Question

A steady laminar boundary layer is formed over a flat plate as shown in the figure. The free stream velocity of the fluid is $U_0$. The velocity profile at the inlet a-b is uniform, while that at a downstream location c-d is given by $u=U_0[2\left(\frac{y}{\delta }\right)-{\left(\frac{y}{\delta }\right)}^2]$.

The ratio of the mass flow rate, ${\dot{m}}_{bd}$ , leaving through the horizontal section b-d to that entering through the vertical section a-b is________

• A. 0.333

Answer 1

The correct option is A 0.333

$U_0$=Velocity of free stream
$\delta$=Boundary layer thickness at section c-d
b=Width of the plate
Velocity profile,

$u=U_0[2\left(\frac{y}{\delta }\right)-{\left(\frac{y}{\delta }\right)}^2]$

Mass flow rate through section a-b,
${\dot{m}}_{ab}=\rho A{U}_0$
= $\rho b\delta U_0$

Mass flow rate through section c-d,
${\dot{m}}_{cd}=\underset{0}{\overset{\delta }{\int }}\rho bdyU$

$=\underset{0}{\overset{\delta }{\int }}\rho b{U}_0[2\frac{y}{\delta }-{\left(\frac{y}{\delta }\right)}^2]dy$

${\dot{m}}_{cd}=\rho b{U}_0{[\frac{2y^2}{2\delta }-\frac{y^3}{3{\delta }^2}]}_0^{\delta }$

$=\rho b{U}_0[\frac{{\delta }^2}{\delta }-\frac{{\delta }^3}{3{\delta }^2}]$

$=\rho b{U}_0[\delta -\frac{\delta }{3}]=\rho b{U}_0\times \frac{2}{3}\delta$

$=\frac{2}{3}\rho b\delta U_0$

Mass flow rate through section b-d,

$\dot{m_{bd}}=\dot{m_{ab}}-\dot{m_{cd}}$

$=\rho b\delta U_0-\frac{2}{3}\rho b\delta U_0$

$=\frac{\rho b\delta U_0}{3}$

$\frac{\dot{m_{bd}}}{\dot{m_{ab}}}=\frac{\rho b\delta U_0}{3\times \rho b\delta U_0}=\frac{1}{3}=0.333$