Page 330 - Intro to Tensor Calculus

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the above equation reduces to the heat equation. Assign units of measurements to each term in the above
equation and make sure the equation is dimensionally homogeneous.

I 37. Show that in spherical coordinates the Navier-Stokes equations of motion for a compressible fluid can
be written
2
V + V 2 ∂p ∂ 1 ∂ ∂
DV ρ θ φ ∂V ρ ~ 1 ∂V ρ
%( − )= %b ρ − + 2µ ∗ + λ ∇· V + µ (ρ (V θ /ρ)+ )
∗
∗
Dt ρ ∂ρ ∂ρ ∂ρ ρ ∂θ ∂ρ ρ ∂θ

1 ∂ 1 ∂V ρ ∂
+ µ ( + ρ (V φ /ρ))
∗
ρ sin θ ∂φ ρ sin θ ∂φ ∂ρ
µ ∗ ∂V ρ 2 ∂V θ 4V ρ 2 ∂V φ 2V θ cot θ ∂ cot θ ∂V ρ
+ (4 − − − − + ρ cot θ (V θ /ρ)+ )
ρ ∂ρ ρ ∂θ ρ ρ sin θ ∂φ ρ ∂ρ ρ ∂θ
2
V cot θ 1 ∂p 1 ∂ 2µ ∗
DV θ V ρ V θ φ ∂V θ ~
∗
%( + − )= %b θ − + ( + V ρ )+ λ ∇· V
Dt ρ ρ ρ ∂θ ρ ∂θ ρ ∂θ

1 ∂ sin θ ∂ 1 ∂V θ ∂ ∂ 1 ∂V ρ
∗
∗
+ µ ( (V φ / sin θ)+ ) + µ (ρ (V θ /ρ)+ )
ρ sin θ ∂φ ρ ∂θ ρ sin θ ∂φ ∂ρ ∂ρ ρ ∂θ

µ ∗ 1 ∂V θ 1 ∂V φ V θ cot θ ∂ 1 ∂V ρ
+ 2 − − cot θ +3 ρ (V θ /ρ)+
ρ ρ ∂θ ρ sin θ ∂φ ρ ∂ρ ρ ∂θ

DV φ V φ V ρ V θ V φ cot θ 1 ∂p ∂ 1 ∂V ρ ∂
% + + = %b φ − + µ ∗ + ρ (V φ /ρ)
Dt ρ ρ ρ sin θ ∂φ ∂ρ ρ sin θ ∂φ ∂ρ

1 ∂ 2µ ∗ 1 ∂V φ
~
∗
+ + V ρ + V θ cot θ + λ ∇· V
ρ sin θ ∂φ ρ sin θ ∂φ

1 ∂ sin θ ∂ 1 ∂V θ
+ µ ∗ (V φ / sin θ)+
ρ ∂θ ρ ∂θ ρ sin θ ∂φ

µ ∗ 1 ∂V ρ ∂ sin θ ∂ 1 ∂V θ
+ 3 + ρ (V φ /ρ) +2 cot θ (V φ / sin θ)+
ρ ρ sin θ ∂φ ∂ρ ρ ∂θ ρ sin θ ∂φ
I 38. Verify all the equations (2.5.28).
I 39. Use the conservation of energy equation (2.5.47) together with the momentum equation (2.5.25) to
derive the equation (2.5.48).
I 40. Verify the equation (2.5.55).
I 41. Consider nonviscous flow and write the 3 linear momentum equations and the continuity equation
and make the following assumptions: (i) The density % is constant. (ii) Body forces are zero. (iii) Steady
state flow only. (iv) Consider only two dimensional flow with non-zero velocity components u = u(x, y)and
v = v(x, y). Show that there results the system of equations
∂u ∂u 1 ∂P ∂v ∂v 1 ∂P ∂u ∂v
u + v + =0, u + v + =0, + =0.
∂x ∂y % ∂x ∂x ∂y % ∂y ∂x ∂y
Recognize that the last equation in the above set as one of the Cauchy-Riemann equations that f(z)= u−iv
be an analytic function of a complex variable. Further assume that the fluid flow is irrotational so that
∂v ∂u 1 2 2 P
− =0. Show that this implies that u + v + = Constant. If in addition u and v are derivable
∂x ∂y 2 %
from a potential function φ(x, y), such that u = ∂φ and v = ∂φ , then show that φ is a harmonic function.
∂x ∂y
By constructing the conjugate harmonic function ψ(x, y) the complex potential F(z)= φ(x, y)+ iψ(x, y)is
0
such that F (z)= u(x, y) − iv(x, y)and F (z) gives the velocity. The family of curves φ(x, y) =constant are
0
called equipotential curves and the family of curves ψ(x, y) = constant are called streamlines. Show that
these families are an orthogonal family of curves.