Page 297 - Intro to Tensor Calculus

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DV x ∂p ∂ h ∂V x i ∂ h ∂V x ∂V y i ∂ h ∂V x ∂V z i
~
∗
% =%b x − + 2µ ∗ + λ ∇· V + µ ∗ + + µ ∗ +
Dt ∂x ∂x ∂x ∂y ∂y ∂x ∂z ∂z ∂x
DV y ∂p ∂ h ∂V y ∂V x i ∂ h ∂V y i ∂ h ∂V y ∂V z i
~
∗
% =%b y − + µ ∗ + + 2µ ∗ + λ ∇· V + µ ∗ +
Dt ∂y ∂x ∂x ∂y ∂y ∂y ∂z ∂z ∂y
h i h i h i
DV z ∂p ∂ ∂V z ∂V x ∂ ∂V z ∂V y ∂ ∂V z
∗
% =%b z − + µ ∗ + + µ ∗ + + 2µ ∗ + λ ∇· V ~
Dt ∂z ∂x ∂x ∂z ∂y ∂y ∂z ∂z ∂z
D ∂() ∂() ∂() ∂()
where () = + V x + V y + V z
Dt ∂t ∂x ∂y ∂z
~
and ∇· V = ∂V x + ∂V y + ∂V z
∂x ∂y ∂z
(2.5.31a)
Table 5.2 Navier-Stokes equations for compressible fluids in Cartesian coordinates.

2 h i h i
DV r V θ ∂p ∂ ∂V r 1 ∂ 1 ∂V r ∂V θ V θ
~
∗
% − =%b r − + 2µ ∗ + λ ∇· V + µ ∗ + −
Dt r ∂r ∂r ∂r r ∂θ r ∂θ ∂r r
h i
∂ ∂V r ∂V z 2µ ∗ ∂V r 1 ∂V θ V r
+ µ ∗ + + − −
∂z ∂z ∂r r ∂r r ∂θ r
h i h i h i
DV θ V r V θ 1 ∂p ∂ 1 ∂V r ∂V θ V θ 1 ∂ 1 ∂V θ V r ~
∗
% + =%b θ − + µ ∗ + − + 2µ ∗ + + λ ∇· V
Dt r r ∂θ ∂r r ∂θ ∂r r r ∂θ r ∂θ r
∂ h 1 ∂V z ∂V θ i 2µ ∗ h 1 ∂V r ∂V θ V θ i
+ µ ∗ + + + −
∂z r ∂θ ∂z r r ∂θ ∂r r
DV z ∂p 1 ∂ h ∂V r ∂V z i 1 ∂ h 1 ∂V z ∂V θ i ∂ h ∂V z i
∗
∗
% =%b z − + µ r + + µ ∗ + + 2µ ∗ + λ ∇· V ~
Dt ∂z r ∂r ∂z ∂r r ∂θ r ∂θ ∂z ∂z ∂z
D ∂() ∂() V θ ∂() ∂()
where () = + V r + + V z
Dt ∂t ∂r r ∂θ ∂z
1 ∂(rV r ) 1 ∂V θ ∂V z
and ∇· ~ V = + +
r ∂r r ∂θ ∂z
(2.5.31b)
Table 5.3 Navier-Stokes equations for compressible fluids in cylindrical coordinates.