Basics of Fluid Mechanics, 2014a

252 CHAPTER 8. DIFFERENTIAL ANALYSIS
rearranging equation (8.92) transforms it into
3 τ x’x’ = τ x’x’ + τ y’y’ + τ z’z’ +6μ ∂U (
x’ ∂Ux’
− 2 μ + ∂U y’
+ ∂U )
z’
∂x’ ∂x’ ∂y’ ∂z’
(8.93)
Dividing the results by 3 so that one can obtained the following
τ x’x’ =
“mechanical” pressure
{ }} {
τ x’x’ + τ y’y’ + τ z’z’
3
+2 μ ∂U x’
∂x’
− 2 3 μ (
∂Ux’
∂x’
+ ∂U y’
∂y’
+ ∂U )
z’
∂z’
(8.94)
The “mechanical” pressure, P m , is defined as the (negative) average value of pressure
in directions of x’–y’–z’. This pressure is a true scalar value of the flow field since
the propriety is averaged or almost 17 invariant to the coordinate transformation. In
situations where the main diagonal terms of the stress tensor are not the same in all
directions (in some viscous flows) this property can be served as a measure of the local
normal stress. The mechanical pressure can be defined as averaging of the normal stress
acting on a infinitesimal sphere. It can be shown that this two definitions are “identical”
in the limits 18 . With this definition and noticing that the coordinate system x’–y’ has no
special significance and hence equation (8.94) must be valid in any coordinate system
thus equation (8.94) can be written as
τ xx = −P m +2μ ∂U x
∂x + 2 3 μ ∇·U (8.95)
Again where P m is the mechanical pressure and is defined as
Mechanical Pressure
P m = − τ xx + τ yy + τ zz
3
(8.96)
It can be observed that the non main (diagonal) terms of the stress tensor are represented
by an equation like (8.72). Commonality engineers like to combined the two difference
expressions into one as
or
(
τ xy = − P m + 2 ) =0
3 μ∇·U
{}}{
( ∂Ux
δ xy +μ
∂y + ∂U )
y
∂x
(
τ xx = − P m + 2 ) =1
3 μ∇·U
{}}{
( ∂Ux
δ xy +μ
∂x + ∂U )
y
∂y
Advance material can be skipped
(8.97)
(8.98)
17 It identical only in the limits to the mechanical measurements.
18 G. K. Batchelor, An Introduction to Fluid Mechanics, Cambridge University Press, 1967, p.141.