r z v v v v θ θ Equations of motion of incompressible Newtonian Fluids in

R z v v v v θ θ equations of motion of

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r z v v v v θ θ = = Equations of motion of incompressible Newtonian Fluids in cylindrical and spherical coordinates: See appendix B, Viscous Fluid Flow , F. White
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Continuity equation: ( ) ( ) ( ) ( ) 1 1 0 0 6 r z V rv v v r r r z θ θ θ θ + + = = ( ) steady flow, 0 axisymmetric flow 0 cylinder infinitely long 0 unidirectional motion v t z v r θ θ θ = = = = Momentum eq. in r - direction ( ) 2 1 v dp dr r θ ρ = balance between centrifugal force & force which is produced by the induced pressure field. 0 p as r dp dr > / / p streamline
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( ) 2 2 0 2 d v v d direction dr dr r θ θ θ + = ( ) 2 viscous dissipation term k equation 0= 3 r dv v d dT Energy r dr dr dr r θ θ µ + ±²²³²²´ BC’s 1 1 1 1 1 2 2 2 2 r=r v & T=T p=p r=r v & T=T At r At r θ θ ω ω = = Integrate eq. (2) ( ) ( ) 1 2 1 1 1 2 2 0 1 2 / dv v dv v d c dr dr r dr r d d r rv c rv c r rv c c r dr dr v cr c r θ θ θ θ θ θ θ θ + = + = = = = = + = +
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( ) 2 2 2 2 2 2 1 1 1 2 r r z r r V V V V V V V v v t r r r z V V V r r z r r V p g r r θ θ θ θ θ θ θ θ θ θ ρ θ µ ρ θ θ θ + + + + = + + + + 1. steady 2. ρ = const. 3. Fully developed in z-dir. 4. axisymmetric in θ -dir. 5. v r = v z = 0 ( ) ( ) ( ) 1 1 1 1 0 rV c rV c r r r r rV r r r θ θ θ = = = Momentum eq. in θ direction
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1 2 2 2 V / 2 r rV c cr c r c θ θ = + = + Energy eq. 2 0 (3) dV V k d dT r r dr dr dr r θ θ µ = + BC’s 1 1 1 1 1 2 2 2 2 r=r v & T=T p=p r=r v & T=T At r At r θ θ ω ω = = using the B.C’s ( ) 2 2 2 2 2 1 2 1 2 2 1 1 1 2 2 2 2 2 1 2 1 , c r r r r c r r r r ω ω ω ω = =
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( ) ( ) ( ) 2 2 2 2 2 1 2 2 1 1 2 1 2 2 2 1 1 ( ) 4 r r V r r r r r r r θ ω ω ω ω = Eq. (1) determines the radial pressure distribution resulting from the motion. p=p(r) obtain this distribution ! Having found v θ (r) , it is substituted into eq. (3) to find temperature distribution ( ) ( ) ( ) ( ) ( ) 2 4 2 2 2 1 1 1 1 1 4 4 2 2 1 2 1 2 1 2 1 1 / ln / ln / Pr 1 1 ln / ln / r r r r r T T r Ec T T r r r r r r r ω ω = + where ( ) 2 2 1 1 2 1 Pr r Ec k T T µ ω = Brinkman number expressing the temp. rise due to dissipation ( ) ( ) 1 1 2 1 2 1 ln / PrEc=0 heat conduction solution ln / r r T T if T T r r =
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SPECIAL CASES : i) case when r 1 0 i.e. in the limit as the inner cylinder vanishes ( α =0)=r 1 /r 2 1 2 .(4) V ( 0) rigid-body rotation Eq r r θ ω = 2 2 r ω i.e. fluid rotates inside the outer cylinder as a rigid body 2 V ω ω = ∇× = JG JG ii) single cylinder rotating in an infinite fluid 2 2 ( , 0) r ω → ∞ = 1 1 r ω 2 1 1 2 ( ) r V r r θ ω → ∞ = flow v . . 0
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