Page 329 - Intro to Tensor Calculus

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2
∗
I 30. Show that for a perfect gas, where λ = − µ and η = µ is a function of position, the vector form
∗
∗
3
of equation (2.5.25) is
D~v 4 2
~
% = %b −∇p + ∇(η∇· ~v)+ ∇(~v ·∇η) − ~v ∇ η +(∇η) × (∇× ~v) − (∇· ~v)∇η −∇ × (∇× (η~v))
Dt 3
Dh Dp ∂Q
I 31. Derive the energy equation % = + −∇· ~ +Φ. Hint: Use the continuity equation.
q
Dt Dt ∂t
I 32. Show that in Cartesian coordinates the Navier-Stokes equations of motion for a compressible fluid
can be written

Du ∂p ∂ ∂u ∂ ∂u ∂v ∂ ∂w ∂u
~
∗
∗
ρ =ρb x − + 2µ ∗ + λ ∇· V + µ ( + ) + µ ( + )
∗
Dt ∂x ∂x ∂x ∂y ∂y ∂x ∂z ∂x ∂z

Dv ∂p ∂ ∂v ∂ ∂v ∂w ∂ ∂w ∂w
~
ρ =ρb y − + 2µ ∗ + λ ∇· V + µ ( + ) + µ ( + )
∗
∗
∗
Dt ∂y ∂y ∂y ∂z ∂z ∂y ∂x ∂y ∂x

Dv ∂p ∂ ∂w ∂ ∂w ∂u ∂ ∂v ∂w
~
∗
∗
∗
ρ =ρb z − + 2µ ∗ + λ ∇· V + µ ( + ) + µ ( + )
Dt ∂z ∂z ∂z ∂x ∂x ∂z ∂y ∂z ∂y
where (V x ,V y ,V z )= (u, v, w).
I 33. Show that in cylindrical coordinates the Navier-Stokes equations of motion for a compressible fluid
can be written
2
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

∂ ∂V r ∂V z 2µ ∗ ∂V r 1 ∂V θ V r
+ µ ( + ) + ( − − )
∗
∂z ∂z ∂r r ∂r r ∂θ r

DV θ V r V θ 1 ∂p 1 ∂ 1 ∂V θ V r ∂ 1 ∂V z ∂V θ
~
∗
% + =%b θ − + 2µ ( + )+ λ ∇· V + µ ( + )
∗
∗
Dt r r ∂θ r ∂θ r ∂θ r ∂z r ∂θ ∂z

∂ 1 ∂V r ∂V θ V θ 2µ ∗ 1 ∂V r ∂V θ V θ
∗
+ µ ( + − ) + ( + − )
∂r r ∂θ ∂r r r r ∂θ ∂r r

∂p ∂ 1 ∂
DV z ∂V z ~ ∂V r ∂V z
% =%b z − + 2µ ∗ + λ ∇· V + µ r( + )
∗
∗
Dt ∂z ∂z ∂z r ∂r ∂z ∂r

1 ∂ 1 ∂V z ∂V θ
+ µ ( + )
∗
r ∂θ r ∂θ ∂z
2
I 34. Show that the dissipation function Φ can be written as Φ = 2µ D ij D ij + λ Θ .
∗
∗
I 35. Verify the identities:
D ∂e t D De D 2
~
(a) % (e t /%)= + ∇· (e t V ) (b) % (e t /%)= % + % V /2 .
Dt ∂t Dt Dt Dt
I 36. Show that the conservation law for heat flow is given by
∂T
+ ∇· (T~v − κ∇T )= S Q
∂t
~
where κ is the thermal conductivity of the material, T is the temperature, J advection = T~v,
~
J conduction = −κ∇T and S Q is a source term. Note that in a solid material there is no flow and so ~v =0and