Fluid flow in curved geometries: Mathematical Modeling and Applications

Transcription

1 Fluid flow in cuved geometies: Mathematical Modeling and Applications D. Muhammad Sajid Theoetical Plasma Physics Division PINSTECH, P.O. Niloe, PAEC, Islamabad Mach 01-06, 010 Islamabad, Paistan

2 Pesentation Layout Analysis of fluid behaviou Govening equations Manifold and Metic opeations Stetching a cuved suface Peistaltic flow in a cuved channel Summay

3 Analysis of fluid behaviou When a constant shea foce is applied: Solid defoms o bends Fluid continuously defoms.

4 Analysis of fluid behaviou Analysis of any poblem in fluid mechanics necessaily includes statement of the basic laws govening the fluid motion. The basic laws, which applicable to any fluid, ae: Consevation of mass Newton s second law of motion The pinciple of angula momentum The fist law of themodynamics The second law of themodynamics

5 Analysis of fluid behaviou NOT all basic laws ae equied to solve any one poblem. On the othe hand, in many poblems it is necessay to bing into the analysis additional elations that descibe the behavio of physical popeties of fluids unde given conditions. Many appaently simple poblems in fluid mechanics that cannot be solved analytically. In such cases we must esot to moe complicated numeical solutions and/o esults of expeimental tests.

6 Govening Equations The flow of a fluid is mainly govened by the laws of consevation of mass and momentum Continuity ρ t ( ρv) = 0, Continuity fo incompessible flow V 0. Equations of motion V ρ t ( V ) V = divσ ρb,

7 Manifold and metic The mathematical model of space is a pai: ( M, g) Diffeentiable Manifold Metic We need to eview these two fundamental concepts

8 Manifold and metic What is a manifold? A manifold is a geometic thing which has open chats, subsets whee a flat set of coodinates is given. In geneal, howeve, they can be built by patching togethe on atlas of open chats. We need to eview open chats

9 Open chats: Manifold and metic The same point is contained in moe than one open chat. Its desciption in both chats is elated by a coodinate tansfomation

10 Manifold and metic Diffeentiable stuctue

11 Manifold and metic Diffeentiable stuctue

12 Manifold Manifold and metic Paallel tanspot A vecto field is paallel tanspoted along a cuve, when it mantains a constant angle with the tangent vecto to the cuve

13 Manifold and metic The metic: a ule to calculate the lenght of cuves!! Connection and covaiant deivative A connection is a map : TM TM TM Fom the poduct of the tangent bundle with itself to the tangent bundle

14 In a basis... Covaiant deivative This defines the covaiant deivative of a (contovaiant) vecto field Fluid Mechanics Goup (FMG) 14

15 Covaiant deivative Covaiant deivative of a geneal tenso Vecto (fist ode tenso, s = 1) i i A i n Ai A, m = Γnm A ; A m i,m = Γ m ξ ξ Second-ode tenso ( s = ) n im A n ij ij T i nj j T,m = ΓnmT Γ T m nm ξ Tij n n Tij,m = ΓimTnj Γ jmt m in ξ i i T j i n n T j,m = ΓnmT j jmt m Γ ξ in i n etc. =, s = 0 = 0, s = = 1, s = 1

16 Fo the velocity field opeations V = i j V ( u ) Gadient of the velocity field is e i L i i = V = V ; j = g g jm m, [( ) ] mm m ll g V Γ g V l l Divegence of the velocity field is i V = V ; i =, ( ) ii i mm g Vi i Γmi g Vm Whee ; epesent the covaiant deivative

17 Divegence of a tenso T = ( ) [( ) ] ii jj i mm jj j ii mm V T = g g g V g g T, Γ g g T Γ g g T ii jj g ii ( ) ii jj g g T Γ j mj (V ) of a vecto ij g ii ij g, j mm m Γ T im ( ) ( ) ii V = g g V g V V, (V ) of a tenso opeations i i mj g mm mj g jj m T mj [ ] mm i Γ g V mi m im

18 Stetching a cuved suface Geomety of the poblem Fig. 1 (a) Fig. 1 (b), v s, u R O (a) a flat stetching sheet, (b) a cuved stetching sheet.

