TheEquation of Continuity and theEquation of Motion in ...

1 The Equation of Continuity and the Equation of Motion in Cartesian,cylindrical, and spherical coordinatesCM3110 Fall 2011 Faith A. MorrisonContinuity Equation, Cartesian coordinates t+(vx x+vy y+vz z)+ ( vx x+ vy y+ vz z)= 0 Continuity Equation, cylindrical coordinates t+1r ( rvr) r+1r ( v ) + ( vz) z= 0 Continuity Equation, spherical coordinates t+1r2 ( r2vr) r+1rsin ( v sin ) +1rsin ( v ) = 0 Equation of Motionfor an incompressible fluid, 3 components in Cartesian coordinates ( vx t+vx vx x+vy vx y+vz vx z)= P x+( xx x+ yx y+ zx z)+ gx ( vy t+vx vy x+vy vy y+vz vy z)= P y+( xy x+ yy y+ zy z)

2 + gy ( vz t+vx vz x+vy vz y+vz vz z)= P z+( xz x+ yz y+ zz z)+ gzEquation of Motionfor an incompressible fluid, 3 components in cylindrical coordinates ( vr t+vr vr r+v r vr v2 r+vz vr z)= P r+(1r (r rr) r+1r r r+ zr z)+ gr ( v t+vr v r+v r v +v vrr+vz v z)= 1r P +(1r2 (r2 r ) r+1r + z z+ r r r)+ g ( vz t+vr vz r+v r vz +vz vz z)= P z+(1r (r rz) r+1r z + zz z)+ gzEquation of Motionfor an incompressible fluid, 3 components in spherical coordinates ( vr t+vr vr r+v r vr +v rsin vr v2 +v2 r)= P r+(1r2 (r2 rr) r+1rsin ( rsin ) +1rsin r + r)+ gr ( v t+vr v r+v r v +v rsin v +vrv r v2 cot r)= 1r P +(1r3 (r3 r ) r+1rsin ( sin ) +1rsin + r r r cot r)+ g ( v t+vr v r+v r v +v rsin v +vrv r+v v cot r)= 1rsin P +(1r3 (r3 r ) r+1rsin ( sin ) +1rsin + r r r+ cot r)

3 + g Equation of Motionfor incompressible, Newtonian fluid ( navier -Stokes equation) 3 components in Cartesiancoordinates ( vx t+vx vx x+vy vx y+vz vx z)= P x+ ( 2vx x2+ 2vx y2+ 2vx z2)+ gx ( vy t+vx vy x+vy vy y+vz vy z)= P y+ ( 2vy x2+ 2vy y2+ 2vy z2)+ gy ( vz t+vx vz x+vy vz y+vz vz z)= P z+ ( 2vz x2+ 2vz y2+ 2vz z2)+ gzEquation of Motionfor incompressible, Newtonian fluid ( navier -Stokes equation), 3 components in cylin-drical coordinates ( vr t+vr vr r+v r vr v2 r+vz vr z)= P r+ ( r(1r (rvr) r)+1r2 2vr 2 2r2 v + 2vr z2)+ gr ( v t+vr v r+v r v +vrv r+vz v z)= 1r P + ( r(1r (rv ) r)+1r2 2v 2+2r2 vr + 2v z2)+ g ( vz t+vr vz r+v r vz +vz vz z)= P z+ (1r r(r vz r)+1r2 2vz 2+ 2vz z2)+ gzEquation of Motionfor incompressible, Newtonian fluid ( navier -Stokes equation), 3 components in sphericalcoordinates ( vr t+vr vr r+v r vr +v rsin vr v2 +v2 r)= P r+ ( r(1r2 r(r2vr))+1r2sin (sin vr )+1r2sin2 2vr 2 2r2sin (v sin ) 2r2sin v )+ gr ( v t+vr v r+v r v +v rsin v +vrv r v2 cot r)= 1r P + (1r2 r(r2 v r))

4 +1r2 (1sin (v sin ))+1r2sin2 2v 2+2r2 vr 2 cot r2sin v )+ g ( v t+vr v r+v r v +v rsin v +vrv r+v v cot r)= 1rsin P + (1r2 r(r2 v r)+1r2 (1sin (v sin ))+1r2sin2 2v 2+2r2sin vr +2 cot r2sin v )+ g Note: ther-component of the navier -Stokes equation in spherical coordinates may be simplified by adding 0 =2r vto the component shown above. This term is zero due to the Continuity equation (mass conservation). SeeBird et. :1. R. B. Bird, W. E. Stewart, and E. N. Lightfoot,Transport Phenomena, 2ndedition, Wiley: NY, R.

5 B. Bird, R. C. Armstrong, and O. Hassager,Dynamics of Polymeric Fluids: Volume 1 Fluid Mechanics,Wiley: NY, 1987.