼❪✮ • ❚❤❡② ❛r❡ ✐♥s❡♥s✐t✐✈❡ t♦ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♣r❡ss✉r❡ ❣r❛❞✐❡♥t ✭❇❛r♦♥ ❡t ❛❧✳ ❬✶❪✮ ❆❧t❤♦✉❣❤ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ r✐❜❧❡ts ✐s ❝♦♥✜r♠❡❞ ❜② ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛✱ ✐ts ♠❡❝❤❛✲ ♥✐s♠ ♦❢ ♦♣❡r❛t✐♦♥ ✇❛s ♥♦t ✇❡❧❧ ✉♥❞❡rst♦♦❞✳ ❲❤❡♥ ❛ t✉r❜✉❧❡♥t ✢♦✇ ❞❡✈❡❧♦♣s ♦♥ ❛ r✐❜❧❡tt❡❞ s✉r❢❛❝❡✱ t❤❡ ❧♦✇❡r ♣❛rt ✭✈❛❧❧❡②s✮ ♦❢ t❤❡ s♠❛❧❧ ♠✐❝r♦✲❣r♦♦✈❡s ❧✐❡ ✐♥t♦ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r✱ ✇❤❡r❡❛s t❤❡ ✉♣♣❡r ♣❛rt ✭t✐♣s✮ ♣r♦tr✉❞❡ ✐♥t♦ t❤❡ ❜✉✛❡r ❧❛②❡r✳ ❙✉❝❤ ❛ ❝♦♥❢❣✉r❛t✐♦♥ ✐s ❛❞✈❛♥t❛❣❡♦✉s ❢♦r t✇♦ ♣r✐♥❝✐♣❛❧ r❡❛s♦♥s✳ ❋✐rst ♦❢ ❛❧❧ t❤❡ s❤❛r♣ ♣❡❛❦s ❞❛♠♣ t❤❡ t✉r❜✉❧❡♥❝❡ ❧❡✈❡❧✱ ✐♥ ❢❛❝t t❤❡② ✐♥t❡r❛❝t ✇✐t❤ t❤❡ ✈♦rt✐❝❛❧ str✉❝t✉r❡s ✭t❤❛t t❛❦❡ ♣❧❛❝❡ ✐♥ t❤❡ ❜✉✛❡r ❧❛②❡r✮ ❛♥❞ r❡❞✉❝❡ t❤❡ ❝r♦ss✲ ✢♦✇✳ ▼♦r❡♦✈❡r✱ t❤❡ ❤✐❣❤ ✈❡❧♦❝✐t② ③♦♥❡s ❣❡♥❡r❛t❡❞ ❜② t❤❡ str❡♠✲✇✐s❡ ✈♦rt❡① ❞♦ ♥♦t r❡❛❝❤ t❤❡ r✐❜❧❡t ✈❛❧❧❡②s✱ ❛♥❞ t❤✐s ♣r♦❞✉❝❡s ❛ ❧♦✇ s❤❡❛r str❡ss✳ ❘❡❝❡♥t ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ❜② ✭❉♦✉❣❧❛s ❡t ❛❧✳ ❬✾❪✮ ❝♦♥✜r♠ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ♠❡❝❤❛♥✐s♠✳ ❚❤❡ ❢♦r❡❣♦✐♥❣ ❛r❣✉♠❡♥t ✐s ✐♥t✉✐t✐✈❡❧② ❝♦♥✈✐♥❝✐♥❣✱ ❜✉t ✐t ✐s ♦♥❧② q✉❛❧✐t❛t✐✈❡ ❛♥❞ ✐t

  10. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✼ ✭❛✮ ❚r✐❛♥❣✉❧❛r r✐❜❧❡t✳ ✭❜✮ ❙♣❛❝❡❞ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ✭❝✮ ❯✲s❤❛♣❡❞ r✐❜❧❡t✳ ✭❞✮ ❇❧❛❞❡ r✐❜❧❡t✳ ❋✐❣✉r❡ ✶✳✹✿ ❋♦✉r ❦✐♥❞ ♦❢ r✐❜❧❡ts s❤❛♣❡✳ ❝❛♥✬t ❜❡ ✉s❡❞ t♦ ❝❛rr② ♦✉t ❛ t❤❡♦r② t❤❛t ❛❧❧♦✇ t♦ ♦❜t❛✐♥ t❤❡ ♦♣t✐♠❛❧ s❤❛♣❡ ♦❢ t❤❡ r✐❜❧❡ts✳ ❚❤❡r❡❢♦r❡✱ ♠❛♥② ✈❛r✐❛t✐♦♥ ✐♥ t❤❡ ❜❛s✐❝ r✐❜❧❡t ❣❡♦♠❡tr② ❤❛✈❡ ❜❡❡♥ t❡st❡s ✐♥ ❛tt❡♠♣ts t♦ ♦❜t❛✐♥ ❛♥ ❤✐❣❤❡r ❞r❛❣ r❡❞✉❝t✐♦♥❀ t❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ❣❡♦♠❡tr✐❡s ❛r❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✶✳✹✿ t❤❡ ✈✲❣r♦♦✈❡ r✐❜❧❡ts ✭s❛✇t♦♦t❤ r✐❜❧❡ts✮ ✇✐t❤ ✈❛r✐♦✉s ❛s♣❡❝t r❛t✐♦s✱ t❤❡ ✉✲❣r♦♦✈❡ r✐❜❧❡ts ✭s❝❛❧❧♦♣❡❞ r✐❜❧❡ts✮✱ ❛ s♣❛❝❡❞ ❣r♦♦✈❡ r✐❜❧❡t✱ ❛♥❞ r❡❝t❛♥❣✉❧❛r r✐❜❧❡ts ✭❜❧❛❞❡ r✐❜❧❡ts✮✳ ❍♦✇❡✈❡r t❤✐s ✐s ♥♦t ❛♥ ❡❛s② ❛❝❤✐❡✈❡♠❡♥t✱ s✐♥❝❡ t❤❡ r✐❜❧❡ts ❣✐✈❡ ❛ s♠❛❧❧ ❞r❛❣ r❡❞✉❝t✐♦♥ ✭♦♥ t❤❡ ♦r❞❡r ♦❢ ✻✲✽ ✪✮✱ t❤❡r❡❢♦r❡✱ t❤❡ ♠❡❛s✉r❡♠❡♥ts r❡q✉✐r❡ ❤✐❣❤ ❛❝❝✉r❛❝②✳ ■t ✐s ❡✈❡♥ ♠♦r❡ ❞✐✣❝✉❧t t♦ ♦❜s❡r✈❡ ❝❤❛♥❣❡s ✐♥ ❞r❛❣ r❡❞✉❝t✐♦♥ ❞✉❡ t♦ ✈❛r✐❛t✐♦♥s ✐♥ ❣❡♦♠❡tr②✱ s✐♥❝❡ t❤❡ ❝❤❛♥❣❡s ✐♥ ❞r❛❣ ♠❛② ❜❡ ✇✐t❤✐♥ t❤❡ s❝❛tt❡r ♦❢ t❤❡ ♠❡❛s✉r❡♠❡♥ts✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ r✐❜❧❡t ♣❡r❢♦r♠❛♥❝❡ ❝❛♥ ❜❡ q✉✐t❡ s❡♥s✐t✐✈❡ t♦ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♠❛❝❤✐♥❡❞ s✉r❢❛❝❡✿ ♣♦♦r q✉❛❧✐t② ♠❛② r❡s✉❧t ✐♥ r♦✉♥❞✐♥❣ ❛t t❤❡ ♣❡❛❦s✱ r✐❜❧❡t ❛s♣❡❝t r❛t✐♦ ✈❛r②✐♥❣ ❛❝r♦ss t❤❡ s✉r❢❛❝❡✱ ❡t❝✳ ❊✈❡♥t✉❛❧❧②✱ t❤❡ ♠❛❝❤✐♥❡s r❡s♦❧✉t✐♦♥ ❞♦❡s ♥♦t ❛❧❧♦✇ ❛ ♣❡r❢❡❝t r❡♣r♦❞✉❝t✐♦♥ ♦❢ t❤❡ ✇❛♥t❡❞ r✐❜❧❡t s❤❛♣❡✳ ❆❧❧ t❤❡s❡ ♣r♦❜❧❡♠s ♠❛② ❤❛✈❡ ❛♥ ✉♥❦♥♦✇♥ ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ ♣❡r❢♦r♠❛♥❝❡s✳ ❆❧t❤♦✉❣❤ t❤❡ ❞✐✣❝✉❧t✐❡s✱ ❛ ❣❡♥❡r❛❧ tr❡♥❞ ♦❢ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ r✐❜❧❡ts s♣❛❝✐♥❣ ❤❛s ❜❡❡♥ ❛❝❤✐❡✈❡❞✳ ❋✐❣✉r❡ ✶✳✺ s❤♦✇s ❛ t②♣✐❝❛❧ ❞r❛❣ r❡❞✉❝t✐♦♥ ✭ DR = − ∆ τ 0 /τ 0 ✮ ❝✉r✈❡ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s♣❛❝✐♥❣ ✭ s + ✮✱ ✐♥ ✇❤✐❝❤ ❞✐✛❡r❡♥t ❞r❛❣ r❡❣✐♠❡s ❝❛♥ ❜❡ ❞❡✜♥❡❞✳ ■♥ t❤❡ ✈✐s❝♦✉s r❡❣✐♠❡✱ ❢♦r♠❛❧❧② s + << 5 ❜✉t ✐♥ ♣r❛❝t✐❝❡ s + < 10 − 15

  11. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✽ ❋✐❣✉r❡ ✶✳✺✿ ❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞r❛❣✲r❡❞✉❝t✐♦♥ r❡❣✐♠❡s ♦❜s❡r✈❡❞ ♦✈❡r tr✐❛♥❣✉❧❛r r✐❜❧❡ts ✇✐t❤ 60 ◦ t✐♣ ❛♥❣❧❡✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣❡❛❦✲t♦✲♣❡❛❦ ❞✐st❛♥❝❡ s + ✳ ✱ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤❡ ♥♦♥❧✐♥❡❛r t❡r♠s t♦ t❤❡ ✢♦✇ ✇✐t❤✐♥ ❛♥❞ ✐♥ t❤❡ ✐♠♠❡❞✐❛t❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ r✐❜❧❡t ❣r♦♦✈❡s ✐s ♥❡❣❧✐❣✐❜❧❡ ❛♥❞✱ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ s + ✳ ❚❤❡ ✈✐s❝♦✉s r❡❣✐♠❡ ❜r❡❛❦s ❞♦✇♥ ♥❡❛r s + = s + opt ✱ t❤❡ ♦♣t✐♠✉♠ s♣❛❝✐♥❣ ❢♦r ✇❤✐❝❤ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐s ♠❛①✐♠✉♠✱ ❛♥❞✱ ❡✈❡♥t✉❛❧❧②✱ t❤❡ r❡❞✉❝t✐♦♥ ❜❡❝♦♠❡s ❛ ❞r❛❣ ✐♥❝r❡❛s❡✱ ❛❞♦♣t✐♥❣ ❛ t②♣✐❝❛❧ ❦✲r♦✉❣❤♥❡ss ❜❡❤❛✈✐♦✉r✳ ❲❤❡♥ r✐❜❧❡t ❛r❡ t♦♦ s♠❛❧❧ t❤❡ t✐♣s ❝❛♥♥♦t ✇❡❧❧ ✐♥t❡r❛❝t ✇✐t❤ t❤❡ ❜✉✛❡r ❧❛②❡r ❛♥❞ t❤❡② ❝❛♥♥♦t ❞❛♠♣ t❤❡ s❡❝♦♥❞❛r② ✢♦✇✳ ❍♦✇❡✈❡r✱ ✐❢ t❤❡② ❛r❡ t♦♦ ❧❛r❣❡✱ t❤❡ str❡❛♠✲✇✐s❡ ✈♦rt❡① ❢❛❧❧ ✐♥s✐❞❡ t❤❡ r✐❜❧❡t ❝❛✈✐t② ❛♥❞ ❣❡♥❡r❛t❡ ❛♥ ❤✐❣❤ s❦✐♥ ❢r✐❝t✐♦♥✳ ❚❤❡ ♥✉♠❡r✐❝❛❧ ❛♣♣r♦❛❝❤ ❛❧s♦ ♣r❡s❡♥ts s♦♠❡ ♣r♦❜❧❡♠s✱ ✐♥ ❢❛❝t t❤❡ s♠❛❧❧ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❣r♦♦✈❡s r❡q✉✐r❡s ❛♥ ❡①tr❡♠❡❧② ✜♥❡ ❛♥❞ ❝♦♠♣❧❡① ❣r✐❞✳ ❚❤❡r❡❢♦r❡ t❤❡ t✐♠❡ ❢♦r t❤❡ s✐♠✉❧❛t✐♦♥ ♠❛② ❜❡❝♦♠❡ ✈❡r② ❧♦♥❣✳

  12. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✾ ✶✳✸ ❚❤❡ ❢r❛❝t❛❧ ❣❡♦♠❡tr✐❡s ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❧❡❛✈❡ ♠♦♠❡♥t❛r✐❧② t❤❡ r✐❜❧❡t t♦♣✐❝ ✐♥ ♦r❞❡r t♦ ❡①♣❧❛✐♥ ❛ ♣❛rt✐❝✉❧❛r ❣❡♦♠❡tr② s❤❛♣❡✿ t❤❡ ❢r❛❝t❛❧✳ ❚❤❡ r❡❛s♦♥ ✇✐❧❧ ❜❡ ❝❧❡❛r ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ ✇❤❡r❡ t❤✐s ✇♦r❦ ♣✉r♣♦s❡ ❛r❡ ❡①♣❧❛✐♥❡❞✳ ❚❤❡ st✉❞② ♦❢ ❢r❛❝t❛❧s ✐s ❛ ✈❡r② ✇✐❞❡ ✜❡❧❞✱ t❤❡r❡❢♦r❡ t❤✐s s❡❝t✐♦♥ ❛✐♠s t♦ ❡①♣❧❛✐♥ ♦♥❧② t❤❡ ♣r✐♥❝✐♣❛❧ ❝♦♥❝❡♣ts✱ ✐♥ ♦r❞❡r t♦ ✉♥❞❡rst❛♥❞ ✇❤② s✉❝❤ ❛ s❤❛♣❡ ✐s t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞✲ ❡r❛t✐♦♥✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ❣❡t ❛ r♦✉❣❤ ✐❞❡❛ ♦❢ ❢r❛❝t❛❧s✱ ❣✐✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥✿ ❛ ❢r❛❝t❛❧ ✐s ❛ ❣❡♦♠❡tr✐❝ ✜❣✉r❡ ✇❤✐❝❤ ❡①❤✐❜✐t s✐♠✐❧❛r ♣❛tt❡r♥s ❛t ✐♥❝r❡❛s✐♥❣❧② s♠❛❧❧ s❝❛❧❡s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ♠❛❣♥✐❢②✐♥❣ s✉❝❤ ❛ ♦❜❥❡❝t✱ ♦♥❡ ❝❛♥ ❛❧✇❛②s ✐❞❡♥t✐❢② t❤❡ s❛♠❡ s❤❛♣❡✳ ❚❤✐s ♣❛rt✐❝✉❧❛r ❢❡❛t✉r❡ t❛❦❡ t❤❡ ♥❛♠❡ ♦❢ s❡❧❢✲s✐♠✐❧❛r✐t②✱ ❛♥❞ ❝❛♥ ❜❡ ❡♠♣❧♦②❡❞ t♦ ❜✉✐❧t ❛ ❢r❛❝t❛❧ ♦❜❥❡❝t✳ ❇❡❢♦r❡ ♣r♦❝❡❡❞✐♥❣ ✇✐t❤ ❛♥ ❡①❛♠♣❧❡✱ ✐t ✐s ✇♦rt❤ t♦ ❡♠♣❛t❤✐③❡ t❤❛t ❢r❛❝t❛❧s ❡①✐st ♦♥❧② ✐♥ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥❝❡♣t✱ ✐♥ ❢❛❝t✱ ❢r❛❝t❛❧s r❡♣❡❛t ✐ts❡❧❢ ❛t ✐♥✜♥✐t❡❧② s♠❛❧❧ ❧❡♥❣t❤ s❝❛❧❡✱ ❣❡♥❡r❛t✐♥❣ ❛♥ ✐♥✜♥✐t❡ ❧♦♥❣ ❝✉r✈❡✱ t❤❛t ❝❛♥ ❡♥❝❧♦s❡ ❛ ✜♥✐t❡ ❛r❡❛✳ ❋✐❣✉r❡ ✶✳✻✿ ❖r✐❣✐♥❛❧ ♠♦❧❞ ❛♥❞ ✜✈❡ ✐t❡r❛t✐♦♥s ♦❢ t❤❡ ❑♦❝❤ s♥♦✇✢❛❦❡✳ ❆ q✉✐t❡ ❢❛♠♦✉s ❢r❛❝t❛❧✱ t❤❡ ❑♦❝❤ ❝✉r✈❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✶✳✻✱ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❜② st❛rt✐♥❣ ✇✐t❤ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡✱ t❤❡♥ r❡❝✉rs✐✈❡❧② ❛❧t❡r✐♥❣ ❡❛❝❤ ❧✐♥❡ s❡❣♠❡♥t ❛s ❢♦❧❧♦✇s✿ ✶✳ ❉✐✈✐❞❡ t❤❡ ❧✐♥❡ s❡❣♠❡♥t ✐♥t♦ t❤r❡❡ s❡❣♠❡♥ts ♦❢ ❡q✉❛❧ ❧❡♥❣t❤✳ ✷✳ ❉r❛✇ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡ t❤❛t ❤❛s t❤❡ ♠✐❞❞❧❡ s❡❣♠❡♥t ❢r♦♠ st❡♣ ✶ ❛s ✐ts ❜❛s❡ ❛♥❞ ♣♦✐♥ts ♦✉t✇❛r❞✳ ✸✳ ❘❡♠♦✈❡ t❤❡ ❧✐♥❡ s❡❣♠❡♥t t❤❛t ✐s t❤❡ ❜❛s❡ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❢r♦♠ st❡♣ ✷✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ t❤❡s❡ st❡♣s t♦ ❝♦♥str✉❝t ❢r❛❝t❛❧ r✐❜❧❡ts✳

  13. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✶✵ ✶✳✹ ❆✐♠ ♦❢ t❤❡ ✇♦r❦ ✶✳✹✳✶ ❈❛s❡ ♦❢ st✉❞② ❚❤✐s ✇♦r❦ ❛✐♠ ❛t ♥✉♠❡r✐❝❛❧❧② st✉❞②✐♥❣ t❤❡ ♣❛rt✐❝✉❧❛r s❤❛♣❡ ♦❢ r✐❜❧❡t s❤♦✇ ✐♥ ✜❣✉r❡ ✶✳✼❜ t❤❛t✱ ✉♥t✐❧ ♥♦✇✱ ❤❛s ♥♦t ❜❡❡♥ t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥✿ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t✳ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❛✮ ❈❧❛ss✐❝❛❧ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ✭❜✮ ❋r❛❝t❛❧ r✐❜❧❡t✳ ❋✐❣✉r❡ ✶✳✼✿ ❙♣❛♥✲✇✐s❡ s❡❝t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝❛❧ tr✐❛♥❣✉❧❛r r✐❜❧❡t s❤❛♣❡ ❛♥❞ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t✱ s✉❜❥❡❝t ♦❢ t❤✐s ✇♦r❦✳ ❚❤✐s ♣❛rt✐❝✉❧❛r s❤❛♣❡ ✐s ❝❛rr✐❡❞ ♦✉t ❢r♦♠ t❤❡ ❝♦♠♠♦♥ tr✐❛♥❣✉❧❛r r✐❜❧❡t t❤r♦✉❣❤ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ s❛♠❡ st❡♣s ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✶✳✸✱ t❤❛t ❤❛✈❡ ❜❡❡♥ ✉s❡❞ t♦ ❜✉✐❧t t❤❡ ❑♦❝❤ s♥♦✇✢❛❦❡✳ ❆t t❤✐s ♣♦✐♥t✱ ✐t ✐s ✈❡r② ✐♠♣♦rt❛♥t t♦ st❛♥❞ ♦✉t t❤❡ r❡❛s♦♥s t❤❛t ❜r✐♥❣ ✉s t♦ ❝❤♦♦s❡ s✉❝❤ ❛ ❣❡♦♠❡tr②✳ ■♥ ✶✾✼✺✱ t❤❡ ♠❛t❤❡♠❛t✐❝✐❛♥ ❇❡♥♦ît ▼❛♥❞❡❧❜r♦t ✐♥✈❡♥t❡❞ t❤❡ ♥❛♠❡ ✧❢r❛❝t❛❧✧✱ ❛♥❞ ✉s❡❞ ✐t t♦ ❡①t❡♥❞ t❤❡ t❤❡♦r❡t✐❝❛❧ ❝♦♥❝❡♣t t♦ t❤❡ ❣❡♦♠❡tr✐❝ ♣❛tt❡r♥s ✐♥ ♥❛t✉r❡✱ ✇❤❡r❡ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✜♥❞ ❛ ❧♦t ♦❢ s❡❧❢✲s✐♠✐❧❛r s❤❛♣❡s✳ ❙✐♥❝❡ t❤❡♥✱ ❢r❛❝t❛❧s s♣r❡❛❞✐♥❣ t❤r♦✉❣❤♦✉t t❤❡ s❝✐❡♥❝❡ ❛♥❞ ❜❡❝❛♠❡ ✉s❡❢✉❧ ✐♥ ❛ ❧♦t ♦❢ ✜❡❧❞s ♦❢ st✉❞②✳ ❚❤❛♥❦s t♦ t❤❡ str♦♥❣ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ❛❧❧ ♥❛t✉r❛❧ s❤❛♣❡s✱ ❛♥❞ ❛❝❝♦r❞✐♥❣ ✇✐t❤ ♦t❤❡rs ✇♦r❦s ❬✼❪✱ ❢r❛❝t❛❧ ❛r❡ ❤❡r❡✐♥ ✉s❡❞ t♦ s✐♠✉❧❛t❡ t❤❡ ♠✐❝r♦ ❛♥❞ ♥❛♥♦s❝♦♣✐❝ str✉❝t✉r❡ ♦❢ t❤❡ r✐❜❧❡t s✉r❢❛❝❡✳ ❚❤❡ ♣r❡s❡♥t st✉❞② ✐s ❜❛s❡❞ ♦♥ t❤❡ ✐❞❡❛ t❤❛t t❤❡ r❡❛❧ s✉r❢❛❝❡ ✐s ♦♥❡ ♦❢ t❤❡ ❦❡② ♣❛r❛♠❡t❡r t♦ ✇❡❧❧ r❡♣r♦❞✉❝❡ t❤❡ r✐❜❧❡t ❡✛❡❝t✳ ❚❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❝♦♥s✐❞❡r❛t✐♦♥ ✐s ❝♦♥✜r♠❡❞ ❜② t❤❡ ❇❛r♦♥ ❡t ❛❧✳ ❬✶❪✱ ✇❤✐❝❤ ❝♦♠♣❛r❡s t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ r❡s✉❧ts ❜❡t✇❡❡♥ t❤❡ ✐❞❡❛❧ ❛♥❞ t❤❡ r❡❛❧ r✐❜❧❡t ❣❡♦♠❡tr②✳

  14. ❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✶✶ ❖✈❡r t❤❡ s❤❛♣❡ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✶✳✼✱ ✇❡ s❤❛❧❧ st✉❞② ♦t❤❡rs s❤❛♣❡s✱ ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ❛ ❝♦♠♣❛r✐s♦♥ ♦❢ r❡s✉❧ts✳ ❆❧❧ t❤❡ ❣❡♦♠❡tr② ✐♥✈♦❧✈❡❞ ✐♥ t❤✐s st✉❞② ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✼✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ✇❡ s❤❛❧❧ ❡①♣❧❛✐♥ ❤♦✇ t❤❡ st✉❞② ♦❢ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t ✐s ♣❡r❢♦r♠❡❞✳ ✶✳✹✳✷ ❉❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ✇♦r❦ ❚❤❡ st✉❞② ❤❡r❡✐♥ ♣❡r❢♦r♠❡❞ ❛✐♠ ❛t ❝❛rr②✐♥❣ ♦✉t t❤❡ s❦✐♥ ❢r✐❝t✐♦♥ ♦✈❡r ❛ ❣r♦♦✈❡❞ ❢r❛❝t❛❧ s✉r❢❛❝❡ t❤r♦✉❣❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✢✉✐❞ ❞②♥❛♠✐❝s ✭❈❋❉✮✳ ❚♦ ❞♦ s♦ ✇❡ ❤❛✈❡ ❞❡❝✐❞❡❞ t♦ t❛❦❧❡ t❤❡ ♣r♦❜❧❡♠ ✐♥ t✇♦ ♣❛rts✿ t❤❡ ♠✐❝r♦s❝♦♣✐❝ ❛♥❞ ♠❛❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠s✳ ❚❤❡ ♠✐❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠ ❞❡❛❧s ✇✐t❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✢♦✇ ✐♥s✐❞❡ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r✱ ✈❡r② ♥❡❛r t❤❡ ✇❛❧❧✳ ■♥ t❤✐s r❡❣✐♦♥ t❤❡ ♠♦♠❡♥t✉♠ ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞ ✐♥ t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥ ❛♥❞ t✇♦ ❢✉♥❞❛♠❡♥t❛❧ q✉❛♥t✐t✐❡s✱ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✱ ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞✳ ❆❢t❡r✇❛r❞s✱ t❤❡ ❝♦♠♣✉t❡❞ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛r❡ ✉s❡❞ ❛s ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♠❛r❝♦s❝♦♣✐❝ ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ❛❧❧♦✇ t♦ ❝❛rr② ♦✉t t❤❡ s❦✐♥ ❢r✐❝t✐♦♥ ❞r❛❣ ♦♥ t❤❡ r✐❜❧❡t ♣❧❛t❡s t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥✳

  15. ❈❤❛♣t❡r ✷ ▼✐❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠ ✶✷

  16. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✶✸ ✷✳✶ ❱✐s❝♦✉s s✉❜❧❛②❡r ❚❤❡ ❡✛❡❝t ♦❢ ❞r❛❣ r❡❞✉❝t✐♦♥ ❣❡♥❡r❛t❡❞ ❜② t❤❡ r✐❜❧❡t s✉r❢❛❝❡s ❤❛s ❜❡❡♥ ❞✐s❝♦✈❡r❡❞ ✐♥ t❤❡ s❡✈❡♥t✐❡s✱ ❤♦✇❡✈❡r t❤❡ ♣❤②s✐❝❛❧ r❡❛s♦♥s t❤❛t ❡①♣❧❛✐♥ t❤✐s ♣❤❡♥♦♠❡♥♦♥ ❤❛✈❡ ❜❡❡♥ ✉♥❦♥♦✇♥ ❢♦r ②❡❛rs✳ ❚❤❡ r♦✉t❡ t♦ ❛ ❞❡❡♣❡r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❡✛❡❝ts ♦❢ ❣r♦♦✈❡❞ s✉r❢❛❝❡s ❤❛s ❜❡❡♥ ♦♣❡♥❡❞ ❜② ❇❡❝❤❡rt✬s ♦❜s❡r✈❛t✐♦♥ t❤❛t t❤❡ t②♣✐❝❛❧ s✐③❡ ♦❢ ❝♦rr✉❣❛t✐♦♥s ✇❤✐❝❤ ❛♣♣❡❛r t♦ ❜❡ ❡①♣❡r✐♠❡♥t❛❧❧② ❡✛❡❝t✐✈❡ ✐s ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡ ❛s t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r ♦❢ t❤❡ t✉r❜✉❧❡♥t str❡❛♠ ❬✷✵❪✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ❝♦♥s✐❞❡r ❛ ❢✉❧❧② ❞❡✈❡❧♦♣❡❞ t✉r❜✉❧❡♥t ❜♦✉♥❞❛r② ❧❛②❡r ✭✜❣✉r❡ ✷✳✶✮✳ ■t ❤❛s ❛ ❢✉♥❞❛♠❡♥t❛❧ r♦❧❡ ✐♥ t❤❡ ♣r❡s❡♥t ✇♦r❦✱ t❤❡r❡❢♦r❡ ✇❡ ❣✐✈❡ ❛ ❜r✐❡✢② ❞❡s❝r✐♣t✐♦♥ ♦❢ ✐t✳ ❚♦ ❞♦ s♦ t❤❡ ❞✐♠❡♥s✐♦♥❧❡ss ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ✇❛❧❧ ✐s ✐♥tr♦❞✉❝❡s y + = y � τ w , ✭✷✳✶✮ ν ρ w t❤❡r❡❜②✱ t❤❡ t✉r❜✉❧❡♥t ✢♦✇ ♥❡❛r ❛ ✢❛t ✇❛❧❧ ❝❛♥ ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ ❢♦✉r r❡❣✐♦♥s✿ • ❚❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r ✭ 0 ≤ y + < 5 ✮ ✿ t❤✐♥ ❧❛②❡r ♥❡❛r t❤❡ ✇❛❧❧ ✇❤❡r❡ t❤❡ ✈❡❧♦❝✐t② ✐s ❧✐♥❡❛r ✇✐t❤ t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ✇❛❧❧ ✭✐❢ s❝❛❧❡❞ ♣r♦♣❡r❧②✮❀ ✐♥ t❤✐s r❡❣✐♦♥ t❤❡ ✈✐s❝♦✉s ❡✛❡❝ts ❛r❡ ❞♦♠✐♥❛♥t✳ • ❇✉✛❡r ❧❛②❡r ✭ 5 ≤ y + < 30 ✮✿ ✐♥ t❤✐s r❡❣✐♦♥ t❤❡ ✢♦✇ ❜❡❣✐♥s t❤❡ tr❛♥s✐t✐♦♥ t♦ t❤❡ t✉r❜✉❧❡♥❝❡ ❛♥❞ t❤❡ ✈✐s❝♦✉s ❡✛❡❝ts ❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ♦❢ t❤❡ ❝♦♥✈❡❝t✐✈❡ ♦♥❡✳ ❚❤❡ ✈❡❧♦❝✐t② ♣r♦✜❧❡ ✐s ♥♦ ❧♦♥❣❡r ❧✐♥❡❛r✳ • ❚❤❡ ▲♦❣✲❧❛✇ r❡❣✐♦♥ ✭ 30 ≤ y + < 500 ✮✿ ❛❢t❡r t❤❡ ❜✉✛❡r r❡❣✐♦♥ t❤❡ tr❛♥s✐t✐♦♥ ✐s ❝♦♠♣❧❡t❡✱ t❤❡ ✢♦✇ ✐s ❢✉❧❧② t✉r❜✉❧❡♥t ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ✢♦✇ ✈❡❧♦❝✐t② ✐s r❡❧❛t❡❞ t♦ t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ t❤❡ ❞✐st❛♥❝❡ t♦ t❤❡ ✇❛❧❧✳ ❚❤✐s ✐s ❦♥♦✇♥ ❛s t❤❡ ❧♦❣✲❧❛✇ r❡❣✐♦♥ ❛♥❞ t❤❡ ❝♦♥✈❡❝t✐✈❡ ❡✛❡❝t ❛r❡ ❞♦♠✐♥❛♥t ♦♥ t❤❡ ✈✐s❝♦✉s✳ ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❛t r✐❜❧❡t ❛r❡ ❛❜♦✉t 10 − 15 ✉♥✐t ✇❛❧❧✱ s♦ t❤❡ ✈❛❧❧❡②s st❛② ✐♥ t❤❡ ✈✐s❝♦✉s s✉❧❛②❡r ❛♥❞ t✐♣s ✐♥ t❤❡ ❜✉✛❡r✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ t♦ t❤❡ ❧❛②❡r ❥✉st ❛❞❥❛❝❡♥t t♦ t❤❡ ✇❛❧❧✱ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r✳ ❲❤❡r❡❛s t❤❡ ♠❛✐♥ ♣❛rt ♦❢ ❛ t✉r❜✉❧❡♥t ❜♦✉♥❞❛r② ❧❛②❡r ❡①❤✐❜✐ts ❝❤❛♦t✐❝ ✢✉✐❞ ♠♦t✐♦♥✱ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r s❤♦✇s q✉✐t❡ r❡❣✉❧❛r ❧♦♥❣✐t✉❞✐♥❛❧ ✈♦rt❡①✱ ✇❤✐❝❤ ♣❧❛② ❛ ❢✉♥❞❛♠❡♥t❛❧ r♦❧❡ ✐♥ t❤❡ ♠♦♠❡♥t✉♠ ❡①❝❤❛♥❣❡ ❛♥❞ t❤❡ t✉r❜✉❧❡♥❝❡ ❣❡♥❡r❛t✐♦♥✳ ❚❤❡s❡ r❡❣✉❧❛r ♣❛tt❡r♥s ❧❡❛❞ t♦ t❤❡ ❣❡♥❡r❛t✐♦♥ ♦❢ t❤❡ s♦ ❝❛❧❧❡❞ ❧♦✇✲s♣❡❡❞

  17. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✶✹ ❋✐❣✉r❡ ✷✳✶✿ ❙tr✉❝t✉r❡ ♦❢ t❤❡ t✉r❜✉❧❡♥t ❜♦✉♥❞❛r② ❧❛②❡r✳ str❡❛❦s✱ t❤❛t ✇✐❧❧ ❜✉rst ✐♥ t❤❡ ✉♣♣❡r ③♦♥❡s ♦❢ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ❛♥❞ ✇✐❧❧ ♣r♦♠♦t❡ t✉r❜✉❧❡♥❝❡✳ ❚❤❡ r✐❜❧❡t ❛r❡ ❝❛♣❛❜❧❡ t♦ ♦❜st❛❝❧❡ t❤❡ ❝r♦ss ✢♦✇ t❤❛t ❣❡♥❡r❛t❡ s✉❝❤ ❛ str✉❝t✉r❡s✱ t❤❡r❡❜② t❤❡② ❝❛♥ ❤❛♠♣❡r t❤❡ t✉r❜✉❧❡♥❝❡ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ✐♥❝r❡❛s❡ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r t❤✐❝❦♥❡ss✱ ❛s ❞❡♠♦♥str❛t❡❞ ❜② ❍♦♦s❤♠❛♥❞ ❬✶✵❪✳

  18. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✶✺ ✷✳✷ ▼❛t❤❡♠❡t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥ ✷✳✷✳✶ ❙t♦❦❡s ❡q✉❛t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ♣r❡s❡♥t t❤❡ ♣❛ss❛❣❡s t❤❛t ❧❡❛❞ ✉s t♦ t❤❡ ❣♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✢♦✇ ✐♥ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r✳ ❋✐rst ♦❢ ❛❧❧ ✇❡ ✇r✐t❡ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✉♥❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❤②♣♦t❤❡s✐s ✿ ◆❡✇t♦♥✐❛♥ ✢✉✐❞✱ ✐♥❝♦♠♣r❡ss✐❜❧❡ ✢✉✐❞ ❛♥❞ ✐s♦t❤❡r♠❛❧ ✢♦✇✳ ❚❤❛♥❦s t♦ t❤❡s❡ ❛ss✉♠♣t✐♦♥ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿  ∇ · u = 0       ✭✷✳✷✮  � ∂ u �   = −∇ p + µ ∇ 2 u .  ρ ∂t + ( u · ∇ ) u   ❍♦✇❡✈❡r✱ ✇❡ ❢♦❝✉s t❤❡ ❛tt❡♥t✐♦♥ ♦♥ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r✱ t❤❡r❡❢♦r❡✱ ❢✉rt❤❡r s✐♠♣❧✐✜❝❛✲ t✐♦♥s ❛r❡ ♣♦ss✐❜❧❡✳ ■♥t♦ s✉❝❤ ❛ r❡❣✐♦♥✱ t❤❡ ❢♦r❝❡s ❞✉❡ t♦ t❤❡ ✈✐s❝♦s✐t② ♦❢ t❤❡ ✢✉✐❞ ❝♦♥tr♦❧ t❤❡ ✢♦✇✱ ❛♥❞ t❤❡ ✐♥❡rt✐❛❧ ❢♦r❝❡s ❜❡❝♦♠❡ ♥❡❣❧✐❣✐❜❧❡✳ ■♥ ♦r❞❡r t♦ ❡①♣❧♦✐t t❤✐s ❢❛❝t ❛♥❞ s✐♠♣❧✐❢② t❤❡ ❡q✉❛t✐♦♥s✱ ✐t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❛ss❡ss t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ✈❛r✐♦✉s t❡r♠s ✐♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s ❣♦❛❧✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ r❡✲✇r✐t❡ t❤❡ ❢♦r♠❡r ❡q✉❛t✐♦♥s ✷✳✷ ✐♥ t❤❡ ❞✐♠❡♥s✐♦♥❧❡ss ❢♦r♠✳ ❋✐rst ♦❢ ❛❧❧ ✇❡ ♥❡❡❞ t♦ ❝❤♦♦s❡ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❞✐♠❡♥s✐♦♥ t❤❛t ❛❧❧♦✇ ✉s t♦ ②✐❡❧❞s ❛❧❧ ✈❛r✐❛❜❧❡s ✐♥ ❞✐♠❡♥s✐♦♥❧❡ss ❢♦r♠❀ ❝❤♦♦s✐♥❣ L ❢♦r t❤❡ ❧❡♥❣t❤ s❝❛❧❡✱ U ❢♦r t❤❡ ✈❡❧♦❝✐t② ❛♥❞ T ❢♦r t❤❡ t✐♠❡✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✐♥tr♦❞✉❝❡ ❞✐♠❡♥s✐♦♥❧❡ss ✈❛r✐❛❜❧❡s ❛♥❞ ♦♣❡r❛t♦rs ❞❡♥♦t❡❞ ❜② ❛♥ ❤❛t✱ ❛♥❞ ❞❡✜♥❡❞ ❛s t ≡ t p ≡ p L u ≡ u x ≡ x ˆ ˆ ˆ U , ˆ L, ∇ ≡ L ∇ , T , ˆ µ U . ✭✷✳✸✮ ❙♦❧✈✐♥❣ ❡q✉❛t✐♦♥s ✷✳✸ ❢♦r t❤❡ ♣❤②s✐❝❛❧ ✈❛r✐❛❜❧❡s ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐♠❡♥s✐♦♥❧❡ss ♦♥❡s ❛♥❞ s✉❜st✐t✉t✐♥❣ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ✷✳✷ r❡s✉❧ts t + ρU 2 ρfU ∂ ˆ u = − µU p + µU u � � u · ˆ L 2 ˆ L 2 ˆ ∇ 2 ˆ ˆ ∇ ˆ ∇ ˆ u . ✭✷✳✹✮ ∂ ˆ L

