Page 306 - Intro to Tensor Calculus

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Using the divergence theorem of Gauss one can derive the general conservation law

∂Q
~
+ ∇· J = S Q (2.5.50)
∂t
The continuity equation and energy equations are examples of a scalar conservation law in the special case
where S Q =0. In Cartesian coordinates, we can represent the continuity equation by letting

~
~
Q = % and J = %V = %(V x ˆe 1 + V y ˆe 2 + V z ˆe 3 ) (2.5.51)
The energy equation conservation law is represented by selecting Q = e t and neglecting the rate of internal
heat energy we let
" #
3
X
~
ˆ
J = (e t + p)v 1 − v i τ xi + q x e 1 +
i=1
" #
3
X
(e t + p)v 2 − v i τ yi + q y ˆ e 2 + (2.5.52)
i=1
" #
3
X
(e t + p)v 3 − v i τ zi + q z ˆ e 3 .
i=1
In a general orthogonal system of coordinates (x 1 ,x 2 ,x 3 ) the equation (2.5.50) is written
∂ ∂ ∂ ∂
((h 1 h 2 h 3 Q)) + ((h 2 h 3 J 1 )) + ((h 1 h 3 J 2 )) + ((h 1 h 2 J 3 )) = 0,
∂t ∂x 1 ∂x 2 ∂x 3
where h 1 ,h 2 ,h 3 are scale factors obtained from the transformation equations to the general orthogonal
coordinates.
The momentum equations are examples of a vector conservation law having the form
∂~a
~
+ ∇· (T )= %b (2.5.53)
∂t
3 3
X X
e
e
where ~a is a vector and T is a second order symmetric tensor T = T jk ˆ j ˆ k . In Cartesian coordinates
k=1 j=1
we let ~a = %(V x ˆe 1 + V y ˆe 2 + V z ˆe 3 )and T ij = %v i v j + pδ ij − τ ij . In general coordinates (x 1 ,x 2 ,x 3 )the
~
momentum equations result by selecting ~a = %V and T ij = %v i v j + pδ ij − τ ij . In a general orthogonal system
the conservation law (2.5.53) has the general form
∂ ∂ ∂ ∂
~
((h 1 h 2 h 3~a)) + (h 2 h 3T · ˆe 1 ) + (h 1 h 3T · ˆe 2 ) + (h 1 h 2T · ˆe 3 ) = %b. (2.5.54)
∂t ∂x 1 ∂x 2 ∂x 3
Neglecting body forces and internal heat production, the continuity, momentum and energy equations
can be expressed in the strong conservative form
∂U ∂E ∂F ∂G
+ + + =0 (2.5.55)
∂t ∂x ∂y ∂z
where
 
ρ
 ρV x 
 
U =  ρV y  (2.5.56)
 
ρV z
e t