Page 298 - Intro to Tensor Calculus

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Observe that for incompressible flow D% = 0 which implies ∇· V =0. Therefore, the assumptions
Dt
of constant viscosity and incompressibility of the flow will simplify the above equations. If on the other
hand the viscosity is temperature dependent and the flow is compressible, then one should add to the above
equations the continuity equation, an energy equation and an equation of state. The energy equation comes
from the first law of thermodynamics applied to a control volume within the fluid and will be considered
in the sections ahead. The equation of state is a relation between thermodynamic variables which is added
so that the number of equations equals the number of unknowns. Such a system of equations is known as
a closed system. An example of an equation of state is the ideal gas law where pressure p is related to gas
density % and temperature T by the relation p = %RT where R is the universal gas constant.

2
h V + V 2 h i h ∗ i
DV ρ θ φ ∂p ∂ ∂V ρ 1 ∂ ∂ V θ µ ∂V ρ
∗
∗
% − = %b ρ − + 2µ ∗ + λ ∇· V ~ + µ ρ +
Dt ρ ∂ρ ∂ρ ∂ρ ρ ∂θ ∂ρ ρ ρ ∂θ
1 ∂ h µ ∗ ∂V ρ ∂ V φ i
+ + µ ρ
∗
ρ sin θ ∂φ ρ sin θ ∂φ ∂ρ ρ
h i
µ ∗ ∂V ρ 2 ∂V θ 4V ρ 2 ∂V φ 2V θ cot θ ∂ V θ cot θ ∂V ρ
+ 4 − − − − + ρ cot θ +
ρ ∂ρ ρ ∂θ ρ ρ sin θ ∂φ ρ ∂ρ ρ ρ ∂θ
2
h V cot θ h ∗ i
DV θ V ρV θ φ 1 ∂p ∂ ∂ V θ µ ∂V ρ
% + − = %b θ − + µ ρ +
∗
Dt ρ ρ ρ ∂θ ∂ρ ∂ρ ρ ρ ∂θ
1 ∂ h 2µ ∗ ∂V θ i
+ + V ρ + λ ∇· ~ V
∗
ρ ∂θ ρ ∂θ
1 ∂ h µ sin θ ∂ V φ µ ∗ ∂V θ i
∗
+ +
ρ sin θ ∂φ ρ ∂θ sin θ ρ sin θ ∂φ
h i
µ ∗ 1 ∂V θ 1 ∂V φ V θ cot θ ∂ V θ 1 ∂V ρ
+ 2cot θ − − +3 ρ +
ρ ρ ∂θ ρ sin θ ∂φ ρ ∂ρ ρ ρ ∂θ
h V θ V φ cot θ i h ∗ i
DV φ V θ V φ 1 ∂p ∂ µ ∂V ρ ∂ V φ
∗
% + + = %b φ − + + µ ρ
Dt ρ ρ ρ sin θ ∂φ ∂ρ ρ sin θ ∂φ ∂ρ ρ
h i
1 ∂ µ sin θ ∂ V φ µ ∗ ∂V θ
∗
+ +
ρ ∂θ ρ ∂θ sin θ ρ sin θ ∂φ
1 ∂ h 2µ ∗ 1 ∂V φ i
∗
+ + V ρ + V θ cot θ + λ ∇· V ~
ρ sin θ ∂φ ρ sin θ ∂φ
µ ∗ h 3 ∂V ρ ∂ V φ sin θ ∂ V φ 1 ∂V θ i
+ +3ρ +2 cot θ +
ρ ρ sin θ ∂φ ∂ρ ρ ρ ∂θ sin θ ρ sin θ ∂φ
D ∂() ∂() V θ ∂() V φ ∂()
where () = + V ρ + +
Dt ∂t ∂ρ ρ ∂θ ρ sin θ ∂φ
2
1 ∂(ρ V ρ) 1 ∂V θ sin θ 1 ∂V φ
and ∇· V = + +
~
ρ 2 ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ
(2.5.31c)
Table 5.4 Navier-Stokes equations for compressible fluids in spherical coordinates.