The Equation of Continuity and the Equation of Motion ...

The Equation of Continuity and the Equation of Motion in Cartesian, cylindrical, and spherical coordinates CM3110 Fall 2011 Faith A. Morrison Continuity Equation, Cartesian coordinates ∂ρ ∂ρ ∂ρ ∂ρ + vx + vy + vz ∂t ∂x ∂y ∂z 



+ρ



∂vz ∂vx ∂vy + + ∂x ∂y ∂z



= 0

Continuity Equation, cylindrical coordinates ∂ρ 1 ∂(ρrvr ) 1 ∂(ρvθ ) ∂(ρvz ) + + + ∂t r ∂r r ∂θ ∂z

= 0

Continuity Equation, spherical coordinates ∂ρ 1 ∂(ρr 2 vr ) 1 ∂(ρvθ sin θ) 1 ∂(ρvφ ) + 2 + + ∂t r ∂r r sin θ ∂θ r sin θ ∂φ

= 0

Equation of Motion for an incompressible fluid, 3 components in Cartesian coordinates ∂ τ˜xx ∂ τ˜yx ∂ τ˜zx ∂vx ∂vx ∂vx ∂P ∂vx + vx + vy + vz + + + = − + ρgx ρ ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z     ∂vy ∂ τ˜xy ∂vy ∂vy ∂vy ∂ τ˜yy ∂ τ˜zy ∂P ρ + vx + vy + vz + + + = − + ρgy ∂t ∂x ∂y ∂z ∂y ∂x ∂y ∂z     ∂ τ˜xz ∂vz ∂vz ∂vz ∂ τ˜yz ∂ τ˜zz ∂P ∂vz + vx + vy + vz + + + = − + ρgz ρ ∂t ∂x ∂y ∂z ∂z ∂x ∂y ∂z 







Equation of Motion for an incompressible fluid, 3 components in cylindrical coordinates ∂vr ∂vr vθ ∂vr v2 ∂vr + vr + − θ + vz ∂t ∂r r ∂θ r ∂z

!

∂P + = − ∂r

∂vθ ∂vθ vθ ∂vθ vθ vr ∂vθ ρ + vr + + + vz ∂t ∂r r ∂θ r ∂z



1 ∂P + = − r ∂θ

∂vz vθ ∂vz ∂vz ∂vz + vr + + vz ∂t ∂r r ∂θ ∂z



= −

ρ 

ρ



∂P + ∂z





1 ∂(r˜ τrr ) 1 ∂ τ˜θr τ˜θθ ∂ τ˜zr + − + r ∂r r ∂θ r ∂z



+ ρgr

1 ∂(r 2 τ˜rθ ) 1 ∂ τ˜θθ ∂ τ˜zθ τ˜θr − τ˜rθ + + + r 2 ∂r r ∂θ ∂z r τrz ) 1 ∂ τ˜θz 1 ∂(r˜ ∂ τ˜zz + + r ∂r r ∂θ ∂z



!

+ ρgz

Equation of Motion for an incompressible fluid, 3 components in spherical coordinates vθ2 + vφ2 vφ ∂vr ∂vr vθ ∂vr ∂vr + vr + + − ρ ∂t ∂r r ∂θ r sin θ ∂φ r ∂P =− + ∂r ρ 1 ∂P + =− r ∂θ

!

1 ∂(r 2 τ˜rr ) τθr sin θ) 1 ∂(˜ 1 ∂ τ˜φr τ˜θθ + τ˜φφ + + − 2 r ∂r r sin θ ∂θ r sin θ ∂φ r vφ2 cot θ vφ ∂vθ ∂vθ ∂vθ vθ ∂vθ vr vθ + vr + + + − ∂t ∂r r ∂θ r sin θ ∂φ r r

!

+ ρgr

!

τ˜φφ cot θ 1 ∂(r 3 τ˜rθ ) τθθ sin θ) 1 ∂(˜ 1 ∂ τ˜φθ τ˜θr − τ˜rθ + + + − 3 r ∂r r sin θ ∂θ r sin θ ∂φ r r

!

+ ρgθ

∂vφ vθ ∂vφ vφ ∂vφ vr vφ vφ vθ cot θ ∂vφ + vr + + + + ρ ∂t ∂r r ∂θ r sin θ ∂φ r r ! 3 τθφ sin θ) 1 ∂(r τ˜rφ ) 1 ∂ (˜ 1 ∂ τ˜φφ τ˜φr − τ˜rφ τ˜φθ cot θ 1 ∂P + + + + + =− + ρgφ r sin θ ∂φ r3 ∂r r sin θ ∂θ r sin θ ∂φ r r 



+ ρgθ

Equation of Motion for incompressible, Newtonian fluid (Navier-Stokes equation) 3 components in Cartesian coordinates ∂vx ∂vx ∂vx ∂vx + vx + vy + vz ρ ∂t ∂x ∂y ∂z



∂ 2 vx ∂ 2 vx ∂ 2 vx ∂P +µ + + = − ∂x ∂x2 ∂y 2 ∂z 2

!

