Conservation Equations of Fluid Dynamics

1 Conservation Equations of Fluid DynamicsA. SalihDepartment of Aerospace EngineeringIndian Institute of Space Science and Technology, Thiruvananthapuram February 2011 This is a summary of Conservation Equations (continuity, Navier Stokes, and energy) that governthe flow of a Newtonian Fluid . Equations in various forms, including vector, indicial, Cartesiancoordinates, and cylindrical coordinates are provided. The nomenclature is listed at the Equations in vector form Compressible flow: t+ ( V) =0(1) DVDt= g p 23 ( V)+ [ ( V+( V)T)](2) cpDTDt= qg+ (k T) + TDpDt+ (3)where the viscous dissipation rate is = : V=( 23 VI+ [ V+( V)T]): VThe foregoing Equations (1),(2), and(3)represent the continuity, Navier Stokes, and energyrespectively.

2 Incompressible flow with constant Fluid properties: V=0(4) DVDt= g p+ 2V(5) cpDTDt= qg+k 2T+ (6)where the viscous dissipation rate is = [ V+( V)T]: VThe foregoing Equations (4),(5), and(6)represent the continuity, Navier Stokes, and Equations in indicial form Compressible flow: t+ ( vi) xi=0(7) ( vi t+vj vi xj)= gi p xi 23 xi( vj xj)+ xj[ ( vi xj+ vj xi)](8) cp( T t+vi T xi)= qg+ xi(k T xi)+ T( p t+vi p xi)+ (9)where the viscous dissipation rate is = i j vi xj=[ 23 vk xk i j+ ( vi xj+ vj xi)] vi xjThe foregoing Equations (7),(8), and(9)represent the continuity, Navier Stokes, and energyrespectively.

3 Incompressible flow with constant Fluid properties: vi xi=0(10) ( vi t+vj vi xj)= gi p xi+ 2vi x2j(11) cp( T t+vi T xi)= qg+k 2T x2i+ (12)where the viscous dissipation rate is = ( vi xj+ vj xi) vi xjThe foregoing Equations (10),(11), and(12)represent the continuity, Navier Stokes, and Equations in Cartesian coordinates Compressible flow: t+ ( u) x+ ( v) y+ ( w) z=0(13)2 ( u t+u u x+v u y+w u z)= gx p x+ x[ ( 23 V+2 u x)]+ y[ ( u y+ v x)]+ z[ ( u z+ w x)] ( v t+u v x+v v y+w v z)= gy p y+ y[ ( 23 V+2 v y)]+ z[ ( v z+ w y)]+ x[ ( v x+ u y)] ( w t+u w x+v w y+w w z)= gz p z+ z[ ( 23 V+2 w z)]+ x[ ( w x+ u z)]+ y[ ( w y+ v z)](14) cp( T t+u T x+v T y+w T z)= qg+ x(k T x)+ y(k T y)+ z(k T z)+ T( p t+u p x+v p y+w p z)+ (15)where the viscous dissipation rate is =2 [( u x)2+( v y)2+( w z)2]+ [( u y+ v x)2+( v z+ w y)]

4 2+( w x+ u z)2] 23 ( u x+ v y+ w z)2 The foregoing Equations (13),(14), and(15)represent the continuity, Navier Stokes, and energyrespectively. Incompressible flow with constant Fluid properties: u x+ v y+ w z=0(16) ( u t+u u x+v u y+w u z)= gx p x+ ( 2u x2+ 2u y2+ 2u z2) ( v t+u v x+v v y+w v z)= gy p y+ ( 2v x2+ 2v y2+ 2v z2) ( w t+u w x+v w y+w w z)= gz p z+ ( 2w x2+ 2w y2+ 2w z2)(17) cp( T t+u T x+v T y+w T z)= qg+k( 2T x2+ 2T y2+ 2T z2)+ (18)3where the viscous dissipation rate is =2 [( u x)2+( v y)2+( w z)2]+ [( u y+ v x)2+( v z+ w y)2+( w x+ u z)2]The foregoing Equations (16),(17), and(18)

