Page 395 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 379

In terms of stress tensors, we have T- '62 = "P '02- That is

In other words, these stress components must be continuous across the fluid interface. Since

the condition 7\j = 1\J gives

Note that this condition means that the slope of the velocity profile is not continuous at
jt'2 = 0. Also

and

so that 722 = 7}j at x 2 = 0 gives Cj = C^ = p 0, the pressure at the interface. Since 7^ = 0
and TJ% - 0, the condition T$ = T$ is clearly satisfied. From Eqs. (vi,vii,viii,xiii), we obtain

and

Thus, the velocity distributions are

and