  19. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✶✻ ❊✈❡♥t✉❛❧❧② ♠✉❧t✐♣❧②✐♥❣ ❢♦r L/ρU 2 t❤❡ ❡q✉❛t✐♦♥ t❛❦❡ t❤❡ ❢♦r♠ St ∂ ˆ u = − 1 P + 1 u � � u · ˆ ∇ ˆ ˆ ˆ ∇ 2 ˆ t + ˆ ∇ ˆ u , ✭✷✳✺✮ ∂ ˆ Re Re ✇❤❡r❡ St = f L Re = ρUL U , µ . ✭✷✳✻✮ St ✐s ❝❛❧❧❡❞ ❙tr♦✉❤❛❧ ♥✉♠❜❡r ❛♥❞ r❡♣r❡s❡♥t t❤❡ ✉♥st❡❛❞② ❡✛❡❝t ❞✉❡ t♦ t❤❡ t✉r❜✉❧❡♥t ✢✉❝t✉❛t✐♦♥s✱ ❛♥❞ Re ✐s t❤❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r✱ ✇❤♦s❡ ♣❤②s✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s t❤❡ r❛t✐♦ ♦❢ t❤❡ ✐♥❡rt✐❛❧ ❢♦r❝❡s ❛♥❞ t❤❡ ✈✐s❝♦✉s ♦♥❡s✳ ❚❤❛♥❦s t♦ t❤✐s ❢♦r♠ ♦❢ t❤❡ ❡q✉❛t✐♦♥✱ ✐s ♥♦✇ ♣♦ss✐❜❧❡ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✇❡✐❣❤t ♦❢ ❡❛❝❤ t❡r♠✳ ❇❡❝❤❡rt ❡t ❛❧✳ ❬✷✼❪ ❝♦♥❝❧✉❞❡✱ ❛❧s♦ ♦♥ t❤❡ ❜❛s✐s ♦❢ ❡①♣❡r✐♠❡♥t❛❧ ♠❡❛s✉r❡♠❡♥ts ❜② ♦t❤❡r ❛✉t❤♦rs✱ t❤❛t t❤❡ ❢r❡q✉❡♥❝✐❡s t②♣✐❝❛❧❧② ❡♥❝♦✉♥t❡r❡❞ ✐♥ t✉r❜✉❧❡♥t ✢♦✇ ❛r❡ ❧♦✇ ❡♥♦✉❣❤ ❢♦r t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r t♦ ❜❡ tr❡❛t❡❞ ❛s q✉❛s✐✲st❡❛❞②✳ ❚❤❡r❡❢♦r❡✱ ✇❡ s✉♣♣♦s❡ t❤❛t St ∼ 0 ❛♥❞ ✇❡ ♥❡❣❧❡❝t t❤❡ ✉♥st❡❛❞② t❡r♠✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❤②♣♦t❤❡s✐s✱ ✇❤✐❝❤ ✐s t❤❡ ❤❡❛rt ♦❢ t❤✐s st✉❞②✱ ❝♦♥❝❡r♥ t❤❡ r❡②♥♦❧❞s ♥✉♠❜❡r✳ ❆s ❡①♣❧❛✐♥❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✶ ✐♥ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r t❤❡ ✈✐s❝♦✉s ❢♦r❝❡s ❛r❡ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♦♥❡s✱ t❤❡♥ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❛r❣✉❡ t❤❛t Re ∼ 0 ♦r 1 /Re → ∞ ✳ ❚❤✐s ❜r✐♥❣ t♦ ❛ ✈❡r② str♦♥❣ s✐♠♣❧✐✜❝❛t✐♦♥✿ ❝♦♥✈❡❝t✐✈❡ t❡r♠ ✐s ♥❡❣❧✐❣✐❜❧❡ r❡s♣❡❝t t♦ t❤❡ ❞✐✛✉s✐♦♥ ❛♥❞ ♣r❡ss✉r❡ t❡r♠ ❧❡❛❞✐♥❣ t♦ ∇ ˆ ˆ P = ˆ ∇ 2 ˆ u . ✭✷✳✼✮ ❚❤❡ r❡❧❛t✐♦♥ ✷✳✼✱ ❝❛❧❧❡❞ ❙t♦❦❡s ❡q✉❛t✐♦♥✱ ✐s t❤❡ ❣♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥ ❢♦r ❛❧❧ t❤❡ ❝r❡❡♣✲ ✐♥❣ ✢♦✇✱ ✐♥ ✇❤✐❝❤ t❤❡ ✈✐s❝♦✉s ❢♦r❝❡s ❛r❡ ❞♦♠✐♥❛♥t✳ ❋r♦♠ ♥♦✇ ♦♥✱ t❤❡ ❤❛t ♦✈❡r t❤❡ ❞✐♠❡♥s✐♦♥❧❡ss q✉❛♥t✐t✐❡s ✐s ❞r♦♣♣❡❞ ❢♦r ❜r❡✈✐t②✱ ❛♥❞ t❤❡ ✈❛r✐❛❜❧❡s ♠✉st ❜❡ ❝♦♥s✐❞❡r❡❞ ❞✐♠❡♥s✐♦♥❧❡ss✳ ✷✳✷✳✷ ❉❡❝♦✉♣❧✐♥❣ ♦❢ t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥ ❲❡ ✇✐s❤ t♦ st✉❞② t❤❡ ❙t♦❦❡s ✢♦✇ ♦❢ ❛ ✈✐s❝♦✉s ✢✉✐❞ ❛❧♦♥❣s✐❞❡ ❛♥ ✐♥✜♥✐t❡ ❝♦rr✉❣❛t❡❞ ✇❛❧❧ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❣✐✈❡♥ s❤❡❛r✱ ♦r ✈❡❧♦❝✐t② ❣r❛❞✐❡♥t✱ ✐♥ t❤❡ r❡❣✐♦♥ ❢❛r ❢r♦♠ t❤❡ ✇❛❧❧✳ ❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✱ ✇❡ s❤❛❧❧ ❛ss✉♠❡ t❤❡ ✇❛❧❧ s✉r❢❛❝❡ t♦ ❜❡ tr❛♥s❧❛t✐♦♥✲✐♥✈❛r✐❛♥t

  20. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✶✼ ❋✐❣✉r❡ ✷✳✷✿ ❲❛❧❧ t❡①t✉r❡ ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♦r❞✐♥❛t❡s✳ ✐♥ t❤❡ z ✲❞✐r❡❝t✐♦♥✱ ❛♥❞ ♣❡r✐♦❞✐❝ ❛❧♦♥❣ t❤❡ x ✲❞✐r❡❝t✐♦♥✳ ❚❤❡ ❞✐r❡❝t✐♦♥ ♥♦r♠❛❧ t♦ t❤❡s❡ ✇✐❧❧ ❜❡ t❤❡ y ✲❛①✐s ❛s s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✷✳ ❚❤❡ ❣❡♦♠❡tr✐❝ ✇❛❧❧ ✐♥✈❛r✐❛♥❝❡ ❛❧♦♥❣ t❤❡ z ✲❛①✐s s✉❣❣❡st t❤❛t ♣❤②s✐❝❛❧ q✉❛♥t✐t✐❡s ❛r❡ t♦♦ ✐♥ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✳ ❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ❛❜♦✈❡✱ t❤❡ ❡q✉❛t✐♦♥s ✷✳✼ ❝❛♥ ❜❡ ❢✉rt❤❡r s✐♠♣❧✐✜❡❞ s❡tt✐♥❣ t♦ ③❡r♦ ❛❧❧ t❤❡ ❞❡r✐✈❛t✐✈❡s ✐♥ z ✓  ∂ 2 u ∂x 2 + ∂ 2 u ∂ 2 u   ∂P  ✓ ∂y 2 + ✓ ∂z 2 ∂x                 ∂P  ✓  ∂x 2 + ∂ 2 v ∂ 2 v ∂ 2 v     ✓ = µ ✭✷✳✽✮ ∂y 2 +     ∂y   ✓ ∂z 2                 ✓ ∂P   ∂ 2 w ∂x 2 + ∂ 2 w ∂ 2 w �   ✓  �  ∂y 2 + ✓ ∂z � ∂z 2 ■t ✐s ❡❛s② t♦ ♦❜s❡r✈❡ t❤❛t t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡ t✇♦ ❞✐♠❡♥s✐♦♥❛❧✱ ❢✉rt❤❡r♠♦r❡✱ t❤❡ ❡q✉❛✲ t✐♦♥ ❢♦r w ❞❡❝♦✉♣❧❡s ❢r♦♠ t❤❡ s②st❡♠ ❛♥❞ ✐t ✐s ❥✉st t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣❡r✐♦❞✐❝✐t② ❛❧❧♦✇s ✉s t♦ r❡❝❛st ❛♥ ✐♥✐t✐❛❧❧② ✸❉ ♣r♦❜❧❡♠ ❣♦✈❡r♥❡❞ ❜② t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥ ✐♥ t✇♦ ❞❡❝♦✉♣❧❡❞ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠s✿ ✶✳ ❚❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♣r♦❜❧❡♠ ❢♦r t❤❡ ✉♥❦♥♦✇♥ ✈❡❧♦❝✐t② w  ∇ 2 w = 0       ✭✷✳✾✮ w ( x, y 0 ) = 0     w y ( x, ∞ ) = 1   ✇❤❡r❡ y 0 ✐s t❤❡ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥ t❤❛t r❡♣r❡s❡♥t t❤❡ ♣r♦✜❧❡ ♦❢ t❤❡ ✇❛❧❧ ✐♥ t❤❡ ♣❧❛♥❡

  21. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✶✽ y x ❋✐❣✉r❡ ✷✳✸✿ ❉♦♠❛✐♥ ❝♦♥t♦✉r ✭r❡❞ ❧✐♥❡✮✳ ❚❤❡ ❛rr♦✇s ♦♥ t❤❡ ✉♣♣❡r ♣❛rt ♦❢ t❤❡ ❞♦♠❛✐♥ ✐♥❞✐❝❛t❡ t❤❛t ✐t ♠✉st ❜❡ ❝♦♥s✐❞❡r❡❞ ❢❛r ❢r♦♠ t❤❡ r✐❜❧❡t ✇❛❧❧✳ ✭ x ✱ y ✮✳ ✷✳ ❚❤❡ tr❛♥s✈❡rs❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ✉♥❦♥♦✇♥s u ✱ v ❛♥❞ P  ∇ p = ∇ 2 u , u = ( u, v )       ✭✷✳✶✵✮ u ( x, y 0 ) = (0 , 0)     u y ( x, ∞ ) = 1   ❚❤❡s❡ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠s ❤❛✈❡ ❜❡❡♥ s♦❧✈❡❞ ✐♥s✐❞❡ t❤❡ ❞♦♠❛✐♥ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✸✳ ❚❤❡ ♥♦ s❧✐♣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❤❛s ❜❡❡♥ ✉s❡❞ ❛t t❤❡ ✇❛❧❧✱ ❛♥❞ ❛ ❝♦♥st❛♥t s❤❡❛r ❤❛s ❜❡❡♥ ✐♠♣♦s❡❞ ❢❛r ❢r♦♠ t❤❡ ❝❛✈✐t②✳

  22. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✶✾ ✷✳✸ Pr♦tr✉s✐♦♥ ❤❡✐❣❤ts ■t ✐s ✐♠♣♦rt❛♥t t♦ ❞❡✜♥❡ t❤❡ ❝♦♥❝❡♣t ♦❢ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❜❡❝❛✉s❡ ✇❡ ✇✐❧❧ ✉s❡ t❤❡♠ ❛s ❜♦✉♥❞❛r✐❡s ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ♠❛❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠✳ ❲❡ ❝❛♥ ❞❡✜♥❡ ♦♥❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❢♦r ❡❛❝❤ ♣r♦❜❧❡♠ ❞❡✜♥❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✷✳✷✿ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♣❛r❛❧❧❡❧ ♣r♦❜❧❡♠ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ♣❛r❛❧❧❡❧ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t✱ ❛♥❞ t❤❛t ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ tr❛♥s✈❡rs❡ ♣r♦❜❧❡♠ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t✳ ◆♦✇✱ ❧❡t ✉s ♣r♦❝❡❡❞ ✇✐t❤ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❢♦❧❧♦✇✐♥❣ t❤❡ ▲✉❝❤✐♥✐✬s ♣❛ss❛❣❡s ❬✷✵❪✳ ❙✐♥❝❡ w ✐s ♣❡r✐♦❞✐❝ ❛❧♦♥❣ x ✲❛①✐s✱ ✇❡ ❝❛♥ ❡①♣r❡ss ✐t ✉s✐♥❣ t❤❡ ❋♦✉r✐❡r s❡r✐❡s + ∞ � W n e inx , w = ✭✷✳✶✶✮ n = −∞ ✇❤❡r❡  W n = a n e −| n | y + b n e | n | y , n � = 0   ✭✷✳✶✷✮ W n = a 0 + b 0 y, n = 0 .   ❚❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t w y = 1 ❢♦r y → + ∞ ✜①❡s ❛❧❧ b ✲❝♦❡✣❝✐❡♥ts ❛s b 0 = 1 ❛♥❞ b n = 0 ❢♦r n � = 0 ✱ ❛♥❞ ❧❡❛✈❡s t❤❡ a ✲❝♦❡✣❝✐❡♥ts t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❝♦♥❞✐t✐♦♥ ❛t t❤❡ ✇❛❧❧ s✉r❢❛❝❡✳ ❙✐♥❝❡ ❛❧❧ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ s❡r✐❡s ✷✳✶✶ ❜✉t W 0 ✈❛♥✐s❤ ❡①♣♦♥❡♥t✐❛❧❧② ❛t ✐♥✜♥✐t②✱ t❤❡ s♦❧✉t✐♦♥ ♦❜t❛✐♥❡❞ ✇✐❧❧ ❛♣♣r♦❛❝❤ t❤❡ ❧✐♥❡❛r ❜❡❤❛✈✐♦✉r w ∼ a 0 + y ✱ ❛♥❞ t❤✉s ✐t ✐♠✐t❛t❡s t❤❡ ❧✐♥❡❛r ✈❡❧♦❝✐t② ♣r♦✜❧❡ ♣r♦❞✉❝❡❞ ❜② ❛ ♣❧❛♥❡ ✇❛❧❧ ❧♦❝❛t❡❞ ❛t y = − a 0 ✳ P❛r❛❧❧❡❧ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ✐s ❞❡✜♥❡❞ ❛s h || = a 0 ✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐t r❡♣r❡s❡♥ts t❤❡ ❞✐st❛♥❝❡ ♦❢ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ✈✐rt✉❛❧ ♣❧❛♥❡ ✇❛❧❧ ❢r♦♠ t❤❡ y ✲❛①✐s ♦r✐❣✐♥ ✭✇❤✐❝❤ ✐s ❧♦❝❛t❡❞ ♦♥ t❤❡ r✐❜❧❡t t✐♣s✮✳ ❚❤✐s q✉❛♥t✐t✐❡② ❤❛✈❡ ❜❡❡♥ ✜rst ❞❡r✐✈❡❞ ❜② ❇❡❝❤❡rt ❡t ❛❧✳ ❬✷✼❪✱ t❤❛t ❞❡✜♥❡❞ ✐t ❛s t❤❡ ❛♣♣❛r❡♥t ♦r✐❣✐♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② ♣r♦✜❧❡ ♦♥ ❛ ❣r♦♦✈❡❞ s✉r❢❛❝❡✳ ▲✉❝❤✐♥✐ s❤♦✇s t❤❛t ❛ s✐♠✐❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ♠❛② ❜❡ ❞❡✜♥❡❞ ❢♦r t❤❡ ❝r♦ss✲✢♦✇ ❛s ✇❡❧❧✱ t❤❡r❡❢♦r❡ ✐t ✐s ❡✈❡♥ ♣♦ss✐❜❧❡ t♦ ✇r✐t❡ w ∼ h || + y ✭✷✳✶✸✮ u ∼ h ⊥ + y ✭✷✳✶✹✮ ❜❡✐♥❣ h ⊥ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t✱ ❝♦♥❝❡♣t✉❛❧❧② ❞❡✜♥❡❞ ❛s t❤❡ ♣❛r❛❧❧❡❧ ♦♥❡✳ ❚❤❡② ❝❛♥ ❜❡ ✉s❡❞ t♦ r❡♣r❡s❡♥t t❤❡ ✐♥✢✉❡♥❝❡ ♦❢ t❤❡ r✐❜❧❡ts ♦♥ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r✳ ■♥

  23. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✵ x y h h || ❋✐❣✉r❡ ✷✳✹✿ ❈r♦ss✲s❡❝t✐♦♥ ♦❢ ❛ r✐❜❧❡t ♣❧❛t❡s ❛♥❞ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✳ ✜❣✉r❡ ✷✳✹ ✐s s❤♦✇♥ t❤❡ ✈✐rt✉❛❧ ♣❧❛♥❡ ✇❛❧❧ ✇❤✐❝❤ ❝r❡❛t❡ t❤❡ ❧✐♥❡❛r ✈❡❧♦❝✐t② ♣r♦✜❧❡✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t t❤❡ ✈❛❧✉❡ ♦❢ ❜♦t❤ ♣❛r❛❧❧❡❧ ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❞❡♣❡♥❞s ♦♥ t❤❡ ♦r✐❣✐♥ ❢ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠✱ ❤❡♥❝❡ ✇❡ ❞❡✜♥❡ ❛ ♥❡✇ q✉❛♥t✐t② ✇❤✐❝❤ ❞♦❡s ♥♦t ❞❡♣❡♥❞s ♦♥ ✐t ∆ h = h || − h ⊥ . ✭✷✳✶✺✮ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ✭ ∆ h ✮ ❤❛s ❜❡❡♥ r❡❧❛t❡❞ t♦ t❤❡ ❡✛❡❝✲ t✐✈❡♥❡ss ♦❢ r✐❜❧❡ts ✭▲✉❝❤✐♥✐ ❡t ❛❧✳ ❬✷✵❪✮✳ ❆♥ ❤✐❣❤ ✈❛❧✉❡ ♦❢ ✐t ✐♠♣❧② ❛♥ ❡✣❝✐❡♥t r✐❜❧❡t✱ ❤❡♥❝❡ ❛♥ ❤✐❣❤ ❞r❛❣ r❡❞✉❝t✐♦♥ ✭ DR ✮✳ ❖♥ t❤❡ ❝♦♥tr❛r②✱ ❛ ❧♦✇ ✈❛❧✉❡ ♦❢ ∆ h ✐s ❛ss♦❝✐❛t❡❞ t♦ ❧♦✇❡r ❞r❛❣ r❡❞✉❝t✐♦♥✳ ❚❤✐s t❤❡♦r② ❝❛♥♥♦t r❡✈❡❛❧s t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ❛♠♦✉♥t✱ ❜✉t ✐t ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ❛ q✉❛❧✐t❛t✐✈❡ ❛♥❛❧②s✐s✳ ■♥ t❤❡ ♣r❡s❡♥t ✇♦r❦ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ❛♥❞ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❞✐✛❡r❡♥❝❡ ✐s ✉s❡❞ t♦ q✉❛❧✐t❛t✐✈❡❧② ❛♥❛❧②③❡ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t ❡✛❡❝t✐✈❡♥❡ss✱ ♠♦r❡♦✈❡r ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✱ ❛ ✇❛② t♦ q✉❛♥t✐t❛t✐✈❡❧② r❡❧❛t❡ ∆ h ❛♥❞ DR ✐s s❤♦✇♥✳

  24. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✶ ✷✳✹ ❇♦✉♥❞❛r② ✐♥t❡❣r❛❧ ♠❡t❤♦❞ ❖✈❡r t❤❡ ❧❛st ✸✵ ②❡❛rs✱ ❛ ✈❡r② ♣♦♣✉❧❛r ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❤❛s ❞❡✈❡❧♦♣❡❞ ❛♥❞ ♠❛t✉r❡❞✳ ❚❤✐s ❡❧❡❣❛♥t ❛♣♣r♦❛❝❤ ✐s t❤❡ ❇♦✉♥❞❛r② ■♥t❡❣r❛❧ ▼❡t❤♦❞ ✭❇■▼✮ ❛♥❞ ✐t ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ✐♥ ♠❛♥② ❛r❡❛s ♦❢ t❤❡ ❡♥❣✐♥❡❡r✐♥❣ ❛♥❞ s❝✐❡♥❝❡✱ ✐♥❝❧✉❞✐♥❣ ✢✉✐❞ ♠❡❝❤❛♥✐❝s✱ ❛❝♦✉st✐❝s✱ ❡❧❡❝tr♦♠❛❣♥❡t✐❝s ❛♥❞ ❢r❛❝t✉r❡ ♠❡❝❤❛♥✐❝s✳ ❚❤❡ ♠❡t❤♦❞ ❛✐♠ ❛t tr❛♥s❢♦r♠✐♥❣ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✐♥ ❛♥ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✱ ❝❛❧❧❡❞ ❇♦✉♥❞❛r② ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥ ✭❇■❊✮✱ ✇❤✐❝❤ r❡❧❛t❡ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ ✐♥t♦ t❤❡ ❞♦♠❛✐♥ t♦ ♦♥❧② t❤❡ ✈❛❧✉❡ ♦♥ t❤❡ ❜♦✉♥❞❛r② ✈❛❧✉❡s✳ ❲❡ ♥♦✇ ❞❡r✐✈❡ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ▲❛♣❧❛❝❡ ❛♥❞ t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥✳ ✷✳✹✳✶ ❇■❊ ❢♦r t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ t✉r♥ t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥✱ ❝♦♥s✐❞❡r❡❞ ✐♥t♦ ❛ ❞♦♠❛✐♥ ❉ ✇✐t❤ ❜♦✉♥❞❛r② ❈✱ ✐♥t♦ t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✳ ❚♦ ❞♦ s♦✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ●r❡❡♥✬s s❡❝♦♥❞ ✐❞❡♥t✐t② ψ ∇ 2 w − w ∇ 2 ψ = ∇ · ( ψ ∇ w − w ∇ ψ ) , ✭✷✳✶✻✮ ✇❤❡r❡ ψ ✐s ❛♥ ❛r❜✐tr❛r② s❝❛❧❛r ❢✉♥❝t✐♦♥✳ ❯s✉❛❧❧② ✇❡ s❡t ψ = G ✱ ✇❤❡r❡ G ✐s t❤❡ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥❀ ❜② ❞❡✜♥✐t✐♦♥ G s❛t✐s❢② t❤❡ s✐♥❣✉❧❛r✐t② ❢♦r❝❡❞ ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ∇ 2 G ( x , x 0 ) + δ ( x − x 0 ) = 0 , ✭✷✳✶✼✮ ✇❤❡r❡ • x = ( x, y ) ✐s t❤❡ ❝♦♦r❞✐♥❛t❡ ✈❡❝t♦r✳ • x 0 = ( x 0 , y 0 ) ✐s ❛ s♣❡❝✐✜❝ ♣♦✐♥t ✐♥t♦ t❤❡ ❞♦♠❛✐♥ ❝❛❧❧❡❞ ♣♦❧❡ ♦r s✐♥❣✉❧❛r✐t② ♣♦✐♥t ♦r ❥✉st t❤❡ s✐♥❣✉❧❛r✐t② ✳ • δ ( x − x 0 ) ✐s t❤❡ ❉✐r❛❝✬s ❞❡❧t❛ ❢✉♥❝t✐♦♥ ✐♥ t✇♦ ❞✐♠❡♥s✐♦♥s✳ ■♥ t❤❡ ♥❡①t st❡♣✱ ✇❡ ✐♥t❡❣r❛t❡ t❤❡ ●r❡❡♥✬s s❡❝♦♥❞ ✐❞❡♥t✐t② ✷✳✶✻ ♦✈❡r t❤❡ ❞♦♠❛✐♥ D ′ = D − D ǫ ✱ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✺❀ ✐♥ ❢❛❝t ✇❡ ❞❡❧❡t❡ ❢r♦♠ D ❛ s♠❛❧❧ ❞✐s❦ D ǫ ♦❢ r❛❞✐✉s ǫ ❛♥❞ ❝❡♥t❡r❡❞ ❛t x 0 ✱ t♦ ❛✈♦✐❞ t❤❡ ❡♥❝❧♦s✐♥❣ ♦❢ t❤❡ ♣♦✐♥t x = x 0 ✱ ✇❤✐❝❤ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ t❤❡ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ❞♦♠❛✐♥✳

  25. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✷ C D C dl D' x 0 D dl n ❋✐❣✉r❡ ✷✳✺✿ ❉♦♠❛✐♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ D ′ ✳ �� �� D ′ ( G ∇ 2 w − w ∇ 2 G ) dS = D ′ ∇ · ( G ∇ w − w ∇ G ) dS. ✭✷✳✶✽✮ ▲❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷✳✶✽ ✐s ❡q✉❛❧ t♦ ③❡r♦✱ s✐♥❝❡ ∇ 2 w = 0 ✱ ❛♥❞ ∇ 2 G = 0 ✇❤❡♥ x � = x 0 ❜❡❝❛✉s❡ ♦❢ t❤❡ ❉✐r❛❝✬s ❞❡❧t❛ ❢✉♥❝t✐♦♥ ♣r♦♣❡rt✐❡s✳ ❆♣♣❧②✐♥❣ t❤❡ ❞✐✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ❛t t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷✳✶✽ ✇❡ ♦❜t❛✐♥ � ( G ∇ w − w ∇ G ) · n dl = 0 , ✭✷✳✶✾✮ C + C ǫ ✇❤❡r❡ C ✐s t❤❡ ❜♦✉♥❞❛r② t♦ D ✱ C ǫ ✐s t❤❡ ❜♦✉♥❞❛r② t♦ D ǫ ✱ ❛♥❞ l ♠❡❛s✉r❡s ❛r❝ ❧❡♥❣t❤ ❛❧♦♥❣ ❡✐t❤❡r C ♦r C ǫ ✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❛❧♦♥❣ C ǫ ❛♥❞ ❞❡✜♥✐♥❣ r = | x − x 0 | ✱ t❤✐s ❡①♣r❡ss✐♦♥ ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞ s✉❜st✐t✉t✐♥❣ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ❜❡t✇❡❡♥ n ❛♥❞ ❜♦t❤ t❤❡ ❣r❛❞✐❡♥ts✱ ✇✐t❤ t❤❡ r ✲❞❡r✐✈❛t✐✈❡ ❛s ❢♦❧❧♦✇ � � ( G∂w � w∂G I ǫ = ( G ∇ w − w ∇ G ) · n dl = ∂r ) dl − ∂r dl, ✭✷✳✷✵✮ C ǫ C ǫ C ǫ ❚❤✐s ♣❛ss❛❣❡ ❝❛♥ ❜❡ ❞♦♥❡ ❜❡❝❛✉s❡ C ǫ ✐s ❛ ❝✐r❝✉♠❢❡r❡♥❝❡ ✇✐t❤ ♥♦r♠❛❧ n ✳ ❲❤❡♥ x � = x 0 ♦r r � = 0 t❤❡ s✐♥❣✉❧❛r ❢♦r❝❡❞ ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❜❡❝♦♠❡ ∇ 2 G = 0 ✱ ❛♥❞ ❛ss✉♠✐♥❣ G ❞❡♣❡♥❞❡♥t ♦♥❧② ♦♥ r ✱ ✐t ❝❛♥ ❜❡ ❢♦✉♥❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ r❡❛❞s G = λ ❧♦❣ r, ✭✷✳✷✶✮ ✇❤❡r❡ λ ✐s ❛♥ ❛r❜✐tr❛r② ❝♦♥st❛♥t✳ ❚❤✐s ❢✉♥❞❛♠❡♥t❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷✳✶✼ ✐s ❝❛❧❧❡❞ ❢r❡❡✲s♣❛❝❡ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ✳ ❲❡ ❝❛♥ ❡♠♣❧♦② t❤❡ ❢r❡❡✲s♣❛❝❡ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ✐♥

  26. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✸ ♦r❞❡r t♦ s✐♠♣❧✐❢② t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ I ǫ ✱ ✐♥ ❢❛❝t✱ s✉❜st✐t✉t✐♥❣ ✷✳✷✶ ✐♥t♦ ✷✳✷✵✱ ✇❡ ♦❜t❛✐♥ � ( λ ❧♦❣ r∂w � wλ I ǫ = ∂r ) dl − r dl. ✭✷✳✷✷✮ C ǫ C ǫ ❙✐♥❝❡ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ✐s ❡q✉❛❧ t♦ ǫ ✱ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ✐s ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ s✉❜st✐t✉t❡ r ✇✐t❤ ǫ ✱ ♠♦r❡♦✈❡r✱ ❧❡tt✐♥❣ ǫ → 0 ✇❡ ✜♥❞ I ǫ = λ ❧♦❣ ǫ∂w ( x 0 ) � dl − w ( x 0 ) λ � dl = ∂r ǫ C ǫ C ǫ = λ ❧♦❣ ǫ∂w ( x 0 ) 2 πǫ − w ( x 0 ) λ ǫ πǫ = − 2 πλw ( x 0 ) . ✭✷✳✷✸✮ ∂r ❘❡t✉r♥✐♥❣ t♦ t❤❡ ❡q✉❛t✐♦♥ ✷✳✶✾✱ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ t❤❡ ✐♥t❡❣r❛❧ ♦♥ C ǫ ✇✐t❤ t❤❡ ✷✳✷✸✱ ❛♥❞ ❝❤♦♦s❡ λ = − 1 / 2 π ✳ ❲❡ ❞❡r✐✈❡ t❤❡ ✈❡❧♦❝✐t② w ❢r♦♠ ❡q✉❛t✐♦♥ ✷✳✶✾✱ t❤❛t ❜❡❝♦♠❡ t❤❡ ❇♦✉♥❞❛r② ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥ ✭❇■❊✮ ❢♦r ♣♦t❡♥t✐❛❧ ✢♦✇ � � w ( x 0 ) = w ( x ) n · ∇ G ( x , x 0 ) dl − G ( x , x 0 ) n · ∇ w ( x ) dl, ✭✷✳✷✹✮ C C ❚❤✐s ✐s t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥❀ t❤❡ ✜rst ✐♥t❡❣r❛❧ ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❝❛❧❧❡❞ ❉♦✉❜❧❡✲▲❛②❡r P♦t❡♥t✐❛❧ ✭❉▲P✮ ❛♥❞ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ❝❛❧❧❡❞ ❙✐♥❣❧❡✲▲❛②❡r P♦t❡♥t✐❛❧ ✭❙▲P✮✳ ❚❤❡ ❡q✉❛t✐♦♥ ✷✳✷✹ ❝❛♥ ❜❡ ✉s❡❞ t♦ s♦❧✈❡ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♣r♦❜❧❡♠✱ ❛♥❞ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ w ✐♥ ❛❧❧ t❤❡ ❞♦♠❛✐♥✳ ❲❡ ♥♦t❡ t❤❛t ♦♥ t❤❡ r✐❣❤t✲ ❤❛♥❞ s✐❞❡✱ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ w ❛♣♣❡❛rs ✐♥s✐❞❡ ✐♥t❡❣r❛❧s ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ♦✈❡r t❤❡ ❜♦✉♥❞❛r② C ✳ ❙♦ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐♥✈♦❧✈❡ ♦♥❧② t❤❡ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♦❢ w ✳ ■♥ ❣❡♥❡r❛❧ ✇❤❡♥ ❛ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❧❡♠ ♠✉st ❜❡ s♦❧✈❡❞✱ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❛r❡ ♣r♦✈✐❞❡❞ ❛♥❞ t❤✐s s✉❣❣❡st t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② t♦ ♣r♦❝❡❡❞✱ t❤❛t r❡♣r❡s❡♥t t❤❡ ❤❡❛rt ♦❢ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ♠❡t❤♦❞✿ ✶✳ ❚❛❦❡ t❤❡ ♣♦✐♥t x 0 t♦ ❧✐❡ ♦♥ t❤❡ ❜♦✉♥❞❛r② C ✳ ❚❤❡♥ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✐♥✈♦❧✈❡s ♦♥❧② ✈❛❧✉❡s ♦♥ t❤❡ ❜♦✉♥❞❛r②✳ ✷✳ ❙♦❧✈❡ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ✉♥❦♥♦✇♥s ❜♦✉♥❞❛r② ✈❛❧✉❡s✱ t❤❛t ❞❡♣❡♥❞s ❜② t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ♣r♦✈✐❞❡❞✳ ✸✳ ❈♦♠♣✉t❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✷✳✷✹ ✐♥ ♦r❞❡r t♦ ✜♥❞ w ❛♥②✇❤❡r❡ ✐♥ t❤❡ ❞♦♠❛✐♥✳

  27. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✹ ❚❤❡ ✜rst st❡♣ r❡q✉✐r❡ ✉s t♦ t❛❦❡ t❤❡ s✐♥❣✉❧❛r✐t② t♦ ❧✐❡ ♦♥ C ✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t t❤❡ ❙▲P ✐s ❝♦♥t✐♥✉♦✉s ❛s t❤❡ ♣♦✐♥t x 0 ❛♣♣r♦❛❝❤❡s ❛♥❞ t❤❡♥ ❝r♦ss❡s C ✱ ✐♥ ❝♦♥tr❛st✱ ❛s x 0 ❝r♦ss❡s C ✱ t❤❡ ❉▲P ✉♥❞❡r❣♦❡s ❛ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t②✳ ■♥ ♦r❞❡r t♦ ✜♥❞ ❛♥ ❡①♣r❡ss✐♦♥ ❢♦r ❉▲P ✇❤❡♥ x 0 ❧✐❡ ♦♥ C ✱ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t x 0 → C ❛♥❞ ✜♥❞ � PV w ( x ) n · ∇ G ( x , x 0 ) dl ± 1 DLP = 2 w ( x 0 ) , ✭✷✳✷✺✮ C ✇❤❡r❡ PV ✐♥❞✐❝❛t❡s t❤❛t ✇❡ ❛r❡ t❛❦✐♥❣ t❤❡ ♣r✐♥❝✐♣❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✳ ■❢ x 0 ❛♣✲ ♣r♦❛❝❤❡s t❤❡ ❜♦✉♥❞❛r② ❢r♦♠ t❤❡ ✐♥s✐❞❡ ♣❧✉s s✐❣♥ ✐s ✈❛❧✐❞✱ ❛♥❞ ✐❢ ✐t ❛♣♣r♦❛❝❤❡s ❢r♦♠ t❤❡ ♦✉ts✐❞❡ ✐t ♠✉st ❜❡ ❝❤♦s❡♥ t❤❡ ♠✐♥✉s s✐♥❣✳ ❙✉♣♣♦s✐♥❣ t♦ ❜❡ ✐♥ t❤❡ ❢♦r♠❡r s✐t✉❛t✐♦♥✱ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ ✷✳✷✺ ✐♥ ✷✳✷✹ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✇✐t❤ x 0 ❧②✐♥❣ ♦♥ t❤❡ ❜♦✉♥❞❛r② � PV 1 � 2 w ( x 0 ) = w ( x ) n · ∇ G ( x , x 0 ) dl − G ( x , x 0 ) n · ∇ w ( x ) dl. ✭✷✳✷✻✮ C C ❚❤✐s ❡q✉❛t✐♦♥ ♦♥❧② ✐♥✈♦❧✈❡s ✈❛❧✉❡s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❚❤❡ s❡❝♦♥❞ st❡♣ ❞❡❛❧s ✇✐t❤ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷✳✷✻ ❛♥❞ t❤❡r❡ ❛r❡ t❤r❡❡ ♣♦ss✐❜❧❡ s✐t✉❛t✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ • ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✿ ❡q✉❛t✐♦♥ ✷✳✷✻ ✐s ✉s❡❞ t♦ ♦❜t❛✐♥ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ∇ w ♦♥ t❤❡ ❜♦✉♥❞❛r②✳ • ◆❡✉♠❛♥♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✿ ❡q✉❛t✐♦♥ ✷✳✷✻ ✐s ✉s❡❞ t♦ ♦❜t❛✐♥ t❤❡ ✈❛❧✉❡s ♦❢ w ♦♥ t❤❡ ❜♦✉♥❞❛r② • ❘♦❜✐♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✿ ❡q✉❛t✐♦♥ ✷✳✷✻ ✐s ✉s❡❞ t♦ ♦❜t❛✐♥ t❤❡ ✈❛❧✉❡ ♦❢ ∇ w ✇❤❡r❡ ✐s ❣✐✈❡♥ t❤❡ ✈❛❧✉❡s ♦❢ w ❛♥❞ ✈✐❝❡✈❡rs❛✳ ❖♥❝❡ t❤❡ st❡♣ t✇♦ ✐s ❞♦♥❡✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ s♦❧✈❡ ❡q✉❛t✐♦♥ ✷✳✷✹ ❢♦r t❤❡ ✉♥❦♥♦✇♥ w ( x 0 ) ✇✐t❤ x 0 ∈ D ✳ ■t ❝❛♥ ❜❡ ❤✐❣❤❧✐❣❤t❡❞ t❤❡ ♣r✐♥❝✐♣❛❧ ❢❡❛t✉r❡ ♦❢ t❤❡ ❇■▼✿ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✐♥ ❛ s♣❡❝✐✜❝ ♣♦✐♥t ✐♥ t❤❡ ❞♦♠❛✐♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✉s✐♥❣ ♦♥❧② t❤❡ ❜♦✉♥❞❛r② ✈❛❧✉❡s✱ ✇✐t❤♦✉t s♦❧✈✐♥❣ ❢♦r ❛❧❧ t❤❡ ❞♦♠❛✐♥✳

  28. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✺ ✷✳✹✳✷ ❇■❊ ❢♦r t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥ ❆s ✇❡❧❧ ❛s ❢♦r ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥✱ ✇❡ ❤❛✈❡ t♦ ✜♥❞ t❤❡ ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥ = µ ∂ 2 u i ∂p . ✭✷✳✷✼✮ ∂x i ∂x j ∂x j ❲❡ ✉s❡ t❤❡ ▲♦r❡♥t③ r❡❝✐♣r♦❝❛❧ r❡❧❛t✐♦♥✱ t❤❛t ❤❛s t❤❡ s❛♠❡ r♦❧❡ ♦❢ t❤❡ s❡❝♦♥❞ ●r❡❡♥✬s ✐❞❡♥t✐t② ✷✳✶✻ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ∂ u ′ i σ ij − u i σ ′ � � = 0 . ✭✷✳✷✽✮ ij ∂x j u ′ i ❛♥❞ σ ′ ij ❛r❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r ❛♥❞ t❤❡ str❡ss t❡♥s♦r ♦❢ ❛ ❙t♦❦❡s ✢♦✇✱ ✇❤❡r❡❛s u i ❛♥❞ σ ij ❛r❡ ❛❣❛✐♥ t❤❡ s❛♠❡ q✉❛♥t✐t✐❡s ❜✉t ❢♦r ❛♥♦t❤❡r ❞✐✛❡r❡♥t ❙t♦❦❡s ✢♦✇✳ ❚♦ ❞❡r✐✈❡ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❙t♦❦❡s ✢♦✇✱ ✇❡ ❛♣♣❧② t❤✐s ✐❞❡♥t✐t② ❢♦r ❛ ♣❛rt✐❝✉❧❛r ❙t♦❦❡s ✢♦✇ ♦❢ ✐♥t❡r❡st ✇✐t❤ ✈❡❧♦❝✐t② ✈❡❝t♦r u i ❛♥❞ str❡ss t❡♥s♦r σ ij ❛♥❞ ❢♦r t❤❡ ✢♦✇ ❞✉❡ t♦ ❛ ♣♦✐♥t ❢♦r❝❡ ( u ′ i , σ ′ ij ) ✳ ❚❤❡ ✈❡❧♦❝✐t② ❛♥❞ str❡ss t❡♥s♦r ♦❢ t❤❡ ❧❛tt❡r ✢♦✇ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ s✐♥❣✉❧❛r✐t② ❢♦r❝❡❞ ❙t♦❦❡s ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ∂p ′ − µ ∂ 2 u ′ i + b i δ ( x − x 0 ) = 0 , ✭✷✳✷✾✮ ∂x i ∂x j ∂x j ✇❤❡r❡ b i st❛♥❞s ❢♦r t❤❡ i ✲t❤ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❝♦♥st❛♥t ✈❡❝t♦r t❤❛t ❞❡t❡r♠✐♥❡s t❤❡ str❡♥❣t❤ ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣♦✐♥t ❢♦r❝❡✳ ❆s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ s✐♥❣✉❧❛r✐t② ❢♦r❝❡❞ ❡q✉❛t✐♦♥ ❛r❡ ❝❛❧❧❡❞ ●r❡❡♥✬s ❢✉♥❝t✐♦♥✱ ❤❡♥❝❡ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ✈❡❧♦❝✐t② ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ♣r❡ss✉r❡ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❛r❡ t❤❡ s♦❧✉t✐♦♥ ✜❡❧❞s ♦❢ ❡q✉❛t✐♦♥ ✷✳✷✾ 1 p ′ ( x ) = 1 u ′ i ( x ) = 4 πµG ij ( x , x 0 ) b j , 4 πp j ( x , x 0 ) b j . ✭✷✳✸✵✮ G ij ✐s ❛ s❡❝♦♥❞ ♦r❞❡r t❡♥s♦r ❝❛❧❧❡❞ ✈❡❧♦❝✐t② ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ❛♥❞ p j ❛ ✈❡❝t♦r ❝❛❧❧❡❞ ♣r❡ss✉r❡ ●r❡❡♥✬s ❢✉♥❝t✐♦♥✳ ❚❤❡ str❡ss t❡♥s♦r ✐♥ ❡q✉❛t✐♦♥ ✷✳✸✵ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✉s✐♥❣ t❤❡ ❝♦♥st✐t✉t✐✈❡ ❧❛✇ ❢♦r ❛ ◆❡✇t♦♥✐❛♥ ✢✉✐❞✱ ✇❤✐❝❤ r❡❛❞s + ∂u ′ � ∂u ′ � ij = − δ ij p ′ + µ σ ′ j i . ✭✷✳✸✶✮ ∂x j ∂x i

