Chapter 32, Problem 5P

Two plates are 10cmapart, as shownin Fig.P32.5. Initially, both plates and the fluid are still. At $t=0$, the top plate is moved at a constant velocity of 8 cm/s. The equations governing the motions of the fluids are

$\frac{\partial v_{\text{oil}}}{\partial t}={\mu }_{\text{oil}}\frac{{\partial }^2{v}_{\text{oil}}}{\partial x^2}\text{ and }\frac{\partial v_{\text{water}}}{\partial t}={\mu }_{\text{water}}\frac{{\partial }^2{v}_{\text{water}}}{\partial x^2}$

and the following relationships hold true at the oil-water interface:

$v_{\text{oil}}=v_{\text{water}}\text{ and }{\mu }_{\text{oil}}\frac{\partial v_{\text{oil}}}{\partial x}={\mu }_{\text{water}}\frac{\partial v_{\text{water}}}{\partial x}$

What is the velocity of the two fluid layers at $t=0.5,1,\text{ and 1}\text{.5 s}$ at distances $t=2,\text{}4,6$, and 8 cm from the bottom plate? Note that ${\mu }_{\text{water}}$ and ${\mu }_{\text{oil}}=1$ and 3 cp, respectively.