Page 303 - Intro to Tensor Calculus

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Surface forces acting upon the control volume involve such things as pressures and viscous forces, while body
forces are due to such things as gravitational, magnetic and electric fields.

Conservation of angular momentum
The conservation of angular momentum states that the time rate of change of angular momentum
(moment of linear momentum) must equal the total moment of all forces and couples acting upon the body.
In symbols,
Z Z Z n
D j k j k j k X j k i
%e ijk x v dτ = e ijk x F (s) dS + %e ijk x F (b) dτ + (e ijk x F + M (α) ) (2.5.44)
Dt (α) (α)
V S V α=1
where M i represents concentrated couples and F k represents isolated forces.
(α) (α)
Conservation of energy
The conservation of energy law requires that the time rate of change of kinetic energy plus internal
energies is equal to the sum of the rate of work from all forces and couples plus a summation of all external
energies that enter or leave a control volume per unit of time. The energy equation results from the first law
of thermodynamics and can be written

D ˙ ˙
(E + K)= W + Q h (2.5.45)
Dt
˙
where E is the internal energy, K is the kinetic energy, W is the rate of work associated with surface and
˙
body forces, and Q h is the input heat rate from surface and internal effects.
Z
Let e denote the internal specific energy density within a control volume, then E = %e dτ represents
V
the total internal energy of the control volume. The kinetic energy of the control volume is expressed as
Z
1
i j
i
K = %g ij v v dτ where v is the velocity, % is the density and dτ is a volume element. The energy (rate
2
V
of work) associated with the body and surface forces is represented
Z Z n
X
˙
j
i
j
i
j
i
j
i
W = g ij F (s) v dS + %g ij F (b) v dτ + (g ij F (α) v + g ij M (α) ω )
S V α=1
j
where ω is the angular velocity of the point x i , F i are isolated forces, and M i are isolated couples.
(α) (α) (α)
i
Two external energy sources due to thermal sources are heat flow q and rate of internal heat production ∂Q
∂t
per unit volume. The conservation of energy can thus be represented
D Z 1 i j Z i j i Z i j ∂Q
%(e + g ij v v ) dτ = (g ij F (s) v − q i n ) dS + (%g ij F (b) v + ) dτ
Dt 2 ∂t
V S V
n (2.5.46)
X
i
j
j
i
+ (g ij F (α) v + g ij M (α) ω + U (α) )
α=1
where U (α) represents all other energies resulting from thermal, mechanical, electric, magnetic or chemical
sources which influx the control volume and D/Dt is the material derivative.
In equation (2.5.46) the left hand side is the material derivative of an integral of the total energy
i j
1
e t = %(e + g ij v v ) over the control volume. Material derivatives are not like ordinary derivatives and so
2