Page 106 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 89
-vi
Y

y2 ------

Fig. 2.28

Example 2.2 For the rectangular region of a two-dimensional flow field depicted in Fig. 2.28,
starting with the definition Eqn (2.81) of circulation, show that it can also be evaluated by
means of the integral around the closed circuit appearing as the last term in Eqn (2.83).
From Eqns (2.76) and (2.81) it follows that

Therefore

But along the lines: C1, q = ui, t = i, ds = dx; C2, q = vj, t = j, ds = dy; C,, q = ui, t = -i,
ds = -dx; and Cq, q = uj, t = -j, ds = -dy. It therefore follows that Eqn (2.84) is equivalent to

2.8 The Navier-Stokes equations

2.8.1 Relationship between rates of strain
and viscous stresses
In solid mechanics the fundamental theoretical model linking the stress and strain
fields is Hooke’s law that states that

Stress 0; Strain (2.85)