u = t ~ t y ~ t y 0 y I.C. I.C. 1 = a = 2 a 1 / 2 - PowerPoint PPT Presentation

Outline Outline 4 4 Plate Suddenly Set in Motion Plate Suddenly Set in Motion 4 Oscillating Plate 4 Oscillating Plate 4 Unsteady Pipe Flows 4 Unsteady Pipe Flows 4 Steady Flows in Noncircular Pipes 4 Steady Flows in Noncircular Pipes 4 Elliptic Cross Section Pipes 4 Elliptic Cross Section Pipes 4 Triangular Cross 4 Triangular Cross- -Section Pipes Section Pipes ME 637 ME 637 G. Ahmadi G. Ahmadi y = u = 0 B.C. U ∂ ∂ 2 B.C. Similarity Solution u u Similarity Solution 0 = ν ∂ ∂ 2 t y 1 = ∞ Let Let a u = t ~ t y ~ t y 0 y I.C. I.C. 1 = a = 2 a 1 / 2 Navier- Navier -Stokes Stokes t = 0 u = 0 Viscous y u ( ) Similarity Similarity η = = f η Fluid x Variables Variables ν U 2 t 0 U o ME 637 ME 637 G. Ahmadi G. Ahmadi 1

1.0 0.8 = f ( 0 ) 1 ′ ′ ′ + η = f 2 f 0 NS NS B.C. B.C. t ν =4 0.6 u/Uo ∞ = f ( ) 0 t ν =1 ′ 2 = − η f ce 0.4 t ν =0.25 2 0.2 t ν =0.062 η ( ) ∫ 2 = − − η η = − η f 1 e d 1 erf 1 t ν =0.0025 1 π 0 0.0 0 0.5 1 1.5 2 2.5 3 ⎛ ⎞ y y = ⎜ ⎟ u U erfc Solution Solution ⎜ ⎟ 0 ν ⎝ 2 t ⎠ Variation of velocity profile with time. Variation of velocity profile with time. ME 637 ME 637 G. Ahmadi G. Ahmadi y = 0 ∂ B.C. = ω u ∂ ∂ ∂ 2 B.C. u U cos t ( ) u ( ( ) ( ) ) u u = − ω − ky ω − U e sin t ay = − ky − ω − + ω − U e k cos t ay a sin t ay 0 0 = ν ∂ ∂ 0 t y ∂ ∂ 2 t y = ∞ u = y Navier- -Stokes Equation Stokes Equation 0 Navier ( ( ) ) − ω θ = ν 2 − 2 θ − θ sin k a cos 2 ak sin ( ) − Let Let = ky ω − u U e cos t ay y 0 Matching Matching = ω = ν = ν 2 2 2 a k 2 ak 2 k Viscous Fluid ω x Solution Solution = k ( ) ν 2 = − ω − ky u U e cos t ky U o cos ω t 0 ME 637 ME 637 G. Ahmadi G. Ahmadi 2

µ ν r t t Navier Navier ∂ ∂ ∂ v ⎛ v ⎞ 1 dP 1 ξ = τ = = = − + ν z ⎜ r z ⎟ Stokes Stokes ρ 2 2 R R R ∂ ρ ∂ ∂ t dz r r ⎝ r ⎠ Nondimensional Nondimensional Variables Variables 1 dP ( ) = u ( R , t ) 0 = − ϕ ξ 2 v R z µ 4 dz = u ( r , 0 ) 0 z ∂ ϕ ∂ ⎛ ∂ ϕ ⎞ 1 = + ⎜ ξ ⎟ 4 Navier- Navier - Stokes Stokes ⎜ ⎟ R ∂ τ ξ ∂ ξ ∂ ξ ⎝ ⎠ ME 637 ME 637 G. Ahmadi G. Ahmadi ξ = ϕ = 1 0 ∂ ψ ∂ ⎛ ∂ ψ ⎞ 1 Boundary Boundary = ξ Navier Navier ⎜ ⎟ ⎜ ⎟ Conditions Conditions Stokes Stokes ϕ = ∂ τ ξ ∂ ξ ∂ ξ τ = 0 0 ⎝ ⎠ ξ = ψ = 1 0 Boundary Boundary φ = − ξ 2 − ψ Changing Changing 1 Conditions Conditions Variable Variable τ = ψ = − ξ 0 2 1 ME 637 ME 637 G. Ahmadi G. Ahmadi 3

