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Course summary
18.354
Course summary 18.354 Dimensional analysis Keplers problem L = r - - PowerPoint PPT Presentation
Course summary 18.354 Dimensional analysis Keplers problem L = r - - PowerPoint PPT Presentation
Course summary 18.354 Dimensional analysis Keplers problem L = r md r dt , Random walks & diffusion x = D 2 n n t = J x x 2 , Mark Haw David Walker (In)stability analysis & pattern formation t
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Kepler’s problem
L = r × mdr dt ,
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Random walks & diffusion
David Walker Mark Haw
∂n ∂t = −∂Jx ∂x = D∂2n ∂x2 ,
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(In)stability analysis & pattern formation
∂tψ = U (ψ) + γ0⇥2ψ γ2(⇥2)2ψ
, U(⇤) = a 2⇤2 + b 3⇤3 + c 4⇤4
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Calculus of variations
I[Y ] Y = lim
✏!0
1 ✏ {I[f(x) + ✏(x − y)] − I[f(x)]} = Z x2
x1
@f @Y (x − y) + @f @Y 0 0(x − y)
- dx
= Z x2
x1
@f @Y − d dx @f @Y 0
- (x − y)dx.
= @f @Y − d dx @f @Y 0
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Surface tension
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Elasticity
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Hydrodynamics
Z
V
∂ρ ∂t dV = Z
S
ρu · ndS = Z
V
r · (ρu)dV.
∂ρ ∂t + r · (ρu) = 0.
Z
V (t)
ρDu Dt dV = Z
V (t)
(rp + ρg)dV
Du Dt = rp ρ + g.
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10 ㎛
Low Re
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Singular perturbations
✏d2u dx2 + du dx = 1.
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Conformal mappings
dw dz = ∂φ ∂x + i∂ψ ∂x = u − iv.
W(Z) = u0 ✓ Ze−iα + R2 Z eiα ◆ iΓ 2π ln Z.
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Rotating flows
∂u ∂t + u · ru + Ω ⇥ (Ω ⇥ r) = 1 ρrpΩ + νr2u 2Ω ⇥ u, r · u = 0.
Taylor columns, etc
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Solitons
KdV equation
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Topological defects
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