  29. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✻ ❙✉❜st✐t✉t✐♥❣ t❤❡ ❡q✉❛t✐♦♥s ✷✳✸✵ ✐♥t♦ t❤❡ ✷✳✸✶✱ ✇❡ ♦❜t❛✐♥ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ str❡ss t❡♥s♦r ✐♥ t❡r♠s ♦❢ ✈❡❧♦❝✐t② ❛♥❞ ♣r❡ss✉r❡ ●r❡❡♥ ❢✉♥❝t✐♦♥ � � ij = 1 − δ ij p k + ∂G ik + ∂G jk σ ′ b k . ✭✷✳✸✷✮ 4 π ∂x j ∂x i ❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ✇❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡♥s♦r T ijk = 1 � − δ ij p k + ∂G ik + ∂G jk � , ✭✷✳✸✸✮ 4 π ∂x j ∂x i t❤✉s σ ′ ij ( x ) = T ijk ( x , x 0 ) b k . ✭✷✳✸✹✮ ❲❡ ❝❛♥ ♥♦✇ s✉❜st✐t✉t❡ t❤❡ r❡❧❛t✐♦♥ ✷✳✸✵ ❛♥❞ ✷✳✸✹ ✐♥t♦ t❤❡ ▲♦r❡♥t③ r❡❧❛t✐♦♥ ✷✳✷✽✱ ❛♥❞ ❞r♦♣♣✐♥❣ t❤❡ ✈❡❝t♦r b j / 4 π ✇❡ ♦❜t❛✐♥ ∂ ( G ik σ ij − µu i T ijk ) = 0 . ✭✷✳✸✺✮ ∂x j ■♥ t❤❡ ♥❡①t st❡♣✱ ✇❡ ✐♥t❡❣r❛t❡ ✷✳✸✺ ♦✈❡r ❛ ❞♦♠❛✐♥ D ′ = D − D ǫ ✱ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✺✱ ❛♥❞ ✉s❡ t❤❡ ❞✐✈❡r❣❡♥❝❡ t❤❡♦r❡♠✱ ♦❜t❛✐♥✐♥❣ �� ∂ ( G ik σ ij − µu i T ijk ) dS = ∂x j D ′ � = C ′ ( G ik σ ij − µu i T ijk ) n j dl = 0 . ✭✷✳✸✻✮ ❙✐♥❝❡ t❤❡ ❜♦✉♥❞❛r② ♦❢ D ′ ✐s C ′ = C + C ǫ ✱ t❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡ ✐♥t❡❣r❛❧ ❛❧❧♦✇ ✉s t♦ s♣❧✐t t❤❡ ❧❡❢t✲❤❛♥❞ ♦❢ ✷✳✸✻ ✐♥ ❛♥ ✐♥t❡❣r❛❧ ♣❡r❢♦r♠❡❞ ♦♥ C ❛♥❞ ❛♥♦t❤❡r ✐♥t❡❣r❛❧ ♣❡r❢♦r♠❡❞ ♦♥ C ǫ ✳ ■♥ ♦r❞❡r t♦ s✐♠♣❧✐❢② ❡q✉❛t✐♦♥ ✷✳✸✻✱ ✇❡ t❛❦❡ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ t❤❡ ❧❛tt❡r ❡①♣r❡ss❡❞ ✐♥ ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s � 2 π I ǫ = ( G ik σ ij − µu i T ijk ) n j rdθ, ✭✷✳✸✼✮ 0 ❛♥❞ ✇❡ t❛❦❡ t❤❡ ❧✐♠✐t ǫ → 0 ✳ ❇❡❢♦r❡ t♦ ♣r♦❝❡❡❞ ✇✐t❤ t❤❡ ❧✐♠✐t ✇❡ ❞♦ s♦♠❡ ❝♦♥s✐❞❡r✲ ❛t✐♦♥s✳ ❚❤❡ ✐♥t❡❣r❛t✐♦♥ ❞♦♠❛✐♥ ♦❢ ✷✳✸✼ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ r❛❞✐✉s ǫ ✱ ❤❡♥❝❡ t❤❡ r❡❧❛t✐♦♥ ǫ = | x − x 0 | ✐s ✈❛❧✐❞✱ ❛♥❞ t❤❡r❡❢♦r❡✱ t❤❡ ❧✐♠✐t ǫ → 0 ♠❡❛♥s t❤❛t x ♠♦✈❡s ❝❧♦s❡r t♦ x 0 ✱ ♦r ♠❛t❤❡♠❛t✐❝❛❧❧②✱ x → x 0 ✳ ■❢ ✇❡ ❡①❛♠✐♥❡ G ij ❛♥❞ T ijk ❛s x → x 0 ✱

  30. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✼ ✇❡ ✜♥❞ ♦✉t t❤❛t t❤❡s❡ ❛❧✇❛②s ❜❡❤❛✈❡ r❡s♣❡❝t✐✈❡❧② ❛s ❛ st♦❦❡s❧❡t ✈❡❧♦❝✐t② ●r❡❡♥✬s ❢✉♥❝✲ t✐♦♥ ❛♥❞ ❛ st♦❦❡s❧❡t str❡ss ●r❡❡♥✬s ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ r❡❛❞ r❡s♣❡❝t✐✈❡❧② ❛s ❢♦❧❧♦✇s G ij = δ ij ❧♦❣ r + ˆ x i ˆ x j T ijk = − 4 ˆ x i ˆ x j ˆ x k r 2 , , ✭✷✳✸✽✮ r 4 ✇❤❡r❡ ˆ x = x − x 0 ❛♥❞ r = | x − x 0 | = ǫ ✳ ❊♠♣❧♦②✐♥❣ ❛❧❧ t❤❡s❡ ❝♦♥s✐❞❡r❛t✐♦♥ ✇❡ ❝❛♥ ✇r✐t❡ �� � � � δ ik ❧♦❣ ǫ + ˆ x i ˆ x k x i ˆ ˆ x j ˆ x k lim ǫ → 0 I ǫ = lim σ ij + 4 µu i n j ǫdrdθ = ǫ 2 ǫ 4 ǫ → 0 C ǫ � δ ik ǫ ❧♦❣ ǫ + ˆ x i ˆ x k � x i ˆ ˆ x k � � = lim σ ij n j + 4 µu i x j n j drdθ. ˆ ✭✷✳✸✾✮ ǫ ǫ 3 ǫ → 0 C ǫ C ǫ ❖t❤❡r s✐♠♣❧✐✜❝❛t✐♦♥s ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ♦❜s❡r✈✐♥❣ t❤❛t lim ǫ → 0 ǫ ❧♦❣ ǫ = 0 , ✭✷✳✹✵✮ x i ˆ ˆ x k x k ∼ ǫ 2 ⇒ lim x i ˆ ˆ = 0 , ✭✷✳✹✶✮ ǫ ǫ → 0 x | 2 = ǫ 2 n = x − x 0 | x − x 0 | = ˆ ǫ ⇒ n j = ˆ x j x j n j = ˆ x j ˆ x j = | ˆ x ǫ ⇒ ˆ ǫ = ǫ. ✭✷✳✹✷✮ ǫ ǫ ❆♣♣❧②✐♥❣ t❤❡s❡ t❤r❡❡ r❡❧❛t✐♦♥s t♦ t❤❡ ❡q✉❛t✐♦♥ ✷✳✸✾✱ ✇❡ ❤❛✈❡ x i ˆ ˆ x k � lim ǫ → 0 I ǫ = lim 4 µu i ǫ 2 dθ. ✭✷✳✹✸✮ ǫ → 0 C ǫ ❲❡ r❡♠❡♠❜❡r✐♥❣ t❤❛t u i ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ x ✱ ❛♥❞ ǫ → 0 ✭✐✳❡✳ x → x 0 ✮✱ t❤❡r❡❢♦r❡ lim ǫ → 0 u i ( x ) = u i ( x 0 ) . ✭✷✳✹✹✮ ❚❤❡ i ✲t❤ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r ❝♦♠♣✉t❡❞ ✐♥ t❤❡ s✐♥❣✉❧❛r✐t② ✐s ❛ ❝♦♥st❛♥t✱ ❛♥❞ ✐t ❝❛♥ ❞r♦♣ t❤❡ ✐♥t❡❣r❛❧ x i ˆ ˆ x k � lim ǫ → 0 I ǫ = lim ǫ → 0 4 µu i ( x 0 ) ǫ 2 dθ ✭✷✳✹✺✮ C ǫ

  31. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✽ ❲❡ ✉s❡ t❤✐s r❡s✉❧t ✐♥ ♦r❞❡r t♦ r❡❛rr❛♥❣❡ t❤❡ r❡❧❛t✐♦♥ ✷✳✸✻ ❛♥❞ ❡①♣r❡ss t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r ❝♦♠♣✉t❡❞ ♦♥ t❤❡ s✐♥❣✉❧❛r✐t② u i ( x 0 ) = 1 � 1 � u i ( x ) T ijk ( x , x 0 ) n j dl − G ik ( x , x 0 ) σ ij ( x ) n j dl. ✭✷✳✹✻✮ 4 π 4 πµ C C ❊q✉❛t✐♦♥ ✷✳✹✻ ✐s t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥✱ ❛♥❞ ✐♥ ❧✐♥❡ ✇✐t❤ t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥✱ ✇❡ ❝❛♥ ❝❛❧❧ t❤❡ t❡r♠s ♦♥ t❤❡ r✐❣❤t✲❡♥❞ s✐❞❡ ❉▲P ❛♥❞ ❙▲P✳ ▼♦r❡♦✈❡r✱ ❛s ✇❡ s❛✇ ✐♥ s❡❝t✐♦♥ ✷✳✹✳✶✱ ✇❡ ❝❛♥ ♣✉t x 0 ♦♥ t❤❡ ❜♦✉♥❞❛r② ❛♥❞ ❞❡r✐✈❡ ❛♥ ❡①♣r❡ss✐♦♥ t❤❛t ✐♥✈♦❧✈❡s ♦♥❧② ❜♦✉♥❞❛r② q✉❛♥t✐t✐❡s � PV 1 2 u i ( x 0 ) = 1 1 � u i ( x ) T ijk ( x , x 0 ) n j dl − G ik ( x , x 0 ) σ ij ( x ) n j dl ✭✷✳✹✼✮ 4 π 4 πµ C C ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛❧❧♦✇ t♦ s♦❧✈❡ ❡q✉❛t✐♦♥ ✷✳✹✼✱ ❛♥❞ s✉❜s❡q✉❡♥t❧② t❤❡ s♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛❧❧ t❤❡ ❞♦♠❛✐♥ ✇✐t❤ t❤❡ ❡q✉❛t✐♦♥ ✷✳✹✻✳

  32. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✷✾ ✷✳✺ ❇♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞ ❚❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ♦❜t❛✐♥❡❞ ❢♦r t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ❛♥❞ t❤❡ tr❛♥s✈❡rs❡ ♣r♦❜❧❡♠s ❛r❡ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡❞✳ ❚❤❡ ♥✉♠❡r✐❝❛❧ ❝♦✉♥t❡r♣❛rt ♦❢ t❤❡ ❇■▼ ✐s ❝❛❧❧❡❞ ❇♦✉♥❞❛r② ❊❧❡♠❡♥t ▼❡t❤♦❞ ✭❇❊▼✮✱ ❛♥❞ ❣❡♥❡r❛t❡ ❛ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ ❚❤❡ ♥✉♠❡r✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❇❊▼ ✐♥✈♦❧✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ st❡♣s✿ ✶✳ ❉✐s❝r❡t✐③❡ t❤❡ ❜♦✉♥❞❛r② ✐♥t♦ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❞✐s❝r❡t❡ ❡❧❡♠❡♥ts✱ ❛♥❞ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧s ✇✐t❤ t❤❡ s✉♠ ♦❢ t❤❡ ✐♥t❡❣r❛❧s ♦✈❡r ❡❛❝❤ t❤❡ ❜♦✉♥❞❛r② s❡❣♠❡♥ts✳ ✷✳ ■♥tr♦❞✉❝❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ ♦✈❡r t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❜♦✉♥❞✲ ❛r② ❡❧❡♠❡♥t✳ ✸✳ ❆♣♣❧② t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❛t s♦♠❡ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥ts ❧♦❝❛t❡❞ ♦✈❡r t❤❡ ❜♦✉♥❞❛r② ❡❧❡♠❡♥ts ✹✳ P❡r❢♦r♠ t❤❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ s✐♥❣❧❡✲ ❛♥❞ ❞♦✉❜❧❡✲❧❛②❡r ♣♦t❡♥t✐❛❧ ♦✈❡r t❤❡ ❜♦✉♥❞✲ ❛r② ❡❧❡♠❡♥ts✳ ✺✳ ❙♦❧✈❡ t❤❡ ❧✐♥❡❛r s②st❡♠ ❢♦r t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ♦❢ ❛t t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥t✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥s✱ ✇❡ s❤❡❧❧ ❞❡s❝r✐❜❡ ✐♥ ❞❡t❛✐❧s ❛❧❧ t❤❡s❡ st❡♣s❀ t❤❡ ✜rst ♦♥❡ ✐s t❤❡ s❛♠❡ ❢♦r ❜♦t❤ t❤❡ ▲❛♣❧❛❝❡ ❛♥❞ ❙t♦❦❡s ❡q✉❛t✐♦♥s ✭t❤❡ ❣❡♦♠❡tr② ♦✈❡r ✇❤✐❝❤ t❤❡ ❡q✉❛t✐♦♥s ❛r❡ ❛♣♣❧✐❡❞ ✐s t❤❡ s❛♠❡✮✱ ❜✉t ❛❧❧ t❤❡ ♥❡①t st❡♣s ❛r❡ s❧✐❣❤t❧② ❞✐✛❡r❡♥t✳ ❲❡ ❤❛✈❡ ❞❡❝✐❞❡❞ t♦ s❤♦✇ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞ ♦♥❧② ❢♦r t❤❡ ♣❛r❛❧❧❡❧ ♣r♦❜❧❡♠❀ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦❜❧❡♠ ✐s ❞♦♥❡ ✐♥ ❛ s✐♠✐❧❛r ❢❛s❤✐♦♥✱ ❤♦✇❡✈❡r ✐t ✐s ♠✉❝❤ ❤❛r❞❡r ❜❡❝❛✉s❡ ♦❢ t❤❡ ✈❡❝t♦r✐❛❧ ♥❛t✉r❡ ♦❢ t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥✳ ✷✳✺✳✶ ❇♦✉♥❞❛r② ❞✐s❝r❡t✐③❛t✐♦♥ ❚❤❡ ❜♦✉♥❞❛r②✲❡❧❡♠❡♥t ♠❡t❤♦❞ ❞❡r✐✈❡s ✐ts ♥❛♠❡ ❢r♦♠ t❤❡ ♣r❛❝t✐❝❡ ♦❢ ❞❡s❝r✐❜✐♥❣ t❤❡ ❜♦✉♥❞❛r② ♦❢ ❛ s♦❧✉t✐♦♥ ❞♦♠❛✐♥ ✇✐t❤ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❡❧❡♠❡♥t❛r② ❣❡♦♠❡tr✐❝❛❧ ✉♥✐ts ❝❛❧❧❡❞ ❜♦✉♥❞❛r② ❡❧❡♠❡♥ts✳ ❆ ✈❛r✐❡t② ♦❢ ❜♦✉♥❞❛r② ❡❧❡♠❡♥ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ✐♥ t✇♦ ❞✐♠❡♥s✐♦♥s✳ ❚❤r❡❡ ♣♦♣✉❧❛r ❝❤♦✐❝❡s ❛r❡ ❧✐♥❡❛r ❡❧❡♠❡♥ts ✇✐t❤ str❛✐❣❤t s❤❛♣❡s✱ ❝✐r❝✉❧❛r ❛r❝s✱ ❛♥❞ ❡❧❡✲ ♠❡♥ts ♦❢ ❝✉❜✐❝ s♣❧✐♥❡s✳ ❙✐♥❝❡ t❤❡ ❧❛st ♦♥❡ ✐s t❤❡ ♠♦st ✢❡①✐❜❧❡ ✐♥ t❡r♠s ♦❢ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t ❣❡♦♠❡tr✐❡s✱ ✇❡ ❤❛✈❡ ❞❡❝✐❞❡❞ t♦ ❡♠♣❧♦② ✐t ✐♥ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥✳

  33. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✵ s i x i +1 x i ❋✐❣✉r❡ ✷✳✻✿ ❊①❛♠♣❧❡ ♦❢ t❤❡ ❞✐s❝r❡t✐③❡❞ ❞♦♠❛✐♥✳ ❚♦ ✐♠♣❧❡♠❡♥t t❤✐s ❞✐s❝r❡t✐③❛t✐♦♥✱ ❧❡t ✉s ❞❡s❝r✐❜❡ t❤❡ ❜♦✉♥❞❛r② ✇✐t❤ ◆ ♥♦❞❡s✱ ❛♥❞ ❞❡✜♥❡ s i ❛s t❤❡ ❝✉rr❡♥t ❧❡♥❣t❤ ♦❢ t❤❡ ♣♦❧②❣♦♥❛❧ ❧✐♥❡ ❝♦♥♥❡❝t✐♥❣ s❡q✉❡♥t✐❛❧ ♥♦❞❡s ❢r♦♠ t❤❡ ✜rst ♥♦❞❡ ✭❛r❜✐tr❛r✐❧② ❝❤♦s❡♥✮ t♦ t❤❡ i ✲t❤ ♥♦❞❡✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❞✐s❝r❡t✐③❡❞ ❞♦♠❛✐♥ ✐s s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✻✳ ❚❤❡ i ✲t❤ ❡❧❡♠❡♥t ♦❢ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✐s ❜❡t✇❡❡♥ t❤❡ i ✲t❤ ❛♥❞ t❤❡ ( i + 1) ✲t❤ ♥♦❞❡✱ ❛♥❞ ✐t ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ t❤❡ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧ ❛s i ( s − s i ) 3 + b x i ( s − s i ) 2 + c x x i ( s ) = a x i ( s − s i ) + x G i , i ( s − s i ) 3 + b y i ( s − s i ) 2 + c y y i ( s ) = a y i ( s − s i ) + y G i , ✭✷✳✹✽✮ ✇✐t❤ x i = ( x G i , y G i ) t❤❡ ❣❧♦❜❛❧ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ i ✲t❤ ♥♦❞❡ ❛♥❞ s i ≤ s ≤ s i +1 t❤❡ ❝✉r✈✐❧✐♥❡❛r ❛❜s❝✐ss❛✳ ❈✉❜✐❝ s♣❧✐♥❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✐♥✈♦❧✈❡s t❤r❡❡ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts ❢♦r ❡❛❝❤ ❝♦♦r❞✐♥❛t❡ ♣❛r❛♠❡tr✐③❛t✐♦♥✱ t❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ s✐① ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts ❢♦r ❡❛❝❤ ❡❧❡♠❡♥ts✳ ❙✐♥❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ✐s N − 1 ✱ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts ✐s 6( N − 1) ✳ ■♥ ♦r❞❡r t♦ ✜① ❛❧❧ t❤❡ ❝♦❡✣❝✐❡♥ts✱ ✇❡ ✐♠♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝♦♥❞✐t✐♦♥s ❢♦r ❜♦t❤ t❤❡ x ✲ ❛♥❞ y ✲❝♦♦r❞✐♥❛t❡s❀ ✇❡ r❡♣♦rt t❤❡♠ ♦♥❧② ❢♦r t❤❡ x ✲❝♦♦r❞✐♥❛t❡✿ • t❤❡ ❣❡♦♠❡tr② ❝♦♥t✐♥✉✐t② x i ( s i +1 ) = x G i +1 ⇒ ⇒ a x i h 3 i + b x i h 2 i + c x i h i + x G i = x G i +1 , i = 1 , ..., N − 1 ✭✷✳✹✾✮ • t❤❡ s❧♦♣❡ ❝♦♥t✐♥✉✐t②

  34. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✶ � � d x i = d x i +1 � � ⇒ � � d s d s � � s i +1 s i +1 ⇒ 3 a x i h 2 i + 2 b x i h i + c x i = c x i +1 , i = 1 , ..., N − 2 ✭✷✳✺✵✮ • t❤❡ ❝✉r✈❛t✉r❡ ❝♦♥t✐♥✉✐t② � � d 2 x i = d 2 x i +1 � � ⇒ � � d s 2 d s 2 � � s i +1 s i +1 ⇒ 6 a x i h i + 2 b x i + = 2 b x i +1 , i = 1 , ..., N − 2 ✭✷✳✺✶✮ ✇❤❡r❡ h i = s i +1 − s i ✳ ❘❡❛rr❛♥❣✐♥❣ ♣r♦♣❡r❧② t❤❡s❡ r❡❧❛t✐♦♥s ✇❡ ❝❛♥ ✜♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s ❢♦r t❤❡ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts✿ a i = b i +1 − b i , ✭✷✳✺✷✮ 3 h i c i = x G i +1 − x G − 1 i 3 h i ( b i +1 + 2 b i ) , ✭✷✳✺✸✮ h i 3 b i +2 = x G i +2 − x G − x G i +1 − x G h i 3 b i + 2 h i + h i +1 b i +1 + h i +1 i +1 i . ✭✷✳✺✹✮ 3 h i +1 h i ❚❤❡ s②st❡♠ ✷✳✺✹ ✐♥✈♦❧✈❡s N − 3 ❡q✉❛t✐♦♥s✱ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ b ✲❝♦❡✣❝✐❡♥ts ✐s N − 1 ✱ t❤❡r❡❢♦r❡✱ t✇♦ ❛❞❞✐t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡ r❡q✉✐r❡❞✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ t❛❦❡ ✉♥❞❡r ❝♦♥✲ s✐❞❡r❛t✐♦♥ ❛ ❝❧♦s❡❞ ❜♦✉♥❞❛r②✱ ❤❡♥❝❡ ✇❡ ✐♠♣♦s❡ ♣❡r✐♦❞✐❝✐t② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ❛t t❤❡ ✜rst ❛♥❞ ❧❛st ♥♦❞❡s ❡①♣r❡ss❡❞✱ r❡s♣❡❝t✐✈❡❧②✱ ❜② t❤❡ ❡q✉❛t✐♦♥s 3 a N − 1 h 2 N − 1 + 2 b N − 1 h N − 1 + c N − 1 = c 1 ✭✷✳✺✺✮ b N = b 1 ✭✷✳✺✻✮ ◆♦✇ ✇❡ ❝❛♥ s♦❧✈❡ t❤❡ s②st❡♠ ✷✳✺✹ ❛♥❞ s✉❜s❡q✉❡♥t❧② ❞❡t❡r♠✐♥❡ a ❛♥❞ b ✲❝♦❡✣❝✐❡♥ts t❤r♦✉❣❤ t❤❡ r❡❧❛t✐♦♥s ✷✳✺✸✳ ❆❢t❡r t❤❡s❡ ♣❛ss❛❣❡s ❛❧❧ t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ❦♥♦✇♥✱ ❛♥❞ ❤❡♥❝❡ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❢♦r ❡❛❝❤ ❡❧❡♠❡♥t ♦❢ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✐s ❦♥♦✇♥✳ ❆❢t❡r t❤❡ ❞✐s✲ ❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r②✱ ✇❤❛t❡✈❡r ❦✐♥❞ ✐t ✐s✱ ✇❡ ❝❛♥ ❡①♣❧♦✐t t❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡ ✐♥t❡❣r❛❧ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛ ❞✐s❝r❡t❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✿

  35. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✷ ✶ N N � � � � w ( x 0 ) = w ( x ) n · ∇ G ( x , x 0 ) dl − G ( x , x 0 ) n · ∇ w ( x ) dl, ✭✷✳✺✼✮ E i E i i =1 i =1 N N 1 � � � � ✭✷✳✺✽✮ 2 w ( x 0 ) = w ( x ) n · ∇ G ( x , x 0 ) dl − G ( x , x 0 ) n · ∇ w ( x ) dl. E i E i i =1 i =1 ◆❡①t st❡♣ ❞❡❛❧ ✇✐t❤ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ✐♥s✐❞❡ t❤❡ ✐♥t❡❣r❛❧s✳ ✷✳✺✳✷ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② q✉❛♥t✐t✐❡s ❖♥❝❡ t❤❡ ❜♦✉♥❞❛r② ❞✐s❝r❡t✐③❛t✐♦♥ ✐s ❝♦♠♣❧❡t❡❞✱ ✇❡ ❝❛♥ ♣❛ss t♦ t❤❡ s❡❝♦♥❞ st❡♣ ♦❢ t❤❡ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞✿ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✉♥❦♥♦✇♥ q✉❛♥t✐t✐❡s ✇❤✐❝❤ ❛♣♣❡❛r ✐♥s✐❞❡ t❤❡ ✐♥t❡❣r❛❧s ✭✐✳❡✳✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥✱ w ( x ) ❛♥❞ ∇ w ( x ) ✮✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛✐♠s ❛t ✇r✐t✐♥❣ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s t❤r♦✉❣❤ t❤❡✐r ✈❛❧✉❡ ❡✈❛❧✉❛t❡❞ t♦ s♣❡❝✐✜❝ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r②✳ ❚❤❡r❡❜②✱ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❜❡❝♦♠❡ ❛ ❧✐♥❡❛r ❛❧❣❡❜r❛✐❝ s②st❡♠✱ ❛♥❞✱ ❛s ✇❡ s❤❡❧❧ ❡①♣❧❛✐♥✱ ✐t ❝❛♥ ❜❡ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡❞✳ ❇❡❢♦r❡ ♣r♦❝❡❡❞✐♥❣ ✇✐t❤ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ♠❛♣ t❤❡ ❣❧♦❜❛❧ ❝♦♦r❞✐♥❛t❡ x ✐♥t♦ t❤❡ ❧♦❝❛❧ ❝♦♦r❞✐♥❛t❡ ξ ∈ [ − 1 , 1] ❞❡✜♥❡❞ ♦✈❡r ❡❛❝❤ ❡❧❡♠❡♥t✳ ❚❤✐s ✐s ❛ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♥✉♠❡r✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧s✳ ❚♦ ❞♦ s♦✱ ✇❡ st❛rt ❜② ❡①♣r❡ss✐♥❣ t❤❡ ✐♥t❡❣r❛♥❞ ❢✉♥❝t✐♦♥ ✐♥ t❡r♠ ♦❢ t❤❡ ❝✉r✈✐❧✐♥❡❛r ❛❜s❝✐ss❛ ❞❡✜♥❡❞ ♦✈❡r ❛♥ ❡❧❡♠❡♥t E i � s i +1 � DLP i = w ( x ) n · ∇ G ( x , x 0 ) dl = w [ x ( s )] n · ∇ G [ x ( s ) , x 0 ] h i ( s ) ds, ✭✷✳✺✾✮ E i s i � s i +1 � SLP i = G ( x , x 0 ) n · ∇ w ( x ) dl = G [ x ( s ) , x 0 ] n · ∇ w [ x ( s )] h i ( s ) ds, ✭✷✳✻✵✮ E i s i ✇❤❡r❡ h i ( s ) ✐s t❤❡ ♠❡tr✐❝ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ i ✲t❤ ❡❧❡♠❡♥t✱ ✇❤✐❧❡ s i ❛♥❞ s i +1 ❛r❡ t❤❡ ♣♦❧②❣♦♥❛❧ ❛r❝❤ ❧❡♥❣t❤s ♦❢ t❤❡ ❡❧❡♠❡♥t ❛t st❛rt✐♥❣ ❛♥❞ ❡♥❞✐♥❣ ♣♦✐♥ts r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❛ ❝✉❜✐❝ s♣❧✐♥❡ ❡❧❡♠❡♥t✱ ❝❛❧❧✐♥❣ ∆ s = s − s i t❤❡ ♠❡tr✐❝ ❝♦❡✣❝✐❡♥ts t❛❦❡s t❤❡ ❢♦r♠✿ �� d x i � 2 � 2 � d y i h i ( s ) = + = d s d s � i ∆ s 2 + 2 b x i ) 2 + (3 a y i ∆ s 2 + 2 b y i ∆ s + c y (3 a x i ∆ s + c x = i ) 2 . ✭✷✳✻✶✮ ✶ ❚❤❡ s❛♠❡ ♣❛ss❛❣❡s ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r t❤❡ st♦❦❡s ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✳

  36. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✸ ❲❡ ❛♣♣❧② ❛♥♦t❤❡r ❝♦♦r❞✐♥❛t❡ tr❛♥s❢♦r♠❛t✐♦♥ ✇❤✐❝❤ ♠❛♣s ❛♥ ❡❧❡♠❡♥t ❢r♦♠ t❤❡ ❣❧♦❜❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❜❛s❡❞ ♦♥ t❤❡ ❝✉r✈✐❧✐♥❡❛r ❛❜s❝✐ss❛ t♦ ❛ ❧♦❝❛❧ ❝♦♦r❞✐♥❛t❡ s②st❡♠ s✉❝❤ t❤❛t t❤❡ i ✲t❤ ❡❧❡♠❡♥t✬s ❜♦✉♥❞❛r② ♣♦✐♥ts ❛r❡ ♠❛♣♣❡❞ ✐♥t♦ t❤❡ ✐♥t❡✈❛❧ [ − 1 , 1] ✳ ❚❤✐s ♠❛♣♣✐♥❣ ✇✐❧❧ ❝❛♥ ❜❡ s✐♠♣❧② ❞♦♥❡ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥ s i ( ξ ) = s i +1 + s i + s i +1 + s i ξ = s m i + s d i ξ, ✭✷✳✻✷✮ 2 2 ❢r♦♠ ✇❤✐❝❤ ✇❡ ❝❛♥ ❡❛s✐❧② ❞❡✜♥❡ t❤❡ ❛ss♦❝✐❛t❡❞ ♠❡tr✐❝ h ξ i = s d i ✳ ■♥tr♦❞✉❝✐♥❣ t❤✐s ♥❡✇ ♣❛r❛♠❡tr✐③❛t✐♦♥ ✐♥t♦ t❤❡ ✐♥t❡❣r❛❧s ✷✳✺✾ ❛♥❞ ✷✳✻✵ ✇❡ ♦❜t❛✐♥✿ � 1 DLP i = h ξ w [ x ( s ( ξ ))] n · ∇ G [ x ( s ( ξ )) , x 0 ] h i ( s ( ξ )) dξ = i − 1 � 1 = h ξ w ( ξ ) n · ∇ G [ ξ, x 0 ] h i ( ξ ) dξ, ✭✷✳✻✸✮ i − 1 � 1 SLP i = h ξ G [ x ( s ( ξ )) , x 0 ] n · ∇ w [ x ( s ( ξ ))] h i ( s ( ξ )) dξ = i − 1 � 1 = h ξ G [ ξ, x 0 ] n · ∇ w ( ξ ) h i ( ξ ) dξ. ✭✷✳✻✹✮ i − 1 ❯♥t✐❧ ♥♦✇✱ ♥♦ ❛ss✉♠♣t✐♦♥ ❛❜♦✉t t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ♦✈❡r t❤❡ ❡❧❡♠❡♥t ❤❛s ❜❡❡♥ ❞♦♥❡✳ ❋♦r ♦✉r ♣✉r♣♦s❡s✱ ✇❡ ♣r♦♣♦s❡ t♦ ✉s❡ ❛ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ✈❛r✐❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❛ ❣♦♦❞ ❝♦♠♣r♦♠✐s❡ ❜❡t✇❡❡♥ ❛❝❝✉r❛❝② ❛♥❞ ♣r♦❣r❛♠♠✐♥❣ ❞✐✣❝✉❧t②❀ t❤✉s ❧❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ w ( ξ ) ❢♦r t❤❡ i ✲t❤ ❡❧❡♠❡♥t ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ w i ( ξ ) ≈ w ( ξ 2 ) − w ( ξ 1 ) ( ξ − ξ 1 ) + w ( ξ 1 ) = ξ 2 − ξ 1 � ξ 1 − ξ � ξ − ξ 1 � � = w i ( ξ 1 ) + 1 + w ( ξ 2 ) = ξ 2 − ξ 1 ξ 2 − ξ 1 = Ψ 1 ( ξ ) w i ( ξ 1 ) + Ψ 2 ( ξ ) w i ( ξ 2 ) , ✭✷✳✻✺✮ ✇✐t❤ ξ 1 ❛♥❞ ξ 2 t❤❡ t✇♦ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡✈❛❧✉❛t❡❞ ♦♥ t❤❡ i ✲t❤ ❡❧❡♠❡♥t✱ ❛♥❞ Ψ 1 ( ξ ) ❛♥❞ Ψ 2 ( ξ ) t❤❡ t✇♦ s❤❛♣❡ ❢✉♥❝t✐♦♥s✳ ❚❤❡ s❛♠❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ f = ∇ w ( x ) f i ( ξ ) ≈ Φ 1 ( ξ ) f i ( ξ 1 ) + Φ 2 ( ξ ) f i ( ξ 2 ) . ✭✷✳✻✻✮

  37. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✹ ❋♦r t❤❡ s❡❡❦ ♦❢ s✐♠♣❧✐❝✐t②✱ ✇❡ ❝❛❧❧ w i 1 = w i ( ξ 1 ) , w i 2 = w i ( ξ 2 ) , f i 1 = f i ( ξ 1 ) , f i 2 = f i ( ξ 2 ) , ✭✷✳✻✼✮ ❛♥❞✱ s✉❜st✐t✉t✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ✷✳✻✺ ❛♥❞ ✷✳✻✻ r❡s♣❡❝t✐✈❡❧② ✐♥t♦ t❤❡ ❞♦✉❜❧❡ ❛♥❞ s✐♥❣❧❡ ❧❛②❡r ♣♦t❡♥t✐❛❧✱ ✇❡ ♦❜t❛✐♥ � 1 � 1 � � DLP i = h ξ w i 1 Ψ 1 n · ∇ Gh i dξ + w i 2 Ψ 2 n · ∇ Gh i dξ . ✭✷✳✻✽✮ i − 1 − 1 � 1 � 1 � � SLP i = h ξ f i 1 Φ 1 Gh i dξ + f i 2 Φ 2 Gh i dξ ✭✷✳✻✾✮ i − 1 − 1 ■t ✐s ♣♦ss✐❜❧❡ t♦ ♥♦t❡ t❤❛t ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✐♥s✐❞❡ t❤❡ ✐♥t❡❣r❛❧s ♦❢ t❤❡ ✷✳✻✽ ❛♥❞ ✷✳✻✾ ❛r❡ ❦♥♦✇♥✱ t❤❡r❡❢♦r❡ t❤❡ ✐♥t❡❣r❛❧s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ♥✉♠❡r✐❝❛❧❧②✱ ❛♥❞ t❤❡ t✇♦ r❡❧❛t✐♦♥s ❜❡❝♦♠❡ ❧✐♥❡❛r✳ ■t ✐s ♥♦✇ ♣♦ss✐❜❧❡ t♦ ❞❡✜♥❡ t❤❡ ✐♥✢✉❡♥❝❡ ❝♦❡✣❝✐❡♥t ❛s ❛s ❢♦❧❧♦✇s � 1 � 1 i = h ξ i = h ξ W 1 W 2 Ψ 1 n · ∇ Gh i dξ, Ψ 2 n · ∇ Gh i dξ, i i − 1 − 1 � 1 � 1 F 1 i = h ξ F 2 i = h ξ Φ 1 Gh i dξ, Φ 2 Gh i dξ, ✭✷✳✼✵✮ i i − 1 − 1 ❛♥❞ s✉❜st✐t✉t✐♥❣ ❡q✉❛t✐♦♥s ✷✳✻✽ ❛♥❞ ✷✳✻✾ ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✷✳✺✽✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✇r✐t❡ t❤❡ ✜♥❛❧ ❡q✉❛t✐♦♥ ❞❡r✐✈❡❞ ❜② t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② ❛♥❞ t❤❡ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ♦♥ t❤❡ ❜♦✉♥❞❛r②✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤❡ ♣r♦❝❡❞✉r❡ ✇❡ ❤❛✈❡ N N 1 � � w 1 i W 1 i + w 2 i W 2 f 1 i F 1 i + f 2 i F 2 � � � � 2 w ( x 0 ) = − . ✭✷✳✼✶✮ i i i =1 i =1 ❚♦ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷✳✼✶ t❤❡ ❢✉♥❝t✐♦♥ w ✐s ❡✈❛❧✉❛t❡❞ ✐♥ x 0 ✱ t❤❛t ✇❡ ❤❛✈❡ ❝❤♦s❡♥ t♦ ❧✐❡ ♦♥ t❤❡ ❜♦✉♥❞❛r②✳ ❚❤✐s ♠❡❛♥s t❤❛t ❛❧❧ t❤❡ ❢✉♥❝t✐♦♥s ✐♥ ❡q✉❛t✐♦♥ ✷✳✼✶ ❛r❡ t♦ ❜❡ ❡✈❛❧✉❛t❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r②✱ ❛♥❞ t❤✐s ✐s t❤❡ s♣❡❝✐✜❝ ❢❡❛t✉r❡ t❤❛t ✇✐❧❧ ❛❧❧♦✇ ✉s t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✳ ■t ✐s ♥♦✇ ✇♦rt❤ t♦ ♥♦t❡ t❤❡ r♦❧❡ ♦❢ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥ts✱ ✇❤✐❝❤ ❛r❡ t❤❡ ♣♦✐♥ts ✐♥ ✇❤✐❝❤ t❤❡ t✇♦ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ✇✐❧❧ ❜❡ ❡✈❛❧✉❛t❡❞ ✭✐✳❡✳ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r s②st❡♠ ✇✐❧❧ ❣✐✈❡ ✉s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ✐♥ ❛❧❧ t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥ts ♦♥ t❤❡ ❜♦✉♥❞❛r②✮✳ ■♥ ❣❡♥❡r❛❧ ✇❡ ✇✐❧❧ ❝❤♦♦s❡ t✇♦ ❝♦❧❧♦❝❛t✐♦♥ ♣♦✐♥ts ❢♦r ❡❛❝❤ ❡❧❡♠❡♥t✱

