❋❧✉✐❞✲str✉❝t✉r❡� st❛❜✐❧✐t②� ❛♥❛❧②s✐s� ♦❢� ✢❡①✐❜❧❡� s♣❧✐tt❡r� ♣❧❛t❡s� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� P❤❉� st✉❞❡♥t� ❛t� ❖◆❊❘❆� ✭▼❡✉❞♦♥✱� ❋r❛♥❝❡✮� ❊✉r♦♣❡❡❛♥� ❘❡s❡❛r❝❤� ●r❛♥t� ✭❊❘❈✮� ❢✉♥❞✐♥❣� ❆❞✈✐s♦r� ✿� ❖❧✐✈✐❡r� ▼❆❘◗❯❊❚� ✭❖◆❊❘❆✮�
■♥tr♦❞✉❝t✐♦♥� ❋❧✉✐❞✲str✉❝t✉r❡� ✈✐❜r❛t✐♦♥s� ❛r❡� ✐♥✈♦❧✈❡❞� ✐♥� ♠❛♥②� ❡♥❣✐♥❡❡r✐♥❣� ✜❡❧❞s✳✳✳� ❈✐✈✐❧� ❡♥❣✐♥❡❡r✐♥❣� ❖✛s❤♦r❡� ✐♥❞✉str②� ❲✐♥❞� ❡♥❡r❣②� ❆❡r♦s♣❛❝❡� ❡♥❣✐♥❡❡r✐♥❣� ✳✳✳t❤❡� ✐❞❡♥t✐✜❝❛t✐♦♥� ♦❢� ✈✐❜r❛t✐♦♥� t❤r❡s❤♦❧❞s� ❞✉❡� t♦� ❝♦✉♣❧✐♥❣� ✐s� ♦❢t❡♥� ❝r✉❝✐❛❧✳� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✭❖◆❊❘❆✮� ✶� ✴� ✶✾� ❊❋▼❈� ❙❡✈✐❧❧❛✱� ✷✵✶✻� ❙❡♣t❡♠❜❡r� ✶✺�
■♥tr♦❞✉❝t✐♦♥� ❚♦� ✜♥❞� ✈✐❜r❛t✐♦♥� t❤r❡s❤♦❧❞s✱� ❛� ❧✐♥❡❛r� ❛♣♣r♦❛❝❤� ✐s� s✉✣❝✐❡♥t� ✿� ❲✐t❤� r❡❞✉❝❡❞� ♠♦❞❡❧s� ✭�❉♦✇❡❧❧✱� ❈❧❛r❦� ❡t� ❈♦①� ✷✵✵✹� ❀� ❊❧♦②� ❡t� ❛❧✳� ✷✵✵✽�✮� ❖♥❧②� ❢❡✇� st✉❞✐❡s� ✇✐t❤� ◆❛✈✐❡r✲❙t♦❦❡s� ✰� ❡❧❛st✐❝✐t②� ✿� ◮ ✓� tr❛♥s♣✐r❛t✐♦♥� ✔� ✭� ❋❡r♥á♥❞❡③� ❡t� ▲❡� ❚❛❧❧❡❝� ✷✵✵✸�✮� ◮ ✓� ❢✉❧❧� ❧✐♥❡❛r✐③❛t✐♦♥� ✔� ✭� P❡❞r❛❣❧✐♦� ✷✵✶✺�✮� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✭❖◆❊❘❆✮� ✷� ✴� ✶✾� ❊❋▼❈� ❙❡✈✐❧❧❛✱� ✷✵✶✻� ❙❡♣t❡♠❜❡r� ✶✺�
■♥tr♦❞✉❝t✐♦♥� ❈♦♥✜❣✉r❛t✐♦♥� ✭� ▲❡❡� ❡t� ❨♦✉� ✷✵✶✸�✮� ✿� D • L U ∞ P❧❛t❡� ❧❡♥❣t❤� L / D = ✷�✱� t❤✐❝❦♥❡ss� E / D = ✵ . ✵✻� ❉❡♥s✐t②� r❛t✐♦� ρ s♦❧✐❞� /ρ ✢✉✐❞� = ✽✹ . ✼� ❘❡②♥♦❧❞s� ♥✉♠❜❡r� R ❡� = U ∞ D /ν = ✶✵✵� ❱❛r②✐♥❣� ♣❛r❛♠❡t❡r� ✿� ♣❧❛t❡� ❜❡♥❞✐♥❣� st✐✛♥❡ss� ✭r✐❣✐❞✐t②✮� K ❇� ▼♦❞❡❧� ✿� ✐♥❝♦♠♣r❡ss✐❜❧❡� ✢♦✇� ✰� ❡❧❛st✐❝✐t②� ✭❙❛✐♥t✲❱❡♥❛♥t� ❑✐r❝❤❤♦✛✮� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✭❖◆❊❘❆✮� ✸� ✴� ✶✾� ❊❋▼❈� ❙❡✈✐❧❧❛✱� ✷✵✶✻� ❙❡♣t❡♠❜❡r� ✶✺�
❖✉t❧✐♥❡� ✶� ◆♦♥✲❧✐♥❡❛r� s✐♠✉❧❛t✐♦♥s� ▲✐♥❡❛r� st❛❜✐❧✐t②� ❛♣♣r♦❛❝❤� ❢♦r� ✢✉✐❞✲str✉❝t✉r❡� ♣r♦❜❧❡♠� ✷� ❙t❛❜✐❧✐t②� r❡s✉❧ts� ✸� ✹� ❈♦♥❝❧✉s✐♦♥�
◆♦♥✲❧✐♥❡❛r� s✐♠✉❧❛t✐♦♥s� ❆♥� ✉♥st❡❛❞②� s✐♠✉❧❛t✐♦♥� ❈♦♥✜❣✉r❛t✐♦♥� ♦❢� ▲❡❡� ✫� ❨♦✉✱� K ❇� = ✵ . ✵✷� ✿� ✷� ✵� − ✷� − ✷� ✵� ✷� ✹� ✻� ✽� ✶✵� ✶✷� ✶✹� ✈❡❧♦❝✐t②� ♠❛❣♥✐t✉❞❡� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✭❖◆❊❘❆✮� ✹� ✴� ✶✾� ❊❋▼❈� ❙❡✈✐❧❧❛✱� ✷✵✶✻� ❙❡♣t❡♠❜❡r� ✶✺�
◆♦♥✲❧✐♥❡❛r� s✐♠✉❧❛t✐♦♥s� ❯♥st❡❛❞②� s✐♠✉❧❛t✐♦♥s✱� ❡✛❡❝t� ♦❢� K ❇� ❘❡s✉❧ts� ❢r♦♠� ▲❡❡� ❡t� ❨♦✉� ✷✵✶✸� ✭❉◆❙✮� ✿� ✈♦rt❡①� s❤❡❞❞✐♥❣� ✐♥� t❤❡� ✇❛❦❡� ✫� s②♠♠❡tr✐❝� str✉❝t✉r❡� ♦s❝✐❧❧❛t✐♦♥s� ❢♦r� ❛❧❧� ❝❛s❡s✱� ❛♥❞� ✵ . ✹� ✵ . ✷� ✵ . ✵� ✵ . ✵✹� ✵ . ✵✻� ✵ . ✵✽� ✵ . ✶✵� P❧❛t❡� ♠❛①✳� ❞✐s♣❧❛❝❡♠❡♥t� ✈s� K ❇� ❚❤r❡s❤♦❧❞� ❛t� K ❇� ∼ ✵ . ✵✺�✳� K ❇� < ✵ . ✵✺� ✿� ❤✐❣❤� str✉❝t✉r❡� ❞❡❢♦r♠❛t✐♦♥� r❡❣✐♠❡� K ❇� > ✵ . ✵✺� ✿� s♠❛❧❧� str✉❝t✉r❡� ❞❡❢♦r♠❛t✐♦♥� r❡❣✐♠❡� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✭❖◆❊❘❆✮� ✺� ✴� ✶✾� ❊❋▼❈� ❙❡✈✐❧❧❛✱� ✷✵✶✻� ❙❡♣t❡♠❜❡r� ✶✺�
❖✉t❧✐♥❡� ✶� ◆♦♥✲❧✐♥❡❛r� s✐♠✉❧❛t✐♦♥s� ▲✐♥❡❛r� st❛❜✐❧✐t②� ❛♣♣r♦❛❝❤� ❢♦r� ✢✉✐❞✲str✉❝t✉r❡� ♣r♦❜❧❡♠� ✷� ❙t❛❜✐❧✐t②� r❡s✉❧ts� ✸� ✹� ❈♦♥❝❧✉s✐♦♥�
▲✐♥❡❛r� st❛❜✐❧✐t②� ❛♣♣r♦❛❝❤� ❢♦r� ✢✉✐❞✲str✉❝t✉r❡� ♣r♦❜❧❡♠� ▲❛❣r❛♥❣✐❛♥✴❊✉❧❡r✐❛♥� ▲❛❣r❛♥❣✐❛♥✴❊✉❧❡r✐❛♥� ❱❡❧♦❝✐t②� ✉� ✱� ❞✐s♣❧❛❝❡♠❡♥t� ξ ✱� r❡❢❡r❡♥❝❡� ❞♦♠❛✐♥� Ω ✱� ♠♦✈✐♥❣� ❞♦♠❛✐♥� Ω( t ) ✉� Ω s� t ξ Ω s� Ω ❢� t ❚❤❡� ❣♦✈❡r♥✐♥❣� ❡q✉❛t✐♦♥s� ❞♦� ♥♦t� ❜❡❧♦♥❣� t♦� t❤❡� s❛♠❡� s♣❛❝❡� ✿� ∂ ✉ ① ∈ Ω ❢� ( t ) B ❋� ∂ t − R ❋� ( ✉ , p ) = ✵ ❋� ❧✉✐❞� ✲� ❊✉❧❡r✐❛♥� ∂ ✷ ξ ❳ ∈ Ω s� ∂ t ✷ − R ❙� ( ξ ) = ✵ ❙� ♦❧✐❞� ✲� ▲❛❣r❛♥❣✐❛♥� R ■� ( ✉ , p )( ① ) + R ■� ( ξ )( ❳ ) = ✵ ■� ♥t❡r❢❛❝❡� ✲� ❇♦t❤� ✦� ❍♦✇� t♦� ❝♦♣❡� ✇✐t❤� ♠♦✈✐♥❣� ❞♦♠❛✐♥s� ❄� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✭❖◆❊❘❆✮� ✻� ✴� ✶✾� ❊❋▼❈� ❙❡✈✐❧❧❛✱� ✷✵✶✻� ❙❡♣t❡♠❜❡r� ✶✺�
▲✐♥❡❛r� st❛❜✐❧✐t②� ❛♣♣r♦❛❝❤� ❢♦r� ✢✉✐❞✲str✉❝t✉r❡� ♣r♦❜❧❡♠� ▲❛❣r❛♥❣✐❛♥✴❊✉❧❡r✐❛♥� ❆r❜✐tr❛r②� ▲❛❣r❛♥❣✐❛♥� ❊✉❧❡r✐❛♥� ❢♦r♠✉❧❛t✐♦♥� ✭� ❉♦♥❡❛� ❡t� ❛❧✳� ✷✵✵✹�✮� ❱❡❧♦❝✐t②� ✉� ✱� ❞✐s♣❧❛❝❡♠❡♥t� ξ ✱� r❡❢❡r❡♥❝❡� ❞♦♠❛✐♥� Ω ✱� ♠♦✈✐♥❣� ❞♦♠❛✐♥� Ω t ✉� ξ Ω s� Ω ❢� ■❞❡❛� ✿� ✐♥tr♦❞✉❝❡� ❛♥� ❛r❜✐tr❛r②� ❡①t❡♥s✐♦♥� ♠❛♣� E ✐♥� t❤❡� ✢✉✐❞� r❡❣✐♦♥� → ❞❡♣❡♥❞❛♥t� ✈❛r✐❛❜❧❡s� ❢✉♥❝t✐♦♥s� ♦❢� ✜①❡❞� r❡❢❡r❡♥❝❡� ❝♦♦r❞✐♥❛t❡s� ❳� ① = ❳ + ξ ( ❳ , t ) ■♥� t❤❡� s♦❧✐❞� ✭❡❧❛st✐❝✐t②✮� ① = E ( ❳ , t ) ■♥� t❤❡� ✢✉✐❞� ✭❡①t❡♥s✐♦♥✮� ❊①t❡♥s✐♦♥� ✭✢✉✐❞� ♠❡s❤✮� ❡q✉❛t✐♦♥� ✿� −R ❊� ( ξ | Ω ❢� , ξ | Ω s� )( ❳ ) = ✵� ✐♥� Ω ❢� ❏❡❛♥✲▲♦✉� P❋■❙❚❊❘� ✭❖◆❊❘❆✮� ✼� ✴� ✶✾� ❊❋▼❈� ❙❡✈✐❧❧❛✱� ✷✵✶✻� ❙❡♣t❡♠❜❡r� ✶✺�
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