= condition condition dS v dS v Then S S - - PowerPoint PPT Presentation

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condition condition ds v ds v then s s proof in happel
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= condition condition dS v dS v Then S S - - PowerPoint PPT Presentation

Jul 20, 2023 637 likes •718 views

Navier Navier- -Stokes Stokes Creeping Flow Creeping Flow Minimum Energy Minimum Energy Point Force Solution Point Force Solution General Solution General Solution Faxen Faxen Law Law

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SLIDE 1

1

ME 637

  • G. Ahmadi

ME 637

  • G. Ahmadi
  • Navier

Navier-

  • Stokes

Stokes

  • Creeping Flow

Creeping Flow

  • Minimum Energy

Minimum Energy

  • Point Force Solution

Point Force Solution

  • General Solution

General Solution

  • Faxen

Faxen Law Law

  • Nonspherical

Nonspherical Particles Particles

  • Oblate Ellipsoids

Oblate Ellipsoids

  • Porolate

Porolate Ellipsoids Ellipsoids

ME 637

  • G. Ahmadi

Reciprocity Reciprocity Theorem Theorem

f p

2

= ρ + ∇ µ + ∇ − v = ⋅ ∇ v p

2

= ∇

( )

τ v ′ ′,

∫ ∫

⋅ ′ ⋅ = ′ ⋅ ⋅

S S

v τ dS v τ dS

( )

τ v,

Let Two Let Two Solution Solution

Then

Proof in Proof in Happel Happel and Brenner and Brenner

ME 637

  • G. Ahmadi

Minimum Energy Minimum Energy Dissipation Theorem Dissipation Theorem Helmhotz Helmhotz

Proof in Proof in Happel Happel and Brenner and Brenner

The dissipation rate in creeping flow is less The dissipation rate in creeping flow is less than any other incompressible, continuous than any other incompressible, continuous motion consistent with the same boundary motion consistent with the same boundary condition condition

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SLIDE 2

2

ME 637

  • G. Ahmadi

( )

τ δ δ + ∇ µ =

ij ij 2 i , j

T P T i

, ij =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + δ πµ =

2 j i ij ij

r r r r 8 1 T

3 i i

r 4 r P π = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + δ − πµ =

2 j i ij ij

r r r r ln 4 1 T

2 i i

r 2 r P π = Stokes Stokes Equation Equation Solution Solution-

  • 3D

3D Solution Solution-

  • 2D

2D

ME 637

  • G. Ahmadi

Minimum Energy Minimum Energy Dissipation Theorem Dissipation Theorem

Then ( ) ( ) ( ) ( ) ( ) ( )

[ ]

∫ ∫

′ ′ − − ′ − ′ τ + ′ ρ ′ − =

S j k ijk ki jk V ij i

dS v R T dV f T v r r r r r r r r r

( )

5 k j i ijk

r 4 r r r 3 R π = r

( ) ( ) ( ) ( ) ( ) ( ) ( )

[ ]

∫ ∫

′ − ′ µ + ′ − ′ τ + ′ ρ ′ − =

S i i , j j j ij V i i

dS P v 2 P dV f P p r r r r r r r r r r

ME 637

  • G. Ahmadi

In an Unbounded Flow, In an Unbounded Flow, For a point Force For a point Force

Then

( )

r δ = ρ

i i

F f

( )

r T F v

ij j i =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⋅ πµ =

2

r r 8 rr I F v

( ) ( )

r P F r p

i i

=

3

r 4 p π ⋅ = r F

Stokeslet Stokeslet

ME 637

  • G. Ahmadi

Lamb Lamb

∑

∞ =

=

n n

P p

( )

( )

φ θ + =

− −

, Y r B r A P

nm 1 n n n n n

( ) ( ) ( )

θ φ + φ = φ θ cos P m sin D m cos C , Y

m n mn mn mn

n ,..., 2 , 1 , m = ,... 2 , 1 , n =

( ) ( ) ( ) ( )( )

∑

∞ =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + µ − ∇ + + Φ ∇ + ξ × ∇ =

n n n 2 n n

3 n 2 1 n 2 P n 2 P r 3 n r v r

Spherical Harmonics Spherical Harmonics

2 2 2 2 2 2 2 2

sin 1 sin sin 1 1 ϕ θ θ θ θ θ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ∇ r r r r r r

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SLIDE 3

3

ME 637

  • G. Ahmadi

Torque Torque Force Force Stresslet Stresslet For For nonuniform nonuniform velocities velocities

( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∇ + − πµ =

∞ ∞

  • 2

2

  • |

R 6 1 | R 6 v U v F ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − × ∇ πµ =

∞

ω v T0

  • 3

| 2 1 R 8

  • 2

2 3

10 R 1 R 3 20 d S ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ + µ π =

ME 637

  • G. Ahmadi

Drag on an Drag on an Axisymmetric Axisymmetric Body Body

s n ρ z U

d I τ µ + − = 2 p

( )

[ ]

T

v v d ∇ + ∇ =

∫

⋅ =

S

τ dS F

( )

∫

⋅ × =

S

T τ dS r

ME 637

  • G. Ahmadi

Drag Force Drag Force

∫ ∫

⋅ ⋅ = ⋅ ⋅ = ⋅ =

S z S z z z

dS F e τ n e τ dS e F

Drag on an Drag on an Axisymmetric Axisymmetric Body Body

ds 2 dS πρ = ψ ρ µ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ψ ∂ ρ ∇ µ − − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ µ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ µ + − = ⋅

