$4-(35)$ For flow of a viscous and incompressible fluid, the governing equations, i$.$e$.$ the Conservation of Mass and the Conservation of Linear Momentum Equations are given below.

Note that grad is a vectoral operator and the bold symbols $V$ and $g$ are vectoral quantities velocity and the gravitational acceleration.

Continuity:

$*V=0$

Momentum:

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

$($a$)$ Write the open form of the continuity equation in $2-$Dimensional $x-y$ plane $(5),$

$($b$)$ Write the open forms of the Momentum Equation in $x-y$ plane $(x$ and $y$ direction equations$)(5),$

$($c$)$ Simplify these equations for a steady, incompressible, unidirectional flow $($i$.$e$.,$ u is the only nonzero velocity component; $v=0)(5),$

$($d$)$ For Couette flow without pressure gradient $d\frac{p}{d}x=0$ between the plates shown in the figure above, determine the velocity distribution between the plates using the simplified equaitons in $($c$)(20),$

$($e$)$ Sketch the velocity distribution $(5).$

The Mass and Linear Momentum Conservation Equations for a fixed control volume are:

${\displaystyle \underset{CV}{\overset{\phantom{}}{}}}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}dV+{\displaystyle \underset{CS}{\overset{\phantom{}}{}}}(V*n)dA=0$

$\frac{dV}{dt}=\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{t}}+u\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{x}}+v\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{y}}+w\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{z}},$grad$=\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}$hat$(i)+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}$hat$(j)+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}$hat$(k),gra{d}^2=$grad$*grad=\frac{de{l}^2}{del{x}^2}+\frac{c^2}{del{y}^2}+\frac{de{l}^2}{del{z}^2}$