Page 397 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 381

It is easily verified that v = r and v = 1/r satisfy the above equation. Thus, the general solution
is

/
where A and B are arbitrary constants
Let r\ and TI denote the radii of the inner and outer cylinders, respectively, Qj and Q>i
their respective angular velocities. Then

and

from which the constants A and B can be obtained to be

so that

and

The shearing stress at the walls is equal to

It can be obtained (see Prob. 6.27) that the torque per unit length which must be applied to
the cylinders (equal and opposite for the two cylinders) to maintain the flow is given by

6.16 Flow Near an Oscillating Plate

Let us consider the following unsteady parallel flow near an oscillating plate: