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

4 Creeping Flow Equation 4 Creeping Flow Equation 4 Stream Function 4 Stream Function 4 Boundary Conditions 4 Boundary Conditions 4 Pressure Variations 4 Pressure Variations 4 Stokes Drag 4 Stokes Drag 4 Oseen 4 Oseen Drag Drag 4 4 Drag on a Droplet Drag on a Droplet ME 637 ME 637 G. Ahmadi G. Ahmadi z U ∞ = θ ϕ ⎧ x r cos cos ⎪ r = θ ϕ y r cos sin ⎨ ⎪ r = θ z r sin ⎩ θ θ y ϕ z Stream Function Stream Function x ∂ ψ ∂ ψ 1 1 = = − v v 2R θ r 2 θ ∂ θ θ ∂ r sin r sin r ME 637 ME 637 G. Ahmadi G. Ahmadi 1

Navier- -Stokes Equation Stokes Equation Navier Navier Navier- -Stokes Equation Stokes Equation 2 ( ) ⎡ ⎤ ∂ 2 θ ∂ ∂ ⎛ ⎞ sin 1 ∂ ∂ ψ ψ ( ) 2 1 E , + ψ = ⎜ ⎟ 0 ⎢ ⎥ 2 ψ + E ( ) ∂ ∂ θ θ ∂ θ 2 2 r r ⎝ sin ⎠ ∂ 2 θ ∂ θ ⎣ ⎦ t r sin r , ψ ∂ ψ ∂ ψ 2 2 E ⎛ 1 ⎞ + θ − θ = ν 4 ψ ⎜ cos sin ⎟ E ⎧ ∂ ψ ⎫ 1 = = = θ ∂ ∂ θ v 0 at r R 2 2 r sin ⎝ r r ⎠ ⎪ ⎪ r ∂ θ 2 θ r sin ⎪ ⎪ ⎪ ∂ ψ ⎪ 1 Boundary Boundary = − = = ⎨ v 0 at r R ⎬ θ θ ∂ r sin r ψ = E 4 ⎪ ⎪ 0 Conditions Conditions 1 ⎪ ⎪ Creeping Flow Creeping Flow ψ = θ → ∞ U r 2 sin 2 as r ⎪ ⎪ ∞ ⎩ 2 ⎭ ME 637 ME 637 G. Ahmadi G. Ahmadi ( ) ψ = θ A 2 f r sin ( ) Solution Solution = + + + Let Let f r Br Cr 2 Dr 4 r ⎛ ⎞ ⎛ ⎞ 2 2 d 2 d 2 ( ) ⎛ ⎞ 3 1 R 3 1 ⎜ ⎟ ⎜ ⎟ − − = Stream f r 0 Stream N N- -S S ⎜ ⎟ ψ = − + 2 2 θ ⎜ ⎟ ⎜ ⎟ Rr r U sin ⎜ ⎟ 2 2 2 2 dr r dr r ∞ ⎝ ⎠ ⎝ ⎠ 4 r 4 2 Function ⎝ ⎠ Function ( ) = m f r Ar ⎡ ⎤ 3 v 3 R 1 ⎛ R ⎞ Solution Solution = − + θ r ⎜ ⎟ ⎢ 1 ⎥ cos U 2 r 2 ⎝ r ⎠ ⎢ ⎥ ⎣ ⎦ Velocity ∞ Velocity [ ] [ ] ( )( ) ( ) − − − − − = m 2 m 3 2 m m 1 2 0 Field Field ⎡ ⎤ 3 v 3 R 1 ⎛ R ⎞ = − θ = − − − θ 1 ⎜ ⎟ sin m 1 , 1 , 2 , 4 ⎢ ⎥ U 4 r 4 ⎝ r ⎠ ⎢ ⎥ ⎣ ⎦ ∞ ME 637 ME 637 G. Ahmadi G. Ahmadi 2

⎛ ⎞ 1 R 3 3 1 Viscous Viscous ψ = ⎜ − + 2 ⎟ 2 θ Rr r U sin ⎜ ⎟ ∞ 4 r 4 2 ⎝ ⎠ Flow Flow Viscous Viscous ⎛ 3 ⎞ 1 R 3 Flow Flow ψ = ⎜ − ⎟ 2 θ Rr U sin ⎜ ⎟ ∞ 4 r 4 ⎝ ⎠ ⎛ − ⎞ 3 1 R Potential Potential ψ = 2 2 θ ⎜ ⎟ U r sin 1 ⎜ ⎟ ∞ 2 r 3 ⎝ ⎠ Flow Flow Potential Potential Flow Flow 3 1 R ψ = − 2 θ U sin ∞ 2 r ME 637 ME 637 G. Ahmadi G. Ahmadi Navier- -Stokes Equation Stokes Equation Drag Navier Drag µ µ ∂ ∂ π ( ) P 3 RU P 3 RU ∫ ∞ cos ∞ sin = − τ θ + θ π θ θ 2 D | sin P | cos 2 R sin d = θ = θ θ = = r r R r R ∂ 3 ∂ θ 2 0 r r 2 r µ = πµ + πµ = πµ 3 RU D 4 U R 2 U R 6 U R = − ∞ θ P P cos ∞ ∞ ∞ ∞ 2 2 r Shear Stress Drag Coefficient Shear Stress Drag Coefficient ∂ ∂ µ θ ⎛ ⎞ D 24 ⎛ ⎞ 3 ( ) 1 v v U sin 3 R 5 R ρ = = ⎜ ⎟ C U 2 R τ = µ ⎜ + θ ⎟ = − ∞ − + r 1 ⎜ ⎟ = ∞ D 1 Re θ r Re ∂ θ ∂ 3 ⎝ r r ⎠ r 4 r 4 r ⎝ ⎠ ρ π 2 2 U R µ ∞ 2 ME 637 ME 637 G. Ahmadi G. Ahmadi 3

∂ v Oseen’s Oseen’s Drag on a Sphere Drag on a Sphere ⋅ ∇ ≈ v U v ∞ ∂ Approximation x Approximation 24 = + 0 . 678 < ≤ × 5 C ( 1 0 . 15 Re ) 0 Re 2 10 ∂ ∂ 1 v v + = − ∇ + ν ∇ D N- -S S 2 N U P Re ∇ v ⋅ = v 0 ∞ ∂ ∂ ρ t x 2 Drag on a Drag on a − 24 ⎡ 3 9 ⎤ Drag ≈ + < < × 5 Drag C 1 10 Re 1 R 2 10 3 = + + 2 + C 1 Re Re ln Re ... ⎢ ⎥ Cylinder Cylinder D D Coefficient Re ⎣ 16 160 ⎦ Coefficient µ 2 + 1 0 π 8 µ 3 Drag on a Drag on a = C D = πµ Drag on a D 6 U R Drag on a d γ = 0 . 577216 ... ⎡ ⎤ ∞ µ ⎛ 8 ⎞ 0 − γ + Re 0 . 5 ln ⎜ ⎟ Droplet Droplet + 1 0 Cylinder ⎢ ⎥ Cylinder ⎝ Re ⎠ µ ⎣ ⎦ d ME 637 ME 637 G. Ahmadi G. Ahmadi 1000 100 Experiment Oseen CD C 10 Oseen D Stokes Eq. (5) Stokes 1 Newton 0 0 1 10 100 1000 10000 Re Re Predictions of various models for drag coefficient for a Predictions of various models for drag coefficient for a spherical particle. spherical particle. ME 637 ME 637 G. Ahmadi G. Ahmadi 4

● 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 ME 637 G. Ahmadi G. Ahmadi 5