Page 403 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 387

Thus,

which gives

where C\ and €2 are constants of integration. Now at y = 0, 0 = 0/ and at y = d, & = Q u,
therefore,

The temperature distribution is therefore given by

6.19 Vorticity Vector

We recall from Chapter 3, Section 3.13 and 14 that the antisymmetric part of the velocity
gradient (Vv) is defined as the spin tensor W. Being antisymmetric, the tensor W is equivalent
to a vector to in the sense that Wx = at x x (see Sect. 2B16). In fact,

Since (see Eq. (3.14.4),

the vector at is the angular velocity vector of that part of the motion, representing the rigid
body rotation in the infinitesimal neighborhood of a material point. Further, o> is the angular
velocity vector of the principal axes of D, which we show below:
Let dx be a material element in the direction of the unit vector n at time t., i.e.,

where ds is the length of dx. Now

But, from Eq. (3.13.6) of Chapter 3, we have