  38. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✺ ❧❛②✐♥❣ r❡s♣❡❝t✐✈❡❧② ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ❛♥❞ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❣♠❡♥t✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ❝♦r♥❡r✱ ❢♦r ✇❤✐❝❤ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡♠ ✐s ❛ ❜✐t ❞✐✛❡r❡♥t✳ ❲❤❡♥ ❛ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ✐s ❛❞❥❛❝❡♥t t♦ ❛ tr✉❡ ❜♦✉♥❞❛r② ❝♦r♥❡r ✭❜✉t ♥♦t ❛♥ ❛rt✐✜❝✐❛❧ ❝♦r♥❡r ❞✉❡ t♦ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥✮✱ t❤❡ ❜♦✉♥❞❛r② ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛♥❞ ✐ts ♥♦r♠❛❧ ❞❡r✐✈❛t✐✈❡ ❛r❡ ❧✐❦❡❧② t♦ ❡①❤✐❜✐t ❞✐s❝♦♥t✐♥✉♦✉s ♦r s✐♥❣✉❧❛r ❜❡❤❛✈✐♦r✱ ❛♥❞ ❛ ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥ ♦❢ t❤♦s❡ q✉❛♥t✐t✐❡s ✐s ♥♦ ♠♦r❡ ❛♣♣r♦♣r✐❛t❡✳ ❚❤❡ ✉s❡❞ ❛♣♣r♦❛❝❤ ✐♥✈♦❧✈❡s ❞✐str✐❜✉t✐♥❣ t❤❡ ♥❡❛r✲❝♦r♥❡r ♥♦❞❡s ❛t ♣♦s✐t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ r♦♦ts ♦❢ t❤❡ ❘❛❞❛✉ ♣♦❧②♥♦♠✐❛❧s ❞❡✜♥❡❞ ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [ − 1 , 1] ✳ ❚❤❡ ✜rst ❢❡✇ ♠❡♠❜❡rs ♦❢ t❤✐s ❢❛♠✐❧② ❛r❡ R 1 ( ξ ) = 1 R 2 ( ξ ) = 1 2(5 ξ 2 − 2 ξ − 1) R 0 ( ξ ) = 1 , 2(3 ξ − 1) R 3 ( ξ ) = 1 8(35 ξ 3 − 15 ξ 2 − 15 ξ + 3) ✭✷✳✼✷✮ ❙✐♥❝❡ ✇❡ ❝❤♦♦s❡ t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ R 1 ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✜① t❤❡ ❛❧❧ t❤❡ q✉❛♥t✐t✐❡s ❝♦♠♣✉t❡❞ ♦♥ t❤❡ ❜♦✉♥❞❛r②✳ ■♥ t❤❡ ❡♥❞✱ ✇❡ ❝❛♥ ❧♦❝❛t❡ t❤❡ s✐♥❣✉❧❛r✐t② ✐♥ ❞✐✛❡r❡♥t ♣♦✐♥ts✱ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ ❡q✉❛t✐♦♥ ❛♥❞ ✉♥❦♥♦✇♥s✳ ❆ s♦❧✈❛❜❧❡ ❧✐♥❡❛r s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ❤❛s ❜❡❡♥ ♦❜t❛✐♥❡❞✳ ✷✳✺✳✸ ❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧s ❆❧❧ t❤❡ ✐♥t❡❣r❛❧ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✐♥✢✉❡♥❝❡ ❝♦❡✣❝✐❡♥ts F ij ❛♥❞ t❤❡ ❦♥♦✇♥ t❡r♠s W j ❛r❡ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡❞ ✇✐t❤ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ●❛✉ss✲▲❡❣❡❞r❡ ♠❡t❤♦❞✳ ❍♦✇❡✈❡r✱ ✇❡ ❤❛✈❡ t♦ ♣❛② ❛tt❡♥t✐♦♥ t♦ s♦♠❡ ♣❛rt✐❝✉❧❛r ✐♥t❡❣r❛❧s✱ t❤❛t r❡s✉❧t t♦ ❜❡ s✐♥❣✉❧❛r✳ ❚❤❡ j ✲t❤ ❧✐♥❡ ♦❢ t❤❡ s②st❡♠ r❡♣r❡s❡♥t t❤❡ ❡q✉❛t✐♦♥ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ♣♦s✐t✐♦♥✐♥❣ ♦❢ t❤❡ s✐♥❣✉❧❛r✐t② ♦♥ t❤❡ j ✲t❤ ❡❧❡♠❡♥t✳ ■♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤✐s j ✲t❤ r♦✇ ♦❢ t❤❡ ♠❛tr✐① F ✱ ✇❡ ❤❛✈❡ t♦ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡ ❢♦r ❡❛❝❤ ❡❧❡♠❡♥t t❤❡ ✐♥t❡❣r❛❧s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥s ✷✳✼✵✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ❛r✐s❡ ✇❤❡♥ t❤❡ ❡❧❡♠❡♥t ♦♥ ✇❤✐❝❤ t❤❡ ✐♥t❡❣r❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞✱ ✐s t❤❡ s❛♠❡ ♦♥ ✇❤✐❝❤ t❤❡ s✐♥❣✉❧❛r✐t② ❧✐❡s✳ ❲❡ ❞❡❛❧ ✇✐t❤ t❤✐s s✐t✉❛t✐♦♥ ✇❤❡♥ t❤❡ s✉❜s❝r✐♣t i ✭t❤❛t ✐❞❡♥t✐❢② t❤❡ ❡❧❡♠❡♥t✮ ✐s ❡q✉❛❧ t♦ j ✭t❤❛t ✐❞❡♥t✐❢② t❤❡ ❡❧❡♠❡♥t ♦♥ ✇❤✐❝❤ t❤❡ s✐♥❣✉❧❛r✐t② ❧✐❡✮✳ ❚❤❡ s✐♥❣✉❧❛r ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ✐♥t❡❣r❛❧s ✐s ❞✉❡ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ●r❡❡♥✬s ❢✉♥❝t✐♦♥ ✭✐♥s✐❞❡ t❤❡ F ❝♦❡✣❝✐❡♥ts✮ ❛♥❞

  39. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✻ ✐ts ❣r❛❞✐❡♥t ✭✐♥s✐❞❡ t❤❡ W ❝♦❡✣❝✐❡♥ts✮ ✐♥s✐❞❡ t❤❡♠✳ ■♥ ❢❛❝t ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✇r✐t❡  G → + ∞   x → x 0 ⇒ ✭✷✳✼✸✮ ∇ G → + ∞   ❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧s ♥❡❡❞ ❛ s♣❡❝✐❛❧ tr❡❛t♠❡♥t❀ t♦ ❝♦♠♣✉t❡ t❤❡♠ ✇❡ ❛❞❞ ❛♥❞ s✉❜tr❛❝t t❤❡ ❢r❡❡✲s♣❛❝❡ ❦❡r♥❡❧s ✐♥ t❤❡ ✐♥t❡❣r❛♥❞ ❛♥❞ s♣❧✐t t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ F ❛♥❞ W ❝♦❡✣❝✐❡♥ts ✐♥ t✇♦ ✐♥t❡❣r❛❧s � G ( x , x 0 ) + 1 � d l − 1 � � F jj = 2 π ln | x − x 0 | ln | x − x 0 | d l, ✭✷✳✼✹✮ 2 π E j E j � PV � PV � n · ∇ G ( x , x 0 ) + 1 n · ( x − x 0 ) � d l − 1 n · ( x − x 0 ) W j = | x − x 0 | 2 . ✭✷✳✼✺✮ | x − x 0 | 2 2 π 2 π E j E j ❚❤❡ ✜rst ✐♥t❡❣r❛❧s ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ✷✳✼✹ ❛♥❞ ✷✳✼✺ ❛r❡ ♥♦♥✲s✐♥❣✉❧❛r ❛♥❞ ♠❛② ❜❡ ❝♦♠♣✉t❡❞ ✇✐t❤ ❤✐❣❤ ❛❝❝✉r❛❝② ✉s✐♥❣ ●❛✉ss✲▲❡❣❡♥❞r❡ q✉❛❞r❛t✉r❡✳ ▼♦r❡♦✈❡r✱ ❡✈❡♥ t❤❡ s❡❝♦♥❞ ✐♥t❡❣r❛❧ ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷✳✼✺ ✐s ♥♦♥✲s✐♥❣✉❧❛r✱ ✐♥ ❢❛❝t✱ ❛s t❤❡ ✐♥t❡❣r❛t✐♦♥ ♣♦✐♥t x ❛♣♣r♦❛❝❤ t❤❡ s✐♥❣✉❧❛r✐t② x 0 ✱ t❤❡ ♥♦r♠❛❧ ✈❡❝t♦r n t❡♥❞s t♦ ❜❡❝❛♠❡ t♦ t❤❡ ♥❡❛r❧② t❛♥❣❡♥t✐❛❧ ✈❡❝t♦r✐❛❧ ❞✐st❛♥❝❡ ( x − x 0 ) ✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♥✉♠❡r❛t♦r ♦❢ t❤❡ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛♥❞ ❜❡❤❛✈❡s q✉❛❞r❛t✐❝❛❧❧② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❝❛❧❛r ❞✐st❛♥❝❡ | x − x 0 | ✱ ❛♥❞ t❤❡ s✐♥❣✉❧❛r✐t② ❞♦❡s ♥♦t ❛♣♣❡❛r✳ ❚❤✉s t❤❡ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ r❡❞✉❝❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ s❡❝♦♥❞ ✐♥t❡❣r❛❧ ♦❢ t❤❡ r✐❣❤t✲ ❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✷✳✼✹✳ ❚❤❡ tr❡❛t♠❡♥t ♦❢ t❤✐s t❡r♠ ❞❡♣❡♥❞ ♦♥ t❤❡ ❦✐♥❞ ♦❢ t❤❡ ❡❧❡♠❡♥t ❞✐s❝r❡t✐③❛t✐♦♥✱ s♦ ✇❡ s❤❡❧❧ ❞❡s❝r✐❜❡ t❤❡ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ ❝✉❜✐❝✲s♣❧✐♥❡ ❡❧❡♠❡♥ts✳ ■♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ❦❡r♥❡❧ ✐♥ F jj ✇❡ ❡①♣r❡ss ✐t ✐♥ t❤❡ ♣❛r❛♠❡tr✐❝ ❢♦r♠ ❛♥ � j +1 � k = ln | x − x 0 | d l = ln | x ( s ) − x ( s 0 ) | h i ( s )d s, ✭✷✳✼✻✮ E j s j ✇❤❡r❡ s 0 r❡♣r❡s❡♥ts t❤❡ s✐♥❣✉❧❛r✐t② ♣♦s✐t✐♦♥ ✭ s j < s 0 < s j +1 ✮✳ ◆♦✇ ✐t ✐s ♣♦ss✐❜❧❡ t♦ s♣❧✐t t❤❡ ✐♥t❡❣r❛❧ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛ss❛❣❡ � j +1 � j +1 � | x ( s ) − x ( s 0 ) | � k = ln h i ( s )d s + ln | s − s 0 | h i ( s ) . d s ✭✷✳✼✼✮ | s − s 0 | s j s j ■♥ t❤✐s ❝❛s❡ t❤❡ ✜rst ♠❡♠❜❡r ❝❛♥ ❜❡ ✐♥t❡❣r❛t❡❞ ✇✐t❤ ♥✉♠❡r✐❝❛❧ ♣r♦❝❡❞✉r❡s✱ ❜✉t t❤❡

  40. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✼ s❡❝♦♥❞ ♦♥❡ r❡♠❛✐♥s s✐♥❣✉❧❛r✳ ❚❤❡r❡❢♦r❡ t❤❡ ❛❢t❡r✇❛r❞s tr❡❛t♠❡♥t ✐s ♥❡❝❡ss❛r②✳ � j +1 ln | s − s 0 | h i ( s )d s = s j � j +1 � j +1 = ln | s − s 0 | ( h i ( s ) − h i ( s 0 )) d s + h i ( s 0 ) ln | s − s 0 | d s. ✭✷✳✼✽✮ s j s j ❚❤❡ ✜rst ♠❡♠❜❡r ✐s ♥✉♠❡r✐❝❛❧❧② ✐♥t❡❣r❛t❡❞✱ ✇❤❡r❡❛s t❤❡ s❡❝♦♥❞ ♦♥❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❡❧❡♠❡♥t❛r② ❛♥❛❧②t✐❝❛❧ ♠❡t❤♦❞✳ ✷✳✺✳✹ ▼❛t❧❛❜ ❛❧❣♦r✐t❤♠ ❞❡s❝r✐♣t✐♦♥ ■♥ ♦r❞❡r t♦ s♦❧✈❡ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ❛♥❞ t❤❡ tr❛♥s✈❡rs❡ ♣r♦❜❧❡♠✱ t❤❡ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞ ❤❛s ❜❡❡♥ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ▼❛t❧❛❜✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❡①♣❧❛✐♥ t❤❡ ❢❡❛t✉r❡s ♦❢ t❤❡ ✐♥♣✉t t❤❛t t❤❡ ❝♦❞❡ r❡❝❡✐✈❡s ❛♥❞ t❤❡ ♦✉t♣✉t t❤❛t ✐t r❡t✉r♥✱ ♠♦r❡♦✈❡r✱ t❤❡ ♠❡t❤♦❞ ❜② ✇❤✐❝❤ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛r❡ ❝♦♠♣✉t❡❞ ✐s ❡①♣❧❛✐♥❡❞✳ ❚❤❡ ❝♦❞❡ t❛❦❡ ✐♥ ✐♥♣✉t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ s❡r✐❡s ♦❢ ♣♦✐♥ts t❤❛t ❞❡s❝r✐❜❡s t❤❡ ❜♦❛r❞ ♦❢ t❤❡ ❞♦♠❛✐♥ t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r ❡❛❝❤ ♣♦✐♥t r❡♣r❡s❡♥ts t❤❡ ❧✐♠✐t ♦❢ ❛ s❡❣♠❡♥t ✉s❡❞ t♦ ❞✐s❝r❡t✐③❡ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ ✜❣✉r❡ ✷✳✼ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♦❜s❡r✈❡ t❤❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ♣♦✐♥ts ✉s❡❞ t♦ ❞✐s❝r❡t✐③❡ ✐t✳ ❋✐rst ♦❢ ❛❧❧ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t ❜♦t❤ t❤❡ ❧❛t❡r❛❧ s✐❞❡s ♦❢ t❤❡ ❞♦♠❛✐♥ ❛r❡ ♥♦t ❝♦✈❡r❡❞ ❜② ♣♦✐♥ts✳ ■♥ ❢❛❝t✱ t❤❡ ❝♦❞❡ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♣❡r✐♦❞✐❝✐t② ♦❢ t❤❡ r✐❜❧❡t ❡①♣❧♦✐t✐♥❣ t❤❡ ♣❡r✐♦❞✐❝ ●r❡❡♥ ❢✉♥❝t✐♦♥ ✷✳✼✾✱ ❛♥❞ ♥♦t t❤r♦✉❣❤ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ G = − 1 4 π log { 2 [cosh ( k ( y − y 0 )) − sinh ( k ( x − x 0 ))] } ✭✷✳✼✾✮ ✇❤❡r❡ k = 2 π/L ✱ ❛♥❞ L ✐s t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ ●r❡❡♥ ❢✉♥❝t✐♦♥✳ ❖❜✈✐♦✉s❧②✱ t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ ●r❡❡♥ ❢✉♥❝t✐♦♥ ♠✉st ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ r✐❜❧❡t✳ ■t ❤❛s ❜❡❡♥ ❝❤♦s❡♥ L = 1 ✱ t❤❡r❡❢♦r❡ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ♣❛r❛♠❡t❡r t❤❛t ②✐❡❧❞s t❤❡ r✐❜❧❡t ❣❡♦♠❡tr② ❞✐♠❡♥s✐♦♥❧❡ss ✐s t❤❡ ♣❡r✐♦❞✳ ▼♦r❡♦✈❡r✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♦❜s❡r✈❡ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣♦✐♥ts ✐♥ ✜❣✉r❡ ✷✳✼ ✐s ♥♦t ✉♥✐❢♦r♠✱ ❛♥❞ t❤❡② ✐♥❝r❡❛s❡ ♥❡❛r t❤❡ ❡❞❣❡s✳ ❚❤✐s ❦✐♥❞ ♦❢ ❞✐str✐❜✉t✐♦♥ ✐s ❢✉♥❞❛♠❡♥t❛❧ ❜❡❝❛✉s❡ ❛❧❧♦✇s t♦ r❡❞✉❝❡ t❤❡ ❡rr♦r ❞✉❡ t♦ t❤❡ ❛❜s❡♥❝❡ ♦❢ t❤❡ ♣♦✐♥ts t❤❛t ❧❛② ♦♥ t❤❡ ❡❞❣❡s✳ ■♥ t❤❡ ❡♥❞✱ ❚❤❡ ✉♣♣❡r ♣❛rt ♦❢ t❤❡ ❞♦♠❛✐♥ ♠✉st ❜❡ t❤❡♦r❡t✐❝❛❧❧② ❧♦❝❛t❡❞ ❛t ❛♥ ✐♥✜♥✐t❡❧② ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ✇❛❧❧✳ ❋r♦♠ ❛ ♣r❛❝t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ t❤❡

  41. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✽ L H ❋✐❣✉r❡ ✷✳✼✿ ❉♦♠❛✐♥ ❜♦r❞❡r ❛♥❞ ❞✐s❝r❡t✐③❛t✐♦♥ ♣♦✐♥ts✳ ✉♣♣❡r s✐❞❡ ♠✉st ♥♦t ✐♥✢✉❡♥❝❡ t❤❡ ✢♦✇ ♥❡❛r t❤❡ ✇❛❧❧✱ ❛♥❞ t♦ ❞♦ s♦ t❤❡ ❝♦❞❡ s❤✐❢t ✐t s✉✣❝✐❡♥t❧② ✉♣✳ ❖♥❝❡ t❤❡ ❛❧❣♦r✐t❤♠ ❤❛s t❛❦❡♥ ✐♥ ✐♥♣✉t s✉❝❤ ❛ ❣❡♦♠❡tr②✱ ✐t ❝♦♠♣✉t❡ t❤❡ ✈❛❧✉❡s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❞♦♠❛✐♥❀ ✐♥ ♣❛rt✐❝✉❧❛r ✐t ❝♦♠♣✉t❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ❧♦✇❡r ✇❛❧❧ ✭✐♥ ❢❛❝t t❤❡ ✈❡❧♦❝✐t② ✐s ❤❡r❡ ❞❡✜♥❡❞ ❜② t❤❡ ♥♦✲s❧✐♣ ❝♦♥❞✐t✐♦♥✮✱ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ✉♣♣❡r ❜♦✉♥❞❛r② ✭✐♥ ❢❛❝t t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ✐s ❤❡❛r ❞❡✜♥❡❞ ❡q✉❛❧ t♦ ♦♥❡✮✳ ❚❤✐s ❧❛st ❣♦❛❧ ❛❧❧♦✇ ✉s t♦ ❝♦♠♣✉t❡ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✳ ❇❡❢♦r❡ t♦ ♣r♦❝❡❡❞ ✇✐t❤ t❤❡ ❡①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✱ ✇❡ ❤❛✈❡ t♦ ♥♦t❡ t❤❛t t❤❡ ❝♦❞❡ ❝❛♥ ❡✈❡♥ ❝♦♠♣✉t❡ t❤❡ ✈❡❧♦❝✐t② ✐♥s✐❞❡ ❛❧❧ t❤❡ ❞♦♠❛✐♥❀ t♦ ❞♦ s♦ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❣✐✈❡♥ ✐♥ ✐♥♣✉t ❛ ❣r✐❞ ♦❢ ♣♦✐♥t ✷ ❛♥❞ t❤❡ ❝♦❞❡ ✇✐❧❧ ❝♦♠♣✉t❡ t❤❡ ✈❡❧♦❝✐t② ♦♥ ✐t✳ ❘❡❣❛r❞✐♥❣ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✱ t❤❡ ♣❛r❛❧❧❡❧ ♦♥❡ ❤❛s ❜❡❡♥ ❝♦♠♣✉t❡❞ t❤r♦✉❣❤ t❤❡ ❡q✉❛t✐♦♥ ✷✳✽✵ ❛♥❞ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ♦♥❡ t❤r♦✉❣❤ t❤❡ ❡q✉❛t✐♦♥ ✷✳✽✶✳ w top − H � ∂w/∂y � � top h � = ✭✷✳✽✵✮ L ✷ ■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❛t t❤❡ ❣r✐❞ ❝❛♥ ❜❡ ❞♦♥❡ ✇✐t❤♦✉t r❡s♣❡❝t ❛♥② ♣r✐♥❝✐♣❧❡✳

  42. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✸✾ v top − H � ∂v/∂y � � top h ⊥ = ✭✷✳✽✶✮ L ✇❤❡r❡ t❤❡ s✉❜s❝r✐♣t ✬t♦♣✬ ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ♣❛r❛♠❡t❡r ✐s ❝♦♠♣✉t❡❞ ♦♥ t❤❡ ✉♣♣❡r s✐❞❡ ♦❢ t❤❡ ❜♦✉♥❞❛r②✱ L ✐s t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ r✐❜❧❡t ❣❡♦♠❡tr② ✭s❡t ❡q✉❛❧ t♦ ♦♥❡✮ ❛♥❞ H ✐s t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ t❤❡ t✐♣ ♦❢ t❤❡ r✐❜❧❡t ❛♥❞ t❤❡ ✉♣♣❡r s✐❞❡ ♦❢ t❤❡ ❜♦✉♥❞❛r②✳ ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ s❡t t♦ ♦♥❡ ❜♦t❤ t❤❡ y ✲❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❝❛♥ ❜❡ ❡❛s✐❧② ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ❣✐✈❡♥ ✐♥ t❤❡ s❡❝t✐♦♥ ✷✳✸✿ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ r✐❜❧❡t t✐♣ ❛♥❞ t❤❡ ♣♦✐♥t ✇❤❡r❡ t❤❡ ✈❡❧♦❝✐t② ❜❡❝♦♠❡ ③❡r♦✳ ❇♦t❤ ❡①♣r❡ss✐♦♥s ✷✳✽✵ ❛♥❞ ✷✳✽✶ ❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ❧♦♦❦✐♥❣ ❛t t❤❡ ✜❣✉r❡ r❡♣♦rt❡❞ ❜❡❧♦✇ w top H y y z z x x h // ❋✐❣✉r❡ ✷✳✽✿ ❈♦♠♣✉t❡❞ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t✳

  43. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✵ ✷✳✻ ❱❛❧✐❞❛t✐♦♥ ♦❢ t❤❡ ❝♦❞❡ ❇❡❢♦r❡ t♦ ♣r♦❝❡❡❞ ✇✐t❤ t❤❡ s✐♠✉❧❛t✐♦♥s✱ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ✈❛❧✐❞❛t❡ t❤❡ ❝♦❞❡❀ ✐♥ ♦t❤❡r ✇♦r❞s ✇❡ ❤❛✈❡ t♦ ❝❤❡❝❦ t❤❛t ♦✉r ❝♦❞❡ ✇♦r❦s✳ ❚♦ ❞♦ s♦✱ ✇❡ ❤❛✈❡ t♦ ❝♦♠♣✉t❡ ❛♥ ❛❧r❡❛❞② s♦❧✈❡❞ ❝❛s❡ ♣r❡s❡♥t ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❛♥❞ ❝♦♠♣❛r❡ t❤❡ r❡s✉❧ts✳ ■❢ ♦✉r ❝❛❧❝✉❧❛t✐♦♥ ✇✐❧❧ ❜❡ ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ t❤❡ ❧✐t❡r❛t✉r❡✱ ♦✉r ❝♦❞❡ ✇✐❧❧ ❜❡ ✈❛❧✐❞❛t❡❞✳ ❋♦r t❤✐s ❣♦❛❧ ✇❡ st✉❞② t❤❡ r✐❜❧❡t s❤❛♣❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✾ ✭❛✮ ❈♦s✐♥✉s r✐❜❧❡t✳ ✭❜✮ P❛r❛❜♦❧✐❝ r✐❜❧❡t✳ ❋✐❣✉r❡ ✷✳✾✿ ❘✐❜❧❡t s❤❛♣❡s ✉s❡❞ ❢♦r t❤❡ ✈❛❧✐❞❛t✐♦♥✳ t❤❡② ❛r❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s y 0 = 2 s π x 2 − 4 sx, y 0 = πs cos x − πs ✭✷✳✽✷✮ ✇❤❡r❡ s r❡♣r❡s❡♥t t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ❞❡♣t❤ ❛♥❞ t❤❡ ♣❡r✐♦❞ ♦❢ t❤❡ r✐❜❧❡t✱ ❛♥❞ 0 ≤ x ≤ 2 π ✳ ❚❤❡s❡ r✐❜❧❡ts ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ❜② ▲✉❝❤✐♥✐ ❬✷✵❪❀ ❤❡ ❤❛s ✐♠♣❧❡♠❡♥t❡❞ t❤❡ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞ ✭t❤♦✉❣❤ ✇✐t❤ ❛ ❞✐✛❡r❡♥t ♠❛t❤❡♠❛t✐❝❛❧ ❛♣♣r♦❛❝❤ ✐♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ tr❛♥s✈❡rs❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t✮✱ ❛♥❞ ❤❡ ❤❛s ❝♦♠♣✉t❡❞ t❤❡ tr❛♥s✈❡rs❡ ❛♥❞ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ✐♥ ❢✉♥❝t✐♦♥ ♦❢ s ✱ ❛s s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✶✵✳ ❋✐rst ♦❢ ❛❧❧ ✇❡ ❝❛♥ ♥♦t❡ t❤❛t t❤❡ r❡s✉❧ts ♦❢ ▲✉❝❤✐♥✐ ♠❛❦❡ s❡♥s❡✳ ■♥ ❢❛❝t✱ ✇❤❡♥ s = 0 t❤❡ r✐❜❧❡t ❝❛✈✐t② ❞✐s❛♣♣❡❛r ❛♥❞ ❛❧❧ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛r❡ ③❡r♦✳ ■♥st❡❛❞✱ ✇❤❡♥ t❤❡ s ♣❛r❛♠❡t❡r ❜❡❝♦♠❡s ❤✐❣❤❡r t❤❡♥ ♦♥❡✱ t❤❡ r✐❜❧❡t ❝❛✈✐t✐❡s ❜❡❝♦♠❡ ❞❡❡♣❡r ❛♥❞ t❤❡ r✐❜❧❡t ❡✛❡❝t st❛rt t♦ ✇♦r❦✳ ❲❤❡♥ t❤❡ ❞❡♣t❤ ♦❢ t❤❡ r✐❜❧❡t ✐s t♦♦ ❤✐❣❤✱ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❞♦ ♥♦t ❞❡♣❡♥❞s ♦♥ s ❛♥②♠♦r❡❀ t❤✐s ❜❡❤❛✈✐♦r ❝❛♥ ❜❡ r❡❝♦❣♥✐③❡❞ ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ tr❡♥❞ ♦❢ ❛❧❧ t❤❡ ❝✉r✈❡s✳

  44. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✶ ■♥ ♦r❞❡r t♦ ✈❛❧✐❞❛t❡ ♦✉r ❝♦❞❡✱ ✇❡ ❤❛✈❡ ❝♦♠♣✉t❡❞ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ❜♦t❤ t❤❡ r✐❜❧❡ts ✐♥ ✜❣✉r❡ ✷✳✾ ❛♥❞ ❢♦r s♦♠❡ ✈❛❧✉❡ ♦❢ s ❀ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♦❜s❡r✈❡ ❢r♦♠ t❤❡ ✜❣✉r❡ ✷✳✶✵ t❤❛t ♦✉r r❡s✉❧ts ❛r❡ ✐♥ ♣❡r❢❡❝t ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ t❤❡ ▲✉❝❤✐♥✐✬s ♦♥❡✳ ❚❤❡ ❝♦❞❡ ✐s t❤❡r❡❢♦r❡ ✈❛❧✐❞❛t❡❞✳ ❋✐❣✉r❡ ✷✳✶✵✿ ❱❛❧✐❞❛t✐♦♥ ❛❣❛✐♥st ▲✉❝❤✐♥✐ ❡t ❛❧✳ ❬✷✵❪

  45. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✷ ✷✳✼ ❘❡s✉❧ts ✷✳✼✳✶ ❈♦♥str✉❝t❡❞ ❣❡♦♠❡tr✐❡s ❚❤✐s ✇♦r❦ ❛✐♠ ❛t ❝♦♠♣✉t✐♥❣ t❤❡ s❦✐♥ ❢r✐❝t✐♦♥ ♦✈❡r ❛ ❢r❛❝t❛❧ r✐❜❧❡t ♣❧❛t❡s ❞✉❡ t♦ ❛ t✉r❜✉❧❡♥t ✢♦✇✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❡✐❣❤t ❞✐✛❡r❡♥t r✐❜❧❡t s❤❛♣❡s ❤❛✈❡ ❜❡❡♥ ❜✉✐❧t ❛♥❞ s✐♠✉❧❛t❡❞✳ ❆❧❧ t❤❡ ❝♦♥str✉❝t❡❞ ❣❡♦♠❡tr✐❡s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡s✳ ❋✐rst ♦❢ ❛❧❧ ✐t ✐s ✇♦rt❤ t♦ r❡♠❡♠❜❡r t❤❛t t❤❡ ❇❊▼ s♦❧✈❡s t❤❡ ❞✐♠❡♥s✐♦♥❧❡ss ❡q✉❛t✐♦♥s ✷✳✶✵ ❛♥❞ ✷✳✾✱ t❤❡r❡❢♦r❡✱ t❤❡ ❣❡♦♠❡tr✐❡s ❣✐✈❡♥ ✐♥ ✐♥♣✉t t♦ t❤❡ ❝♦❞❡ ♠✉st ❜❡ ❝♦♥s✐❞❡r❡❞ ❞✐♠❡♥s✐♦♥❧❡ss✳ ▼♦r❡♦✈❡r✱ ✇❡ ❤❛✈❡ ❝❤♦s❡♥ t♦ ❜✉✐❧❞ r✐❜❧❡t ❣❡♦♠❡tr✐❡s ✇✐t❤ ♣❡r✐♦❞ ❡q✉❛❧ t♦ ♦♥❡✱ ❛♥❞ t❤✐s ✐♠♣❧② t❤❛t ❙t♦❦❡s ❡q✉❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ ❞✐♠❡♥s✐♦♥❛❧✐③❡❞ ✇✐t❤ t❤❡ ♣❡r✐♦❞ ✭ L ✮✳ ❚✇♦ ❢r❛❝t❛❧ ❣❡♦♠❡tr✐❡s ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❜② t❤❡ t✇♦ ❞✐✛❡r❡♥t tr✐❛♥❣✉❧❛r ♠♦❧❞ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✶✶❛ ❛♥❞ ✜❣✉r❡ ✷✳✶✶❜❀ t❤❡ ✜rst ♦♥❡ ❤❛s ❛♥ ❛♥❣❧❡ t♦ t❤❡ ✈❡rt❡① ♦❢ 120 ◦ ✭t❤❛t ✐♠♣❧② L/H = 0 . 29 ✱ ✇❤❡r❡ H ✐♥❞✐❝❛t❡s t❤❡ r✐❜❧❡t ❝❛✈✐t② ❞❡♣t❤✮✱ ❛♥❞ t❤❡ s❡❝♦♥❞ ♦❢ 90 ◦ ✭t❤❛t ✐♠♣❧② L/H = 0 . 5 ✮✳ ❚❤❡s❡ ❣❡♦♠❡tr✐❡s ❤❛✈❡ ❜❡❡♥ ❜✉✐❧t ✇✐t❤ ❛ s✉✐t❛❜❧❡ ❈❆❉ s♦❢t✇❛r❡ ❛♥❞ ❢♦❧❧♦✇✐♥❣ t❤❡ s❛♠❡ st❡♣s ✉s❡❞ t♦ ❜✉✐❧❞ t❤❡ ❑♦❝❤ s♥♦✇✢❛❦❡ ✐♥ s❡❝t✐♦♥ ✶✳✸✳ ❋♦r ❡①❛♠♣❧❡s✱ ✐♥ ♦r❞❡r t♦ ❜✉✐❧t t❤❡ ✜rst ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ❜♦t❤ t❤❡ s✐❞❡s ♦❢ t❤❡ tr✐❛♥❣✉❧❛r ♠♦❧❞ ❤❛✈❡ ❜❡❡♥ ❞✐✈✐❞❡❞ ✐♥ t❤r❡❡ ❡q✉❛❧ ♣❛rts✱ ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ❝❡♥tr❛❧ s❡❣♠❡♥t ❤❛s ❜❡❡♥ s✉❜st✐t✉t❡❞ ❜② ❛♥ ❛♣♣r♦♣r✐❛t❡❧② s❝❛❧❡❞ ♦r✐❣✐♥❛❧ ♠♦❧❞s✳ ❚❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡ ❤❛s ❜❡❡♥ ❢♦❧❧♦✇❡❞ ❢♦r t❤❡ ❛❢t❡r✇❛r❞s ✐t❡r❛t✐♦♥s✳ ❚❤✐s t②♣❡ ♦❢ ❝♦♥str✉❝t✐♦♥ ✐s ♣♦ss✐❜❧❡ t❤❛♥❦s t♦ t❤❡ s❡❧❢✲s✐♠✐❧❛r✐t② ♦❢ t❤❡ ❢r❛❝t❛❧s✳ ❖♥❝❡ ❛❧❧ t❤❡ r✐❜❧❡t s❤❛♣❡ ❤❛✈❡ ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t✱ ✇❡ ❤❛✈❡ ✉s❡❞ ❛ s✉✐t❛❜❧❡ ♠❡s❤✐♥❣ s♦❢t✇❛r❡ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛♥ ♦♣t✐♠❛❧ ♣♦✐♥ts ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❢r❛❝t❛❧ s❤❛♣❡✳ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♠✉st ✐♥ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡ r✉❧❡s ❞❡s❝r✐❜❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✺✳✹✱ t❤❡r❡❜② ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛♥ ❡①❝❡❧❧❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t✐❡s ✐♥ t❤❡ ❝♦r♥❡rs✳ ❉♦✐♥❣ s♦ t❤❡ ♣♦✐♥ts t❤❛t ❞❡s❝r✐❜❡ ❡❛❝❤ ❣❡♦♠❡tr② ❛r❡ ♦❜t❛✐♥❡❞ ❛♥❞ ❣✐✈❡♥ ✐♥ ✐♥♣✉t t♦ t❤❡ ▼❛t❧❛❜ ❝♦❞❡ t❤❛t ✐♠♣❧❡♠❡♥t t❤❡ ❇❊▼✳

  46. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✸ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❛✮ 120 ◦ tr✐❛♥❣✉❧❛r ♠♦❧❞✳ ✭❜✮ 90 ◦ tr✐❛♥❣✉❧❛r ♠♦❧❞✳ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❝✮ ❋✐rst ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ ✭❞✮ ❋✐rst ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❡✮ ❙❡❝♦♥❞ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ ✭❢✮ ❙❡❝♦♥❞ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❣✮ ❚❤✐r❞ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ ✭❤✮ ❚❤✐r❞ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ ❋✐❣✉r❡ ✷✳✶✶✿ ❋r❛❝t❛❧s ❣❡♦♠❡tr✐❡s ✉♥❞❡r ✐♥✈❡st✐❣❛t✐♦♥✳

  47. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✹ 120 ◦ tr✐❛♥❣❧❡ ✷✳✼✳✷ ■♥ t❤✐s s❡❝t✐♦♥ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❢♦r t❤❡ tr✐❛♥❣✉❧❛r r✐❜❧❡t ✭✇❤✐❝❤ ❤❛s ❛♥ ❛♥❣❧❡ t♦ t❤❡ ✈❡rt❡① ❡q✉❛❧ t♦ 120 ◦ ✮ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✶✷❛ ❛r❡ s❤♦✇♥✳ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❛✮ ❚r✐❛♥❣✉❧❛r r✐❜❧❡t✳ ✭❜✮ ❇❧❛❞❡ r✐❜❧❡t✳ ❋✐❣✉r❡ ✷✳✶✷✿ ●❡♦♠❡tr✐❡s ✉♥❞❡r ✐♥✈❡st✐❣❛t✐♦♥✳ ■♥ ♦r❞❡r t♦ ❤❛✈❡ ❛ r❡❢❡r❡♥❝❡ ✈❛❧✉❡ ❢♦r t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡♠✱ ✇❡ ❤❛✈❡ t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ t❤❡ ❜❧❛❞❡ r✐❜❧❡t s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✶✷❜✱ ✇✐t❤ t❤❡ s❛♠❡ ❞❡♣t❤ ♦❢ t❤❡ tr✐❛♥❣✉❧❛r ♦♥❡✳ ■♥ t❛❜❧❡ ✷✳✶ ❛r❡ r❡♣♦rt❡❞ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❝♦♠♣✉t❡❞ ❢♦r t❤❡ tr✐❛♥❣✉❧❛r ❛♥❞ t❤❡ r❡❝t❛♥❣✉❧❛r r✐❜❧❡t❀ t❤❡ ❞❛t❛ s❤♦✇s t❤❛t t❤❡ ∆ h ✐s ❤✐❣❤❡r ✐♥ t❤❡ ❜❧❛❞❡ r✐❜❧❡t ❝❛s❡✳ ❈❛s❡ h || h ⊥ ∆ h ❇❧❛❞❡ r✐❜❧❡t 0 . 1723 0 . 0801 0 . 0922 ❚r✐❛♥❣✉❧❛r r✐❜❧❡t 0 . 1026 0 . 0704 0 . 0322 ❚❛❜❧❡ ✷✳✶✿ Pr♦tr✉s✐♦♥ ❤❡✐❣❤ts ♦❢ ❜❧❛❞❡✲r✐❜❧❡t ❛♥❞ 120 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ■♥ ❧✐♥❡ ✇✐t❤ ▲✉❝❤✐♥✐✬s t❤❡♦r②✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✉♥❞❡rst❛♥❞ t❤❛t ❞r❛❣ r❡❞✉❝t✐♦♥ ❛❝❤✐❡✈❡❞ ✇✐t❤ t❤❡ tr✐❛♥❣✉❧❛r r✐❜❧❡t ✇✐❧❧ ❜❡ s♠❛❧❧❡r t❤❡♥ t❤❛t ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❡ ❜❧❛❞❡ ♦♥❡✳ ❚❤❡ r❡s✉❧t ✐s ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ ❛ ❧♦t ♦❢ ❡①♣❡r✐♠❡♥t ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ t✇♦ r✐❜❧❡t s❤❛♣❡s ❛♥❞ ❡✈❡♥ ✇✐t❤ t❤❡ t❤❡♦r❡t✐❝❛❧ ✇♦r❦ ❜② ❇❡❝❤❡rt ❛♥❞ ❇❛rt❡✇❡r❢❡r ❬✷✼❪✳ ❋r♦♠ ❛ ♣❤②s✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ t❤❡ ✈✐s❝♦✉s ✢♦✇ ♦✈❡r tr✐❛♥❣✉❧❛r r✐❜❧❡t ❣❡♦♠❡tr② ❣❡♥❡r❛t❡s ❛♥ ❤✐❣❤❡r s❦✐♥ ❢r✐❝t✐♦♥ t❤❡♥ t❤❛t ♦✈❡r t❤❡ ❜❧❛❞❡ r✐❜❧❡t ❬✶❪✳ ■t ✐s ❡①tr❡♠❡❧② ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❛t t❤❡

