Consider that special case of a viscous fluid near a wall that is set suddenly in motion as shown in Figure 1. The unsteady Navier-Stokes reduces to

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1 Exact Solutios to the Navier-Stokes Equatio Ustead Parallel Flows (Plate Suddel Set i Motio) Cosider that special case of a viscous fluid ear a wall that is set suddel i motio as show i Figure. The ustead Navier-Stokes reduces to t u ν () Viscous Fluid x U o Figure. Schematics of flow ear a wall suddel set i motio. The boudar coditios are: At u U () at, u (3) The crespodig iitial coditio f the fluid that starts from rest is give as at t u. (4) Similarit Solutio (Group The) Let t ~ t, a ~ t, (5) Equatio () implies that a, a, (6)

2 Thus, ~ t (7) Now itroducig the similarit variables we fid νt u U η, f ( η), (8) η νt, u u η 4νt (9) t η νt t η. () η t Substitutig (9) ad () i Equatio (), we fid η f νf, () t 4νt f + ηf () Boudar ad iitial coditios ()-(4) i terms of the similarit variables become f (), f ( ). (3) From Equatio (), it follows that f η, l f l c η (4) f η f ce, ad f c η η e dη +, (5)

3 where the first boudar coditio i (3) is used. The secod boudar coditio implies that f ( ) η + c e dη c (6) η e dη π Equatio (5) the becomes f η ( η) π η η e d erf (7) f erfcη, u U erfc (8) νt Time variatios of the velocit profile as predicted b Equatio (8) are show i Figure...8 tν4 u/uo.6.4. tν tν.5 tν.6 tν.5. Trasfm Method Figure. Time variatios of velocit profile. 3

4 A alterative is to use the trasfm method. Takig Laplace trasfm of Equatio (), it follows that u su ν (9) s u u ν () The solutio to () is s ν s ν u Ae + Be () Boudar coditios () ad (3) impl that U A, B () s Thus, the solutio i the trasfm domai is give b s ν U u e (3) s Iverse Laplace trasfm of (3) gives u U erfc. (4) νt Oscillatig Plate Cosider that case of a viscous fluid ear a oscillatig wall as show i Figure 3. The ustead Navier-Stokes reduces to t u ν (5) 4

5 Viscous Fluid U o cosωt x Figure. Schematics of flow ear a oscillatig wall. The boudar coditios are: u U cosωt at (6) u at (7) Let k u U e cos( ωt a). (8) The t k ωu e si( ωt a) (9) u k U e ( k cos( ωt a) + a si( ωt a )) (3) ( k cosθ ka si θ a cosθ), θ ωt a (3) u k U e Substitutig (9)-(3) ito Equatio (5) it follows that ω si θ ν ( k a ) cosθ ak si θ) (3) 5

6 a k (33) ω akν k ν (34) ω k a (35) ν Thus, the velocit profile is give as ( ωt k) k u U e cos, ω k. (36) ν Ustead Flow i a Tube Cosider flow i a circular tube subject to a step chage i pressure as show i Figure 4. The Navier-Stokes equatio reduces to v t z ρ dp dz v + ν r r r r z (37) Itroducig dimesioless variables, we fid r µ t νt ξ, R ρr R dp v z, (38) 4µ dz τ, R ϕ( ξ) ϕ ϕ 4 + ξ. (39) τ ξ ξ ξ R z Figure 4. Schematics of flow i a tube subject to step chage i pressure. 6

7 The boudar coditio is ϕ at ξ, (4) with the iitial coditios ϕ at τ. (4) Let φ ξ Equatio (39) reduces to ψ, (4) ψ τ ξ ξ ψ ξ ξ (43) The boudar ad iitial coditios (4) ad (4) ow become At ξ, ψ. (44) At τ, ψ ξ. (45) To fid the solutio the method of separatio of variable is used. That is let ψ F ( ξ) T( τ) (46) Equatio (43) the becomes T& T d df ξ α Fξ dξ dξ. (47) From Equatio (47), it follows that T& + α T, (48) 7

8 d F df ξ + ξ + α ξ F. (49) dξ dξ The solutios to Equatios (48) ad (49) are give as T Ce α τ (5) F AJ ( αξ) + BY (αξ), (5) ) (αξ) where J (αξ ad Y are Bessel fuctio of first ad secod kid of zeroth der. The boudar coditios are ad ( ) F ~ fiite B sice Y ( ). (5) () J ( α). (53) F Equatio (53) is a characteristic equatio. The crespodig eigevalues, α, are give as α.45, α 5.5, α , (54) The geeral solutio f Equatio (43) the is give b ψ A e τ ( α ξ) α J (55) Usig the iitial coditio the ξ A J ( α ξ ) (55) A ( ξ ) ξj ( α ξ) ξj ( α ξ) dξ dξ 4J ( α )/ α ( α ).5J 3 (56) A 8 α J. (57) 3 ( α ) Hece, 8

9 ψ 8 e α τ α 3 J J ( α ξ) ( α ), (58) ad ( α ξ) J ( α ) J ϕ ξ 8 e (59) ατ 3 α Variatio of the velocit profile i the pipe is show schematicall i Figure 5. τ Figure 5. Variatios of velocit field i a tube subject to a step chage i pressure. Nocircular Pipe Flows Cosider stead state viscous flows i a pipe with arbitrar cross sectio uder a costat pressure gradiet as show i Figure 6. The Navier-Stokes equatio is give as dp W cost. (6) µ dz The crespodig boudar coditio is W o S. (6) 9

10 S Figure 6. A arbitrar cross-sectio pipe subject to a costat pressure gradiet. Elliptical Pipes Cosider a elliptical cross-sectio pipe show i Figure 7 with its boudar give as x a + b. (6) We assume that the velocit field is give b x w A. (63) a b x b a z Figure 7. Elliptical cross-sectio pipe subject to a costat pressure gradiet.

11 ( + b ) A a dp w A + (64) a b a b µ dz Hece dp a b x A (65) µ dz a + b a b The flow rate is give as Q wdxd. (66) After itegratio, it follows that 3 3 π dp a b Q. (67) 4µ dz a + b Triagular Pipes Cosider a pipe as show i Figure 8 whose cross sectio is a equilateral triagle. The equatio of the sectio is give as ( x,) ( x a)( x 3 + a)( x a) f. (68) Assumig w Af ( x, ) (69) The dp w A f ( x, ) aa. (7) µ dz

12 x 3 + a a x x a a Figure 8. A triagular pipe subject to a costat pressure gradiet. Thus, dp A (7) µ a dx Hece, dp w + (7) µ a dx ( x a)( x 3 + a)( x + 3 a)