Page 185 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
The above quantity, stored within the control volume, is equal to the rate of change of
the total mass within the control volume, which is given as
∂ρ 177
x 1 x 2 (7.7)
∂t
We can therefore write
∂ρ ∂(ρu 1 ) ∂(ρu 2 )
x 1 x 2 =− x 1 x 2 + (7.8)
∂t ∂x 1 ∂x 2
or
∂ρ ∂(ρu 1 ) ∂(ρu 2 )
+ + = 0 (7.9)
∂t ∂x 1 ∂x 2
The above equation is known as the equation of conservation of mass, or the continuity
equation for two-dimensional flows. In three dimensions, the continuity equation is
∂ρ ∂(ρu 1 ) ∂(ρu 2 ) ∂(ρu 3 )
+ + + = 0 (7.10)
∂t ∂x 1 ∂x 2 ∂x 3
If the density is assumed to be constant, then the above equation is reduced to

∂u 1 ∂u 2 ∂u 3
+ + = 0 (7.11)
∂x 1 ∂x 2 ∂x 3
Using vector notation, the above equation is written as (divergence-free velocity field)
∇.u = 0 (7.12)

or, using an indicial notation,

∂u i
= 0 (7.13)
∂x i
where i = 1, 2 for a two-dimensional case and i = 1, 2, 3 for three-dimensional flows.

7.2.2 Conservation of momentum

The conservation of momentum equation can be derived in a fashion similar to the con-
servation of mass equation. Here, the momentum equations are derived on the basis of the
conservation of momentum principle, that is, the total force generated by the momentum
transfer in each direction is balanced by the rate of change of momentum in each direction.
The momentum equation has directional components and is therefore a vector equation. In
order to derive the conservation of momentum equation, let us consider the control volume
shown in Figure 7.4.
The momentum entering the control volume in the x 1 direction is given as
2
ρu 1 x 2 u 1 = ρu x 2 (7.14)
1