explain thoroughly please

OuestionA$1[30$ marks$]$

The flow of a layer of viscous fluid is induced between a fixed lower plate and an upper plate moving at constant velocity Vas shown in Figure QA$1($a$).$ The clearance between plates is $\backslash ($ h $\backslash ).$ The fluid is Newtonian and the fluid flow is in the steady state. Assume the fluid does not slip at either plate. If the plates are infinitely large, the $\backslash ($ z $\backslash )$ and $\backslash ($ y $\backslash )$ component of fluid velocity can be assumed to be zero. The fluid can only flow in the $\backslash ($ x $\backslash )$ direction, and the fluid velocity $\backslash ($ u $\backslash )$ is only a function of $\backslash ($ y $\backslash ).$ The fluid flow is fully developed and the dynamic viscosity of the fluid is $\backslash (\backslash$mu $\backslash ).$ Assume the fluid flow is laminar. $\backslash ($ y$=0\backslash )$ is at the inner surface of the lower plate.

$($a$)$ Derive the mathematical expressions for the velocity profile of fluid, i$.$e$.\backslash ($ u $\backslash )$ as a function of $\backslash ($ y $\backslash ),$ by using $\backslash ($ V$,$ h $\backslash )$ and $\backslash (\backslash$mu $\backslash )$ as input parameters.

$($b$)$ Schematically sketch the expected fluid velocity profile $\backslash ($ u$($y$)\backslash )$ between the plates. $[4]$

Extend the problem to a case of two layers of different immiscible viscous Newtonian fluids flowing between a fixed lower plate and an upper plate moving steadily at velocity V as shown in Figure QA$1($b$).$ The dynamic viscosity of the two different fluids is $\backslash (\backslash$mu$\_\{1\}\backslash )$ and $\backslash (\backslash$mu$\_\{2\}\backslash )$ respectively. The layer thickness of the two different fluids $\backslash ($ h$\_\{1\}\backslash )$ and $\backslash ($ h$\_\{2\}\backslash )$ are unequal. The fluid velocity and the fluid shear stress at the interface between the two different fluids are continuous, i$.$e$.\backslash ($ u$\_\{1\}=$u$\_\{2\}\backslash )$ and $\backslash (\backslash$tau$\_\{1\}=\backslash$tau$\_\{2\}\backslash )$ at the inter$-$fluid interface.

$($c$)$ Derive the mathematical expressions for the velocity profile of fluid $(1)$ and the velocity profile of fluid $\backslash (\backslash$quad $\backslash )$ as a function of $\backslash ($ y $\backslash ),$ by using $\backslash ($ V$,$ h$\_\{1\}\backslash )$ and $\backslash ($ h$\_\{2\}\backslash ),$ and $\backslash (\backslash$mu$\_\{1\}\backslash )$ and $\backslash (\backslash$mu$\_\{2\}\backslash )$ as input parameters.