Basics of Fluid Mechanics, 2014a

8.5. DERIVATIONS OF THE MOMENTUM EQUATION 251
Equation (8.84) describing in Lagrangian coordinates a single particle. Changing it to
the Eulerian coordinates transforms equation (8.84) into
Dɛ x’
Dt − Dɛ y’
Dt = Dγ xy
(8.85)
Dt
From (8.69) it can be observed that the right hand side of equation (8.85) can be
replaced by τ xy /μ to read
Dɛ x’
Dt − Dɛ y’
Dt = τ xy
μ
(8.86)
From equation (8.76) τ xy be substituted and equation (8.86) can be continued
and replaced as
Figure 8.12 depicts the approximate
linear deformation of the element. The linear
deformation is the difference between
the two sides as
Dɛ x’
Dt
= ∂U x’
∂x’
Dɛ x’
Dt − Dɛ y’
Dt = 1
2 μ (τ x’x’ − τ y’y’ ) (8.87)
(8.88)
y’
U y’dt
⎛
⎜
⎝U x’ + ∂Ux ⎞
’
dx
∂x ’ ⎟
⎠ dt
’
x’
⎛
⎞
⎜
⎝U y’ + ∂Uy’ dy
∂y ’ ⎟
’ ⎠ dt
The same way it can written for the y’ coordinate.
Fig. -8.12. Linear strain of the element purple
denotes t and blue is for t + dt. Dashed
Dɛ y’
squares denotes the movement without the linear
change.
Dt = ∂U y’
(8.89)
∂y’
Equation (8.88) can be written in the y’ and is similar by substituting the coordinates.
The rate of strain relations can be substituted by the velocity and equations (8.88) and
(8.89) changes into
(
∂Ux’
τ x’x’ − τ y’y’ =2μ − ∂U )
y’
∂x’ ∂y’
(8.90)
Similar two equations can be obtained in the other two plans. For example in y’–z’ plan
one can obtained
(
∂Ux’
τ x’x’ − τ z’z’ =2μ − ∂U )
z’
(8.91)
∂x’ ∂z’
Adding equations (8.90) and (8.91) results in
2
4
{ }} {
{ }} {
(3 − 1) τ x’x’ − τ y’y’ − τ z’z’ = (6 − 2) μ ∂U (
x’ ∂Uy’
− 2 μ + ∂U )
z’
∂x’ ∂y’ ∂z’
(8.92)