4/5/018 Diffusion & Viscosiy: Navier-Sokes Equaion 1
4/5/018 Diffusion Equaion Imagine a quaniy C(x,) represening a local propery in a fluid, eg. - hermal energy densiy - concenraion of a polluan - densiy of phoons propagaing diffusively hrough a scaering medium For a fluid a res, V=0, he diffusive ranspor of he quaniy C in he fluid is described by he Diffusion Equaion, C DC In his expression, D is he diffusion coefficien, D v 3 wih v σ he velociy of he diffusing paricles, and λ he mean free pah.
4/5/018 Navier-Sokes Equaion Viscous Force In general, he viscous force f visc includes differen aspecs, ha of -shearviscosiy η - bulk viscosiy ζ enailing he following full viscous force visc 1 f v( ) v 3 which for incompressible flow, f visc v 0, is resriced o v 3
4/5/018 Navier-Sokes Equaion For a fluid wih (shear) viscosiy η, he equaion of moion is called he Navier-Sokes equaion. In is mos basic form, ie.for incompressible media v v v p v Wihou any discussion, his is THE mos imporan equaion of hydrodynamics. While he Euler equaion did sill allow he descripion of many analyically racable problems, he nonlinear viscosiy erm in he Navier-Sokes equaion makes he solving of he NS equaion very complicaed. There are only a few siuaions ha allow analyical soluions for he NS equaion, he remainder needs o be solved numerically/compuaionally. Navier-Sokes Equaion The general and full Navier Sokes equaion, for a fluid wih -shearviscosiyη - bulk viscosiy ζ is given by v 1 v v p v ( ) ( v) 3 4
4/5/018 Reynolds Number The Reynolds number is he measure of he imporance of viscous effecs of a flow hereby assumming he bulk viscosiy ζ=0 and is defined as he raio of he magniude of he inerial force - magniude of he viscous force Re magniude inerial force ( v ) v magniude viscous force v For large Reynolds number, he flow ges unsable, and finally becomes urbulen. Reynolds Number The Reynolds number is he raio of he magniude of he inerial force o he magniude of he viscous force Re magniude inerial force ( v ) v magniude viscous force v We can find an order of magniude rough esimae for he Reynolds number. Wih U he characerisic magniude of he velociy in a sysem of characerisic size L, we have U ( v ) v L U L v Re UL 5
4/5/018 Navier-Sokes Equaion: analyical soln s Due o he high level of nonlineariy and complexiy of he full compressible Navier-Sokes equaions, here are hardly any analyical soluions known of he Navier-Sokes equaion. v v v p v One may ry o find some specific configuraions ha would allow an analyical reamen. This involves simplifying he equaions by making he following assumpions: - abou he fluid - abou he flow - geomery of he problem Typical assumpions are: - laminar flow - -D configuraion - seady flow - flow beween plaes - incompressible flow Examples are: - parallel flow in a channel - Couee flow - Hagen-Poiseuille flow, ie. flow in a cylindrical pipe. Consider he following configuraion: - flow of a fluid hrough a channel -seady flow - incompressible flow - axisymmeric geomery (-D problem) v v v p v - he -D flow field is represened by a -D velociy field, wih u he componen in he x-direcion, v in he y-direcion u v v 6
4/5/018 - he -D flow field is represened by a -D velociy field, wih u he componen in he x-direcion, v in he y-direcion - he flow of he sysem is hen described by he (a) coninuiy equaion (b) Navier-Sokes equaion v v v p v - which for he sysem a hand simplify o: u v coninuiy equaion: 0 x y (noice: incompressibiliy) u u 1 p u u x-momenum (NS): u v x y x x y v v 1 p v v y-momenum (NS): u v x y y x y - Boundary condiion: he flow is consrained by fla parallel walls of he channel, v v0 y v v v v 0 y x y x - Coninuiy equaion: u v u 0; 0 x y x - Using hese relaions, we end up wih he Navier-Sokes equaions: 1 p u 0 x y 1 p 0 y 7
4/5/018 - Given ha u 0 x we immediaely infer ha u(x,y) mus be independen of x. Hence u y can only be a funcion of y, i.e u(x,y)=u(y). This implies, via he relaion, 1 p u p dp 0 ha, cs. x y x dx and ha he general soluion for u(y) is given by 1 1 p x uy ( ) y AyB - The general soluion for u(y) is given by 1 1 p x uy ( ) y AyB - Using he boundary condiions ha he velociy u=0 a he border of he channel, ie. u(±r)=0, he consans A and B ge fixed 1 R dp A0; B dx which yields he complee soluion for he flow velociy u(y) hrough he channel: 1 R dp y uy ( ) 1 dx R 8
4/5/018-1 R dp y uy ( ) 1 dx R - Flow hrough a channel hus displays a parabolic velociy disribuion, summeric abou he cenral axis. The maximum velociy u max is aained along he cenral axis, u max 1 R dp dx 9