A fluid is in laminar motion between two parallel plates separated by a distance 'b' under the cation of one of the plates and also under the pressure of a pressure gradient in such a way that the net forward discharge across any section is zero. Consider 'v' be the velocity of the moving plate.

i) Find the point where the minimum velocity occurs and its magnitude.

ii) Draw a rough sketch of velocity distribution across any section.

Data:-

Here, fluid is in laminar motion between two parallel plates separated by distance 'b' and under the action of one of the plates

Hence, it is case of couette flow

$\therefore$ vol.of moving plate=0

$\theta=0$

To find:-

i) Point where minimum velocity occur=?

ii) Velocity distribution across any section=?

Solution:-

Velocity distribution of couette flow is

$u=\frac{U}{b}\cdot y+\frac{1}{2u}(-\frac{\partial P}{\partial x})(by-y^2)$........(1)

and discharge per unit width Q

$Q=\frac{Ub}{2}-\frac{b^3}{12\mu}\cdot \frac{\partial P}{\partial x}$

but Net forward discharge

Q=0

$0=\frac{Ub}{2}-\frac{b^3}{12\mu}\cdot \frac{\partial P}{\partial x}$

$\frac{\partial P}{\partial x}=\frac{Ub}{2}\times \frac{12\mu}{b^3}$

$\frac{\partial P}{\partial x}=\frac{6\mu U}{b^2}$

Minimum velocity occurs when,

$\frac{\partial u}{\partial y}=0$

$\therefore$ Differentiating equation (1) w.e.t y and equating to zero

$\therefore \frac{d}{dy}[u=\frac{U}{b}\cdot y-\frac{1}{2u} \frac{\partial P}{\partial x}(by-y^2)]=0$

$\frac{U}{b}-\frac{1}{2u} \frac{\partial P}{\partial x}(by-y^2)=0$

$\frac{U}{b}=\frac{1}{2u} \frac{\partial P}{\partial x}(b-2y)$

$(b-2y)=\frac{U}{b}\times \frac{2u}{\frac{\partial P}{\partial x}}$

$(b-2y)=\frac{U}{b}\times \frac{2\cdot u\cdot b^2}{6\cdot \mu\cdot U}$.......$\because \frac{\partial P}{\partial x}=\frac{6\mu U}{b^2}$

$\therefore 2y=b-\frac{b}{3}$

$\therefore y=\frac{b}{3}$

Hence minimum velocity occurs at $\frac{b}{3}$ from fixed plate.

The magnitude of minimum velocity can be obtained by putting $y=\frac{b}{3}$ in equation (1)

$\therefore u=\frac{U}{b}\cdot y-\frac{1}{2u}\frac{\partial P}{\partial x}(by-y^2)$

$u_{min}=\frac{U}{b}\times \frac{b}{3}-\frac{1}{2u}(\frac{6\mu U}{b^2})[b\times \frac{b}{2}-\frac{b^2}{g}]$

$u_{min}=\frac{U}{3}-\frac{3U}{b^2}[\frac{3b^2}{g}-\frac{b^2}{g}]$

$=\frac{U}{3}-\frac{3U}{b^2}[\frac{2b^2}{g}]$

$=\frac{U}{3}-\frac{2U}{3}$

$u_{min}=-\frac{U}{3}$

ii) Velocity distribution

$u=\frac{U}{b}\cdot y-\frac{1}{2u}\frac{\partial P}{\partial x}(by-y^2)$

$=\frac{U}{b}\cdot y-\frac{1}{2u}(\frac{6\mu U}{b^2})(by-y^2)$

$u=\frac{U}{b}\cdot y-\frac{3U}{b^2}(by-y^2)$

The velocity distribution may be drawn by substituting values of 'y' such as 0.1b, 0.2b, 0.3b etc