State assumptions made at each stage. Plot the dimensionless velocity profile for different value of $\frac{dp}{dx}$.

Coutte flow:

Navier-stokes equation in x-direction

$u\frac{du}{dx}+v\frac{du}{dy}+w\frac{du}{dz}+\frac{du}{dt}=F_x-\frac{1}{S}\frac{\partial P}{\partial x}+\frac{u}{S}[\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}+\frac{\partial ^2u}{\partial z^2}]$...........(1)

Assumptions:

i) Fluid is incompressible (S=constant)

ii) Flow is one dimensional (x-direction)

$\therefore u\ne 0, v=0,w=0$

iii) Flow is steady $\therefore \frac{\partial u}{\partial t}=0$

iv) The flow is independent of any variation in z-diection

v) The body forces per unit mass are zero. $\therefore F_x=F_y=F_x=0$

$\therefore$ From continuity equation

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$

$\therefore \frac{\partial u}{\partial x}=0$

$\therefore u=f(y)$

Considering above all assumptions equation (1) becomes

$0=-\frac{1}{S}\frac{\partial P}{\partial x}+\frac{u}{S}\frac{\partial ^2u}{\partial y^2}$

$\therefore \frac{\partial ^2u}{\partial y^2}=\frac{1}{u}\frac{\partial P}{\partial x}$

Integrating,

$\frac{\partial u}{\partial y}=\frac{1}{u}\frac{\partial P}{\partial x}\cdot y+c_1$

Again integrating,

$u=\frac{1}{2u}\frac{\partial P}{\partial x}y^2+c_1\cdot y+c_2$............(2)

where $c_1$ & $c_2$ are constants of integration.

Boundary conditions are

i) at y=0,u=0 $\therefore c_2=0$

ii) at y=b,u=U...........Velocity of moving plate

Using boundary conditions

$U=\frac{1}{2u}\cdot \frac{\partial P}{\partial x}b^2 + c_1\cdot b$

$\therefore c_1=\frac{U-\frac{1}{2u}\cdot \frac{\partial P}{\partial x}\cdot b^2}{b}$

$c_1=\frac{U}{b}-\frac{1}{2u}\cdot \frac{\partial P}{\partial x}\cdot b$

Putting $c_1$ and $c_2$ in equation (2)

$U=\frac{1}{2u}\cdot \frac{\partial P}{\partial x}y^2+[\frac{U}{b}-\frac{1}{2u}\cdot \frac{\partial P}{\partial x}\cdot b]y+0$

$u=\frac{U}{b}\cdot y+\frac{1}{2u}(-\frac{\partial P}{\partial x})(by-y^2)$..........Equation for velocity profile in coutte flow

Velocity profile for different $\frac{\partial P}{\partial x}$