  48. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✺ tr✐❛♥❣✉❧❛r r✐❜❧❡t ✇❛❧❧ s✉r❢❛❝❡ ✐s s♠❛❧❧❡r t❤❡♥ t❤❡ ❜❧❛❞❡ r✐❜❧❡t ♦♥❡✱ s♦ t❤❛t ♦♥❡ ❡①♣❡❝t ❛♥ ❤✐❣❤❡r ❞r❛❣ ♦♥ t❤❡ ❜❧❛❞❡ r✐❜❧❡t✳ ❚❤❡r❡❢♦r❡ ❛ ✈✐s❝♦✉s ♠❡❝❤❛♥✐s♠ ✇❤✐❝❤ ❝♦♠♣❡♥s❛t❡ t❤❡ ❛r❡❛ ✐♥❝r❡❛s❡ ❡①✐st✳ ◆♦✇ ❧❡t ✉s ❧♦♦❦ t♦ t❤❡ r❡s✉❧ts r❡♣♦rt❡❞ ✐♥ t❛❜❧❡ ✷✳✷✱ t❤❡② ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❛♣♣❧②✐♥❣ t❤❡ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞ ❛t ❡❛❝❤ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡ s❤♦✇ ✐♥ ✜❣✉r❡ ✷✳✶✶❛✳ ■t❡r❛t✐♦♥ h || h ⊥ ∆ h 0 0 . 1026 0 . 0704 0 . 0322 1 0 . 0819 0 . 0604 0 . 0215 2 0 . 0755 0 . 0554 0 . 0201 3 0 . 0737 0 . 0541 0 . 0196 ❚❛❜❧❡ ✷✳✷✿ Pr♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ❛❧❧ ❢r❛❝t❛❧s ✐t❡r❛t✐♦♥s ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ ❋✐rst ♦❢ ❛❧❧ ✐t ✐s ✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t | ∆ h 3 − ∆ h 0 | < | ∆ h blade − ∆ h 0 | ✭✷✳✽✸✮ ∆ h 0 ∆ h 0 ✇❤❡r❡ ∆ h 3 ❛♥❞ ∆ h 0 ❛r❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts r❡s♣❡❝t✐✈❡❧② ♦❢ t❤❡ ✐t❡r❛t✐♦♥ ✸ ❛♥❞ t❤❡ ✐t❡r❛t✐♦♥ ✵✱ ✇❤❡r❡❛s ∆ h blade ✐s t❤❛t ♦❢ t❤❡ ❜❧❛❞❡ r✐❜❧❡t✳ ❚❤❡ ❜❧❛❞❡ r✐❜❧❡t ❛♥❞ t❤❡ tr✐❛♥❣✉❧❛r ♦♥❡ ✭✐t❡r❛t✐♦♥ ✵✮ ❣❡♥❡r❛t❡ ❛ ❞r❛❣ r❡❞✉❝t✐♦♥ t❤❛t ❞✐✛❡rs ♦❢ ❛ ❢❡✇ ♣❡r❝❡♥t ♣♦✐♥ts ✭❛❜♦✉t 5 ✮✱ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡✐r ∆ h ✐s | ∆ h blade − ∆ h 0 | / ∆ h 0 = 1 . 86 ✳ ❙✐♥❝❡ ✇❡ ❤❛✈❡ | ∆ h 3 − ∆ h 0 | / ∆ h 0 = 0 . 39 ✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❛r❣✉❡ t❤❛t t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ❞r❛❣ ♦❜t❛✐♥❡❞ ♣❛ss✐♥❣ ❢r♦♠ t❤❡ tr✐❛♥❣✉❧❛r t♦ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t ✇✐❧❧ ❜❡ s♠❛❧❧✱ ❛♥❞ ❝❡rt❛✐♥❧② ♥♦t ❤✐❣❤❡r t❤❡♥ ❛ ❢❡✇ ♣❡r❝❡♥t ♣♦✐♥ts✳ ❚❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡♠ ❞❡❝r❡❛s❡ ❣♦✐♥❣ ♦♥ ✇✐t❤ t❤❡ ✐t❡r❛t✐♦♥s✳ ❚❤❡ ❧❛st tr❡♥❞ ✐s ❝❡rt❛✐♥❧② ❞✉❡ t♦ ❛ ✈✐s❝♦✉s ❡✛❡❝t✱ ✐♥ ❢❛❝t t❤❡ ❡q✉❛t✐♦♥s t❤❛t ❛❧❧♦✇ ✉s t♦ ❝♦♠♣✉t❡ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❞♦ ♥♦t ♣r❡s❡♥t t❤❡ ❝♦♥✈❡❝t✐✈❡ t❡r♠✳ ■♥ ♦t❤❡r ✇♦r❞s t❤❡ ❡✛❡❝t ♦❢ t❤❡ r✐❜❧❡t s❤❛♣❡ ♦♥ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❝❛♥♥♦t ❜❡ ❛s❝r✐❜❡❞ t♦ t❤❡ t✉r❜✉❧❡♥❝❡ r❡❞✉❝t✐♦♥✳ ❚❤❡r❡❢♦r❡ t❤❡ ♠✐❝r♦s❝♦♣✐❝ ❢❡❛t✉r❡ ♦❢ t❤❡ r✐❜❧❡t s✉r❢❛❝❡ ❛r❡ ❝❛♣❛❜❧❡ t♦ ♠♦❞✐❢② t❤❡ ✈✐s❝♦✉s ✢♦✇ ♥❡❛r t❤❡ ✇❛❧❧✳ ■t ✐s ❡✈❡♥ ✇♦rt❤ t♦ ❝❧❛r✐❢② t❤❛t t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ✈✐s❝♦✉s ✢♦✇ ♠❛② ❛✛❡❝t t❤❡ t✉r❜✉❧❡♥❝❡ ❧❡✈❡❧✱ ❜✉t ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❝❛♥♥♦t

  49. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✻ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t s✉❝❤ ❛ ♠♦❞✐✜❝❛t✐♦♥✳ ❍♦✇❡✈❡r✱ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❝♦♥s✐❞❡r❛t✐♦♥ ❞♦❡s ♥♦t ❡①❝❧✉❞❡ ❛ ♣♦ss✐❜❧❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♥❞ t❤❡ t✉r❜✉❧❡♥❝❡ ♠♦❞✐✜❝❛t✐♦♥✳ ❋✐❣✉r❡ ✷✳✶✸ ❛♥❞ ✜❣✉r❡ ✷✳✶✹ r❡✈❡❛❧ t❤❛t t❤❡ ♣r♦tr✉s✐♦♥s ❤❡✐❣❤ts ❤❛✈❡ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❜❡❤❛✈✐♦r✱ ❛♥❞ t❤❡② ❛♣♣r♦❛❝❤ ❛ ✜♥✐t❡ ✈❛❧✉❡✳ ❚❤❡ ♠♦t✐✈❛t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❛❞❞✐t✐♦♥❛❧ tr✐❛♥❣✉❧❛r ♣✐❡❝❡s t❤❛t ❛r❡ ✉s❡❞ t♦ ❜✉✐❧❞ ❡❛❝❤ ♥❡①t ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥✳ ■♥ ❢❛❝t✱ ❣♦✐♥❣ ♦♥ ✇✐t❤ t❤❡ ✐t❡r❛t✐♦♥ t❤❡s❡ ♣✐❡❝❡s ❜❡❝♦♠❡ ✐♥❝r❡❛s✐♥❣❧② s♠❛❧❧✱ ❛s ❧♦♥❣ ❛s t❤❡② ❞♦ ♥♦t ❛✛❡❝t t❤❡ ✢♦✇ ❛♥②♠♦r❡✳ ■♥ ♦r❞❡r t♦ ❤❛✈❡ ❛♥ ✐❞❡❛ ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡ ♦r❞❡r ♦❢ t❤❡ ❞✐st✉r❜❛♥❝❡ ✇❤✐❝❤ ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ✢♦✇✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ❜❛s❡ B 3 ❛♥❞ t❤❡ ❤❡✐❣❤t H 3 ♦❢ t❤❡ ♠♦st ❧✐tt❧❡ tr✐❛♥❣❧❡ t❤❛t ♠❛❦❡ ✉♣ t❤❡ t❤✐r❞ ✐t❡r❛t✐♦♥✳ ❚❤❡ ❜❛s❡ s❝❛❧❡ ❡✈❡r② ✐t❡r❛t✐♦♥ ✇✐t❤ ❛ ❢❛❝t♦r ❡q✉❛❧ t♦ 1 / 3 ✱ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ B 3 = B 0 / 3 3 ✱ ✇❤❡r❡ B 0 r❡♣r❡s❡♥t t❤❡ ❜❛s❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ tr✐❛♥❣❧❡✱ ✇❤✐❝❤ t②♣✐❝❛❧❧② ❤❛s ❛ ✈❛❧✉❡ ♦❢ 100 µm ❛♥❞ t❤❡r❡❢♦r❡ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♦❜t❛✐♥ B 3 ≈ 4 µm ✳ ❊✈❡♥t✉❛❧❧② ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝♦♠♣✉t❡ √ t❤❡ ❤❡✐❣❤t ❡①♣❧♦✐t✐♥❣ ✐ts r❡❧❛t✐♦♥ ✇✐t❤ t❤❡ ❜❛s❡ H 3 = B 3 3 / 6 ≈ 1 µm ✳ ●❛r❝í❛✲▼❛②♦r❛❧ ❡t ❛❧✳ ♣♦✐♥t❡❞ ♦✉t t❤❛t r✐❜❧❡t ❛r❡❛ ✐s t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣❛r❛♠❡t❡r ❢♦r t❤❡ r✐❜❧❡t ❡✛❡❝t ❡✈❛❧✉❛t✐♦♥✱ t❤❡r❡❢♦r❡ ✐t ✐s ❡✈❡♥ ♣♦ss✐❜❧❡ t♦ ❝♦♥❝❧✉❞❡ t❤❛t ❛ r✐❜❧❡t ❝❛✈✐t② ✈❛r✐❛t✐♦♥ ♦❢ ❛❜♦✉t 4 µm 2 ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ✈✐s❝♦✉s ✢♦✇✱ ❛♥❞ ❝♦♥s❡q✉❡♥t❧② t❤❡ t✉r❜✉❧❡♥t ♦♥❡✳ ❚❤✐s ✐s ♦♥❧② ❛ q✉❛❧✐t❛t✐✈❡ r❡s✉❧t✱ ❜✉t ✐t ✐s ✐♠♣♦rt❛♥t ❜❡❝❛✉s❡ r❡✈❡❛❧s t❤❛t t❤❡ ♥❛♥♦s❝♦♣✐❝ str✉❝t✉r❡ ♦❢ t❤❡ s✉r❢❛❝❡ ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ✢♦✇ ❛♥❞ ❤❡♥❝❡ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥❀ t❤❡r❡❢♦r❡ t❤❡ s✉r❢❛❝❡ ❝❛♥ ❜❡ ♠❛❝❤✐♥❡❞ ✉♥t✐❧ t❤❡ ♥❛♥♦s❝♦♣✐❝ ❞✐♠❡♥s✐♦♥ ✭✐t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡r❧✐❡ t❤❛t t❤❡ ♥❛♥♦s❝♦♣✐❝ str✉❝t✉r❡ ♦❢ t❤❡ s✉r❢❛❝❡ ❝❛♥ ❜❡ ♥❡❣❧❡❝t❡❞ ❢♦r t❤❡ ❢✉❧❧② ✇❡t ❝❛s❡✱ ❜✉t ♥♦t ❢♦r t❤❡ s✉♣❡r❤②❞r♦♣❤♦❜✐❝ ♦♥❡ ❛s ✇❡ ✇✐❧❧ s❡❡ ✐♥ t❤❡ ❧❛st ❝❤❛♣t❡r✮✳ ❚❤❡ ❢r❛❝t❛❧ r✐❜❧❡ts ✉♥t✐❧ ♥♦✇ t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t t❤r♦✉❣❤ t❤❡ ❛❞❞✐t✐♦♥ ♦❢ tr✐❛♥❣❧❡s ✇✐t❤ t❤❡ ✈❡rt❡① t❤❛t ♣♦✐♥t ♦✉t✇❛r❞ t❤❡ ✇❛❧❧❀ ❛s t❤❡ ❞❛t❛ s❤♦✇♥✱ s✉❝❤ ❛ ❢r❛❝t❛❧✐③❛t✐♦♥ ❤❛s ❛ ♥❡❣❛t✐✈❡ ❡✛❡❝t ♦♥ t❤❡ ❞r❛❣✳ ■t ✐s ♥♦✇ ✐♥t❡r❡st✲ ✐♥❣ t♦ st✉❞② t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❛ ❢r❛❝t❛❧ r✐❜❧❡t ❝♦♥str✉❝t❡❞ ❡①♣❧♦✐t✐♥❣ t❤❡ s❛♠❡ tr✐❛♥❣✉❧❛r ♠♦❧❞ ❜✉t ✇✐t❤ ❡❛❝❤ s♠❛❧❧ tr✐❛♥❣❧❡ t❤❛t ♣♦✐♥t ✐♥✇❛r❞✳ ❚❤❡ ❣❡♦♠❡tr✐❡s ♦❜t❛✐♥❡❞ ❛r❡ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✶✺✳

  50. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✼ 0.11 h || data 0.1 fit h data Protrusion height fit 0.09 0.08 0.07 0.06 0.05 0 1 2 3 Number of iteration ❋✐❣✉r❡ ✷✳✶✸✿ P❛r❛❧❧❡❧ ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ 0.034 h data 0.032 fit 0.03 0.028 h 0.026 0.024 0.022 0.02 0.018 0 1 2 3 Number of iteration ❋✐❣✉r❡ ✷✳✶✹✿ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r t❤❡ 120 ◦ tr✐❛♥❣❧❡✳

  51. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✽ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❛✮ 120 ◦ tr✐❛♥❣✉❧❛r ♠♦❧❞✳ ✭❜✮ ❋✐rst ✐♥✇❛r❞ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❝✮ ❙❡❝♦♥❞ ✐♥✇❛r❞ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ ✭❞✮ ❚❤✐r❞ ✐♥✇❛r❞ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ ❋✐❣✉r❡ ✷✳✶✺✿ ❘✐❜❧❡t ❣❡♦♠❡tr✐❡s ✇✐t❤ ✐♥✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥✳ ❚❤❡ r❡s✉❧ts r❡♣♦rt❡❞ ✐♥ t❛❜❧❡ ✷✳✸ r❡✈❡❛❧s t❤❛t ❡❛❝❤ ✐t❡r❛t✐♦♥ ❣❡♥❡r❛t❡ ❛♥ ✐♥❝r❡♠❡♥t ♦❢ ❛❧❧ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts❀ ✐♥ ✜❣✉r❡ ✷✳✶✻ ❛♥❞ ✜❣✉r❡ ✷✳✶✼ ❛r❡ s❤♦✇♥ t❤❡✐r tr❡♥❞s✳ ■t❡r❛t✐♦♥ h || h ⊥ ∆ h 0 0 . 1026 0 . 0704 0 . 0322 1 0 . 1112 0 . 0715 0 . 0397 2 0 . 1138 0 . 0723 0 . 0415 3 0 . 1146 0 . 0725 0 . 0421 ❚❛❜❧❡ ✷✳✸✿ Pr♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ❛❧❧ ✐♥✇❛r❞ ❢r❛❝t❛❧s ✐t❡r❛t✐♦♥s ♦❢ t❤❡ ✶✷✵ ◦ tr✐❛♥❣❧❡✳ ❈♦♠♣❛r✐♥❣ t❤❡s❡ ❡①♣♦♥❡♥t✐❛❧ ❝✉r✈❡s ✇✐t❤ ✜❣✉r❡ ✷✳✶✸ ❛♥❞ ✜❣✉r❡ ✷✳✶✹✱ ✐t st❛♥❞s ♦✉t t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ s❧♦♣❡✳ ❲❡ ❝❛♥ t❤❡r❡❢♦r❡ ❛r❣✉❡ t❤❛t t❤❡ ✐♥✇❛r❞ ❢r❛❝t❛❧✐③❡❞ ❝❛✈✐t② ❛✛❡❝t

  52. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✹✾ 0.12 0.11 h || data Protrusion height fit h data 0.1 fit 0.09 0.08 0.07 0 1 2 3 Number of iteration ❋✐❣✉r❡ ✷✳✶✻✿ P❛r❛❧❧❡❧ ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ✐♥✇❛r❞ ❢r❛❝t❛❧s ✐t❡r❛t✐♦♥s ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ 0.044 0.042 0.04 h 0.038 0.036 0.034 h data fit 0.032 0 1 2 3 Number of iteration ❋✐❣✉r❡ ✷✳✶✼✿ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ✐♥✇❛r❞ ❢r❛❝t❛❧s ✐t❡r❛t✐♦♥s ♦❢ t❤❡ 120 ◦ tr✐❛♥❣❧❡✳

  53. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✵ t❤❡ ✈✐s❝♦✉s ✢♦✇ s♦ t❤❛t t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐s ✐♥❝r❡❛s❡❞✳ ■t ❡♠❡r❣❡s t❤❛t t❤❡ ❡♥❤❛♥❝❡ ♦❢ t❤❡ ❝r♦ss✲s❡❝t✐♦♥❛❧ r✐❜❧❡t ❛r❡❛ ✭❞✉❡ t♦ t❤❡ ♦✉t✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥✮ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✐♥❝r❡❛s❡ ♦❢ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ♣♦t❡♥t✐❛❧✱ ✇❤❡r❡❛s t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ✭❞✉❡ t♦ t❤❡ ✐♥✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥✮ ✐s r❡❧❛t❡❞ t♦ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ♣♦t❡♥t✐❛❧✳ ❚❤❡ r❡s✉❧ts ✐s ✐♥ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❛t ♦❜t❛✐♥❡❞ ❜② ●❛r❝✐❛✲▼❛②♦r❝❛❧ ❬✷✷❪✱ ✐♥ ✇❤✐❝❤ t❤❡② ✐♥❞❡♥t✐❢② t❤❡ ❝r♦ss✲s❡❝t✐♦♥❛❧ ❛r❡❛ ❛s t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣❛r❛♠❡t❡r ✐♥ t❤❡ r✐❜❧❡t ❡✛❡❝t✐✈❡♥❡ss ❡✈❛❧✉❛t✐♦♥✳ ❆s ✇❡❧❧ ❛s t❤❡ ♦✉t✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♣♣r♦❛❝❤ ❛ ✜♥✐t❡ ✈❛❧✉❡✱ t❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ♦❜s❡r✈❡ ❛❣❛✐♥ t❤❛t ❛ t♦♦ s♠❛❧❧ ❞✐st✉r❜❛♥❝❡ ♦❢ t❤❡ s✉r❢❛❝❡ ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♥❞ ❡✈❡♥t✉❛❧❧② t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥✳ ❆♥♦t❤❡r ♦♥❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❢r❛❝t❛❧✐③❛t✐♦♥ ❝❛♥ ❜❡ ♣♦✐♥t❡❞ ♦✉t❀ t❤♦✉❣❤ t❤❡ ❢r❛❝t❛❧✐③❛t✐♦♥s ❛r❡ s②♠♠❡tr✐❝ r❡s♣❡❝t t♦ t❤❡ ♦r✐❣✐♥❛❧ ♠♦❧❞✱ t❤❡ ✈❛❧✉❡ ♦❜t❛✐♥❡❞ ❢♦r t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛r❡ ♥♦t✳ ❈♦♠♣❛r✐♥❣ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ❞❡❝r❡♠❡♥ts ♦❢ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ♦✉t✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ ✇✐t❤ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✐♥❝r❡♠❡♥t ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ✐♥✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥✱ ✐t ❝❛♥ ❜❡ ♥♦t✐❝❡❞ t❤❛t t❤❡ ✜rst ♦♥❡ ✐s ❧❛r❣❡r t❤❡♥ t❤❡ s❡❝♦♥❞ ♦♥❡✳ ❋r♦♠ t❤✐s ♦❜s❡r✈❛t✐♦♥ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ✈✐s❝♦✉s ✢♦✇ ✐s ♠♦r❡ s✉s❝❡♣t✐❜❧❡ t♦ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ r✐❜❧❡t ❝❛✈✐t② r❛t❤❡r t❤❡♥ t❤❡ ✐♥❝r❡♠❡♥t ♦❢ ✐t❀ t❤❡r❡❢♦r❡ t❤❡ r✐❜❧❡t ❡✛❡❝t ❝❛♥ ❜❡ ❡❛s✐❡r r❡❞✉❝❡❞ t❤❡♥ ✐♥❝r❡❛s❡❞✳ ❚❤❡ ✇❡tt❡❞ s✉r❢❛❝❡ ✐s ❡q✉❛❧ ✐♥ ❜♦t❤ t❤❡ ✐♥✇❛r❞ ❛♥❞ ♦✉t✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥✱ ❜✉t t❤❡ ✜rst ♦♥❡ s❡❡♠s t♦ ♣r♦♠✐s❡ ❛ ❧♦✇❡r ❞r❛❣✳ ■t ✐s t❤❡r❡❢♦r❡ ♣♦ss✐❜❧❡ t♦ ❛r❣✉❡ ✐♥ t❤❛t ❝❛s❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠❡❝❤❛♥✐s♠ ✇❤✐❝❤ ❝♦♠♣❡♥s❛t❡ t❤❡ ✇❡tt❡❞ ✇❛❧❧ s✉r❢❛❝❡✳ ❚❤❡ ✐♠♣❧❡♠❡♥t❡❞ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ❝♦❞❡ ❝❛♥ ❡✈❡♥ ❝♦♠♣✉t❡ t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ✐♥s✐❞❡ t❤❡ ❞♦♠❛✐♥✳ ❚❤❡ ✈✐s✉❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ❝♦♥t♦✉r ❛❧❧♦✇s t♦ ❛♣♣r❡❝✐❛t❡ t❤❡ ❡✛❡❝t ♦❢ t❤❡ r✐❜❧❡t ♦♥ t❤❡ ♥❡❛r ✇❛❧❧ ✢♦✇✳ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡r❧✐❡ t❤❛t t❤❡ ❇❊▼ s♦❧✈❡s t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ❞♦❡s ♥♦t ♣r❡s❡♥t t❤❡ ❝♦♥✈❡❝t✐✈❡ t❡r♠✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❛r❡ ❛ss✉♠✐♥❣ t❤❛t t❤❡ ❝♦♥✈❡❝t✐✈❡ t❡r♠ ✐s ♥❡❣❧✐❣✐❜❧❡ ❡✈❡♥ ❛✇❛② t❤❡ ✇❛❧❧✱ ❜✉t s✉❝❤ ❛ s✐♠♣❧✐✜❝❛t✐♦♥ s❤♦✉❧❞ ❜❡ ♦♥❧② ✈❛❧✐❞ ♥❡❛r t❤❡ ✇❛❧❧✱ ✐♥ t❤❡ ✈✐s❝♦✉s s✉❜❧❛②❡r✳ ■♥ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤✐s✱ t❤❡ ❝♦♠♣✉t❡❞ ✈❡❧♦❝✐t② ✜❡❧❞s ♠✉st ❜❡ ✉s❡❞ t♦ ❡✈❛❧✉❛t❡ t❤❡ ✢♦✇ ✐♥ t❤❡ ♣r♦①✐♠✐t② ♦❢ t❤❡ ✇❛❧❧✳ ❙✉❝❤ ❛ ❧✐♠✐t❛t✐♦♥ ❞♦❡s ♥♦t ❛❧❧♦✇ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❡✛❡❝t ♦♥ t❤❡ ❜✉✛❡r ❛♥❞ ❧♦❣❛r✐t❤♠✐❝ ❧❛②❡r ❛s ✇❛❧❧ ❛s t❤❡ t✉r❜✉❧❡♥t ❧❡✈❡❧✱ ❜✉t ✐t ❝❛♥ ❜❡ ✉s❡❢✉❧ ❢♦r t❤❡ ♥❡❛r ✇❛❧❧ ✢♦✇ str✉❝t✉r❡ ✉♥❞❡rst❛♥❞✐♥❣✳ ❚❤❡ str❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞s ❛r❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✶✽✱ ✜❣✉r❡ ✷✳✶✾ ❛♥❞ ✜❣✉r❡ ✷✳✷✵✳ ■♥ ❛❧❧ ❝❛s❡s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ♦♥ t❤❡ ♣❡❛❦s ✐s ❞✐✛❡r❡♥t ❜② t❤❛t ♦✈❡r t❤❡ ✈❛❧❧❡②s✱ s♦ t❤❛t ✐♥ t❤❡ ❧❛st r❡❣✐♦♥ t❤❡ ❧♦✇ ✈❡❧♦❝✐t② ③♦♥❡

  54. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✶ ❋✐❣✉r❡ ✷✳✶✽✿ ❙tr❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ ✭ w ✮ ❢♦r t❤❡ s✐♠♣❧❡ 120 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ❋✐❣✉r❡ ✷✳✶✾✿ ❙tr❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ ✭ w ✮ ❢♦r t❤❡ t❤✐r❞ ♦✉t✇❛r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳

  55. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✷ ❋✐❣✉r❡ ✷✳✷✵✿ ❙tr❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ w ❢♦r t❤❡ t❤✐r❞ ✐♥✇❛r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ✐s ♠♦r❡ ✇✐❞t❤✳ ❚❤✐s ✐s ❛ ♣♦s✐t✐✈❡ ❡✛❡❝t ❜❡❝❛✉s❡ ❦❡❡♣ t❤❡ ✈✐s❝♦✉s ✢♦✇ ❛✇❛② ❢r♦♠ t❤❡ t✉r❜✉❧❡♥t s♣♦ts✱ ✇❤✐❝❤ ❣❡♥❡r❛t❡ ❤✐❣❤ ✈❡❧♦❝✐t② ❛♥❞ ❤❡♥❝❡ ❤✐❣❤ s❦✐♥ ❢r✐❝t✐♦♥✳ ▼♦r❡♦✈❡r✱ t❤❡ ❤✐❣❤ ❣r❛❞✐❡♥t ♦♥ t❤❡ ♣❡❛❦s r❡✈❡❛❧ t❤❡ ✐♠♣♦rt❛♥❝❡ t♦ ❤❛✈❡ ❛ ♣❡r❢❡❝t s❤❛r♣ ✭✐t r❡❞✉❝❡ t❤❡ s✉r❢❛❝❡ ✇❤❡r❡ t❤❡ ❤✐❣❤ ❣r❛❞✐❡♥t ✐s ♣r❡s❡♥t✮✳ ■♥ ❡❛❝❤ ❝❛s❡ t❤❡ ❜❧❛❝❦ ❜❛♥❞ ❧✐❡ ♦♥ t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ r✐❜❧❡t ❝❛✈✐t② ❛♥❞ ✐t ❢♦❧❧♦✇ t❤❡ ❜♦✉♥❞❛r②✳ ❚❤❡r❡❢♦r❡ t❤❡ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② s❤❛♣❡ ❡✈❡♥ ❛✛❡❝t t❤❡ ❜❧❛❝❦ ❜❛♥❞ s❤❛♣❡✳ ■♥ ❜♦t❤ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t ✭✜❣✉r❡ ✷✳✶✾ ❛♥❞ ✜❣✉r❡ ✷✳✷✵✮ t❤❡ ❣❡♦♠❡tr② ♠♦❞✐✜❝❛t✐♦♥ ❞✉❡ t♦ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ s❡❡♠s t♦ ❛✛❡❝t t❤❡ ✢♦✇✱ ❜❡❝❛✉s❡ t❤❡ ❧♦✇ ✈❡❧♦❝✐t② r❡❣✐♦♥ ✭r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ❜❛♥❞ ❛tt❛❝❤❡❞ t♦ t❤❡ r✐❜❧❡t ❜♦tt♦♠✮ ✐s ♠♦❞✐✜❡❞✳ ❚❤❡ t❤✐r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ✢♦✇ ✭❛s ❛❧r❡❛❞② s❡❡♥ t❤❛♥❦ t♦ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✮✳ ■♥ t❤❡ ♦✉t✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ t❤❡ ❜❧❛❝❦ ❜❛♥❞ ✐s s❤✐❢t❡❞ ✉♣ ❛♥❞ t❤✐s ♠❛❦❡ ❢♦r ❛ ❧♦✇❡r ❧♦♥❣✐t✉❞✐♥❛❧ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t✳ ❖♥ t❤❡ ❝♦♥tr❛r② t❤❡ ✐♥✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ s❤✐❢t ❞♦✇♥ t❤❡ ❜❧❛❝❦ ❜❛♥❞✳ ■♥ ✜❣✉r❡ ✷✳✷✶✱ ✜❣✉r❡ ✷✳✷✷ ❛♥❞ ✜❣✉r❡ ✷✳✷✸ ❛r❡ r❡♣♦rt❡❞ t❤❡ y ✲✈❡❧♦❝✐t② ❝♦♠♣♦♥❡♥t ❝♦♥t♦✉r ❛♥❞ t❤❡ str❡❛♠✲❧✐♥❡s ✐♥ t❤❡ z ✲ y ♣❧❛♥❡✳ ❚❤❡ tr❛♥s✈❡rs❡ ✢♦✇ ♦✈❡r t❤❡ r✐❜❧❡t ✐s s✉st❛✐♥❡❞ ❜② t❤❡ ✐♠♣♦s❡❞ ❧✐♥❡❛r ✈❡❧♦❝✐t②✱ ✇❤✐❝❤ ❛♥✐♠❛t❡ t❤❡ ✢♦✇ ❢r♦♠ t❤❡ ❧❡❢t t♦ t❤❡ r✐❣❤t ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❙✉❝❤ ❛ ✐♠♣♦s✐t✐♦♥ ❣❡♥❡r❛t❡ ❛ ❞♦✇♥✲ ✇❛s❤ ♦❢ t❤❡ ♥♦r♠❛❧ ✈❡❧♦❝✐t② ♦♥ t❤❡ ❧❡❢t r✐❜❧❡t t✐♣✱ ❛♥❞ ❛♥ ✉♣✲✇❛s❤ ♦♥ t❤❡ r✐❣❤t ♦♥❡✳ ❚❤✐s ❡✛❡❝t ❝❤❛♥❣❡ t❤❡ str❡❛♠✲❧✐♥❡s ♣❛t❤✱ ❛♥❞ ❢♦r❝❡ t❤❡♠ t♦ t❤❡ ❜♦tt♦♠ ♦❢ t❤❡ r✐❜❧❡t✳

  56. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✸ ❋✐❣✉r❡ ✷✳✷✶✿ ❙♣❛♥✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ ✭ v ✮ ❛♥❞ str❡❛♠❧✐♥❡s ❢♦r t❤❡ s✐♠♣❧❡ 120 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ❋✐❣✉r❡ ✷✳✷✷✿ ❙♣❛♥✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ ✭ v ✮ ♥❛❞ str❡❛♠❧✐♥❡s ❢♦r t❤❡ t❤✐r❞ ♦✉t✇❛r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳

  57. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✹ ❋✐❣✉r❡ ✷✳✷✸✿ ❙♣❛♥✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ ✭ v ✮ ❛♥❞ str❡❛♠❧✐♥❡s ❢♦r t❤❡ t❤✐r❞ ✐♥✇❛r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 120 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ❚❤❡ v ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ❜r✐♥❣ ❞♦✇♥ ❡✈❡♥ t❤❡ str❡❛♠✲✇✐s❡ ❝♦♠♣♦♥❡♥t✱ ❛♥❞ ❝♦♥tr✐❜✉t❡ t♦ ❤♦❧❞ t❤❡ ✢♦✇ ✐♥ t❤❡ r✐❜❧❡t ❝❛✈✐t②✳ ❚❤❡ ✐♥✇❛r❞ ❢r❛❝t❛❧ r✐❜❧❡t ❣❡♥❡r❛t❡ t❤❡ ❤✐❣❤❡st ♥♦r♠❛❧ ✈❡❧♦❝✐t② ♠❛❣♥✐t✉❞❡✱ s♦ t❤❛t ✐t ✐s ❜❡tt❡r ❝❛♣❛❜❧❡ t♦ ❤♦❧❞ ❞♦✇♥ t❤❡ ✢♦✇ ✐♥ t❤❡ r✐❜❧❡t ❝❛✈✐t②✱ ❛♥❞ t❤✐s ❝❛♥ ♠❛② ❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❤✐❣❤❡r ❞r❛❣ r❡❞✉❝t✐♦♥✳ ❚❤❡ str❡❛♠✲❧✐♥❡s ♦♥ t❤❡ r✐❜❧❡t ❜♦tt♦♠ r❡✈❡❛❧ ✐♥ ❛❧❧ t❤r❡❡ ❝❛s❡s ❛ ❝♦r♥❡r ❡❞❞② ✭▼♦✛❛tt ❡t ❛❧✳ ❬✶✾❪✮ ✇✐t❤ ✈❛r✐♦✉s ❞✐♠❡♥s✐♦♥✳ ❚❤❡ ♦✉t✇❛r❞ ❢r❛❝t❛❧ r✐❜❧❡t ♣r❡s❡♥t t❤❡ ❜✐❣❣❡st ✈♦rt❡①✱ ❛♥❞ ✐t ❝❛♥ ♠❛② ❜❡ r❡❧❛t❡❞ t♦ t❤❡ ❜❛❞ ❞r❛❣ r❡❞✉❝t✐♦♥ ♣❡r❢♦r♠❛♥❝❡s✳

  58. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✺ 90 ◦ tr✐❛♥❣❧❡ ✷✳✼✳✸ ■♥ ♦r❞❡r t♦ st✉❞② t❤❡ ♣❡r✐♦❞ ♦♥ ❞❡♣t❤ r❛t✐♦♥ ❡✛❡❝t✱ ❛♥♦t❤❡r tr✐❛♥❣✉❧❛r r✐❜❧❡t s❤❛♣❡ ❤❛s ❜❡❡♥ st✉❞✐❡❞ ✭✜❣✉r❡ ✷✳✷✹❛✮✳ ■t ❤❛s t❤❡ ❛♥❣❧❡ t♦ t❤❡ ✈❡rt❡① ♦❢ 90 ◦ ✭ L/H = 0 . 5 ✮ ✐♥st❡❛❞ ♦❢ 120 ◦ ✭ L/H = 0 . 29 ✮✱ t❤❡r❡❢♦r❡✱ s✐♥❝❡ t❤❡ ♣❡r✐♦❞ ♠✉st ❜❡ ❡q✉❛❧ t♦ ♦♥❡ ✐♥ ❜♦t❤ ❝❛s❡s ✭❛s ❡①♣❧❛✐♥❡❞ ✐♥ s❡❝t✐♦♥ ✷✳✼✳✶✮✱ t❤❡ ♥❡✇ ❣❡♦♠❡tr② ✐s ❞❡❡♣❡r✱ ❛♥❞ ❝♦♥s❡q✉❡♥t❧② ✐t ❤❛s ❛♥ ❤✐❣❤❡r ❝r♦ss✲s❡❝t✐♦♥❛❧ ❛r❡❛✳ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❛✮ ❚r✐❛♥❣✉❧❛r r✐❜❧❡t✳ ✭❜✮ ❇❧❛❞❡ r✐❜❧❡t✳ ❋✐❣✉r❡ ✷✳✷✹✿ ●❡♦♠❡tr✐❡s ✉♥❞❡r ✐♥✈❡st✐❣❛t✐♦♥✳ ❙✐♥❝❡ r✐❜❧❡t ❡✛❡❝t ❞❡♣❡♥❞s ♦♥ ✭❜✉t ♥♦t ♦♥❧②✮ t❤❡ ❝r♦ss✲s❡❝t✐♦♥❛❧ ❛r❡❛ ♦❢ t❤❡ ✇❛❧❧ ❝❛✈✐t②✱ ✐t ✐s r❡❛s♦♥❛❜❧❡ t♦ s✉♣♣♦s❡ t❤❛t t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ✇✐❧❧ ❜❡ ❤✐❣❤❡r t❤❡♥ t❤❛t ♦❜t❛✐♥❡❞ ❢♦r t❤❡ 120 ◦ tr✐❛♥❣❧❡✳ ❚❤❡ ❞❛t❛ r❡♣♦rt❡❞ ✐♥ t❛❜❧❡ ✷✳✹ s❤♦✇ t❤❡ ✈❛❧✉❡ ♦❢ ∆ h ❀ ✐t ✐s ❤✐❣❤❡r t❤❛♥ t❤❛t ❢♦✉♥❞ ❢♦r t❤❡ ♣r❡✈✐♦✉s s❤❛♣❡✱ ❛♥❞ t❤✐s ♠❡❛♥s ❛ ❜❡tt❡r ❞r❛❣ r❡❞✉❝t✐♦♥ ❛s ❛r❣✉❡❞ ✇✐t❤ t❤❡ ❝♦♥s✐❞❡r❛t✐♦♥ ❛❜♦✉t t❤❡ r✐❜❧❡t ❛r❡❛✳ ❈❛s❡ h || h ⊥ ∆ h ❇❧❛❞❡ r✐❜❧❡t 0 . 207 0 . 0834 0 . 1236 ❚r✐❛♥❣✉❧❛r r✐❜❧❡t 0 . 1396 0 . 0788 0 . 0608 ❚❛❜❧❡ ✷✳✹✿ ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ r❡❝t❛♥❣✉❧❛r ❛♥❞ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ❆❧❧ t❤❡ 90 ◦ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ❛r❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✶✶✱ ❛♥❞ t❤❡ ❝♦♠♣✉t❡❞ ♣r♦tr✉s✐♦♥

  59. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✻ ❤❡✐❣❤ts ❛r❡ r❡♣♦rt❡❞ ✐♥ t❛❜❧❡ ✷✳✺✱ ✇❤❡r❡❛s t❤❡✐r tr❡♥❞ ❛r❡ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✷✺ ❛♥❞ ✜❣✉r❡ ✷✳✷✻✳ ■t❡r❛t✐♦♥ h || h ⊥ ∆ h 0 0 . 1396 0 . 0788 0 . 0608 1 0 . 1117 0 . 0744 0 . 0373 2 0 . 0989 0 . 068 0 . 0309 3 0 . 0949 0 . 0659 0 . 029 ❚❛❜❧❡ ✷✳✺✿ Pr♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ❛❧❧ t❤❡ ❢r❛❝t❛❧s ✐t❡r❛t✐♦♥s ♦❢ t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ ■t ❝❛♥ ❜❡ ✐♠♠❡❞✐❛t❡❧② ✉♥❞❡rst♦♦❞ t❤❡ s✐♠✐❧❛r✐t② ✇✐t❤ t❤❡ 120 ◦ tr✐❛♥❣✉❧❛r ❝❛s❡✳ ❚❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❞❡❝r❡❛s❡ r❛♣✐❞❧② ❣♦✐♥❣ ♦♥ ✇✐t❤ t❤❡ ✐t❡r❛t✐♦♥ ❛♥❞ ❛♣♣r♦❛❝❤ ❛ ✜♥✐t❡ ✈❛❧✉❡✳ ❆❣❛✐♥ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝♦♥✜r♠ t❤❛t ❛ s♠❛❧❧ ❞✐st✉r❜❛♥❝❡ ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♥❞ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥✳ ❆♥♦t❤❡r s✐♠✐❧❛r✐t② ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ ∆ h ✈❛❧✉❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♠♦❧❞ ❛♥❞ t❤❛t ♦❢ t❤❡ ✜rst ✐t❡r❛t✐♦♥❀ ❜♦t❤ ❛r❡ ❛✛❡❝t❡❞ ❜② t❤❡ ❜✐❣❣❡st ❞❡❝r❡❛s❡✳ ❚❤❡ ❞❡❝r❡❛s❡ ♠❛❣♥✐t✉❞❡ ✐s s❧✐❣❤t❧② ❞✐✛❡r❡♥t ✭✐t ✐s ❜✐❣❣❡r ❢♦r t❤❡ 120 ◦ ❝❛s❡✮✱ ❜❡❝❛✉s❡ t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r ♠♦❧❞ ❜r✐♥❣s t♦ ❛ str♦♥❣❡r r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❝❛✈✐t②✳ ❆s ✇✐t❤ ❛s t❤❡ 120 ◦ tr✐❛♥❣❧❡✱ ❛♥ ✐♥✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ ❤❛s ❜❡❡♥ ❝♦♥str✉❝t❡❞✳ ❚❤❡ s❤❛♣❡s ♦❜t❛✐♥❡❞ ❛r❡ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✷✼✳

