cylindrical, and spherical coordinates. CM3110 Fall 2011 Faith A. Morrison. Continuity Equation, Cartesian coordinates. âÏ. ât. + (vx. âÏ. âx. + vy. âÏ. ây.
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.