2 n s n n

E s 1 2 p s v n v s v n v 2 p s n s s n n τ n

∫

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ρ ψ ∂ ∂ ρ µπ = dS E n F

2 2 3 z

ME 637

  • G. Ahmadi

Fluid not at Fluid not at rest far away rest far away

Flow for a Flow for a Point Force Point Force

r 8 F

2 z ρ

πµ = ψ

3 z

r z 4 F p π − =

2 2 2

z r + ρ =

2 r z

r lim 8 F ρ ψ πµ =

∞ →

( )

2 r Z

r lim 8 F ρ ψ − ψ πµ =

∞ ∞ →

Point Point Particle Particle

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SLIDE 4

4

ME 637

  • G. Ahmadi

z U a b ξ= ξo

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ θ ξ = φ θ ξ = φ θ ξ = cos sinh c z sin sin cosh c y cos sin cosh c x ξ = λ sinh θ = ζ cos

2 2 2 2

1 1 c sin cosh c y x ζ − + λ = θ ξ = + = ρ λζ = c z

Let Let

≥ λ > ∞

1 1 − ≥ ζ ≥

ME 637

  • G. Ahmadi

E4 = ψ

= ψ

( )

ξ = ξ λ = λ = λ ∂ ψ ∂

( )

ξ = ξ λ = λ

( )( )

2 2 2 2

1 1 2 Uc U 2 1 ζ − + λ = ρ → ψ

∞ → ξ λor

Boundary Conditions Boundary Conditions

ME 637

  • G. Ahmadi

λ ∂ ∂ + λ = ξ ∂ ∂ 1

2

ζ ∂ ∂ ζ − − = θ ∂ ∂

2

1

2 2 2 2 2

z 1 E ∂ ∂ + ρ ∂ ∂ ρ − ρ ∂ ∂ =

( ) ( ) ( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ζ ∂ ∂ ζ − + λ ∂ ∂ + λ ζ + λ =

2 2 2 2 2 2 2 2 2 2

1 1 c 1 E

Noting Noting Leads To Leads To

ME 637

  • G. Ahmadi

( ) ( )

λ ζ − = ψ g 1

2

( ) ( ) [ ] ( )⎟

⎠ ⎞ ⎜ ⎝ ⎛ + λ + λ + λ − λ + λ − ζ − = ψ

−

1 C cot 1 C 2 1 C 2 1 1

2 3 1 2 2 1 2

( ) ( ) ( ) ( ) ( ) ( )

⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ λ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + λ − λ − + λ λ λ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + λ − λ − + λ λ − ρ = ψ

− − 1 2 2 2 1 2 2 2 2

cot 1 1 1 cot 1 1 1 1 U 2 1 sinhξ = λ Assumed Assumed Solution Solution

Boundary Conditions Boundary Conditions

Solution Solution

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SLIDE 5

5

ME 637

  • G. Ahmadi

( ) ( ) ( ) ( ) ( ) ( )

1 2 2 2 1 2 2 2 2

cot 1 1 1 cot 1 1 1 U 2 1 λ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + λ − λ − + λ λ λ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + λ − λ − + λ λ ρ − = ψ

− − 2 z

lim c 8 F ρ λψ πµ =

∞ → λ

( )

1 2 z

cot 1 cU 8 F λ − λ − λ πµ − =

−

aUK 6 F

Z

πµ − =

Solution for Solution for Oblate Oblate Spheroids Spheroids moving in a moving in a stationary fluid stationary fluid

Drag Drag

ME 637

  • G. Ahmadi

( ) [ ]

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ λ − λ − λ + λ =

− 1 2 2

cot 1 1 4 3 1 K

aUK 6 F

Z

πµ − =

Drag Drag

2 2

b a c − = 1 ) b / a ( 1 c b

2

  • −

= = λ

ME 637

  • G. Ahmadi

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ + + λ λ π ρ − = ψ

−1 2 3

cot 1 U a U

0 →

λ

aU 16 Fz µ − =

Drag Drag

ME 637

  • G. Ahmadi

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ θ ξ = ψ θ ξ = ψ θ ξ = cos cosh c z sin sin sinh c y cos sin sinh c x a b τ = τ0 z U

ξ = τ cosh θ = ζ cos

const = τ const = ξ

Let Let

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SLIDE 6

6

ME 637

  • G. Ahmadi

( ) ( ) ( ) ( ) ( ) ( )⎥

⎦ ⎤ ⎢ ⎣ ⎡ − τ τ − τ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − τ + τ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − τ τ − τ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − τ + τ ρ − = ψ

− −

1 coth 1 1 1 coth 1 1 U 2 1

2 1 2 2 2 1 2 2 2

( )

1 2 z

coth 1 cU 8 F τ − τ + τ πµ − =

−

2 2

b a c − =

2

  • )

a / b ( 1 1 c a cosh − = = ξ = τ

Solution Solution Drag Drag

ME 637

  • G. Ahmadi

bUK 6 F

z

πµ − =

( ) [ ]

1 1 2 2

coth 1 1 4 3 K

− −

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ τ − τ + τ − τ = 2 1 2 ln b a ln U 4 F

a Z

− + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ πµ − = Drag Drag Elongated Elongated Rode Rode

ME 637

  • G. Ahmadi
  • Navier

Navier-

  • Stokes

Stokes

  • Creeping Flow

Creeping Flow

  • Minimum Energy

Minimum Energy

  • Point Force Solution

Point Force Solution

  • General Solution

General Solution

  • Faxen

Faxen Law Law

  • Nonspherical

Nonspherical Particles Particles

  • Oblate Ellipsoids

Oblate Ellipsoids

  • Porolate

Porolate Ellipsoids Ellipsoids

ME 637

  • G. Ahmadi