  60. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✼ 0.14 h || data 0.13 fit h data 0.12 Protrusion height fit 0.11 0.1 0.09 0.08 0.07 0.06 0 1 2 3 Number of iteration ❋✐❣✉r❡ ✷✳✷✺✿ P❛r❛❧❧❡❧ ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ 0.065 h data 0.06 fit 0.055 0.05 h 0.045 0.04 0.035 0.03 0.025 0 1 2 3 Number of iteration ❋✐❣✉r❡ ✷✳✷✻✿ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r t❤❡ 90 ◦ tr✐❛♥❣❧❡✳

  61. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✽ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❛✮ 90 ◦ tr✐❛♥❣✉❧❛r ♠♦❧❞✳ ✭❜✮ ❋✐rst ✐t❡r❛t✐♦♥ ❢♦r t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ 0.5 0.5 0 0 -0.5 -0.5 0 0.5 1 0 0.5 1 ✭❝✮ ❙❡❝♦♥❞ ✐t❡r❛t✐♦♥ ❢♦r t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ ✭❞✮ ❚❤✐r❞ ✐t❡r❛t✐♦♥ ❢♦r t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ ❋✐❣✉r❡ ✷✳✷✼✿ ❈♦♥str✉❝t❡❞ ✐♥✇❛r❞ ❢r❛❝t❛❧s ❣❡♦♠❡tr✐❡s✳ ❊✈❡♥ ✐♥ t❤✐s ❝❛s❡ t❤❡ ❞❛t❛ r❡♣♦rt❡❞ ✐♥ t❛❜❧❡ ✷✳✻ ❛♥❞ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✷✽ ❛♥❞ ✜❣✉r❡ ✷✳✷✾ ❝♦♥✜r♠s t❤❡ ♣r❡✈✐♦✉s r❡s✉❧ts✳ ■t❡r❛t✐♦♥ h || h ⊥ ∆ h 0 0 . 1396 0 . 0788 0 . 0608 1 0 . 1464 0 . 0779 0 . 0685 2 0 . 1495 0 . 0782 0 . 0713 3 0 . 1506 0 . 0783 0 . 0723 ❚❛❜❧❡ ✷✳✻✿ Pr♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ❛❧❧ ✐♥✇❛r❞ ❢r❛❝t❛❧s ✐t❡r❛t✐♦♥s ♦❢ t❤❡ 90 ◦ tr✐❛♥❣❧❡✳

  62. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✺✾ 0.16 0.14 h || data Protrusion height fit 0.12 h data fit 0.1 0.08 0.06 0 1 2 3 Number of iteration ❋✐❣✉r❡ ✷✳✷✽✿ P❛r❛❧❧❡❧ ❛♥❞ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ✐♥✇❛r❞ ❢r❛❝t❛❧s ✐t❡r❛t✐♦♥s ❢♦r t❤❡ 90 ◦ tr✐❛♥❣❧❡✳ 0.074 0.072 0.07 0.068 h 0.066 0.064 0.062 h data fit 0.06 0 1 2 3 Number of iteration ❋✐❣✉r❡ ✷✳✷✾✿ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ✐♥✇❛r❞ ❢r❛❝t❛❧s ✐t❡r❛t✐♦♥s ❢♦r t❤❡ 90 ◦ tr✐❛♥❣❧❡✳

  63. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✵ ❋✐❣✉r❡ ✷✳✸✵✿ ❙tr❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ ❢♦r t❤❡ s✐♠♣❧❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ❚❤❡ ✈❡❧♦❝✐t② ✜❡❧❞s ❤❛✈❡ ❜❡❡♥ ❝♦♠♣✉t❡❞✳ ❚❤❡ str❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ✐s s❤♦✇♥ ✐♥ ✜❣✉r❡ ❈♦♠♣❛r✐♥❣ t❤❡ 90 ◦ ✢♦✇ ✜❡❧❞s ✇✐t❤ t❤❛t ♦❜t❛✐♥❡❞ ✷✳✸✵✱ ✜❣✉r❡ ✷✳✸✶ ❛♥❞ ✜❣✉r❡ ✷✳✸✷✳ ✐♥ t❤❡ ♣❛r❛❣r❛♣❤✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♣♦✐♥t ♦✉t s♦♠❡ ❞✐✛❡r❡♥❝❡s✳ ❚❤❡ ✈❡❧♦❝✐t② ❣r❛❞✐❡♥t ♦♥ t❤❡ ✈❛❧❧❡② ✐s s♠❛❧❧❡r✱ ❛♥❞ t❤✐s r❡❞✉❝❡s ✈✐s❝♦✉s s❦✐♥ ❢r✐❝t✐♦♥✳ ❚❤❡ ❡✛❡❝t ✐s ❞✉❡ t♦ t❤❡ ❡♥❤❛♥❝❡ ♦❢ t❤❡ r✐❜❧❡t ❝❛✈✐t②✱ t❤❛t ❛❧❧♦✇ ❛ ❞❡❡♣❡r ♣❡♥❡tr❛t✐♦♥ ♦❢ t❤❡ ✢♦✇✳ ❚❤❡ ❧♦✇ ✈❡❧♦❝✐t② ③♦♥❡ ✐s ✇✐❞❡r✱ s♦ ✐t ✐s ♣♦ss✐❜❧❡ t♦ s✉♣♣♦s❡ t❤❛t t❤❡ ❤✐❣❤ ✈❡❧♦❝✐t② s♣♦ts ❞✉❡ t♦ t❤❡ t✉r❜✉❧❡♥❝❡ ❛r❡ ❞✉♠♣❡❞ ✐♥ ❛ ❜❡tt❡r ✇❛②✳ ❚❤❡ ♥♦r♠❛❧ ✈❡❧♦❝✐t② ❝♦♥t♦✉r✱ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✷✳✸✸✱ ✜❣✉r❡ ✷✳✸✹ ❛♥❞ ✜❣✉r❡ ✷✳✸✺✱ r❡✈❡❛❧ ❛ ❜✐❣❣❡r ❝♦r♥❡r ❡❞❞② ✐♥ ❛❧❧ t❤❡ ❝❛s❡s✳

  64. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✶ ❋✐❣✉r❡ ✷✳✸✶✿ ❙tr❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ ❢♦r t❤❡ t❤✐r❞ ♦✉t✇❛r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ❋✐❣✉r❡ ✷✳✸✷✿ ❙tr❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ✜❡❧❞ ❢♦r t❤❡ t❤✐r❞ ✐♥✇❛r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳

  65. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✷ ❋✐❣✉r❡ ✷✳✸✸✿ y ✈❡❧♦❝✐t② ❝♦♠♣♦♥❡♥t ✜❡❧❞ ❢♦r t❤❡ s✐♠♣❧❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ❋✐❣✉r❡ ✷✳✸✹✿ y ✈❡❧♦❝✐t② ❝♦♠♣♦♥❡♥t ✜❡❧❞ ❢♦r t❤❡ t❤✐r❞ ♦✉t✇❛r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳

  66. ❈❍❆P❚❊❘ ✷✳ ▼■❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✸ ❋✐❣✉r❡ ✷✳✸✺✿ y ✈❡❧♦❝✐t② ❝♦♠♣♦♥❡♥t ✜❡❧❞ ❢♦r t❤❡ t❤✐r❞ ✐♥✇❛r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳

  67. ❈❤❛♣t❡r ✸ ▼❛❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠ ✻✹

  68. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✺ ✸✳✶ ◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ❚❤❡ ♠❛❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠ r❡♣r❡s❡♥t t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤✐s ✇♦r❦✳ ■t ❛✐♠ ❛t ❝❛rr②✐♥❣ ♦✉t t❤❡ ❞r❛❣ ♦✈❡r ❛ ❢r❛❝t❛❧ r✐❜❧❡t ♣❧❛t❡s✱ ❛♥❞ ❝♦♠♣❛r❡ ✐t ✇✐t❤ t❤❛t ♦❜t❛✐♥❡❞ ✇✐t❤ s✐♠♣❧❡ tr✐❛♥❣✉❧❛r r✐❜❧❡ts✳ ❚♦ ❞♦ s♦✱ ❛ r❡❝t❛♥❣✉❧❛r ❝❤❛♥♥❡❧ ✐s st✉❞✐❡❞ t❤r♦✉❣❤ ❞✐r❡❝t ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥✳ ✸✳✶✳✶ ❋✐♥✐t❡ ✈♦❧✉♠❡ ♠❡t❤♦❞ ❚❤❡ ❋✐♥✐t❡ ❱♦❧✉♠❡ ▼❡t❤♦❞ ✭❋❱▼✮ ✐s ❛ ❞✐s❝r❡t✐③❛t✐♦♥ ❛♣♣r♦❛❝❤ ✐♥ ✇❤✐❝❤ t❤❡ s♦❧✉t✐♦♥ ❞♦♠❛✐♥ ✐s s✉❜❞✐✈✐❞❡❞ ✐♥t♦ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦♥t✐❣✉♦✉s ❈♦♥tr♦❧ ❱♦❧✉♠❡s ✭❈❱s✮✱ ❛♥❞ t❤❡ ◆✲❙ ❡q✉❛t✐♦♥s ❛r❡ ❛♣♣❧✐❡❞ t♦ ❡❛❝❤ ❈❱✳ ❆t t❤❡ ❝❡♥tr♦✐❞ ♦❢ ❡❛❝❤ ❈❱ ❧❛②s ❛ ❝♦♠♣✉✲ t❛t✐♦♥❛❧ ♥♦❞❡ ❛t ✇❤✐❝❤ t❤❡ ✈❛r✐❛❜❧❡s ✈❛❧✉❡s ❛r❡ t♦ ❜❡ ❡✈❛❧✉❛t❡❞✳ ❇❡❢♦r❡ ♣r♦❝❡❡❞✐♥❣ t♦ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♠❡t❤♦❞✱ ✇❡ s❤♦✇ t❤❡ ❡q✉❛t✐♦♥s ✸✳✶✱ t❤❛t ❛r❡ t❤❡ ❞✐♠❡♥s✐♦♥❧❡ss ❝♦♥t✐♥✉✐t② ❛♥❞ ♠♦♠❡♥t✉♠ ❡q✉❛t✐♦♥s ❤❡r❡✐♥ ✉s❡❞ t♦ ♣❡r❢♦r♠ t❤❡ ❞✐r❡❝t ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥✳  ∂u i  = 0   ∂x i     ✭✸✳✶✮  ∂ 2 u i ∂u i ∂u i = − ∂p + 1    ∂t + u j + f i .   ∂x j ∂x j Re ∂x j ∂x j  ❚❤❡ ✈❛r✐❛❜❧❡s ✉s❡❞ t♦ ②✐❡❧❞s ❞✐♠❡♥s✐♦♥❧❡ss ❡q✉❛t✐♦♥s ✸✳✶ ❛r❡ δ ❛♥❞ u τ ✳ ❚❤❡ ✜rst ♦♥❡ r❡♣r❡s❡♥ts t❤❡ ❤❛❧❢ ✇✐❞t❤ ♦❢ t❤❡ ❝❤❛♥♥❡❧ ✐♥ t❤❡ y ❞✐r❡❝t✐♦♥✱ ✇❤❡r❡❛s t❤❡ s❡❝♦♥❞ ✐s t❤❡ � s❤❡❛r ✈❡❧♦❝✐t② ❞❡✜♥❡❞ ❛s u τ = τ wall /ρ ✳ Re ❞❡♥♦t❡s t❤❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r ❞❡✜♥❡❞ ❛s Re = U 0 δ/ν ✳ ❚❤❡ ✜rst st❡♣ ♦❢ t❤❡ ♠❡t❤♦❞ ❞❡❛❧s ✇✐t❤ t❤❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤✐s ❞✐♠❡♥s✐♦♥❧❡ss ◆✲ ❙ ❡q✉❛t✐♦♥s ♦✈❡r ❛ s✐♥❣❧❡ ❝♦♥tr♦❧ ✈♦❧✉♠❡ ❝❛❧❧❡❞ Ω ✳ ❲❡ ✉s❡ ❛ ❈❛rt❡s✐❛♥ ❣r✐❞ ❛♥❞ ❛ ❝♦❧❧♦❝❛t❡❞ ❛rr❛♥❣❡♠❡♥t✱ t❤❡r❡❢♦r❡ ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡ ❛r❡ ❝❛❧❝✉❧❛t❡❞ ✐♥ t❤❡ ❝❡❧❧ ❝❡♥t❡r✳ ■♥ ✜❣✉r❡ ✸✳✶ ✐s s❤♦✇♥ ❛ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❈❛rt❡s✐❛♥ ❝♦♥tr♦❧ ✈♦❧✉♠❡ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ♥♦t❛t✐♦♥ ✉s❡❞✳ ❚❤❡ ❈❱ ❝♦♥s✐st ♦❢ s✐① ♣❧❛♥❡ ❢❛❝❡s✱ ❞❡♥♦t❡❞ ✇✐t❤ ❧♦✇❡r ❝❛s❡ ❧❡tt❡rs ✭❡✱✇✱♥✱s✱t✱❜✮ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡✐r ♦r✐❡♥t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❡♥tr❛❧ ♥♦❞❡ P✳

  69. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✻ T t E N e n P s w S W b z x y B ❋✐❣✉r❡ ✸✳✶✿ ❚❤r❡❡ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♥tr♦❧ ✈♦❧✉♠❡ ❢♦r ❛ ❈❛rt❡s✐❛♥ ❣r✐❞✳ ❖♥❝❡ t❤❡ ✐♥t❡❣r❛t✐♦♥ ♦♥ t❤❡ ❝♦♥tr♦❧ ✈♦❧✉♠❡ ✸✳✶ ✐s ♣❡r❢♦r♠❡❞✱ t❤❡ ♠❡t❤♦❞ ❡①♣❧♦✐ts t❤❡ ●❛✉ss✬s t❤❡♦r❡♠ ✭❛♥❞ t❤❡ ✐♥❝♦♠♣r❡ss✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❞✐✛✉s✐✈❡ t❡r♠✮ t♦ ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s✿ • ❚❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥ ❜❡❝❛♠❡ � � ∇ · u ❞ Ω = u · n ❞ S = 0 ✭✸✳✷✮ S • ❚❤❡ ❝♦♥✈❡❝t✐✈❡ t❡r♠ ♦❢ t❤❡ ♠♦♠❡♥t✉♠ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡ � ∂u i � u j ❞ Ω = u i u · n ❞ S ✭✸✳✸✮ ∂x j Ω S • ❚❤❡ ❞✐✛✉s✐✈❡ t❡r♠ ♦❢ t❤❡ ♠♦♠❡♥t✉♠ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡ ∂ 2 u i � 1 � 1 ❞ Ω = Re ∇ u i · n ❞ S ✭✸✳✹✮ Re ∂x j ∂x j Ω S ❆❢t❡r t❤❡ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❛❜♦✈❡ tr❛♥s❢♦r♠❛t✐♦♥✱ t❤❡ ◆✲❙ ❡q✉❛t✐♦♥s

  70. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✼ r❡❛❞  � u · n ❞ S = 0     S    ✭✸✳✺✮   ∂u i ∂p 1 � � � � �   ∂t dΩ + u i u · n d S = − dΩ + Re ∇ u i · n + f i dΩ .   ∂x j  Ω S Ω S Ω ❚❤❡ ❡q✉❛t✐♦♥ ✸✳✺ ♣r❡s❡♥ts ✐♥t❡❣r❛❧s ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ ❈❱✬s s✉r❢❛❝❡ S ♦r ♦♥ t❤❡ ✈♦❧✉♠❡ Ω ✱ ❛♥❞ ❜♦t❤ ♠✉st ❜❡ ❛♣♣r♦①✐♠❛t❡❞✳ ❋♦r t❤❡ ✜rst t②♣❡✱ ✇❡ ❝❛♥ ❝❛❧❧ t❤❡ ❣❡♥❡r✐❝ q✉❛♥t✐t② t❤❛t ✢♦✇s t❤r♦✉❣❤ t❤❡ s✉r❢❛❝❡ ❛s φ ❛♥❞ t❤❛♥❦s t♦ t❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡ ✐♥t❡❣r❛❧ ✇❡ ❝❛♥ ✇r✐t❡ 6 � � � φ d S = φ d S, ✭✸✳✻✮ S S k k =1 ✇❤❡r❡ ✻ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥tr♦❧ ✈♦❧✉♠❡ ❢❛❝❡s ❛♥❞ φ ❝❛♥ ❜❡ ❜♦t❤ t❤❡ ❝♦♥✈❡❝t✐✈❡ ♦r ❞✐✛✉s✐✈❡ ✢✉①✳ ❊q✉❛t✐♦♥ ✸✳✻ ✐s st✐❧❧ ❡①❛❝t✱ ❜✉t ✇❡ ♥❡❡❞ ♥♦✇ t♦ ✐♥tr♦❞✉❝❡ t❤❡ ✜rst ❛♣♣r♦①✐♠❛t✐♦♥✳ ❚❤❡ s✉r❢❛❝❡ ✐♥t❡❣r❛❧ ♦❢ φ ✱ ❢♦r ❛♥ ❛r❜✐tr❛r② ❢❛❝❡ k ✐s ❡st✐♠❛t❡❞ ✇✐t❤ t❤❡ ♠✐❞✲♣♦✐♥t r✉❧❡✱ t❤❛t ✐s t❤❡ ✐♥t❡❣r❛❧ ♦♥ t❤❡ sq✉❛r❡ s✉r❢❛❝❡ ✐s ❛♣♣r♦①✐♠❛t❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣✿ � φ d S ≈ φ k S k , ✭✸✳✼✮ S k ✇❤❡r❡ S k ✐s t❤❡ ❛r❡❛ ♦❢ t❤❡ k ✲t❤ sq✉❛r❡ ❢❛❝❡ ♦❢ t❤❡ ❝✉❜❡✱ ❛♥❞ φ k ✐s t❤❡ ✈❛❧✉❡ ♦❢ φ ❛t t❤❡ ❝❡♥t❡r ♦❢ t❤❡ sq✉❛r❡ ❢❛❝❡✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ s❤♦✇ t❤❛t t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛❝❝✉r❛❝②✳ ❲❤❡r❡❛s S k ✐s ❛ ❣❡♦♠❡tr✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ ✐t ❝❛♥ ❜❡ ❡❛s✐❧② ❝♦♠♣✉t❡❞✱ φ k ❝♦♥t❛✐♥s t❤❡ ✉♥❦♥♦✇♥s t❤❛t ✇❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡✳ ▼♦r❡♦✈❡r✱ s✐♥❝❡ φ k ✐s t❛❦❡♥ ♦♥ t❤❡ sq✉❛r❡ ❢❛❝❡ ❝❡♥t❡r✱ t❤❡ ✉♥❦♥♦✇♥s ✐♥s✐❞❡ ✐t ❛r❡ t♦♦✱ t❤❡r❡❢♦r❡ ✇❡ ♠✉st ❡①♣r❡ss φ k ✐♥ t❡r♠s ♦❢ t❤❡ ♥♦❞❛❧ ✈❛❧✉❡✳ ❚♦ ❞♦ s♦✱ ✇❡ ❝❛♥ ❞♦ ❛♥ ✐♥t❡r♣♦❧❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ♥✉♠❡r✐❝❛❧ ♥♦❞❡s❀ ❢♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡ t❤❡ ✈❛❧✉❡ ♦❢ φ k ♦♥ t❤❡ ❡❛st s✐❞❡✱ ✇❡ ❞♦ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ P ♥♦❞❡ ❛♥❞ t❤❡ ❊ ♥♦❞❡✳ ❍❡♥❝❡✱ r❡❢❡rr✐♥❣ t♦ t❤❡ ✜❣✉r❡ ✸✳✶✱ t❤❡ ✈❛❧✉❡ ♦❢ φ ❛t ❈❱✲❢❛❝❡ ❝❡♥t❡r ✐s ♦❜t❛✐♥ ❜② ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♥❡❛r❡st ♥♦❞❡s✱ ❛s ❢♦❧❧♦✇s ✭❡✳❣ ❡ ❢❛❝❡✮✿ λ = x e − x E φ e = λφ P + (1 − λ ) φ E , . ✭✸✳✽✮ x E − x P ❚❤✐s s❝❤❡♠❡ ✐s ❝❛❧❧❡❞ ❝❡♥tr❛❧ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ ✭❈❉❙✮ ❛♥❞ ✐t ✐s ♦❢ s❡❝♦♥❞ ♦r❞❡r ❛❝❝✉r❛❝②✳

  71. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✽ ❋♦r ❝❧❛r✐t②✱ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥✈❡❝t✐✈❡ ✢✉①❡s ✇✐❧❧ ❜❡ s❤♦✇♥ ♦♥❧② ❢♦r t❤❡ ❡✲❢❛❝❡ ♦❢ t❤❡ ❈❛rt❡s✐❛♥ ❈❱ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✸✳✶ ❛♥❞ ♦♥❧② ❢♦r t❤❡ ✜rst ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r✳ ❚❤❡ ✢✉①❡s ❛t t❤❡ ♦t❤❡r s✉r❢❛❝❡s ❝❛♥ ❜❡ tr❡❛t❡❞ ✐♥ t❤❡ s❛♠❡ ❢❛s❤✐♦♥ ❛♣♣❧②✐♥❣ t❤❡ ❛♣♣r♦♣r✐❛t❡ s✉❜s❝r✐♣ts ♣❡r♠✉t❛t✐♦♥✳ ❇❡❢♦r❡ t♦ ❛♣♣❧② t❤❡ ❛❜♦✈❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✇❡ ❤❛✈❡ t♦ ❡①♣❧❛✐♥ t❤❡ ♣❛rt✐❝✉❧❛r tr❡❛t♠❡♥t ♦❢ t❤❡ ♥♦♥✲❧✐♥❡❛r ❝♦♥✈❡❝t✐✈❡ t❡r♠❀ ✐♥s✐❞❡ ✐t t❤❡ ✈❡❧♦❝✐t② ✉♥❦♥♦✇♥ ❛♣♣❡❛r t✇♦ t✐♠❡s✱ ❛♥❞ ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ ♣r♦❜❧❡♠s✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ t❡r♠ u · n ✇✐t❤ t❤❡ ✈❛❧✉❡ ❛t t❤❡ ♣r❡✈✐♦✉s t✐♠❡✲st❡♣✳ ❉♦✐♥❣ s♦✱ u · n ✐s ❛ ❝♦♥st❛♥t ❛♥❞ ✐t ❝❛♥ ❞r♦♣ t❤❡ ✐♥t❡❣r❛❧✳ ❚❤❡ ❞✐s❝r❡t❡ ❝♦♥✈❡❝t✐✈❡ ✢✉① r❡❛❞✿ � � � u i u · n d S = u · n u i d S ≈ u · n [ λφ P + (1 − λ ) φ E ] d S = S e S e S � = u · n d S [ λφ P + (1 − λ ) φ E ] = ˙ m e [ λφ P + (1 − λ ) φ E ] , ✭✸✳✾✮ S ❛♥❞ t❤❡ ❞✐✛✉s✐✈❡ ♦♥❡ � Re ∇ u i · n d S ≈ S ( u E − u P ) 1 ✭✸✳✶✵✮ Re ( x E − x P ) S � m e = u · n ˙ d S ✐s t❤❡ ♠❛ss ✢♦✇ r❛t❡ t❤r♦✉❣❤ t❤❡ s✉r❢❛❝❡ ❡✳ ❚❤❡ s♣❛t✐❛❧ ❞✐s❝r❡t✐③❛t✐♦♥ S ✐s ❝♦♠♣❧❡t❡❞ ❜② t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ♣r❡ss✉r❡ ❣r❛❞✐❡♥t ❛♥❞ ❜♦❞② ❢♦r❝❡s✱ ❛❧s♦ ❝❛❧❧❡❞ s♦✉r❝❡ t❡r♠s✳ ❋♦r ❛♥ ❛r❜✐tr❛r② s♦✉r❝❡ t❡r♠ q t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❛♣♣❧✐❡❞✿ � q dΩ = ∆Ω q P ✭✸✳✶✶✮ Ω ❚❤❡ ✐♥t❡❣r❛❧ ✐s ❡st✐♠❛t❡❞ ❜② t❤❡ ♣r♦❞✉❝t ❜❡t✇❡❡♥ t❤❡ ❝❡♥tr❛❧ ✈❛❧✉❡ ♦❢ q ✐♥ t❤❡ ❝♦♥tr♦❧ ✈♦❧✉♠❡ ❝❡♥t❡r ❛♥❞ t❤❡ ❝❡❧❧ ✈♦❧✉♠❡✳ ❚❤✐s ✐s ❛❣❛✐♥ ❛ s❡❝♦♥❞ ♦r❞❡r ❛❝❝✉r❛t❡ ❛♣♣r♦①✐♠❛✲ t✐♦♥✳ ❆❢t❡r t❤❡ s♣❛❝❡ ❞✐s❝r❡t✐③❛t✐♦♥✱ ❛ t✐♠❡ ❛❞✈❛♥❝❡♠❡♥t str❛t❡❣② ✐s ♥❡❝❡ss❛r②✱ ❛♥❞ ✇❡ ✇✐❧❧ ❛♥❛❧②③❡ t❤❡ ♠❡t❤♦❞ ❤❡r❡✐♥ ❛❞♦♣t❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ✸✳✶✳✷ ❋r❛❝t✐♦♥❛❧ st❡♣ ❚❤❡ ❢r❛❝t✐♦♥❛❧ st❡♣ ♠❡t❤♦❞ ✐s ❛ t❡❝❤♥✐q✉❡ t♦ ❛❞✈❛♥❝❡ ✐♥ t✐♠❡ t❤❡ ✢✉✐❞ ✢♦✇ ❣♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥s ❛♥❞ ✇❛s ✜rst❧② ❞❡✈❡❧♦♣❡❞ ❜② ❈❤♦r✐♥ ❬✻❪ ❛♥❞ t❤❡♥ ✐♠♣r♦✈❡❞ ❜② ♦t❤❡r ❛✉t❤♦rs✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✐s ❜❛s❡❞ ♦♥ ❍❡❧♠❤♦❧t③✕❍♦❞❣❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦r s✐♠♣❧② ❍♦❞❣❡ ❞❡❝♦♠✲ ♣♦s✐t✐♦♥❀ ✐t st❛t❡s t❤❛t t❤❡ ✈❡❝t♦r ✜❡❧❞ u ❞❡✜♥❡❞ ♦♥ ❛ s✐♠♣❧② ❝♦♥♥❡❝t❡❞ ❞♦♠❛✐♥ ❝❛♥

  72. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✻✾ ❜❡ ✉♥✐q✉❡❧② ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ❛ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡ ✭s♦❧❡♥♦✐❞❛❧✮ ♣❛rt ❛♥❞ ❛♥ ✐rr♦t❛t✐♦♥❛❧ ♣❛rt✳ ❚②♣✐❝❛❧❧②✱ t❤❡ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ♦❢ t✇♦ st❛❣❡s✿ ♣r❡❞✐❝t✐♦♥ ❛♥❞ ❝♦rr❡❝t✐♦♥✳ ■♥ t❤❡ ♣r❡❞✐❝t✐♦♥ st❡♣✱ t❤❡ ♠♦♠❡♥t✉♠ ❡q✉❛t✐♦♥ ✐s s♦❧✈❡❞ ✇✐t❤♦✉t t❤❡ ♣r❡ss✉r❡ t❡r♠s✱ ❜✉t t❤❡ r❡s✉❧t✐♥❣ ✈❡❧♦❝✐t✐❡s ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✳ ■♥ t❤❡ ❝♦rr❡❝t✐♦♥ st❡♣ t❤❡ ♣r❡✈✐♦✉s s♦❧✉t✐♦♥ ✐s ❝♦rr❡❝t❡❞ ❛♥❞ t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ ✐s ♣r♦❥❡❝t❡❞ ♦♥t♦ ❛ ❞✐✈❡r❣❡♥❝❡✲ ❢r❡❡ ✜❡❧❞✳ ❙❡✈❡r❛❧ ♥✉♠❡r✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❛r❡ ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱❛♥❞ ✇❡ ✇✐❧❧ ♣r❡s❡♥t t❤❡ ❢r❛❝t✐♦♥❛❧ st❡♣ ✈❡rs✐♦♥ ♣r♦♣♦s❡❞ ❜② ❑✐♠ ❛♥❞ ▼♦✐♥ ❬✶✶❪✳ ❚❤❡ ♠❡t❤♦❞ ✐s s❡♠✐✲✐♠♣❧✐❝✐t ❛♥❞ ♥♦t ❛❧❧ t❤❡ t❡r♠s ♦❢ t❤❡ ♠♦♠❡♥t✉♠ ❡q✉❛t✐♦♥ ❛r❡ ❞✐s❝r❡t✐③❡❞ ✐♥ t✐♠❡ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡ s❡❝♦♥❞ ♦r❞❡r ❈r❛♥❦✲◆✐❝♦❧s♦♥ s❝❤❡♠❡ ✐s ✉s❡❞ ❢♦r t❤❡ ✇❛❧❧✲♥♦r♠❛❧ ❞✐✛✉s✐✈❡ t❡r♠ ❛♥❞ t❤❡ s❡❝♦♥❞ ♦r❞❡r ❆❞❛♠s✲❇❛s❤❢♦rt❤ s❝❤❡♠❡ ❢♦r ❛❧❧ ♦❢ t❤❡ ♦t❤❡r t❡r♠s ✐♥ ♠♦♠❡♥t✉♠ ❡q✉❛t✐♦♥✳ ❚❤❡ ✜rst st❡♣ ♦❢ t❤❡ ♠❡t❤♦❞ ❛✐♠ ❛t s♦❧✈✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ u i − u n ˆ ) + 1 i ) + δp i = C ( u n i , u n − 1 u i , u n ReD (ˆ δx, ✭✸✳✶✷✮ i ∆ t ✇❤❡r❡ ❈ ❛♥❞ ❉ r❡♣r❡s❡♥t r❡s♣❡❝t✐✈❡❧② t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥✈❡❝t✐✈❡ ❛♥❞ ❞✐✛✉s✐✈❡ t❡r♠s✳ ■♥ t❤❡ ♣r❡❞✐❝t♦r st❡♣ ✇❡ s♦❧✈❡ t❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✸✳✶✷ ❢♦r ˆ u i ✱ ❜✉t ✐t ✇✐❧❧ ♥♦t s❛t✐s❢② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✜❡❧❞ ˆ u i ❤❛s t♦ ❜❡ ♣r♦❥❡❝t❡❞ ♦♥t♦ ❛ ❞✐✈❡r❣❡♥❝❡✲❢r❡❡ ✜❡❧❞ u n +1 ❛♥❞ t♦ ❞♦ s♦ ✇❡ ❡♠♣❧♦② t❤❡ s♦ ❝❛❧❧❡❞ ♣r♦❥❡❝t✐♦♥ st❡♣ t❤❛t i ❢♦❧❧♦✇s✳ ❲❡ ✇r✐t❡ t❤❡ ❝♦♠♣❧❡t❡ ❞✐s❝r❡t✐③❡❞ ◆✲❙ ❡q✉❛t✐♦♥ u n +1 − u n ) + 1 i ) − δp i , u n − 1 i i = C ( u n u i , u n ReD (ˆ ✭✸✳✶✸✮ i ∆ t δx i ❛♥❞ ✇❡ s✉❜st✐t✉t❡ t❤❡ s✉♠ ♦❢ t❤❡ ❝♦♥✈❡❝t✐✈❡ ❛♥❞ ❞✐✛✉s✐✈❡ t❡r♠s ✇✐t❤ t❤❡ ❡q✉❛t✐♦♥ ✸✳✶✷ u n +1 − ˆ u i = − δp i ✭✸✳✶✹✮ ∆ t δx i ❚❛❦✐♥❣ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✸✳✶✹ δ δp = 1 δ ˆ u i ✭✸✳✶✺✮ δx i δx i ∆ t δx i ✇❡ ♦❜t❛✐♥ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ♣r❡ss✉r❡✲❧✐❦❡ ✈❛r✐❛❜❧❡✱ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❝❛♥ ❝♦♠✲ ♣✉t❡ φ ✱ ❛♥❞ ❛❢t❡r✇❛r❞s ✇❡ ❝❛♥ ✜♥❞ t❤❡ ✈❡❧♦❝✐t② ✜❡❧❞ u n +1 s♦❧✈✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ✸✳✶✹✳ i

  73. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✵ ✸✳✷ ❈♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥ ❛♥❞ ❣r✐❞ s♣❛❝✐♥❣ ❚❤❡ ✢♦✇ ❣❡♦♠❡tr② ❛♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❛r❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✸✳✷✳ ❚❤❡ ❛rr♦✇ s❤♦✇ t❤❡ ✢♦✇ ❞✐r❡❝t✐♦♥✱ ❛♥❞ t❤❡ ✉♣♣❡r ❛♥❞ t❤❡ ❧♦✇❡r ♣❧❛t❡s r❡♣r❡s❡♥t t❤❡ ✇❛❧❧✱ t❤❡r❡❢♦r❡ t❤❡ ♥♦ s❧✐♣ ❝♦♥❞✐t✐♦♥ ✐s ❤❡r❡ ❛♣♣❧✐❡❞✳ y z x ❋✐❣✉r❡ ✸✳✷✿ ●❡♦♠❡tr② t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ❢♦r t❤❡ ❞✐r❡❝t ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥✳ ■♥ t❤❡ str❡❛♠✇✐s❡ ❛♥❞ s♣❛♥✇✐s❡ ❞✐r❡❝t✐♦♥ t❤❡ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ✉s❡❞✳ ❚❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❝♦♥❞✐t✐♦♥ ❝❛♥ ❜❡ ❥✉st✐✜❡❞ ✐❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❜♦① ✭♣❡r✐♦❞✮ ✐s ❝❤♦s❡♥ t♦ ✐♥❝❧✉❞❡ t❤❡ ❧❛r❣❡st ❡❞❞✐❡s ✐♥ t❤❡ ✢♦✇✳ ■♥ ♦r❞❡r t♦ ❝❤♦♦s❡ ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡r ❞♦♠❛✐♥✱ ✇❡ ❡①❛♠✐♥❡ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ t✇♦✲♣♦✐♥t ❝♦rr❡❧❛t✐♦♥ ♠❡❛s✉r❡♠❡♥ts ✉s❡❞ ✐♥ ❑✐♠ ❡t ❛❧✳ ❬✶✷❪ ❛♥❞ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✸✳✸ ❛♥❞ ✜❣✉r❡ ✸✳✹✳ ❋✐❣✉r❡ ✸✳✸✿ ❚✇♦✲♣♦✐♥ts ❝♦rr❡❧❛t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r ❛❧♦♥❣ t❤❡ str❡❛♠✲✇✐s❡ ❞✐r❡❝✲ t✐♦♥✳

  74. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✶ ❋✐❣✉r❡ ✸✳✹✿ ❚✇♦✲♣♦✐♥ts ❝♦rr❡❧❛t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✈❡❝t♦r ❛❧♦♥❣ t❤❡ s♣❛♥✲✇✐s❡ ❞✐r❡❝t✐♦♥✳ ▲♦♦❦✐♥❣ ❛t t❤❡s❡ ❝♦rr❡❧❛t✐♦♥s✱ ✐t ❤❛s ❜❡❡♥ ❡st❛❜❧✐s❤❡❞ t❤❡ ❝❤❛♥♥❡❧ ❞✐♠❡♥s✐♦♥ ❛s 2 π ① π ✐♥ ❞✐♠❡♥s✐♦♥❧❡ss ✉♥✐t✱ t❤❡r❡❜② t❤❡ ❡♥❞ ♦❢ t❤❡ ❝❤❛♥♥❡❧ ✐♥ t❤❡ ♣❡r✐♦❞✐❝ ❞✐r❡❝t✐♦♥ ❧❛② ✇❤❡r❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♦❢ t❤❡ ✈❡❧♦❝✐t✐❡s ✐s ❛❧♠♦st ③❡r♦✳ ❚❤❡r❡❢♦r❡ t❤❡ ❞♦♠❛✐♥ ❡♥❝❧♦s❡ t❤❡ ❧❛r❣❡st ❡❞❞✐❡s ✐♥ t❤❡ ✢♦✇✱ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ s❝❛❧❡ ✐s t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✳ ■♥ t❡r♠s ♦❢ ✉♥✐t ✇❛❧❧ t❤❡ ❞♦♠❛✐♥ ✐s 1106 × 352 × 552 ✶ ✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ❝❛rr✐❡❞ ♦✉t ✇✐t❤ ❛ r❡s♦❧✉t✐♦♥ ♦❢ ❣r✐❞ ♣♦✐♥ts 161 × 201 × 129 ✭♠♦r❡ t❤❡♥ 4 ♠✐❧❧✐♦♥ ♣♦✐♥ts✮ ❢♦r ❛ ❘❡②♥♦❧❞s ♥✉♠❜❡r ♦❢ ✷✽✵✵✱ ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❜✉❧❦ ✈❡❧♦❝✐t②✱ ❛♥❞ t❤❡ ❝❤❛♥♥❡❧ ❤❛❧❢✲✇✐❞t❤ δ ✭❛ ❘❡②♥♦❧❞s ♥✉♠❜❡r ♦❢ ✶✼✻ ❜❛s❡❞ ♦♥ t❤❡ ✇❛❧❧ s❤❡❛r ✈❡❧♦❝✐t② u τ ✮✳ ❲✐t❤ t❤✐s ❣r✐❞✱ t❤❡ s♣❛❝✐♥❣ ✐♥ t❤❡ str❡❛♠✇✐s❡ ❛♥❞ s♣❛♥✇✐s❡ ❞✐r❡❝t✐♦♥s ❛r❡ r❡s♣❡❝t✐✈❡❧② ∆ x + = 7 ✉✳✇✳ ❛♥❞ ∆ z + = 4 ✉✳✇✳✳ ▼♦r❡♦✈❡r✱ ❛ ♥♦♥✲✉♥✐❢♦r♠ ♠❡s❤ ✐s ✉s❡❞ ✐♥ t❤❡ ♥♦r♠❛❧ ❞✐r❡❝t✐♦♥❀ ✐t ✐s str❡t❝❤❡❞ t❤r♦✉❣❤ ❛ ❤②♣❡r❜♦❧✐❝ t❛♥❣❡♥t ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ✜rst ♣♦✐♥t ♥❡❛r t❤❡ ✇❛❧❧ ✐s ❧♦❝❛t❡❞ ❛t y + min = 0 . 052 ✳ ❙✉❝❤ ❛ ❣r✐❞ ✐s s✉✣❝✐❡♥t❧② ✜♥❡ t♦ s♦❧✈❡ t❤❡ ❡ss❡♥t✐❛❧ t✉r❜✉❧❡♥t s❝❛❧❡✱ ❛♥❞ ♥♦ s✉❜❣r✐❞ ♠♦❞❡❧ ✐s ✉s❡❞✳ ✶ ❚❤❡ ❝❤❛♥♥❡❧ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❡✈❡♥ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ ✐t ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥ ❛r❡ ❤✐❣❤❡r t❤❡♥ t❤❡ ♠✐♥✐♠❛❧ ❝❤❛♥♥❡❧ ❧❡♥❣t❤s ❬✶✸❪✳

