Page 194 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
186

Hot, vertical plate Fluid circulation

g

x 2

x 1
Figure 7.7 Natural convective flow near a hot, vertical plate

On introducing the above non-dimensional scales into the governing equations, we
obtain the non-dimensional form of the equations as follows:

Continuity equation
∂u ∗ ∂u ∗
1 2
+ = 0 (7.64)
∂x ∗ ∂x ∗
1 2
x 1 momentum equation
!
2 ∗
2 ∗
∂u ∗ ∂u ∗ ∂u ∗ ∂p ∗ ∂ u ∂ u
1 + u ∗ 1 + u ∗ 1 =− + Pr 1 + 1 (7.65)
∂t ∗ 1 ∂x ∗ 2 ∂x ∗ ∂x ∗ ∂x ∗2 ∂x ∗2
1 2 1 1 2
x 2 momentum equation
!
2 ∗
2 ∗
∂u ∗ ∂u ∗ ∂u ∗ ∂p ∗ ∂ u ∂ u
2 + u ∗ 2 + u ∗ 2 =− + Pr 2 + 2 + GrP r T ∗ (7.66)
2
∂t ∗ 1 ∂x ∗ 2 ∂x ∗ ∂x ∗ ∂x ∗2 ∂x ∗2
1 2 2 1 2
Energy equation
!
2
2
∂T ∗ ∂T ∗ ∂T ∗ ∂ T ∗ ∂ T ∗
+ u ∗ 1 + u ∗ 2 = + (7.67)
∂t ∗ ∂x ∗ ∂x ∗ ∂x ∗2 ∂x ∗2
1 2 1 2