Page 187 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
12
+ ....
∂t
t +∆x
2
12
∂x
2
∂p 179
p p +∆x 1 + ....
∂x 1
11
∆x t +∆x + ....
∂t
t 11 2 11 1
∂x 1
∆x
1
t 12
Figure 7.5 Infinitesimal control volume in a flow field. Derivation of conservation of
momentum in x 1 direction. Viscous and pressure forces
Similarly, the total force due to the deviatoric stress (viscosity or friction) acting on the
control volume in the x 1 direction is written as (see Figure 7.5)

∂τ 11 ∂τ 12
τ 11 + x 1 x 2 − τ 11 x 2 + τ 12 + x 2 x 1 − τ 12 x 1 (7.21)
∂x 1 ∂x 2
Simplifying, we obtain the net force due to the deviatoric stress as
∂τ 11 ∂τ 12
x 1 x 2 + x 2 x 2 (7.22)
∂x 1 ∂x 2
The total force acting on the control volume in the x 1 direction is

∂p ∂τ 11 ∂τ 12

x 1 x 2 − + + (7.23)
∂x 1 ∂x 1 ∂x 2
As mentioned before, for equilibrium, the net momentum in the x 1 direction should be
equal to the total force acting on the control volume in the x 1 direction, that is,

2
∂(ρu ) ∂(ρu 1 u 2 ) ∂(ρu 1 ) ∂p ∂τ 11 ∂τ 12
1
x 1 x 2 + + = x 1 x 2 − + + (7.24)
∂x 1 ∂x 2 ∂t ∂x 1 ∂x 1 ∂x 2
Simplifying, we obtain
2
∂(ρu 1 ) ∂(ρu ) ∂(ρu 1 u 2 ) ∂p ∂τ 11 ∂τ 12
1
+ + =− + + (7.25)
∂t ∂x 1 ∂x 2 ∂x 1 ∂x 1 ∂x 2
Note that the external and body forces (buoyancy) are not included in the above force
balance. In the above equations, the deviatoric stresses τ ij are expressed in terms of the
velocity gradients and dynamic viscosity as

∂u i ∂u j 2 ∂u k
τ ij = µ + − δ ij (7.26)
∂x j ∂x i 3 ∂x k