19 Stetching a cuved suface Mathematical Fomulation ), ( s R, ) ( 1 1 = R u u R u s p R R R uv s u R Ru u v ν ρ, 1 p R u = ρ ( ) = R g ab Cuvilinea coodinates metic { }, 0 ) ( = s u R v R continuity equation components of equation of motion

20 Stetching a cuved suface The bounday conditions applicable to the pesent flow ae Using the similaity tansfomation The poblem taes the fom. as 0 0, 0, at 0, = = = u u v as u ( ) ( ). ), (,, a P s a p f a R R v asf u ν η η ρ η ν η = = = = f P = η η, ) ( ) ( f f ff f f f f P = η η η η η η, as 0 0, 0, at 1 0, = = = η η f f f f

21 Stetching a cuved suface Eliminating pessue f iv f η f f ( f f ff ) ( f ff ) ff = ( η ) ( η ) η ( η ) ( η ) The sin fiction coefficient f (0) 1 Re 1 / x C f = f (0) = f (0) It is impotant to point out that the Cane s poblems can be ecoveed by taing. The numeical solution of the poblem is developed by a shooting method using a Runge-Kutta algoithm.

22 Stetching a cuved suface = 1000 ; = 0 ; = 10 ; = f 'HhL 0.4 = 1000 ; = 0 ; = 10 ; = 5 f HhL h h -component of velocity, s-component of velocity

23 Stetching a cuved suface PHhL = 1000 ; = 0 ; = 10 ; = Dimensionless pessue h 1/ C f Re x Sin fiction coefficient

24 Peistaltic flow in a cuved channel Geomety of the poblem π ( ) = ( ) H X, t a bsin X ct, Uppe wall λ π ( ) ( ) H X, t = a bsin X ct. Lowe wall λ

25 Peistaltic flow in a cuved channel Mathematical Fomulation Govening equations in fixed fame: R {( ) } R R V R U * * = X * V V R U V U p 1 V V = R R * * * * ν ( ) t R R R X R R R R R R R * * R V V R U, * R R * * X ( R R) ( R R) X * * U U R U U UV R p 1 * U V ν = * * * * ( R R ) t R R R X R R R R X R R R R * * R U U R V. * R R * * X ( R R ) ( R R ) X 0,

26 Peistaltic flow in a cuved channel Govening equations in wave fame: {( ) } R v R ( ) ( ) * R u c u c u * * = v v v p 1 * v c v = ν * * * ( R ) x R x R R * * R v v R u, * R * * x ( R ) ( R ) x ( ) ( ) * * u u R u c u u c v R p 1 * u c v = ν * * * * ( R ) x R R R x R R R R x R * * R u u R v. * v R * * x ( R ) ( R ) x x 0,

27 Peistaltic flow in a cuved channel Non-dimensional vaiables: πx u v ρca x=, η =, u =, v=, Re =, λ a c c µ * π a H R P = p, h=, =, λµ c a a Steam function: ψ ψ u =, v= δ whee δ = η η x Unde the long wavelength appoximation we have P = 0, η P 1 ψ 1 ψ ( η ) 1 0. = x η η ( η ) η π a. λ

28 Peistaltic flow in a cuved channel Eliminating pessue 1 = 0. ( ) 1 ψ 1 ψ ( η ) η η η η η q ψ ψ =, = 1 at η = h= 1 φsin x, η q ψ ψ =, = 1 at η = h= 1 φsin x, η Solution of the poblem C ( ) ( ( ) ) ( ) ( ) C 1 η η ψ = η ln η 1 C C ln ( η ) C C η η C3 u = ( η ) { ( η ) } C η 1 1 ln 1. ( ) 8h h q = 4h h ln h ln h h ln h ln h ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1

29 Peistaltic flow in a cuved channel dp = C C ( ) ( ) ( ) ( ) ( ) ( h q) h ( h) ( h) ( h) ( h) ln ln = 4h ( h ) ln( h) ln( h) h ln h ln h h ( h ) ( h q) ln h = 4h h ln h ln h h ln h ln h ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 h ( h h h q q) ( h ) 3 h ln h ( h q ( ( h ) q) ln ( h) ( h q) ln ( h) ) C4 = ( 4h ( h ) ( ln ( h) ln ( h) ) ( h ) ln ( h) ln ( h) ) The axial pessue gadient tuned out to be ( ) 8h h q ( ) ( ) ( ) ( ) ( ) ( ) ( ) dx 4 h h ln h ln h h ln h ln h The dimensionless pessue ise ove one wavelength is defined by P λ π = 0 dp dx dx.

30 Peistaltic flow in a cuved channel

31 Peistaltic flow in a cuved channel = 3.5 = 5 =10

32 Summay The opeations fo cuvilinea coodinates have been discussed. Flow of a viscous fluid due to Stetching a cuved suface is analyzed. Peistaltic flow in a cuved channel is investigated.

33 Than You!