P$1\mathrm{.}48$A thin plate is separated from two fixed plates by very viscous liquids $1$ and $2,$ respectively, as in Fig. P$1\mathrm{.}48.$ The plate spacings $h1$ and $h2$ are unequal, as shown. The contact area is A between the center plate and each fluid. $($a$)$ Assuming a linear velocity distribution in each fluid, derive the force $F$ required to pull the plate at velocity $V.($b$)$ Is there a necessary relation between the two viscosities, $1$ and $2?$