  75. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✷ ✸✳✸ ◆❛✈✐❡r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❯♥t✐❧ ♥♦✇✱ ✇❡ ❤❛✈❡ ❞❡s❝r✐❜❡❞ t❤❡ ♠❡t❤♦❞s ✉s❡❞ t♦ ✐♠♣❧❡♠❡♥t t❤❡ ❞✐r❡❝t ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥✱ ❜✉t ✇❡ ❤❛✈❡ ♥♦t ②❡t ✐♥tr♦❞✉❝❡❞ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s❀ t❤✐s ✐s ❛ ❢✉♥❞❛✲ ♠❡♥t❛❧ ♣♦✐♥t ❜❡❝❛✉s❡ ❛❧❧♦✇ ✉s t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ tr✐❛♥❣✉❧❛r r✐❜❧❡t ✇❛❧❧ ❛♥❞ ✐ts ❢r❛❝t❛❧✐③❛✐t✐♦♥ ✐♥ ❛ s✐♠♣❧❡ ✇❛②✳ ■♥ ❢❛❝t✱ t❤❡ ❣r♦♦✈❡ ♦❢ t❤❡ r✐❜❧❡t s✉r❢❛❝❡ ❛r❡ ❡①tr❡♠❡❧② s♠❛❧❧✱ ❛♥❞ t❤❡ ❣r✐❞ ✉s❡❞ t♦ st✉❞② ✐t s❤♦✉❧❞ ❜❡ t♦♦ ✜♥❡ ❛♥❞ ❝♦♠♣❧❡①✱ t❤❡r❡❢♦r❡ ✇❡ ❝❛♥✲ ♥♦t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t s✉❝❤ ❛ ❣❡♦♠❡tr②✳ ■♥ ♦r❞❡r t♦ ❛✈♦✐❞ t❤❡ ♣r♦❜❧❡♠✱ ✇❡ s✐♠✉❧❛t❡ t❤❡ r✐❜❧❡t ♣❧❛t❡ t❤r♦✉❣❤ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ◆❛✈✐❡r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✸✳✶✻ ✭✐♥ ❜♦t❤ t❤❡ str❡❛♠✲ ❛♥❞ s♣❛♥✲✇✐s❡ ❞✐r❡❝t✐♦♥✮✱ ✐♥ ✇❤✐❝❤ t❤❡ s❧✐♣ ❧❡♥❣t❤s ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✳ ■♥ ✜❣✉r❡ ✸✳✺ ❛r❡ s❦❡t❝❤❡❞ t❤❡ ✈❡❧♦❝✐t② ♣r♦✜❧❡ ❢r♦♠ ✇❤✐❝❤ t❤❡ s❧✐♣ ❧❡♥❣t❤s ❛r✐s❡✳ ❋✐❣✉r❡ ✸✳✺✿ ❱❡❧♦❝✐t② ♣r♦✜❧❡ ❢r♦♠ ✇❤✐❝❤ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛r❡ ❝♦♠♣✉t❡❞✳ ❚❤✐s ♣❛rt✐❝✉❧❛r ❝♦♥❞✐t✐♦♥ ❝❛♥ ❜❡ s✐♠✉❧❛t❡❞ ❛♣♣❧②✐♥❣ t❤❡ t✇♦ s❧✐♣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✸✳✶✻ � � ∂u ∂w � � u wall = kh || w wall = kh ⊥ ✭✸✳✶✻✮ � � ∂y ∂y � � wall wall ✇❤❡r❡ u wall ❛♥❞ w wall ❛r❡ r❡s♣❡❝t✐✈❡❧② t❤❡ str❡❛♠✲✇✐s❡ ❛♥❞ s♣❛♥✲✇✐s❡ ✈❡❧♦❝✐t✐❡s ❛t t❤❡ ✇❛❧❧ ✭t❤❡✐r ✈❛❧✉❡ ✐s ♥♦♥✲③❡r♦✮✱ k ✐s ❛ s❝❛❧❡ ❝♦❡✣❝✐❡♥t✱ h || ❛♥❞ h ⊥ ❛r❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♥❞ ∂u/∂y ❛♥❞ ∂w/∂y ❛r❡ t❤❡ ❞❡r✐✈❛✲ t✐✈❡s ♦❢ t❤❡ ✈❡❧♦❝✐t✐❡s ❝♦♠♣✉t❡❞ ♦♥ t❤❡ ✇❛❧❧✳ ❚❤❡ k ❝♦❡✣❝✐❡♥t ✐s ♥❡❡❞❡❞ ❜❡❝❛✉s❡ t❤❡ q✉❛♥t✐t✐❡s ✉s❡❞ t♦ ②✐❡❧❞s ❞✐♠❡♥s✐♦♥❧❡ss t❤❡ ❙t♦❦❡s ❡q✉❛t✐♦♥ ✭t❤❡ ❣♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥ ♦❢

  76. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✸ t❤❡ ♠✐❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❝❛rr✐❡❞ ♦✉t t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✮ ❛r❡ ❞✐✛❡r❡♥t ❢r♦♠ t❤❛t ✉s❡❞ ❢♦r t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s s♦❧✈❡❞ ✐♥ t❤❡ ❉◆❙✳ ❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥♥♦t ❞✐r❡❝t❧② ❣✐✈❡ ✐♥ ✐♥♣✉t t❤❡ ✈❛❧✉❡ ❝❛rr✐❡❞ ♦✉t ✇✐t❤ t❤❡ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞✱ ❜✉t ✇❡ ♠✉st ❛❞❥✉st ✐t s♦ t❤❛t ✐t r❡s✉❧ts ❛❞✐♠❡♥s✐♦♥❛❧✐③❡❞ t❤r♦✉❣❤ t❤❡ ❤❛❧❢ ❝❤❛♥♥❡❧ ✇✐❞t❤ δ ❛s ❢♦❧❧♦✇ h || δ = b k = b δh BEM = kh BEM , ✭✸✳✶✼✮ || || δ ✇❤❡r❡ b ✐♥❞✐❝❛t❡ t❤❡ r✐❜❧❡t s♣❛❝✐♥❣✱ h || ❛♥❞ h BEM r❡♣r❡s❡♥t r❡s♣❡❝t✐✈❡❧② t❤❡ ❞✐♠❡♥s✐♦♥❛❧ || ❧♦♥❣✐t✉❞✐♥❛❧ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❛♥❞ t❤❛t ❝♦♠♣✉t❡❞ ✇✐t❤ t❤❡ ❇❊▼✳ ❚❤❡ ✈❛❧✉❡ ♦❢ k ♠✉st ❜❡ ✜①❡❞✱ ❛♥❞ ♦♥❝❡ ✐t ❤❛s ❜❡❡♥ ❝❤♦s❡♥✱ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ r✐❜❧❡t ♣❡r✐♦❞ r❡s♣❡❝t t♦ t❤❡ ❝❤❛♥♥❡❧ ❤❛s ❜❡❡♥ ✜①❡❞✳ ❲❡ ❤❛✈❡ ❝❤♦s❡♥ k = 0 . 05 ✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ ❛ b + = 10 ✳ ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✸✳✶✻ ❛ss✐❣♥ ❛ ♥♦♥✲③❡r♦ ✈❛❧✉❡ t♦ t❤❡ str❡❛♠✇✐s❡ ❛♥❞ s♣❛♥✲ ✇✐s❡ ✈❡❧♦❝✐t✐❡s ❛t t❤❡ ✇❛❧❧✳ ❚❤❡ ❡✛❡❝t ♦❢ t❤❡s❡ t✇♦ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❤❛s ❜❡❡♥ st✉❞✐❡❞ ❜② ❚✳ ▼✐♥ ❛♥❞ ❏✳ ❑✐♠ ❬✷✹❪❀ t❤❡② ❤❛✈❡ t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ t❤r❡❡ ❞✐✛❡r❡♥t ❝❛s❡s✿ ♦♥❧② str❡❛♠✇✐s❡ s❧✐♣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✱ ♦♥❧② s♣❛♥✇✐s❡ s❧✐♣ ❝♦♥❞✐t✐♦♥ ❛♥❞ ❜♦t❤ t❤❡ ❝♦♥❞✐t✐♦♥ t♦❣❡t❤❡r✳ ❆s ❡①♣❡❝t❡❞ ✐♥ t❤❡ ✜rst ❝❛s❡ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ✇❛s ♦❜t❛✐♥❡❞✱ ✇❤❡r❡❛s ✐♥ t❤❡ s❡❝♦♥❞ ❝❛s❡✱ ✇❤❡♥ t❤❡ s❧✐♣✲❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ✉s❡❞ ✐♥ t❤❡ s♣❛♥✇✐s❡ ❞✐r❡❝t✐♦♥✱ t❤❡ ♠❡❛♥ ❞r❛❣ ✐♥❝r❡❛s❡s✳ ❚❤❡ ♣❤②s✐❝❛❧ r❡❛s♦♥ ❜❡❤✐♥❞ t❤❡ ❧❛st ❜❡❤❛✈✐♦r ✐s t❤❛t t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ✈♦rt❡① ❛r❡ ❞✉♠♣❡❞ ✇✐t❤ ❛ ♥♦ s❧✐♣ ❝♦♥❞✐t✐♦♥✱ ❛♥❞ ❛r❡ ❧❡ss ❞✉♠♣❡❞ ✇✐t❤ ❛ s❧✐♣ ❝♦♥❞✐t✐♦♥✱ s♦ t❤❛t t❤❡ t✉r❜✉❧❡♥t ♦s❝✐❧❧❛t✐♦♥ ❛r❡ ❢r❡❡ t♦ ❞❡✈❡❧♦♣✳ ❘❡❣❛r❞✐♥❣ t❤❡ ❧❛st ❝❛s❡✱ t❤❡r❡ ✐s ❛ ❜❛❧❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❞r❛❣ ❞✉❡ t♦ t❤❡ str❡❛♠✇✐s❡ s❧✐♣ ❝♦♥❞✐t✐♦♥ ❛♥❞ t❤❡ ✐♥❝r❡❛s❡ ♦❢ ✐t ❞✉❡ t♦ t❤❡ s❧✐♣ s♣❛♥✇✐s❡ ❝♦♥❞✐t✐♦♥ s♦ t❤❛t t❤❡ ❡✛❡❝t ♦❢ t❤❡ r✐❜❧❡t ✐s r❡♣r♦❞✉❝❡❞✳ ❊✈❡♥t✉❛❧❧② ✇❡ ♠✉st ❡♠♣❛t❤✐③❡ t❤❛t t❤✐s ❦✐♥❞ ♦❢ ❛♣♣r♦❛❝❤ ✐♥ t❤❡ st✉❞② ♦❢ t❤❡ r✐❜❧❡t ❤❛s ♥❡✈❡r ❜❡❡♥ t❡st❡❞ ❛♥❞ ✈❛❧✐❞❛t❡❞✱ ❤❡♥❝❡ t❤❡ r❡s✉❧ts s❤♦✉❧❞ ❜❡ ❝♦♥✜r♠❡❞ t❤r♦✉❣❤ ❡①♣❡r✐♠❡♥t✳ ◆❡✈❡rt❤❡❧❡ss t❤✐s ♠❡t❤♦❞ s❡❡♠s t♦ ❜❡ ❝♦rr❡❝t ❢r♦♠ ❛ ❝♦♥❝❡♣t✉❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐♥ ❢❛❝t t❤❡ ❡✛❡❝t ♦❢ t❤❡ s♣❛♥✇✐s❡ s❧✐♣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ✢♦✇ ✐s s✐♠✐❧❛r t♦ t❤❛t ♣❡r❢♦r♠❡❞ ❜② t❤❡ r✐❜❧❡t✱ ❜♦t❤ ❛❝t ♦♥ t❤❡ ❝r♦ss ✢♦✇✳

  77. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✹ ✸✳✹ ❱❛❧✐❞❛t✐♦♥ ♦❢ t❤❡ ❝♦❞❡ ■♥ ♦r❞❡r t♦ ✈❛❧✐❞❛t❡ t❤❡ ❝♦❞❡ ✐♠♣❧❡♠❡♥t❡❞ ❢♦r t❤❡ ❉◆❙✱ ✇❡ ❤❛✈❡ t❛❦❡♥ ❛s r❡❢❡r❡♥❝❡ t❤❡ st✉❞② ♣❡r❢♦r♠❡❞ ❜② ❑✐♠✱ ▼♦✐♥ ❛♥❞ ▼♦s❡r ✐♥ ✶✾✽✻ ❬✶✷❪ ❛t t❤❡ ◆❆❙❆ ❆♠❡s ❘❡s❡❛r❝❤ ❈❡♥t❡r✱ ✐♥ ✇❤✐❝❤ t❤❡② ❝❛rr✐❡❞ ♦✉t ♦♥❡ ♦❢ t❤❡ ✜rst ❞✐r❡❝t ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ♦❢ ❛ t✉r❜✉❧❡♥t ❝❤❛♥♥❡❧ ✢♦✇✳ ❚❤❡② ❝❤♦s❡ ❛ 4 πδ × 2 δ × 2 πδ ❝❤❛♥♥❡❧✱ ✇❤❡r❡ δ ✐s t❤❡ ❝❤❛♥♥❡❧ ❤❛❧❢ ✇✐❞t❤ ✐♥ t❤❡ y ✲❞✐r❡❝t✐♦♥✳ ❚❤❡② ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡❞ t❤❡ ✉♥st❡❛❞② ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s ✇✐t❤ ❛ ❘❡②♥♦❧❞s ♥✉♠❜❡r ♦❢ 2800 ✱ ❜❛s❡❞ ♦♥ t❤❡ ❜✉❧❦ ✈❡❧♦❝✐t② ❛♥❞ t❤❡ ❝❤❛♥♥❡❧ ❤❛❧❢✲✇✐❞t❤ δ ✭❛ ❘❡②♥♦❧❞s ♥✉♠❜❡r ♦❢ ✶✽✵ ❜❛s❡❞ ♦♥ t❤❡ ✇❛❧❧ s❤❡❛r ✈❡❧♦❝✐t② u τ ✮✱ ✇✐t❤ ❛ ❣r✐❞ r❡s♦❧✉t✐♦♥ ♦❢ 192 × 129 × 160 ✐♥ x ✱ y ✱ z ✳ ❲✐t❤ t❤✐s ❣r✐❞✱ t❤❡ s♣❛❝✐♥❣ ✐♥ t❤❡ str❡❛♠✲✇✐s❡ ❛♥❞ s♣❛♥✲✇✐s❡ ❞✐r❡❝t✐♦♥s ❛r❡ r❡s♣❡❝t✐✈❡❧② ∆ x + = 12 ✉✳✇✳ ❛♥❞ ∆ z + = 7 ✉✳✇✳✳ ▼♦r❡♦✈❡r✱ ❛ ♥♦♥✲✉♥✐❢♦r♠ ♠❡s❤ ✐s ✉s❡❞ ✐♥ t❤❡ ♥♦r♠❛❧ ❞✐r❡❝t✐♦♥✱ ❛♥❞ t❤❡ ✜rst ♠❡s❤ ♣♦✐♥t ❛✇❛② ❢r♦♠ t❤❡ ✇❛❧❧ ✐s ❛t y + = 0 . 05 ✳ ❉♦✐♥❣ s♦ t❤❡② s♦❧✈❡❞ ❛❧❧ ❡ss❡♥t✐❛❧ t✉r❜✉❧❡♥❝❡ s❝❛❧❡s ♦♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❣r✐❞ ❛♥❞ ♥♦ s✉❜❣r✐❞ ♠♦❞❡❧ ✇❛s ✉s❡❞✳ ▼♦r❡♦✈❡r✱ t❤❡② ❝♦♠♣✉t❡❞ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ t✉r❜✉❧❡♥❝❡ st❛t✐st✐❝s t❤❛t ❛r❡ ❡✈❡♥ ♥♦✇ ✐♥ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛✳ ❚❤❡ ❝❛s❡ s♦❧✈❡❞ ✐♥ t❤❡ ♣r❡s❡♥t ✇♦r❦ ❤❛✈❡ s♦♠❡ ❞✐✛❡r❡♥❝❡s r❡s♣❡❝t t♦ t❤❡ ❑✐♠ ❛♥❞ ▼♦✐♥ ❝❛s❡✿ t❤❡ ❣r✐❞ ❤❡r❡✐♥ ✉s❡❞ ✐s ✜♥❡r t❤❡♥ ❑✐♠ ❛♥❞ ▼♦✐♥ ♦♥❡ ❛♥❞ t❤❡ ♠❡t❤♦❞ ✉s❡❞ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥s ✐s ❞✐✛❡r❡♥t ✷ ✳ ■♥ ♦r❞❡r t♦ ✈❛❧✐❞❛t❡ t❤❡ ❞♥s ❝♦❞❡✱ ✇❡ ❤❛✈❡ ❝❛rr✐❡❞ ♦✉t t❤❡ ✈❡❧♦❝✐t② r♦♦t ♠❡❛♥ sq✉❛r❡ ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥ ❛♥❞ ✇❡ ❤❛✈❡ ❝♦♠♣❛r❡❞ t❤❡♠ ✇✐t❤ t❤❛t ♦❜t❛✐♥❡❞ ❜② ❑✐♠ ❛♥❞ ▼♦✐♥✱ ❛s ♣r❡s❡♥t❡❞ ✐♥ ✜❣✉r❡ ✸✳✻✱ ✸✳✼ ❛♥❞ ✸✳✽ ✷ ■♥ ❬✶✷❪ ❛ s♣❡❝tr❛❧ ❝♦❞❡ ✇❛s ❡♠♣❧♦②❡❞✳

  78. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✺ 3 Our code results Kim and Moin results 2.5 2 u rms 1.5 1 0.5 0 0 0.5 1 1.5 2 y ❋✐❣✉r❡ ✸✳✻✿ ❘♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ str❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② u ′ ✢✉❝t✉❛t✐♦♥✳ 3 Our code results Kim and Moin results 2.5 2 v rms 1.5 1 0.5 0 0 0.5 1 1.5 2 y ❋✐❣✉r❡ ✸✳✼✿ ❘♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ ♥♦r♠❛❧ ✈❡❧♦❝✐t② v ′ ✢✉❝t✉❛t✐♦♥✳

  79. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✻ 3 Our code results Kim and Moin results 2.5 2 w rms 1.5 1 0.5 0 0 0.5 1 1.5 2 y ❋✐❣✉r❡ ✸✳✽✿ ❘♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ s♣❛♥✇✐s❡ ✈❡❧♦❝✐t② w ′ ✢✉❝t✉❛t✐♦♥✳ ❚❤❡ ✜❣✉r❡s s❤♦✇ ❛ ✈❡r② s❛t✐s❢②✐♥❣ ❛❣r❡❡♠❡♥t ♦❢ t❤❡ r❡s✉❧ts✳ ✸✳✺ ❘❡s✉❧ts ■♥ ♦r❞❡r t♦ ♣❡r❢♦r♠ t❤❡ ❞✐r❡❝t ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s✱ ✇❡ ❤❛✈❡ t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t ❛♥❞ ✐ts t❤✐r❞ ✐♥✇❛r❞ ❢r❛❝t❛❧③❛t✐♦♥✱ ❜♦t❤ s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✷✳✷✼✳ ▲✉❝❤✐♥✐✬s t❤❡♦r② r❡❧❛t❡ ❛♥ ❤✐❣❤ ✈❛❧✉❡ ♦❢ ∆ h t♦ ❛♥ ❤✐❣❤ ❞r❛❣ r❡❞✉❝t✐♦♥❀ s✐♥❝❡ t❤❡ 90 ◦ s❤♦✇ ❛♥ ❤✐❣❤❡r ∆ h t❤❡♥ t❤❡ 120 ◦ r✐❜❧❡t✱ ✐t s❡❡♠s t♦ ❜❡ t❤❡ ♠♦r❡ ♣r♦♠✐s✐♥❣ ❣❡♦♠❡tr✐❡s ✐♥ t❡r♠s ♦❢ ❞r❛❣ r❡❞✉❝t✐♦♥✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ ❝❤♦s❡♥ ✐t✳ ▼♦r❡♦✈❡r✱ t❤✐s s❤❛♣❡ s❤♦✇ t❤❡ ❤✐❣❤❡st ✐♥❝r❡❛s❡ ♦❢ t❤❡ ∆ h ✈❛❧✉❡ ✇❤❡♥ t❤❡ ❢r❛❝t❛❧✐③❛t✐♦♥ ♣r♦❝❡❞✉r❡ ✐s ❛♣♣❧✐❡❞✳ ❚❤✐s ✐s ❛♥ ✐♠♣♦rt❛♥t ❢❡❛t✉r❡✱ ❜❡❝❛✉s❡ t❤❡ ❢r❛❝t❛❧✐③❛t✐♦♥ ♠♦❞✐❢② t❤❡ r✐❜❧❡t ❣❡♦♠❡tr② ♦♥ t❤❡ s❝❛❧❡ ♦❢ ♥❛♥♦♠❡t❡rs✱ ❤❡♥❝❡ ✇❡ ❝❛♥ s✉♣♣♦s❡ t♦ ❤❛✈❡ s♠❛❧❧ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ✢♦✇✳ ❚❤❡r❡❢♦r❡✱ ❤✐❣❤❡r ✐s t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ∆ h ❛♥❞ ❜❡tt❡r ✇❡ ❝❛♥ ❛♣♣r❡❝✐❛t❡ t❤❡ ❝❤❛♥❣✐♥❣ ✐♥ t❤❡ ✢♦✇✳ ❲❡ ❤❛✈❡ ♦♥❧② st✉❞✐❡❞ t❤❡ t❤✐r❞ ✐t❡r❛t✐♦♥ ❜❡❝❛✉s❡ ✐t r❡♣r❡s❡♥t t❤❡ ❧✐♠✐t ❛t ✇❤✐❝❤ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤t ❞♦ ♥♦t ✈❛r② ❛♥②♠♦r❡✳ ■♥ t❛❜❧❡ ✸✳✶ ❛r❡ s❤♦✇♥ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r t❤❡ s❡❧❡❝t❡❞ ❝❛s❡s✳

  80. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✼ ■t❡r❛t✐♦♥ h || h ⊥ ∆ h 0 0 . 1396 0 . 0788 0 . 0608 3 0 . 1506 0 . 0783 0 . 0723 ❚❛❜❧❡ ✸✳✶✿ Pr♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t ❛♥❞ ❢♦r ✐ts t❤✐r❞ ❢r❛❝t❛❧✐③❛t✐♦♥✳ ■♥ t❤✐s s❡❝t✐♦♥ t❤❡ ❉◆❙ r❡s✉❧ts ❛r❡ s❤♦✇♥ ❛♥❞ ❝♦♠♠❡♥t❡❞✳ ❚❤r❡❡ ❞✐✛❡r❡♥t ❝❤❛♥♥❡❧s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞✿ t❤❡ ❝♦♠♣❧❡t❡❧② s♠♦♦t❤ ❝❤❛♥♥❡❧ ✭t❤❡ s❛♠❡ ✉s❡❞ ❢♦r t❤❡ ✈❛❧✐❞❛t✐♦♥✮✱ t❤❡ ❝❤❛♥♥❡❧ ✇✐t❤ s✐♠♣❧❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t ❛♥❞ t❤❡ ❝❤❛♥♥❡❧ ✇✐t❤ t❤❡ t❤✐r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ❢♦r t❤❡ 90 ◦ r✐❜❧❡t✳ ■t ✐s ✇♦rt❤ t♦ r❡♠❡♠❜❡r t❤❛t t❤❡ ❧♦✇❡r ♣❧❛t❡s ♦❢ t❤❡ r✐❜❧❡t ❝❤❛♥♥❡❧s ✐s t❤❡ ♦♥❡ ✇✐t❤ t❤❡ ❣r♦♦✈❡❞ ✇❛❧❧✳ ❚❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❛r❡ r❡♣♦rt❡❞ ✐♥ t❛❜❧❡ ✸✳✷✳ ❈❛s❡ Re | ∂p/∂x | u τ τ wall Re τ DR ❙♠♦♦t❤ 2777 0 . 00397 0 . 0634 0 . 0040 177 − ❚r✐❛♥❣✉❧❛r t✐❜❧❡t 2777 0 . 00376 0 . 0613 0 . 0038 170 . 4 5 . 3% ❋r❛❝t❛❧ r✐❜❧❡t 2777 0 . 00370 0 . 0608 0 . 037 168 . 9 6 . 8% ❚❛❜❧❡ ✸✳✷✿ ❘❡s✉❧ts ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❉◆❙ s✐♠✉❧❛t✐♦♥✳ Re ✐s t❤❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r ❜❛s❡❞ ♦♥ t❤❡ ❜✉❧❦ ✈❡❧♦❝✐t②✱ ∂p/∂x ✐s t❤ ♠❡❛♥ ♣r❡ss✉r❡ ❣r❛❞✐❡♥t✱ u τ ✐s t❤❡ ❢r✐❝t✐♦♥ ✈❡❧♦❝✐t②✱ τ wall ✐s t❤❡ s❦✐♥ ❢r✐❝t✐♦♥✱ Re τ ✐s t❤❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r ❜❛s❡❞ ♦♥ t❤❡ ❢r✐❝t✐♦♥ ✈❡❧♦❝✐t②✱ DR ✐s t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ♣❡r❝❡♥t❛❣❡✳ ❚❤❡ ✜rst ♣❛r❛♠❡t❡r t❛❦❡♥ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ✐s t❤❡ ♣r❡ss✉r❡ ❣r❛❞✐❡♥t ❜❡t✇❡❡♥ t❤❡ ♦✉t❧❡t ❛♥❞ t❤❡ ✐♥❧❡t✳ ■t ✐s ❢✉♥❞❛♠❡♥t❛❧ ❜❡❝❛✉s❡ ❛❧❧♦✇ ✉s t♦ ❝♦♠♣✉t❡ t❤❡ ❞r❛❣ ❛♣♣❧✐❡❞ ❜② t❤❡ ✢♦✇ ♦♥ t❤❡ ♣❧❛t❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧✳ ■t ✐s ✇♦rt❤ t♦ ✉♥❞❡r❧✐❡ t❤❛t t❤❡ ❝♦❞❡ ❝♦♠♣✉t❡s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r❡ss✉r❡ ❣r❛❞✐❡♥t ❡✈❡r② ✐t❡r❛t✐♦♥✱ ❛♥❞ ✐t ❛✐♠ ❛t ♠❛✐♥t❛✐♥✐♥❣ t❤❡ ❜✉❧❦ ✈❡❧♦❝✐t②✱ ❝♦♠♣✉t❡❞ ❛s s❤♦✇♥ ✐♥ ❡q✉❛t✐♦♥ ✸✳✶✽✱ ❡q✉❛❧ t♦ ♦♥❡✳ � 2 U m = 1 ¯ u d( y/δ ) ✭✸✳✶✽✮ 2 0

  81. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✽ 10 -3 -3 -3.5 dp/dx -4 -4.5 -5 0 100 200 300 400 500 Time step ❋✐❣✉r❡ ✸✳✾✿ Pr❡ss✉r❡ ❣r❛❞✐❡♥t t✐♠❡ ❤✐st♦r② ❢♦r t❤❡ ❝❤❛♥♥❡❧ ✇✐t❤ t❤❡ s♠♦♦t❤ ♣❧❛t❡s✳ ❚❤❡r❡❢♦r❡✱ ❣♦✐♥❣ ♦♥ ✇✐t❤ t❤❡ t✐♠❡st❡♣ t❤❡ ♣r❡ss✉r❡ ❣r❛❞✐❡♥t ♦s❝✐❧❧❛t❡ ✐♥ ♦r❞❡r t♦ ♣✉rs✉❡ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❝♦♥❞✐t✐♦♥✳ ❋✐❣✉r❡s ✸✳✾✱ ✸✳✶✵ ❛♥❞ ✸✳✶✶ s❤♦✇ t❤❡ ✢✉❝t✉❛t✐♦♥ ♦❢ t❤❡ ♣r❡ss✉r❡ ❣r❛❞✐❡♥t ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ t✐♠❡✳ ■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ❢♦r ❡❛❝❤ ❝❛s❡✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ♠❡❛♥ ✈❛❧✉❡ ♦❢ t❤❡ ♣r❡ss✉r❡ ❣r❛❞✐❡♥t❀ ✐♥ ❢❛❝t✱ ✐t ✐s r❡❧❛t❡❞ t♦ t❤❡ ❞r❛❣ ♣❡r❢♦r♠❡❞ ❢r♦♠ t❤❡ t✇♦ ♣❧❛t❡s ♦♥ t❤❡ ✢♦✇✳ ❚❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ❢♦r ❡❛❝❤ ❝❛s❡ ❤❛s ❜❡❡♥ ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇✱ � � ∂p − ∂p � � � � ∂x ∂x � � 0 i DR = x100 , ✭✸✳✶✾✮ � ∂p � � ∂x � 0 ✇❤❡r❡ t❤❡ s✉❜s❝r✐♣ts 0 r❡❢❡rs t♦ t❤❡ ✈❛❧✐❞❛t✐♦♥ ❝❛s❡✱ ❛♥❞ t❤❡ s✉❜s❝r✐♣ts i ✐♥❞✐❝❛t❡s t❤❡ ❝❛s❡ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥✳ ❉❛t❛ r❡♣♦rt❡❞ ✐♥ t❛❜❧❡ ✸✳✷ r❡✈❡❛❧ t❤❛t t❤❡ s✐♠♣❧❡ tr✐❛♥❣✉❧❛r r✐❜❧❡t s✉r❢❛❝❡s ♣r♦❞✉❝❡ ❛ ♥♦t✐❝❡❛❜❧❡ ❞r❛❣ r❡❞✉❝t✐♦♥ r❡s♣❡❝t t♦ t❤❡ s♠♦♦t❤ ♣❧❛t❡s✳ ▼♦r❡✲ ♦✈❡r✱ ❛s ❡①♣❡❝t❡❞ ❢r♦♠ t❤❡ ✈❛❧✉❡ ♦❢ ∆ h ✱ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t ❣❡♥❡r❛t❡ ❛♥ ❡✈❡♥ ❤✐❣❤❡r ❞r❛❣ r❡❞✉❝t✐♦♥✳ ■t s❡❡♠s r❡❛s♦♥❛❜❧❡ t❤❡ ❤②♣♦t❤❡s✐s ♦❢ ▲✉❝❤✐♥✐ t❤❛t ❛♥ ❤✐❣❤❡r ∆ h ♣r♦❞✉❝❡s ❛♥ ❤✐❣❤❡r ❞r❛❣ r❡❞✉❝t✐♦♥✳ ❲❡ r❡❝❛❧❧ t❤❛t t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ s❧✐♣ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ str❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t②✱ t❤❛t ❣✐✈❡ t❤❡ ♠❡❛♥ ✈❡❧♦❝✐t② ♣r♦✜❧❡s s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✸✳✶✷ ❛♥❞ ✜❣✉r❡ ✸✳✶✸✳ ■t ✐s ❝❧❡❛r ✐♥ ✜❣✉r❡ ✸✳✶✷ ❛♥❞ ✜❣✉r❡ ✸✳✶✸✱ t❤❛t t❤❡ str❡❛♠✲ ✇✐s❡ s❧✐♣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❛✛❡❝ts t❤❡ ♠❡❛♥ ✈❡❧♦❝✐t② ♣r♦✜❧❡ ❛s ❡①♣❡❝t❡❞✱ ✐♥ ♣❛rt✐❝✉❧❛r

  82. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✼✾ 10 -3 5 0 dp/dx -5 -10 -15 0 100 200 300 400 500 Time step ❋✐❣✉r❡ ✸✳✶✵✿ Pr❡ss✉r❡ ❣r❛❞✐❡♥t t✐♠❡ ❤✐st♦r② ❢♦r t❤❡ ❝❤❛♥♥❡❧ ✇✐t❤ t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t ♣❧❛t❡✳ 10 -3 5 0 dp/dx -5 -10 -15 0 100 200 300 400 500 Time step ❋✐❣✉r❡ ✸✳✶✶✿ Pr❡ss✉r❡ ❣r❛❞✐❡♥t t✐♠❡ ❤✐st♦r② ❢♦r t❤❡ ❝❤❛♥♥❡❧ ✇✐t❤ t❤❡ t❤✐r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t ♣❧❛t❡✳

  83. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✽✵ 20 15 u mean 10 5 Smooth case Triangular riblet Fractal riblet 0 0 0.5 1 1.5 2 y ❋✐❣✉r❡ ✸✳✶✷✿ ▼❡❛♥ str❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ♣r♦✜❧❡s✳ t❤❡② ❛r❡ ✉♣✇❛r❞ s❤✐❢t❡❞✳ ❆❧❧ t❤❡ r✐❜❧❡tt❡❞ s✉r❢❛❝❡s ❛❧t❡r t❤❡ ❧♦❣❛r✐t❤♠✐❝ ❧❛✇ ❛s u + = 1 klogy + + C + ∆ U + , ✭✸✳✷✵✮ ✇❤❡r❡ k ≈ 0 . 41 ✐s t❤❡ ✈♦♥ ❑ár♠á♥ ❝♦♥st❛♥t ❛♥❞ C ≈ 5 . 2 ❛♥❞ ∆ U + ❝❤❛r❛❝t❡r✐③❡s t❤❡ s❤✐❢t ♦❢ t❤❡ ❧♦❣❛r✐t❤♠✐❝ r❡❣✐♦♥✳ ❆ ♣♦s✐t✐✈❡ s❤✐❢t ❝♦rr❡s♣♦♥❞s t♦ ❞r❛❣ r❡❞✉❝t✐♦♥✱ ✇❤❡r❡❛s ❛ ♥❡❣❛t✐✈❡ s❤✐❢t ❝♦rr❡s♣♦♥❞s t♦ ❛ ❞r❛❣ ✐♥❝r❡❛s❡✱ ❛s ♦♥ st❛♥❞❛r❞ r♦✉❣❤ s✉r❢❛❝❡s✳ ❚❤✐s ✈❡❧♦❝✐t② s❤✐❢t ✐s t❤❡ ❦❡② ♣❛r❛♠❡t❡r t♦ ❝❤❛r❛❝t❡r✐③❡ r✐❜❧❡t ❡✛❡❝ts ✭❆✉♣♦✐① ❬✷❪✮✱ ❛♥❞ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❛❞♦♣t❡❞ ❛r❡ ❛❜❧❡ t♦ r❡♣r♦❞✉❝❡ ✐t✳ ▲♦♦❦✐♥❣ ❛t ✜❣✉r❡ ✸✳✶✸ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♥♦t❡ t❤❛t t❤❡ ♣r♦✜❧❡ ♦❢ t❤❡ r✐❜❧❡t ❝❛s❡s ❛r❡ ♥♦t ♦♥❧② ✉♣✇❛r❞ s❤✐❢t❡❞✱ ✐♥ ❢❛❝t t❤❡ ❧♦❣❛r✐t❤♠✐❝ ③♦♥❡s ♦❢ t❤❡ ♣r♦✜❧❡s ❛r❡ ♥❡❛r❡r t♦ t❤❡ ✈❛❧✐❞❛t✐♦♥ ❝❛s❡ t❤❡♥ t❤❡ ♥❡❛r t❤❡ ✇❛❧❧ ♣r♦✜❧❡✳ ❆s ❝❛rr✐❡❞ ♦✉t ❜② ❚✳ ▼✐♥ ❛♥❞ ❏✳ ❑✐♠ ❬✷✹❪✱ t❤❡ ♣✉r❡ str❡❛♠✲✇✐s❡ s❧✐♣ ❝♦♥❞✐t✐♦♥ r❡✈❡❛❧s ❛ ♣❡r❢❡❝t❧② ✉♣✇❛r❞ s❤✐❢t❡❞ ♠❡❛♥ ✈❡❧♦❝✐t② ♣r♦✜❧❡✱ ✇❤❡r❡❛s ❢r♦♠ t❤❡ ♣✉r❡ s♣❛♥✲✇✐s❡ s❧✐♣ ❝♦♥❞✐t✐♦♥ ❛✛❡❝t t❤❡ ❧♦❣❛r✐t❤♠✐❝ r❡❣✐♦♥ ✇✐t❤ ❛ ❞♦✇♥✇❛r❞ s❤✐❢t✳ ❊✈❡♥t✉❛❧❧②✱ ✇❤❡♥ ❜♦t❤ t❤❡ s❧✐♣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❛♣♣❧✐❡❞✱ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❜❡❤❛✈✐♦r ❛r✐s❡✱ ❛s ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ ✐♥ ✜❣✉r❡ ✸✳✶✸✳ ❚❤❡ ✉♣✇❛r❞ s❤✐❢t ✐s ♥♦t✐❝❡❛❜❧❡ ❜❡t✇❡❡♥ t❤❡ s♠♦♦t❤ ❝❤❛♥♥❡❧ ❛♥❞ t❤❡ r✐❜❧❡t ❝❤❛♥♥❡❧s✱ ✇❤❡r❡❛s ✐t ✐s ❛❧♠♦st ③❡r♦ ❜❡t✇❡❡♥ t❤❡ ❢r❛❝t❛❧ ❛♥❞ t❤❡ s✐♠♣❧❡ r✐❜❧❡t ❝❤❛♥♥❡❧✳ ❚❤❡ q✉❛♥t✐t② t❤❛t r❡❛❧❧② ❝♦♥tr✐❜✉t❡ t♦ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡

