Page 305 - Intro to Tensor Calculus

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terms of the temperature T as ~ = −κ∇ T ,where κ is the thermal conductivity. Consequently, the energy
q
equation can be written as
De ∂Q
~
% + p(∇· V )= +Φ+ ∇(k∇T ). (2.5.49)
Dt ∂t
In Cartesian coordinates (x, y, z)we use

D ∂ ∂ ∂ ∂
= + V x + V y + V z
Dt ∂t ∂x ∂y ∂z
~
∇· V = ∂V x + ∂V y + ∂V z
∂x ∂y ∂z

∂ ∂T ∂ ∂T ∂ ∂T
∇· (κ∇T )= κ + κ + κ
∂x ∂x ∂y ∂y ∂z ∂z
In cylindrical coordinates (r, θ, z)

D ∂ ∂ V θ ∂ ∂
= + V r + + V z
Dt ∂t ∂r r ∂θ ∂z
1 ∂ 1 ∂V θ ∂V z
~
∇· V = (rV r )+ +
2
r ∂r r ∂θ ∂z

1 ∂ ∂T 1 ∂ ∂T ∂ ∂T
∇· (κ∇T )= rκ + κ + κ
2
r ∂r ∂r r ∂θ ∂θ ∂z ∂z
and in spherical coordinates (ρ, θ, φ)
D ∂ ∂ V θ ∂ V φ ∂
= + V ρ +
Dt ∂t ∂ρ ρ ∂θ ρ sin θ ∂φ
1 ∂ 1 ∂ 1 ∂V φ
~
∇· V = (ρV ρ )+ (V θ sin θ)+
2
ρ ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ

1 ∂ 2 ∂T 1 ∂ ∂T 1 ∂ ∂T
∇· (κ∇T )= ρ κ + κ sin θ + 2 κ
2
2
2
ρ ∂ρ ∂ρ ρ sin θ ∂θ ∂θ ρ sin θ ∂φ ∂φ
The combination of terms h = e + p/% is known as enthalpy and at times is used to express the energy
equation in the form
Dh Dp ∂Q
% = + −∇· ~ +Φ.
q
Dt Dt ∂t
The derivation of this equation is left as an exercise.
Conservative Systems
Let Q denote some physical quantity per unit volume. Here Q can be either a scalar, vector or tensor
field. Place within this field an imaginary simple closed surface S which encloses a volume V. The total
RRR
amount of Q within the surface is given by Qdτ and the rate of change of this amount with respect
V
RRR
to time is ∂ Qdτ. The total amount of Q within S changes due to sources (or sinks) within the volume
∂t
~
and by transport processes. Transport processes introduce a quantity J, called current, which represents a
RR
~
flow per unit area across the surface S. The inward flux of material into the volume is denoted −J · ˆndσ
S
(ˆn is a unit outward normal.) The sources (or sinks) S Q denotes a generation (or loss) of material per unit
RRR
volume so that S Q dτ denotes addition (or loss) of material to the volume. For a fixed volume we then
V
have the material balance
ZZZ ZZ ZZZ
∂Q ~
dτ = − J · ˆndσ + S Q dτ.
∂t
V S V