Basics of Fluid Mechanics, 2014a

254 CHAPTER 8. DIFFERENTIAL ANALYSIS
The significance of the difference between the thermodynamic pressure and the
mechanical pressure associated with fluid dilation which connected by ∇·U. The
physical meaning of ∇·U represents the relative volume rate of change. For simple gas
(dilute monatomic gases) it can be shown that λ vanishes. In material such as water,
λ is large (3 times μ) but the net effect is small because in that cases ∇·U −→ 0.
For complex liquids this coefficient, λ, can be over 100 times larger than μ. Clearly for
incompressible flow, this coefficient or the whole effect is vanished 20 . In most cases,
the total effect of the dilation on the flow is very small. Only in micro fluids and small
and molecular scale such as in shock waves this effect has some significance. In fact
this effect is so insignificant that there is difficulty in to construct experiments so this
effect can be measured. Thus, neglecting this effect results in
( ∂Ui
τ ij = −Pδ ij + μ + ∂U )
j
(8.103)
∂x j ∂x i
To explain equation (8.103), it can be written for specific coordinates. For example, for
the τ xx it can be written that
τ xx = −P +2 ∂U x
∂x
(8.104)
and the y coordinate the equation is
however the mix stress, τ xy ,is
τ yy = −P +2 ∂U y
∂y
τ xy = τ yx =
( ∂Uy
∂x + ∂U )
x
∂y
(8.105)
(8.106)
in
ρ
For the total effect, substitute equation (8.102) into equation (8.61) which results
( ) DUx
= − ∂ ( P + ( 2
3 μ − λ) ∇·U ) ( ∂ 2 )
U x
+ μ
Dt
∂x
∂x 2 + ∂2 U x
∂y 2 + ∂2 U x
∂z 2
or in a vector form as
+f Bx
(8.107)
N-S in stationary Coordinates
ρ DU
( ) 1
Dt = −∇P + 3 μ + λ ∇ (∇·U)+μ ∇ 2 U + f B
(8.108)
20 The reason that the effect vanish is because ∇·U =0.