  84. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✽✶ 20 15 Smooth case 10 u + Triangular riblet Fractal riblet 5 0 10 0 10 1 10 2 10 3 y + ❋✐❣✉r❡ ✸✳✶✸✿ ▼❡❛♥ str❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ♣r♦✜❧❡s ♣❧♦tt❡❞ ♦♥ t❤❡ y + ❝♦♦r❞✐♥❛t❡ ✭❡①♣r❡ss ✐♥ ✇❛❧❧ ✉♥✐ts✮ ✐♥ s❡♠✐✲❧♦❣❛r✐t♠✐❝ s❝❛❧❡✳ ✈❡❧♦❝✐t② ✐♥ t❤❡ y ❞✐r❡❝t✐♦♥✱ ✐♥ ❢❛❝t ✐t ✐s r❡❧❛t❡❞ t♦ t❤❡ s❦✐♥ ❢r✐❝t✐♦♥ ❜② ✇❡❧❧ ❦♥♦✇♥ r❡❧❛t✐♦♥ ✸✳✷✶ � τ wall = ρν ∂u � + ρu ′ v ′ , ✭✸✳✷✶✮ � ∂y � wall ✇❤❡r❡ ∂u/∂y r❡♣r❡s❡♥t t❤❡ ♠❡❛♥ ♣r♦✜❧❡ s❧♦♣❡ ❛t t❤❡ ✇❛❧❧✳ ❆s s❦❡t❝❤❡❞ ✐♥ ✜❣✉r❡ ✸✳✶✹ ❢r❛❝t❛❧ r✐❜❧❡t s❤♦✇s t❤❡ ❧♦✇❡r s❧♦♣❡✳ ❆❣❛✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ r✐❜❧❡t ❝❤❛♥♥❡❧ ✐s q✉✐t❡ s♠❛❧❧✱ ❛♥❞ ✐t ❝❛♥♥♦t ❛❧♦♥❡ ❥✉st✐❢② t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❛ ❢r❛❝t❛❧ r✐❜❧❡t✱ t❤❡r❡❢♦r❡ ✐t ✐s ✐♠♣♦rt❛♥t ❧♦♦❦✐♥❣ ❛t t❤❡ t✉r❜✉❧❡♥t ❡✛❡❝t✳ ❚❤❡ s❦✐♥ ❢r✐❝t✐♦♥ ✐s ❞✐r❡❝t❧② r❡❧❛t❡❞ t♦ t❤❡ s❤❡❛r ✈❡❧♦❝✐t②✱ τ wall = u 2 τ , ✭✸✳✷✷✮ ρ ✇❤✐❝❤ ❞❡❝r❡❛s❡ ✐♥ t❤❡ s❛♠❡ ✇❛②✳ ❚❤❡ s❤❡❛r ✈❡❧♦❝✐t② tr❡♥❞ ✐s s❤♦✇♥ ✐♥ ✜❣✉r❡ ✸✳✶✺ ❛♥❞ ✜❣✉r❡ ✸✳✶✻✳ ❚❤❡ r♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ ✐st❛♥t❛♥❡♦✉s ✈❡❧♦❝✐t✐❡s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ✜❣✉r❡ ✸✳✶✼✱ ✜❣✉r❡ ✸✳✶✽ ❛♥❞ ✜❣✉r❡ ✸✳✶✾✳ ❚❤❡ ✜❣✉r❡s ♣r❡s❡♥t t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ r♦♦t ♠❡❛♥ sq✉❛r❡ ♦♥ ♦♥❧② t❤❡ ♦♥❡ ❤❛❧❢ ♦❢ t❤❡ ❝❤❛♥♥❡❧ ✇✐t❤ t❤❡ r✐❜❧❡ts ✇❛❧❧✱ ✐♥ ❢❛❝t t❤❡ ♦t❤❡r ❤❛❧❢ r❡♠❛✐♥ ❛❧♠♦st ✉♥❝❤❛♥❣❡❞✳ ❚❤❡ str❡❛♠✲✇✐s❡ ❝♦♠♣♦♥❡♥t✱ ♣r❡s❡♥t❡❞ ✐♥ ✜❣✉r❡ ✸✳✶✼✱ s❤♦✇ t❤❛t ❡❛❝❤ ❞✐✛❡r❡♥t ❝❛s❡ r❡❛❝❤ t❤❡ ♠❛①✐♠✉♠ ♦❢ t❤❡ r♦♦t ♠❡❛♥ sq✉❛r❡ ❛t t❤❡ s❛♠❡ y ❝♦♦r❞✐♥❛t❡✱ ❤♦✇❡✈❡r✱ ✐♥ ❜♦t❤ t❤❡ s✐♠♣❧❡ r✐❜❧❡t ❛♥❞ ❢r❛❝t❛❧ ❝❛s❡✱ t❤❡ ♠❛①✐♠✉♠ ❤❛s

  85. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✽✷ 20 Smooth case Triangular riblet Fractal riblet 15 10 5 0 0 0.002 0.004 0.006 0.008 0.01 y ❋✐❣✉r❡ ✸✳✶✹✿ ◆❡❛r ✇❛❧❧ t❛♥❣❡♥ts t♦ t❤❡ ♠❡❛♥ str❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t② ♣r♦✜❧❡s✳ 0.07 0.065 Shear velocity 0.06 0.055 0.05 0 100 200 300 400 500 Time step ❋✐❣✉r❡ ✸✳✶✺✿ u τ t✐♠❡ ❤✐st♦r② ❢♦r t❤❡ s✐♠♣❧❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳

  86. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✽✸ 0.07 0.065 Shera velocity 0.06 0.055 0.05 0 100 200 300 400 500 Time step ❋✐❣✉r❡ ✸✳✶✻✿ u τ t✐♠❡ ❤✐st♦r② ❢♦r t❤❡ t❤✐r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ ❢♦r t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ 3 Smooth case Triangular riblet 2.5 Fractal riblet 2 u rms 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 y ❋✐❣✉r❡ ✸✳✶✼✿ ❘♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ str❡❛♠✲✇✐s❡ ✈❡❧♦❝✐t②✳ ❊❛❝❤ ❝❛s❡ ✐s ♥♦r♠❛❧✐③❡❞ ❜② t❤❡ ❛❝t✉❛❧ ✈❛❧✉❡ ♦❢ u τ ✳

  87. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✽✹ 1 0.8 0.6 v rms Smooth case 0.4 Triangular riblet Fractal riblet 0.2 0 0 0.2 0.4 0.6 0.8 1 y ❋✐❣✉r❡ ✸✳✶✽✿ ❘♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ ♥♦r♠❛❧ ✈❡❧♦❝✐t②✳ ❊❛❝❤ ❝❛s❡ ✐s ♥♦r♠❛❧✐③❡❞ ❜② t❤❡ ❛❝t✉❛❧ ✈❛❧✉❡ ♦❢ u τ ✳ 1.2 1 0.8 w rms 0.6 Smooth case 0.4 Triangular riblet Fractal riblet 0.2 0 0 0.2 0.4 0.6 0.8 1 y ❋✐❣✉r❡ ✸✳✶✾✿ ❘♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ s♣❛♥✲✇✐s❡ ✈❡❧♦❝✐t②✳ ❊❛❝❤ ❝❛s❡ ✐s ♥♦r♠❛❧✐③❡❞ ❜② t❤❡ ❛❝t✉❛❧ ✈❛❧✉❡ ♦❢ u τ ✳

  88. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✽✺ ❋✐❣✉r❡ ✸✳✷✵✿ ❈♦♥t♦✉r ♦❢ t❤❡ str❡❛♠✲✇✐s❡ ✈♦rt✐❝✐t② ✐♥ t❤❡ z − y ♣❧❛♥❡ ❢♦r t❤❡ s✐♠♣❧❡ r✐❜❧❡t ❝❤❛♥♥❡❧✳ ❛♥ ❤✐❣❤❡r ✈❛❧✉❡✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ❛r❣✉❡ t❤❛t t❤❡ ◆❛✈✐❡r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❣❡♥❡r❛t❡ ❛♥ ✐♥❝r❡♠❡♥t ♦❢ t❤❡ t✉r❜✉❧❡♥t ✐♥t❡♥s✐t② ♦❢ t❤❡ str❡❛♠✲✇✐s❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❧♦❝✐t②✳ ❚❤✐s s❡❡♠s t♦ ❜❡ ✐♥ ❝♦♥tr❛st ✇✐t❤ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥✱ ❤♦✇❡✈❡r ✐❢ ✇❡ ❧♦♦❦ ❛t t❤❡ ✜❣✉r❡ ✸✳✶✽ ❛♥❞ ✜❣✉r❡ ✸✳✶✾ ✇❡ ❝❛♥ ✜❣✉r❡ ♦✉t t❤❡ ♣r♦❜❧❡♠✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ♥♦t❡ ✭✐♥ ❜♦t❤ t❤❡ ✜❣✉r❡✮ t❤❛t t❤❡ t❤r❡❡ ❝✉r✈❡s ♦❢ t❤❡ r♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ s♣❛♥✲✇✐s❡ ✈❡❧♦❝✐t✐❡s ❝♦✐♥❝✐❞❡ ♥❡❛r t❤❡ ✇❛❧❧✱ ❜✉t ❛t ❝❡rt❛✐♥ y ❝♦♦r❞✐♥❛t❡s t❤❡ s✐♠♣❧❡ r✐❜❧❡t r♦♦t ♠❡❛♥ sq✉❛r❡ ❣r♦✇ ✉♣ s❧♦✇❧② t❤❡♥ t❤❡ s♠♦♦t❤ ❝❛s❡✱ ❛♥❞ t❤❡ r♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t ❣r♦✇ ✉♣ ❡✈❡♥ ♠♦r❡ s❧♦✇❧② t❤❡♥ t❤❡ s✐♠♣❧❡ tr✐❛♥❣✉❧❛r r✐❜❧❡t✳ ❚❤❡ ✈❡❧♦❝✐t✐❡s ❝♦♠♣♦♥❡♥ts ♣❧♦tt❡❞ ✐♥ ✜❣✉r❡ ✸✳✶✽ ❛♥❞ ✸✳✶✾ ❛r❡ r❡s♣♦♥s✐❜❧❡ ♦❢ t❤❡ str❡❛♠✲✇✐s❡ ✈♦rt❡①✱ t❤❡r❡❢♦r❡ ❛ r❡❞✉❝t✐♦♥ ♦❢ ❜♦t❤ t❤❡ r♦♦t ♠❡❛♥ sq✉❛r❡ ✐♥❞✐❝❛t❡s ❛ ❞✉♠♣✐♥❣ ♦❢ t❤❡ ❝r♦ss ✢♦✇✳ ❚❤❡ r♠s ❢❡❛t✉r❡s r❡✈❡❛❧ t❤❛t t❤❡ s❧✐♣ ❝♦♥❞✐t✐♦♥ ❛r❡ ❛❜❧❡ t♦ r❡♣r♦❞✉❝❡ t❤❡ r✐❜❧❡t ❡✛❡❝t✱ ✇❤✐❝❤ ✇♦r❦s t❤r♦✉❣❤ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❝r♦ss✲✢♦✇✳ ❚❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ t✉r❜✉❧❡♥❝❡ ❧❡✈❡❧ ❝❛♥ ❜❡ ❝♦♥✜r♠❡❞ ❜② t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ str❡❛♠✲ ✇✐s❡ ❛♥❞ t❤❡ s♣❛♥✲✇✐s❡ ✈♦rt✐❝✐t②✳ ■♥ ✜❣✉r❡ ✸✳✷✵ ✐s ♣r❡s❡♥t❡❞ t❤❡ ❝♦♥t♦✉r ♦❢ t❤❡ str❡❛♠✲ ✇✐s❡ ✈♦rt✐❝✐t② ✐♥ t❤❡ z − y ♣❧❛♥❡❀ t❤❡ ❜♦tt♦♠ s✐❞❡ ♦❢ t❤❡ ❝❤❛♥♥❡❧ ♣r❡s❡♥t t❤❡ ❣r♦♦✈❡❞ s✉r❢❛❝❡ ✭❤❡r❡ ✐s ❛♣♣❧✐❡❞ t❤❡ s❧✐♣ ❝♦♥❞✐t✐♦♥✮✱ ❛♥❞ t❤❡ t♦♣ s✐❞❡ t❤❡ s♠♦♦t❤ ✇❛❧❧ ✭❤❡r❡ ✐s ❛♣♣❧✐❞❡ t❤❡ ♥♦✲s❧✐♣ ❝♦♥❞✐t✐♦♥✮✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ♥♦t❡ t❤❛t ♥❡❛r t❤❡ ♥♦✲s❧✐♣ ✇❛❧❧ ❛r❡ ❧♦❝❛t❡❞ ❛ ❧♦t ♦❢ s♣♦ts ♦❢ ❤✐❣❤ ✈♦rt✐❝✐t②✱ ✇❤❡r❡❛s t❤❡ r✐❜❧❡t s✉r❢❛❝❡s s❤♦✇ ❛ ❧♦✇❡r ❣❡♥❡r❛t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ s♣♦ts ❛♥❞ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ ✈♦rt✐❝✐t② ✐s s♠❛❧❧❡r✳ ❚❤❡ ❜❡❤❛✈✐♦r ✐s ❞✉❡ t♦ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❜✉rst✐♥❣ ❝②❝❧❡✳✳ ❋✐❣✉r❡ ✸✳✷✵ ❝♦♥✜r♠s t❤❛t t❤❡ t✉r❜✉❧❡♥t ✐♥t❡♥s✐t② ✐s r❡❞✉❝❡❞ ❜② t❤❡ r✐❜❧❡t ✇❛❧❧✳ ■♥ ✜❣✉r❡ ✸✳✷✶ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♦❜s❡r✈❡ t❤❡ s♣❛♥✲

  89. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✽✻ ❋✐❣✉r❡ ✸✳✷✶✿ ❈♦♥t♦✉r ♦❢ t❤❡ s♣❛♥✲✇✐s❡ ✈♦rt✐❝✐t② ✐♥ t❤❡ x − y ♣❧❛♥❡ ❢♦r t❤❡ s✐♠♣❧❡ r✐❜❧❡t ❝❤❛♥♥❡❧✳ ✇✐s❡ ✈♦rt✐❝✐t② ✐♥ t❤❡ x − y ♣❧❛♥❡❀ ❛❣❛✐♥✱ t❤❡ ❧♦✇❡r ✇❛❧❧ ♣r❡s❡♥ts t❤❡ r✐❜❧❡t ❛♥❞ t❤❡ ✉♣♣❡r ♦♥❡ ✐s s♠♦♦t❤✳ ■t ✐s ❝❧❡❛r t❤❛t ✐♥ t❤❡ ✉♣♣❡r ③♦♥❡ ♦❢ t❤❡ ❝❤❛♥♥❡❧ ❛♥ ❤✐❣❤❡r ❞✐str✐❜✉t✐♦♥ ♦❢ ✈♦rt❡① ✐s ♣r❡s❡♥t✳ ❚❤✐s ❡✛❡❝t ❝❛♥ ❜❡ r❡❧❛t❡❞ t♦ t❤❡ ❣❡♥❡r❛t✐♦♥ ♦❢ ✈♦rt✐❝❡s ✐♥ t❤❡ ♥❡❛r ✇❛❧❧ r❡❣✐♦♥✳ ■♥ ♦r❞❡r t♦ ❛ss❡s t❤❡ ❡✛❡❝t ♦❢ t❤❡ r❡②♥♦❧❞s ♥✉♠❜❡r✱ t❤❡ ❉◆❙ ❤❛s ❜❡❡♥ ♣❡r❢♦r♠❡❞ ❛t Re = 2200 ❛♥❞ Re = 3500 ✳ ❚❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❛❜❧❡ ✸✳✸ ❛♥❞ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✸✳✷✷✳ ❈❛s❡ Re | ∂p/∂x | u τ Re τ ❙♠♦♦t❤ 2200 0 . 00395 0 . 0628 142 ❙♠♦♦t❤ 2800 0 . 00397 0 . 0634 177 ❙♠♦♦t❤ 3500 0 . 00370 0 . 0607 217 ❘✐❜❧❡t 2200 0 . 00385 0 . 0621 141 ❘✐❜❧❡t 2800 0 . 00376 0 . 0631 170 ❘✐❜❧❡t 3500 0 . 00351 0 . 0652 232 ❚❛❜❧❡ ✸✳✸✿ ❉◆❙ r❡s✉❧ts ❢♦r t❤❡ s♠♦♦t❤ ❣❡♦♠❡tr② ❛♥❞ t❤❡ 90 ◦ tr✐❛♥❣✉❧❛r r✐❜❧❡t ❛t ❞✐❢✲ ❢❡r❡♥t r❡②♥♦❧❞s ♥✉♠❜❡r✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ♥♦t❡ t❤❡ ❧✐♥❡❛r tr❡♥❞ ❢♦r ❜♦t❤ t❤❡ s♠♦♦t❤ ❛♥❞ t❤❡ r✐❜❧❡t ♣❧❛t❡s✱ ✇❤✐❝❤

  90. ❈❍❆P❚❊❘ ✸✳ ▼❆❈❘❖❙❈❖P■❈ P❘❖❇▲❊▼ ✽✼ 10 -3 4.2 Smooth 4.1 fit RIblet 4 fit 3.9 dp/dx 3.8 3.7 3.6 3.5 3.4 140 160 180 200 220 240 Re ❋✐❣✉r❡ ✸✳✷✷✿ Pr❡ss✉r❡ ❣r❛❞✐❡♥t ❛t ❞✐✛❡r❡♥t r❡②♥♦❧❞s ♥✉♠❜❡r✳ ✐♥❞✐❝❛t❡ ❛ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❞r❛❣ ✇❤❡♥ t❤❡ ❘❡②♥♦❧❞s ♥✉♠❜❡r ✐♥❝r❡❛s❡✳ ❚❤❡ r✐❜❧❡t ♣❧❛t❡s r❡s✉❧ts s❤✐❢t❡❞ ❞♦✇♥✱ ❛♥❞ t❤✐s ❝♦♥✜r♠ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♦❜t❛✐♥ ❛ ❞r❛❣ r❡❞✉❝t✐♦♥ ❢♦r ❡❛❝❤ r❡②♥♦❧❞s ♥✉♠❜❡r ❤❡r❡✐♥ ❝♦♥s✐❞❡r❡❞✳

  91. ❈❤❛♣t❡r ✹ ❈♦♥❝❧✉s✐♦♥ ✽✽

  92. ❈❍❆P❚❊❘ ✹✳ ❈❖◆❈▲❯❙■❖◆ ✽✾ ✹✳✶ ❈♦♥❝❧✉s✐♦♥ ■♥ t❤✐s ✇♦r❦ ✇❡ ❤❛✈❡ st✉❞✐❡❞✱ t❤r♦✉❣❤ ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s✱ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ♣♦✲ t❡♥t✐❛❧✐t② ♦❢ ❛ s❡r✐❡s ♦❢ r✐❜❧❡t s✉r❢❛❝❡s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢r❛❝t❛❧✐③❛t✐♦♥ ♦❢ t✇♦ ❞✐✛❡r❡♥t tr✐❛♥❣✉❧❛r r✐❜❧❡t ♠♦❧❞s✳ ■♥ ♦r❞❡r t♦ ♣❡r❢♦r♠ s✉❝❤ s✐♠✉❧❛t✐♦♥s✱ ✇❡ ❤❛✈❡ s✉❜❞✐✈✐❞❡❞ t❤❡ st✉❞② ✐♥ t✇♦ ♣❛rts✿ t❤❡ ♠✐❝r♦s❝♦♣✐❝ ❛♥❞ t❤❡ ♠❛❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠✳ ❚❤❡ ✜rst ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ st✉❞✐❡❞ t❤r♦✉❣❤ t❤❡ ❜♦✉♥❞❛r② ❡❧❡♠❡♥t ♠❡t❤♦❞✳ ■t ✐s ✇❡❧❧ ❦♥♦✇♥ ❜❡❝❛✉s❡ ♦❢ ✐ts ❝❛♣❛❜✐❧✐t② t♦ s♦❧✈❡ ❡q✉❛t✐♦♥s ✐♥s✐❞❡ ❝♦♠♣❧❡① ❞♦♠❛✐♥s✳ ■t ✐s t✉r♥❡❞ ♦✉t t♦ ❜❡ ❡①tr❡♠❡❧② ❡✣❝✐❡♥t ❢♦r t❤❡ st✉❞② ♦❢ ❛ t✇♦ ❞✐♠❡♥s✐♦♥❛❧ ❢r❛❝t❛❧ ❣❡♦♠✲ ❡tr②✳ ❚❤❡ t✐♠❡ r❡q✉✐r❡❞ ❢♦r t❤❡ s✐♠✉❧❛t✐♦♥s ✇❛s ♦❢ t❤❡ ♦r❞❡r ♦❢ ♠✐♥✉t❡s❀ t❤❡ ❧♦♥❣❡st s✐♠✉❧❛t✐♦♥ ❤❛s r❡q✉✐r❡❞ ❛ t✐♠❡ ♦❢ ♦♥❧② ✜✈❡ ♠✐♥✉t❡s✳ ❚❤❡ ♠✐❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ s♦❧✈❡❞ ❢♦r t❤❡ t✇♦ tr✐❛♥❣✉❧❛r r✐❜❧❡t ❛♥❞ t❤❡✐r ❢r❛❝✲ t❛❧✐③❛t✐♦♥✱ ✐♥ ♦r❞❡r t♦ ❝❛rr② ♦✉t t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❢♦r ❡❛❝❤ s❤❛♣❡✳ ❚❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ✈❛❧✉❡s r❡✈❡❛❧❡❞ ❛♥ ❡①♣♦♥❡♥t✐❛❧ tr❡♥❞✱ ❛♥❞ t❤❛t ❛❧❧ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♣♣r♦❛❝❤ ❛ ✜♥✐t❡ ❧✐♠✐t ✇❤❡♥ t❤❡ ❢r❛❝t❛❧✐③❛t✐♦♥ ❣♦ ♦♥✱ ✐♥ ♣❛rt✐❝✲ ✉❧❛r✱ ❛❢t❡r t❤❡ t❤✐r❞ ✐t❡r❛t✐♦♥ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❞✐❞♥✬t ❝❤❛♥❣❡ ❛♥②♠♦r❡✳ ❚❤✐s ✐s ❛ ♠❡❛♥✐♥❣❢✉❧ ❣♦❛❧ ❜❡❝❛✉s❡ ❝♦♥✜r♠ t❤❛t t❤❡ r✐❜❧❡t ❡✛❡❝t ✐s ♥♦t ❛✛❡❝t❡❞ ❜② t❤❡ ♥❛♥♦s❝♦♣✐❝ ❢❡❛t✉r❡ ♦❢ t❤❡ s✉r❢❛❝❡✱ t❤❡r❡❢♦r❡✱ t❤❡ r♦✉❣❤♥❡ss ♦❢ t❤❡ s✉r❢❛❝❡ ❝❛♥ ❜❡ ♥❡❣❧❡❝t❡❞ ✇❤❡♥ ✐t ✐s t♦♦ ✜♥❡✳ ❆♥♦t❤❡r ✐♥t❡r❡st✐♥❣ r❡s✉❧ts ❤❛s ❜❡❡♥ ♦❜t❛✐♥❡❞ t❤❛♥❦s t♦ t❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t✇♦ ❦✐♥❞ ♦❢ ❢r❛❝t❛❧✐③❛t✐♦♥✱ ♦♥❡ t❤❛t ♣r♦tr✉❞❡ t❤❡ ❣❡♦♠❡tr② ✐♥s✐❞❡ t❤❡ ✢✉✐❞ ❞♦♠❛✐♥✱ ❛♥❞ t❤❡ ♦t❤❡r t❤❛t ♣r♦tr✉❞❡ ✐t ✐♥s✐❞❡ t❤❡ ✇❛❧❧✳ ❚❤❡ ✐♥✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ ❣❡♥❡r❛t❡s ❛♥ ✐♥❝r❡✲ ♠❡♥t ♦❢ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡♠❀ ✐♥st❡❛❞ t❤❡ ♦✉t✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ ❣❡♥❡r❛t❡s ❛ r❡❞✉❝t✐♦♥ ♦❢ ❛❧❧ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ q✉❛♥t✐t✐❡s✳ ❆❧t❤♦✉❣❤ t❤❡ ✐♥✇❛r❞ ❛♥❞ t❤❡ ♦✉t✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ ❛r❡ s②♠♠❡tr✐❝ r❡s♣❡❝t t♦ t❤❡ tr✐❛♥❣✉❧❛r ♠♦❧❞✱ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ✈❛r✐❛t✐♦♥ ♠❛❣♥✐t✉❞❡s ❛r❡ ♥♦t✳ ❚❤✐s ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ✢♦✇ ♦✈❡r ❛ tr✐❛♥❣✉❧❛r r✐❜❧❡t ♣❧❛t❡s ✐s ♠♦r❡ ❛✛❡❝t❡❞ ❜② ❛ ❞✐st✉r❜❛♥❝❡ t❤❛t ♣r♦tr✉❞❡ ✐♥t♦ t❤❡ ✢✉✐❞ r❛t❤❡r t❤❡♥ ♦♥❡ t❤❛t ♣r♦tr✉❞❡ ✐♥t♦ t❤❡ ✇❛❧❧✳ ▼♦r❡♦✈❡r✱ t❤❡ ✐♥✇❛r❞ ❢r❛❝t❛❧✐③❛t✐♦♥ ❣❡♥✲ ❡r❛t❡ ❛♥ ✐♥❝r❡❛s❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✱ ❛♥❞ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❛r❣✉❡✱ ✐♥ ❧✐♥❡ ✇✐t❤ t❤❡ ▲✉❝❤✐♥✐✬s t❤❡♦r②✱ t❤❛t t❤❡ ✐♥✇❛r❞ ❢r❛❝t❛❧ r✐❜❧❡t ❣❡♥❡r❛t❡ ❛♥ ❤✐❣❤❡r ❞r❛❣ r❡❞✉❝t✐♦♥ t❤❡♥ t❤❡ s✐♠♣❧❡ tr✐❛♥❣✉❧❛r ♦♥❡✳ ❚❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤✐s ✇♦r❦ ❝♦♥❝❡r♥❡❞ t❤❡ ♠❛❝r♦s❝♦♣✐❝ ♣r♦❜❧❡♠❀ ✐t ❛❧❧♦✇❡❞ ✉s t♦ ❝♦♠♣❧❡t❡ t❤❡ st✉❞② ♦❢ t❤❡ r✐❜❧❡t ♣❧❛t❡s ❛♥❞ ❝❛rr② ♦✉t t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥✳ ❚❤❡ tr✐❛♥❣✉❧❛r

  93. ❈❍❆P❚❊❘ ✹✳ ❈❖◆❈▲❯❙■❖◆ ✾✵ r✐❜❧❡t ✇✐t❤ t❤❡ ❤✐❣❤❡st ∆ h ❛♥❞ ✐ts t❤✐r❞ ❢r❛❝t❛❧ ✐t❡r❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ s♦❧✈❡❞ t❤r♦✉❣❤ t❤❡ ❞✐r❡❝t ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥✳ ■♥ ♦r❞❡r t♦ ❛✈♦✐❞ ❛❧❧ t❤❡ ❞✐✣❝✉❧t✐❡s t❤❛t ❛r✐s❡ ❢r♦♠ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ r✐❜❧❡t ♣❧❛t❡s✱ t❤❡ s♣❛♥✲✇✐s❡ ❛♥❞ t❤❡ str❡❛♠✲✇✐s❡ ◆❛✈✐❡r ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❤❛✈❡ ❜❡❡♥ ✉s❡❞✳ ❚❤❡ s✐♠✉❧❛t✐♦♥ r❡✈❡❛❧❡❞ t❤❛t t❤❡ s✐♠♣❧❡ tr✐❛♥❣✉❧❛r r✐❜❧❡t ♣❧❛t❡s ❛♥❞ t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t ♣❧❛t❡s ❛r❡ ❛✛❡❝t❡❞ r❡s♣❡❝t✐✈❡❧② ❜② ❛ ❞r❛❣ t❤❛t ✐s 5 . 3% ❛♥❞ 6 . 8% s♠❛❧❧❡r t❤❡♥ t❤❡ s♠♦♦t❤ ♣❧❛t❡s✳ ❆s ❡①♣❡❝t❡❞ ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡ t❤❡ s✐♠♣❧❡ r✐❜❧❡t ♣❧❛t❡s ❤❛✈❡ ❛ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐♥ t❤❡ r❛♥❣❡ ♦❢ 4 − 7% ❀ ✐t ✐s ❝❧❡❛r t❤❛t ✐t ✇♦✉❧❞ ❜❡ ♣♦ss✐❜❧❡ ♦❜t❛✐♥ ❛ ❜❡tt❡r ❞r❛❣ r❡❞✉❝t✐♦♥ ✇✐t❤ t❤❡ tr✐❛♥❣✉❧❛r r✐❜❧❡t✱ ❤♦✇❡✈❡r t❤✐s ✇♦r❦ ❞♦❡s ♥♦t ❛✐♠ ❛t ♦♣t✐♠✐③✐♥❣ t❤❡ s❤❛♣❡ ♦❢ t❤❡ r✐❜❧❡t✱ r❛t❤❡r ✐t ❛✐♠ ❛t st✉❞②✐♥❣ t❤❡ ❡✛❡❝t ♦❢ ♥❛♥♦✲ ❢❡❛t✉r❡s ♦❢ t❤❡ r✐❜❧❡t s✉r❢❛❝❡s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✐♠♣♦rt❛♥t ❛❝❤✐❡✈❡♠❡♥t ❝♦♥s✐sts ✐♥ t❤❡ ♣❡r❝❡♥t ♦❢ ❞r❛❣ r❡❞✉❝t✐♦♥ ♦❜t❛✐♥❡❞ ❢♦r t❤❡ ❢r❛❝t❛❧ r✐❜❧❡t✱ t❤❛t ✐s ❛❧♠♦st 2 ♣❡r❝❡♥t ♣♦✐♥ts ❤✐❣❤❡r t❤❡♥ t❤❛t ♦❜t❛✐♥❡❞ ❢♦r t❤❡ s✐♠♣❧❡ r✐❜❧❡t✳ ❚❤✐s r❡s✉❧ts s❡❡♠s t♦ ❝♦♥✜r♠ t❤❡ r♦❧❡ ♦❢ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✿ t❤❡② ❝❛♥ ❜❡ ✉s❡❞ t♦ ♣r❡❞✐❝t t❤❡ ❡✣❝✐❡♥❝② ♦❢ ❛ r✐❜❧❡t s✉r❢❛❝❡✳ ❚❤❡ ❧❛st st❛t❡♠❡♥t ♠✉st ❜❡ ②❡t ❝❛r❡❢✉❧❧② ❡✈❛❧✉❛t❡❞✱ ✐♥ ❢❛❝t t❤❡r❡ ❛r❡ ♥♦t ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ✐♥ s✉♣♣♦rt ♦❢ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ r✐❜❧❡t t❤r♦✉❣❤ t❤❡ s❧✐♣ ❝♦♥❞✐t✐♦♥s✳ ❍♦✇❡✈❡r✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ r♦♦t ♠❡❛♥ sq✉❛r❡ ♦❢ t❤❡ ✈❡❧♦❝✐t② ❝♦♠♣♦♥❡♥ts s❡❡♠s t♦ r❡✈❡❛❧ t❤❛t t❤❡ ♣❤②s✐❝❛❧ ♠❡❝❤❛♥✐s♠ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ s❧✐♣ ❝♦♥❞✐t✐♦♥s ✐s t❤❡ s❛♠❡ t❤❛t ❛❧❧♦✇ r✐❜❧❡t t♦ ✇♦r❦s✳ ❚❤❡ st✉❞② ❤❡r❡✐♥ ♣❡r❢♦r♠❡❞ s❡❡♠s t♦ ❝♦♥✜r♠ t❤❡ r♦❧❡ ♦❢ t❤❡ ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts✱ ❛♥❞ t❤❛t t❤❡ s❧✐♣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❝❛♣❛❜❧❡ t♦ r❡♣r❡s❡♥t t❤❡ r✐❜❧❡t s✉r❢❛❝❡s✳ ❍♦✇❡✈❡r✱ t❤❡ st✉❞② ♦❢ ❚❛❡❣❡❡ ▼✐♥ ❛♥❞ ❏♦❤♥ ❑✐♠ ❬✶✽❪ ❝❛rr② ♦✉t t❤❛t t♦✉❣❤ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ♣r♦tr✉s✐♦♥ ❤❡✐❣❤ts ✭♠♦r❡ ♣r❡❝✐s❡❧② t❤❡ str❡❛♠✲✇✐s❡ s❧✐♣ ❧❡♥❣t❤ ❛♥❞ t❤❡ s♣❛♥✲✇✐s❡ s❧✐♣ ❧❡♥❣t❤✮ ✐s ❡q✉❛❧ t♦ ③❡r♦✱ ✐t ✐s ❡✈❡♥ ♣♦ss✐❜❧❡ t♦ ♦❜t❛✐♥ ❛ ❞r❛❣ r❡❞✉❝t✐♦♥✳

  94. ❈❍❆P❚❊❘ ✹✳ ❈❖◆❈▲❯❙■❖◆ ✾✶ ✹✳✷ ❋✉t✉r❡ ❞❡✈❡❧♦♣❡♠❡♥ts ■♥ t❤❡ ❧❛st ②❡❛rs t❤❡ r❡s❡❛r❝❤❡rs ❤❛✈❡ ❢♦❝✉s❡❞ t❤❡✐r ❛tt❡♥t✐♦♥ ♦♥ ❛ ♥❡✇ ❦✐♥❞ ♦❢ s✉r❢❛❝❡s✱ ✇❤✐❝❤ ❛❧❧♦✇ ❛♥ ❤✐❣❤❡r ❞r❛❣ r❡❞✉❝t✐♦♥ t❤❡♥ r✐❜❧❡ts✿ t❤❡ ❙✉♣❡r✲❍②❞r♦♣❤♦❜✐❝ s✉r❢❛❝❡s ✭❙❍ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✮✳ ❚❤❡ ♠❡❝❤❛♥✐s♠ ❜② ✇❤✐❝❤ s✉❝❤ ❛ s✉r❢❛❝❡ ♦♣❡r❛t❡ ✐s ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t ❜② t❤❡ r✐❜❧❡t ♦♥❡✱ ❛♥❞ ✐t ✐s ❜❛s❡❞ ♦♥ t❤❡ ❤✐❣❤ ❝❛♣❛❜✐❧✐t② ✐♥ r❡♣❡❧ ✇❛t❡r✳ ❚❤✐s ♣❛rt✐❝✉❧❛r ❢❡❛t✉r❡ ✐s ❞✉❡ t♦ t❤❡ ♥❛♥♦✲str✉❝t✉r❡s ♣r❡s❡♥t ♦♥ t❤❡ s✉r❢❛❝❡✱ t❤❛t ❛❧❧♦✇ t❤❡ s✉r❢❛❝❡ t♦ r❡♠❛✐♥ ✐♥ t❤❡ ✉♥✇❡tt❡❞ st❛t❡✱ s♦ t❤❛t t❤❡ ✢✉✐❞ ✢♦✇ ♦✈❡r t❤❡ tr❛♣♣❡❞ ❛✐r✳ ❙❍ s✉r❢❛❝❡s ❛r❡ ♣r❡s❡♥t ✐♥ ♥❛t✉r❡✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ t✇♦✲s❝❛❧❡ str✉❝t✉r❡ ♦❢ t❤❡ ❜✐r❞✬s ❢❡❛t❤❡r ✭✜❣✉r❡ ✹✳✶✮ ✇❤✐❝❤ r❡♥❞❡rs ✐t ❡①❝❡♣t✐♦♥❛❧❧② ✇❛t❡r♣r♦♦❢✱ ✇✐t❤ ❛✐r tr❛♣♣❡❞ ✉♥❞❡r♥❡❛t❤ t❤❡ ♥❡t✇♦r❦ ♦❢ ❜❛r❜s ❛♥❞ ❜❛r❜✉❧❡s✳ ❋✐❣✉r❡ ✹✳✶✿ ❇✐r❞✬s ❢❡❛t❤❡r ❛t ♠✐❝r♦s❝♦♣❡✳ ❙❍ ♠❛t❡r✐❛❧s ❤❛✈❡ r❡❝❡✐✈❡❞ tr❡♠❡♥❞♦✉s ❛tt❡♥t✐♦♥ ✐♥ r❡❝❡♥t ②❡❛rs ❢♦r t❤❡ ❧❛r❣❡ ✈❛r✐❡t② ♦❢ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ✇❤✐❝❤ t❤❡② ❝♦✉❧❞ ❜❡ ♣✉t t♦ ♣r♦✜t❛❜❧❡ ✉s❡✱ r❛♥❣✐♥❣ ❢r♦♠ ❛♥t✐✇❡tt✐♥❣ t♦ ❛♥t✐✲✐❝✐♥❣✱ ❢r♦♠ s❡❧❢✲❝❧❡❛♥✐♥❣ t♦ ❛♥t✐✲❝♦rr♦s✐♦♥✱ ❛♥❞ ♦❜✈✐♦✉s❧② t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ❝❛♣❛❜✐❧✐t✐❡s✳ ❊❛r❧② ❡①♣❡r✐♠❡♥ts s✉❣❣❡st❡❞ t❤❛t t❤❡② r❡❞✉❝❡ ❞r❛❣ ✐♥ ❜♦t❤ ❧❛♠✐♥❛r ❛♥❞ t✉r❜✉❧❡♥t ❜♦✉♥❞❛r② ❧❛②❡r ✢♦✇s✳ ❚❤❡ ❙❍ s✉r❢❛❝❡s ❛r❡ st✉❞✐❡❞ ❜② ❛ ❧♦t ♦❢ r❡s❡❛r❝❤❡rs ❢♦r t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ t❤❡ ♥❛✈❛❧ ✜❡❧❞✱ ✐♥ ✇❤✐❝❤ ❛ ❞r❛❣ r❡❞✉❝t✐♦♥ ✉♣ t♦ 40% ✐s ♣♦ss✐❜❧❡✳ ❚❤❡ ♠❛❥♦r ♣r♦❜❧❡♠ ✇✐t❤ t❤❡ ❙❍ s✉r❢❛❝❡s ❝♦♥❝❡r♥ t❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ t❤❡ ✉♥✇❡tt❡❞ st❛t❡ t♦ t❤❡ ✇❡tt❡❞ ♦♥❡✱ ✐♥ ✇❤✐❝❤ t❤❡ ❞r❛❣ r❡❞✉❝t✐♦♥ ✐s ❝♦♠♣r♦♠✐s❡❞✳ ❚❤✐s ♣❤❡♥♦♠❡♥♦♥ ✐s ❞✉❡ t♦ t❤❡ ❞✐s♣❡rs✐♦♥ ♦❢ t❤❡ ❛✐r ✐♥ t❤❡ ✉♣♣❡r ✢✉✐❞✳ ◆♦✇❛❞❛②s✱ t❤❡ ♣r✐♥❝✐♣❛❧ s♦❧✉t✐♦♥

  95. ❈❍❆P❚❊❘ ✹✳ ❈❖◆❈▲❯❙■❖◆ ✾✷ ❋✐❣✉r❡ ✹✳✷✿ ❯♥✇❡tt❡❞ st❛t❡ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ✇❡tt❡❞ st❛t❡ ♦♥ t❤❡ r✐❣❤t✳ ❝♦♥s✐sts ✐♥ ❣❡♥❡r❛t✐♥❣ ❛ ❢r❛❝t❛❧✲❧✐❦❡ ♥❛♥♦✲str✉❝t✉r❡❞ s✉r❢❛❝❡s✱ s♦ t❤❛t t❤❡ ❛✐r ✐s tr❛♣♣❡❞ ❛♥❞ t❤❡ tr❛♥s✐t✐♦♥ ❜❡❝♦♠❡ s❧♦✇❡r✳ ❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ❜❡t✇❡❡♥ ♠❛t❡r✐❛❧s ❛♥❞ ♥❛♥♦✲t❡❝❤♥♦❧♦❣✐❡s r❡♣r❡s❡♥t ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ✜❡❧❞ ♦❢ st✉❞②✳ ■t ❛❧❧♦✇s t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❝♦♠♣❧❡① s✉r❢❛❝❡s ✇❤✐❝❤ ❛r❡ ❛❜❧❡ t♦ r❡❞✉❝❡ t❤❡ ❞r❛❣ ❛♥❞ ❝♦♥tr✐❜✉t❡ t♦ t❤❡ ❣❧♦❜❛❧ ❡♥❡r❣② s❛✈✐♥❣✳

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