Separation of Variables Separation of Variables ( ) ( ) → ∞ ⇒ = Y 0 0 F 0 ~ finite B 0 Boundary Boundary & ⎛ ⎞ T 1 d dF ( ) ( ) ψ = ξ τ ⎜ ⎟ ( ) ( ) F T = ξ = − α 2 Conditions Conditions = ⇒ α = ⎜ ⎟ F 1 0 J 0 ξ ξ ξ T F d d ⎝ ⎠ 0 & + α = 2 T T 0 2 = − α τ T Ce Eigenvalues Eigenvalues α 1 = α 2 = α 3 = 2 . 405 5 . 52 8 . 654 Bessel Equation Bessel Equation Bessel Functions Bessel Functions ( ) ∑ General 2 General 2 d F dF − α τ ψ = α ξ A e J ( ) ( ) n ξ 2 + ξ + α 2 ξ 2 = F 0 = αξ + αξ F AJ BY n 0 n Solution Solution ξ ξ 2 d 0 0 d n ME 637 ME 637 G. Ahmadi G. Ahmadi Solution Solution ( ) ∑ Initial Initial − ξ 2 = α ξ 1 A J n 0 n Condition Condition ( ) n α ξ J ∑ ϕ = − ξ − − α τ 2 1 8 0 n e n ( ) ( ) α α ( ) 3 1 J ∫ − ξ 2 ξ α ξ ξ 1 J d 8 n 0 n n 1 n = = A 0 ( ) n ( ) 1 α 3 2 α ∫ J ξ 2 α ξ ξ J d n 1 n 0 n 0 ( ) 2 − α τ α ξ e J n ∑ τ ψ = Solution Solution 0 n 8 ( ) α α 3 J n n 1 n ME 637 ME 637 G. Ahmadi G. Ahmadi 4

Ellipse Ellipse 1 dP Navier Navier ∇ = = 2 W const 2 2 ⎛ ⎞ 2 2 ⎛ ⎞ ⎛ ⎞ x y x y Stokes Stokes µ = ⎜ − − ⎟ dz + = w A 1 ⎜ ⎟ ⎜ ⎟ 1 Let Let ⎜ ⎟ 2 2 a b ⎝ a ⎠ ⎝ b ⎠ ⎝ ⎠ y W = 0 S x z a b ME 637 ME 637 G. Ahmadi G. Ahmadi ) ( )( ) ( ) ( ) ( + = − − + + + = ⎛ ⎞ 2 2 f x , y x a x 3 y 2 a x 3 y 2 a 0 2 2 2 A a b 1 dP NS NS ∇ 2 = − + = − = w A ⎜ ⎟ 2 2 2 2 µ ⎝ a b ⎠ a b dz y ( ) − + = x 3 y 2 a 0 ⎛ ⎞ 2 2 2 2 1 dP a b x y ⎜ ⎟ = − − − A 1 ⎜ ⎟ a µ + 2 dz 2 2 2 2 a b a b ⎝ ⎠ x Flow Rate Flow Rate ( ) + + = x 3 y 2 a 0 π dP a 3 b 3 = − ∫∫ Q = Q wdxdy µ 2 + 2 4 dz a b 2a ME 637 ME 637 G. Ahmadi G. Ahmadi 5

( ) ● ● Plate Suddenly Set in Motion Plate Suddenly Set in Motion w = Af x , y Let Let ● Oscillating Plate ● Oscillating Plate 1 dP ( ) ∇ 2 = ∇ 2 = = NS w A f x , y 12 aA NS ● Unsteady Pipe Flows ● Unsteady Pipe Flows µ dz ● Steady Flows in Noncircular ● Steady Flows in Noncircular 1 dP = A Pipes Pipes µ Solution Solution 12 a dx ● ● Elliptic Cross Section Pipes Elliptic Cross Section Pipes ) ( )( ) 1 dP ( = − − + + + w x a x 3 y 2 a x 3 y 2 a ● Triangular Cross- -Section Pipes Section Pipes µ ● Triangular Cross 12 a dx ME 637 ME 637 G. Ahmadi G. Ahmadi ME 637 G. Ahmadi 6