Question $(100$ points$).$ Consider a fluid bounded by two parallel plates extended to infinity much

that no end effects are encountered. The planar walls and the fluid are initially at rest. The lower

wall is suddenly accelerated in the x$-$direction, as shown in Figure $3-6($Hoffman$,2000).$ A spatial

coordinate system is selected such that the lower wall includes the xz plane to which the $y-$axis is

perpendicular. The spacing between the two plates is denoted by h$.$ The Navier$-$Stokes equations

for this problem can be written as

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}=$

$u\frac{de{l}^{2}u}{del{y}^2}$

where

$u$ is the kinematic viscosity of the fluid.

The initial and boundary conditions for this problem are stated as follows:

a$.$ Initial Condition:

hmm$0\mathrm{.}000217\frac{m^2}{s}\frac{1}{3}y=0\mathrm{.}001mu=u(t,y)$