Basics of Fluid Mechanics, 2014a

250 CHAPTER 8. DIFFERENTIAL ANALYSIS
coordinates). The linear deformations in the x’ and y’ directions which is rotated 45 ◦
relative to the x and y axes can be expressed in both coordinates system. The angular
strain rate in the (x, y) is frame related to the strain rates in the (x’, y’) frame. Figure
8.11(a) depicts the deformations of the triangular particles between time t and t + dt.
The small deformations a , b, c, and d in the Figure are related to the incremental linear
strains. The rate of strain in the x direction is
dɛ x = c
(8.78)
dx
The rate of the strain in y direction is
dɛ y = a
(8.79)
dx
The total change in the deformation angle is related to tan θ, in both sides (d/dx+b/dy)
which in turn is related to combination of the two sides angles. The linear angular
deformation in xy direction is
dγ xy = b + d
(8.80)
dx
Here, dɛ x is the linear strain (increase in length divided by length) of the particle
in the x direction, and dɛ y is its linear strain in the y-direction. The linear strain in the x ′
direction can be computed by observing Figure 8.11(b). The hypotenuse of the triangle
is oriented in the x’ direction (again observe Figure 8.11(b)). The original length of the
hypotenuse √ √
2dx. The change in the hypotenuse length is (c + b) 2 +(a + d) 2 . It
can be approximated that the change is about 45 ◦ because changes are infinitesimally
small. Thus, cos 45 ◦ or sin 45 ◦ times the change contribute as first approximation to
change. Hence, the ratio strain in the x’ direction is
dɛ x’ =
√
(c + b) 2 +(a + d) 2
√ ≃
2dx
∼0
(c + b) (c + b) { }} {
√ + √ + f (dx’)
2
√ 2
(8.81)
2dx
Equation (8.81) can be interpreted as (using equations (8.78), (8.79), and (8.80)) as
dɛ x’ = 1 ( )
a + b + c + d
= 1 2 dx 2 (dɛ y + dɛ y + dγ xy ) (8.82)
In the same fashion, the strain in y’ coordinate can be interpreted to be
dɛ y’ = 1 2 (dɛ y + dɛ y − dγ xy ) (8.83)
Notice the negative sign before dγ xy . Combining equation (8.82) with equation (8.83)
results in
dɛ x’ − dɛ y’ = dγ xy (8.84)