5 Represent the continuity, Navier Stokes, and Equations in cylindrical coordinates Compressible flow: t+1r ( rur) r+1r ( u ) + ( uz) z=0(19) ( ur t+ur ur r+u r ur +uz ur z u2 r)= gr p r+ r[ ( 23 V+2 ur r)]+1r [ (1r ur + u r)]+ z[ ( ur z+ uz r)]+2 r( ur r 1r u urr) ( u t+ur u r+u r u +uz u z+uru r)= g 1r p +1r [ ( 23 V+2r u +2urr)]+ z[ ( u z+1r uz )]+ r[ ( u r u r+1r ur )]+2 r(1r ur + u r u r) ( uz t+ur uz r+u r uz +uz uz z)= gz p z+ z[ ( 23 V+2 uz z)]+ r[ ( uz r+ uz z)]+1r [ (1r uz + u z)]+ r( ur z+ uz r)(20) cp( T t+ur T r+u r T +uz T z)

6 = qg+1r r(kr T r)+1r (kr T )+ z(k T z)+ T( p t+ur p r+u r p +uz p z)+ (21)4where the viscous dissipation rate is =2 [( ur r)2+(1r u +urr)2+( uz z)2]+ [(1r ur + u r u r)2+( u z+1r uz )2+( uz r+ ur z)2] 23 (1r (rur) r+1r u + uz z)2 The foregoing Equations (19),(20), and(21)represent the continuity, Navier Stokes, and energyrespectively. Incompressible flow with constant Fluid properties:1r (rur) r+1r u + uz z=0(22) ( ur t+ur ur r+u r ur +uz ur z u2 r)= gr p r+ [1r r(r ur r)+1r2 2ur 2+ 2ur z2 2r2 u urr2] ( u t+ur u r+u r u +uz u z+uru r)= g 1r p + [1r r(r u r)+1r2 2u 2+ 2u z2+2r2 ur u r2] ( uz t+ur uz r+u r uz +uz uz z)= gz p z+ [1r r(r uz r)+1r2 2uz 2+ 2uz z2](23) T t+ur T r+u r T +uz T z= qgcp+ [1r r(r T r)+1r2 2T 2+ 2T z2]+ cp(24)where the viscous dissipation rate is =2 [( ur r)2+(1r u +urr)2+( uz z)2]+ [(1r ur + u r u r)]

7 2+( u z+1r uz )2+( uz r+ ur z)2]The foregoing Equations (22),(23), and(24)represent the continuity, Navier Stokes, and Navier Stokes Equations in stress formIt is sometimes convenient to write the Navier Stokes Equations in terms of stresses. Below wegive the stress form of the Navier Stokes Equations in both Cartesian and cylindrical coordinates. Cartesian coordinates: DuDt= gx p x+ xx x+ yx y+ zx z DvDt= gy p y+ xy x+ yy y+ zy z DwDt= gz p z+ xz x+ yz y+ zz z(25)where the deviatoric stress components are given by Stokes law of viscosity xx= 23 V+2 u x yy= 23 V+2 v y zz= 23 V+2 w z xy= yx= ( u y+ v x) yz= zy= ( v z+ w y) zx= xz= ( w x+ u z) Cylindrical coordinates.

8 (DurDt u2 r)= gr p r+1r (r rr) r+1r r + zr z r (Du Dt+uru r)= g 1r p +1r2 (r2 r ) r+1r + z z DuzDt= gz p z+1r (r rz) r+1r z + zz z(26)where the deviatoric stress components are given by Stokes law of viscosity rr= 23 V+2 ur r = 23 V+2 (1r u +urr) zz= 23 V+2 uz z6 r = r= (1r ur + u r u r) z= z = ( u z+1r uz ) zr= rz= ( uz r+ ur z)Nomenclature thermal diffusivity thermal expansion coefficient dynamic viscosity kinematic viscosity density viscous dissipation ratecp specific heat at constant pressurek thermal conductivityp pressure qg rate of heat generation per unit volumer, ,z cylindrical coordinate variablest timeT temperatureu,v,w cartesian velocity componentsur,u ,uz cylindrical velocity componentsx,y,z cartesian coordinate variables i j Kronecker delta i j (i j)

9 Thcomponent of stress tensorgi ithcomponent of gravitational accelerationvi ithcomponent of velocity vectorxi ithcartesian coordinate variableg gravitational accelerationV velocity vector deviatoric stress tensorI unit tensor7 References1. Aris, R.,Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice Hall,Englewood Cliffs, NJ (1962).2. Bird, R. B., W. E. Stewart, and E. N. Lightfoot,Transport Phenomena, 2nded., JohnWiley, New York (2002).3. White, F. M.,Viscous Fluid Flow,3rded.

10 , McGraw-Hill, New York (2006).8