Page 102 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 85

TB

Fig. 2.25

Therefore, if we neglect the higher-order terms,

6X {
avsx avsy ( avsx aVsy)}st = -st av
YCt -YA‘ -
a= - (vc- VA)-= v+ ------ v ------ -
6X St ax2 ay2 ax 2 ay 2 sx ax
(2.7 1 a)

(2.71b)

The rate of shear strain in the xy plane is given by

In much the same way, for three-dimensional flows it can be shown that there are
two other components of the rate of shear strain

)
aw
d7XL- -+- au , (2.72b, c)
1
-- ( ax az
dt
2
2.7.3 Rate of direct strain
Following an analogous process we can also calculate the direct strains and their
corresponding rates of strain, for example
Exx = xF‘-xE’ - (UF‘-UE’)stz (.+--- (u-g$)}-=-&t (2.73)
st au
au sx
-
XF - XE SX ax 2 sx ax