4 Creeping Flow Equation 4 Creeping Flow Equation 4 Stream Function - - PowerPoint PPT Presentation

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ME 637

• G. Ahmadi

ME 637

• G. Ahmadi

4 4Creeping Flow Equation Creeping Flow Equation 4 4Stream Function Stream Function 4 4Boundary Conditions Boundary Conditions 4 4Pressure Variations Pressure Variations 4 4Stokes Drag Stokes Drag 4 4Oseen Oseen Drag Drag 4 4Drag on a Droplet Drag on a Droplet

ME 637

• G. Ahmadi

2R

θ

r U∞ z

ME 637

• G. Ahmadi

⎪ ⎩ ⎪ ⎨ ⎧ θ = ϕ θ = ϕ θ = sin r z sin cos r y cos cos r x

r θ ϕ x y z

θ ∂ ψ ∂ θ = sin r 1 v

2 r

r sin r 1 v ∂ ψ ∂ θ − =

θ

Stream Function Stream Function

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ME 637

• G. Ahmadi

( ) ( )

( )

ψ ν = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ θ θ ∂ ψ ∂ − θ ∂ ψ ∂ θ ψ + θ ∂ ψ ψ ∂ θ + ψ ∂ ∂

4 2 2 2 2 2 2

E sin r 1 cos r sin r E 2 , r , E sin r 1 E t

Navier Navier-

• Stokes Equation

Stokes Equation

E4 = ψ

Creeping Flow Creeping Flow

ME 637

• G. Ahmadi

sin 1 r sin r

2 2 2 2

= ψ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ θ ∂ ∂ θ θ ∂ ∂ θ + ∂ ∂

⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ∞ → θ = ψ = = ∂ ψ ∂ θ − = = = θ ∂ ψ ∂ θ =

∞ θ

r as sin r U 2 1 R r at r sin r 1 v R r at sin r 1 v

2 2 2 r

Boundary Boundary Conditions Conditions Navier Navier-

• Stokes Equation

Stokes Equation

ME 637

• G. Ahmadi

N N-

• S

S

( )

θ = ψ

2

sin r f

Let Let

( )

r f r 2 dr d r 2 dr d

2 2 2 2 2 2

= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −

( )

m

Ar r f =

( )( ) [ ] ( ) [ ]

2 1 m m 2 3 m 2 m = − − − − −

4 , 2 , 1 , 1 m − =

Solution Solution

ME 637

• G. Ahmadi

Solution Solution

( )

4 2

Dr Cr Br r A r f + + + = θ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ψ

∞ 2 2 3

sin U r 2 1 Rr 4 3 r R 4 1

θ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − =

∞

cos r R 2 1 r R 2 3 1 U v

3 r

θ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =

∞ θ

sin r R 4 1 r R 4 3 1 U v

3

Stream Stream Function Function Velocity Velocity Field Field

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Potential Potential Flow Flow

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − θ = ψ

∞ 3 3 2 2

r R 1 sin r U 2 1

Viscous Viscous Flow Flow

θ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ψ

∞ 2 2 3

sin U r 2 1 Rr 4 3 r R 4 1

ME 637

• G. Ahmadi

Potential Potential Flow Flow

θ − = ψ

∞ 2 3

sin U r R 2 1

Viscous Viscous Flow Flow

θ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ψ

∞ 2 3

sin U Rr 4 3 r R 4 1

ME 637

• G. Ahmadi

Shear Stress Shear Stress Navier Navier-

• Stokes Equation

Stokes Equation

θ µ = ∂ ∂

∞ cos

r RU 3 r P

3

θ µ = θ ∂ ∂

∞ sin

r 2 RU 3 P

2

θ µ − =

∞ ∞

cos r 2 RU 3 P P

2

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − θ µ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + θ ∂ ∂ µ = τ

∞ θ θ 3 3 r r

r 4 R 5 r 4 R 3 1 r sin U r v v r 1

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Drag Coefficient Drag Coefficient Drag Drag

( )

∫

π = = θ

θ θ π θ + θ τ − =

2 R r R r r

d sin R 2 cos | P sin | D

R U 6 R U 2 R U 4 D

∞ ∞ ∞

πµ = πµ + πµ =

Re 24 R U 2 1 D C

2 2 D

= π ρ =

∞

( )

µ ρ =

∞

R 2 U Re

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ME 637

• G. Ahmadi

Drag Drag Coefficient Coefficient Oseen’s Oseen’s Approximation Approximation

x v U v ∂ ∂ ≈ ∇ ⋅

∞

v v v v

2

P 1 x U t ∇ ν + ∇ ρ − = ∂ ∂ + ∂ ∂

∞

= ⋅ ∇ v ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + = ... Re ln Re 160 9 Re 16 3 1 Re 24 C

2 D

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + γ − π = Re 8 ln 5 . Re 8 CD

... 577216 . = γ Drag on a Drag on a Cylinder Cylinder

N N-

• S

S

ME 637

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Drag on a Drag on a Droplet Droplet

5 678 . D

10 2 Re ) Re 15 . 1 ( Re 24 C × ≤ < + =

5 3 2 D

10 2 R 1 Re 10 1 C × < < + ≈

−

d d

1 3 2 1 R U 6 D µ µ + µ µ + πµ =

∞

Drag on a Drag on a Cylinder Cylinder Drag on a Sphere Drag on a Sphere

ME 637

• G. Ahmadi

D

C

Re

Stokes Oseen

ME 637

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1 10 100 1000

CD

1 10 100 1000 10000

Re

Stokes

• Eq. (5)

Oseen Newton Experiment Predictions of various models for drag coefficient for a Predictions of various models for drag coefficient for a spherical particle. spherical particle.

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ME 637

• G. Ahmadi
• Creeping Flows

Creeping Flows

• Stream Function

Stream Function

• Pressure Variations

Pressure Variations

• Stokes Drag

Stokes Drag

• Oseen

Oseen Drag Drag

• Drag on a Droplet

Drag on a Droplet

ME 637

• G. Ahmadi