Page 188 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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where δ ij is the Kroneker delta, which is equal to unity if i = j and equal to zero if i = j.
From the previous expression, τ 11 is expressed as
CONVECTION HEAT TRANSFER

∂u 1 ∂u 1 2 ∂u 1 2 ∂u 2
τ 11 = µ + − − (7.27)
∂x 1 ∂x 1 3 ∂x 1 3 ∂x 2
Note that i = j = 1 in the above equation and k = 1, 2 for a two-dimensional flow.
The above equation may be simplified as follows:

4 ∂u 1 2 ∂u 2
τ 11 = µ − (7.28)
3 ∂x 1 3 ∂x 2
Similarly, τ 12 is

∂u 1 ∂u 2
τ 12 = µ + (7.29)
∂x 2 ∂x 1
Substituting Equations 7.28 and 7.29 into Equation 7.25, we obtain the x 1 component
of the momentum equation as
2
∂(ρu 1 ) ∂(ρu ) ∂(ρu 1 u 2 )
1
+ + =
∂t ∂x 1 ∂x 2
∂p ∂ 4 ∂u 1 2 ∂u 2
− + µ −
∂x 1 ∂x 3 ∂x 1 3 ∂x 2
∂ ∂u 2 ∂u 1
+ µ + (7.30)
∂x 2 ∂x 1 ∂x 2
The momentum component in the x 2 direction can be derived by the following steps,
which are similar to the derivation of the x 1 component of the momentum equation. The
x 2 momentum equation is

2
∂(ρu 2 ) ∂(ρu 1 u 2 ) ∂(ρu )
2
+ + =
∂t ∂x 1 ∂x 2
∂p ∂ ∂u 1 ∂u 2
− + µ +
∂x 2 ∂x 1 ∂x 2 ∂x 1
∂ 4 ∂u 2 2 ∂u 1
+ µ − (7.31)
∂x 2 3 ∂x 2 3 ∂x 1
For a constant density flow (incompressible flow), the momentum equations can be fur-
ther reduced by taking the density term out of the differential signs. In addition, substitution
of the conservation of mass equation (Equation 7.11) into the momentum equation leads to
a further simplification of the momentum equation. After simplification (see Appendix D
for the detailed derivation), the momentum equations are

2 2
∂u 1 ∂u 1 ∂u 1 ∂p ∂ u 1 ∂ u 1
ρ + u 1 + u 2 =− + µ + (7.32)
∂t ∂x 1 ∂x 2 ∂x 1 ∂x 2 ∂x 2
1 2