+ ρgx

∂vy ∂vy ∂vy ∂vy ρ + vx + vy + vz ∂t ∂x ∂y ∂z



∂ 2 vy ∂ 2 vy ∂ 2 vy ∂P +µ + + = − ∂y ∂x2 ∂y 2 ∂z 2

!

+ ρgy

∂vz ∂vz ∂vz ∂vz + vx + vy + vz ρ ∂t ∂x ∂y ∂z



∂ 2 vz ∂ 2 vz ∂ 2 vz ∂P +µ + + = − ∂z ∂x2 ∂y 2 ∂z 2

!

+ ρgz







Equation of Motion for incompressible, Newtonian fluid (Navier-Stokes equation), 3 components in cylindrical coordinates ∂vr ∂vr vθ ∂vr v2 ∂vr + vr + − θ + vz ∂t ∂r r ∂θ r ∂z

!

∂ ∂P +µ = − ∂r ∂r

∂vθ vθ ∂vθ vr vθ ∂vθ ∂vθ + vr + + + vz ρ ∂t ∂r r ∂θ r ∂z



1 ∂P +µ = − r ∂θ

∂vz ∂vz vθ ∂vz ∂vz ρ + vr + + vz ∂t ∂r r ∂θ ∂z



1 ∂ ∂P ∂vz +µ = − r ∂z r ∂r ∂r

ρ 





∂ ∂r

1 ∂(rvr ) r ∂r  



2 ∂vθ ∂ 2 vr 1 ∂ 2 vr − + + 2 r ∂θ 2 r 2 ∂θ ∂z 2

!

∂ 2 vθ 2 ∂vr 1 ∂(rvθ ) 1 ∂ 2 vθ + + + 2 r ∂r r ∂θ 2 r 2 ∂θ ∂z 2 



∂ 2 vz 1 ∂ 2 vz + + 2 r ∂θ 2 ∂z 2

!

+ ρgr !

+ ρgθ

+ ρgz

Equation of Motion for incompressible, Newtonian fluid (Navier-Stokes equation), 3 components in spherical coordinates vθ2 + vφ2 ∂vr vθ ∂vr vφ ∂vr ∂vr + vr + + − ρ ∂t ∂r r ∂θ r sin θ ∂φ r

!

   ∂ 2 vr ∂ 1 ∂  2  1 ∂vr 1 + r v sin θ + r 2 r 2 ∂r r 2 sin θ ∂θ ∂θ r 2 sin θ ∂φ2  ∂ 2 ∂vφ 2 (vθ sin θ) − 2 − 2 + ρgr r sin θ ∂θ r sin θ ∂φ ! vφ2 cot θ vφ ∂vθ ∂vθ ∂vθ vθ ∂vθ vr vθ ρ + vr + + + − ∂t ∂r r ∂θ r sin θ ∂φ r r

=−

∂P +µ ∂r

1 ∂P +µ =− r ∂θ

ρ



∂vφ vθ ∂vφ ∂vφ + vr + ∂t ∂r r ∂θ =−

1 ∂P r sin θ ∂φ

∂ ∂r



∂ 2 vθ 1 1 ∂ (vθ sin θ) + 2 2 sin θ ∂θ r sin θ ∂φ2  2 ∂vr 2 cot θ ∂vφ + 2 − 2 + ρgθ r ∂θ r sin θ ∂φ  vφ ∂vφ vr vφ vφ vθ cot θ + + + r sin θ ∂φ r r     ∂ 2 vφ ∂vφ 1 ∂ 1 1 ∂ 1 ∂ +µ 2 (vφ sin θ) + 2 2 r2 + 2 r ∂r ∂r r ∂θ sin θ ∂θ r sin θ ∂φ2  2 cot θ ∂vθ 2 ∂vr + 2 + 2 + ρgφ r sin θ ∂φ r sin θ ∂φ 1 ∂ ∂vθ r2 r 2 ∂r ∂r 



1 ∂ + 2 r ∂θ





Note: the r-component of the Navier-Stokes equation in spherical coordinates may be simplified by adding 0 = 2 r ∇ · v to the component shown above. This term is zero due to the continuity equation (mass conservation). See Bird et. al. References: 1. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd edition, Wiley: NY, 2002. 2. R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Fluids: Volume 1 Fluid Mechanics, Wiley: NY, 1987.