$3-$ A unidirectional, steady flow i$.$e$.u=u(y)$;$v=0$ with pressure gradient $d\frac{p}{d}x<mn\; id="EditorElement\_754538">0</mn>$ between two parallel

plates a distance $H$ apart, as in the figure below, is to be determined. The viscous liquid is of constant density

$$ and viscosity $.$ The lower plate is fixed while the upper plate is moving with constant velocity $U.$

$($a$)$ In the below, show simplification of the Continuity Equation with the given flow conditions,

$($b$)$ Show simplifications of the $x$ and $y$ scalar components of the Navier Stokes equations, with the given

flow conditions and also considering the continuity equation,

$($c$)$ Solve the resulting $x$ equation for the velocity profile $u(y),$ and sketch it in the figure below.

Continuity:

grad$*V=0$

Momentum:

$\frac{dV}{dt}=g-$gradp$+gra{d}^{2}V$

Continuity Equation grad$*vec(V)=0=>$del$\frac{v}{d}$elx$+$del$\frac{v}{d}$ely$+$del$\frac{w}{d}$elz$=0$

Navior$-$Stokes Equation:

$x$ component $2\frac{u}{d}$elt$+$udel$\frac{u}{d}$elx$+$vdel$\frac{u}{d}$ely$+w^2\frac{u}{d}$elz$=g-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{x}}+(\frac{\text{d}\text{e}\text{l}\text{v}}{del{x}^2}+\frac{de{l}^{2}u}{del{y}^2}+\frac{de{l}^{2}u}{del{z}^2})$

g component del$\frac{v}{d}$elt$+$udel$\frac{v}{d}$elx$+$vdel$\frac{v^2}{d}$ely$+$wdel$\frac{v}{d}$elz$=-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{y}}+(\frac{de{l}^{2}v}{del{x}^2}+\frac{de{l}^{2}v}{del{y}^2}+\frac{de{l}^{2}v}{del{z}^2})$