Question

Consider a steady, fully-developed, uni-directional laminar flow of an incompressible Newtonian fluid (viscosity $\mu$) between two infinitely long horizontal plates separated by a distance $2H$ as shown in the figure. The flow is driven by the combined action of a pressure gradient and the motion of the bottom plate at $y = -H$ in the negative $x$ direction. Given that $\Delta P/L = (P_1 - P_2)/L > 0$, where $P_1$ and $P_2$ are the pressures at two $x$ locations separated by a distance $L$. The bottom plate has a velocity of magnitude $V$ with respect to the stationary top plate at $y = H$. Which one of the following represents the $x$-component of the fluid velocity vector?

• A. $\frac{\Delta{P} H^2}{L 2\mu} \left( 1 - \frac{y^2}{H^2} \right) + \frac{V}{2} \left( \frac{y}{H} - 1 \right)$
• B. $\frac{\Delta{P} H^2}{L 2\mu} \left( \frac{y^2}{H^2} - 1 \right) + \frac{V}{2} \left( \frac{y}{H} - 1 \right)$
• C. $\frac{\Delta{P} H^2}{L 2\mu} \left( \frac{y^2}{H^2} - 1 \right) - \frac{V}{2} \left( \frac{y}{H} - 1 \right)$
• D. $\frac{\Delta{P} H^2}{L 2\mu} \left( 1 - \frac{y^2}{H^2} \right) - \frac{V}{2} \left( \frac{y}{H} - 1 \right)$

Answer 1

Answer: (A)

$\frac{\Delta{P} H^2}{L 2\mu} \left( 1 - \frac{y^2}{H^2} \right) + \frac{V}{2} \left( \frac{y}{H} - 1 \right)$

Answer 2

The correct option is (A):$\frac{\Delta{P} H^2}{L 2\mu} \left( 1 - \frac{y^2}{H^2} \right) + \frac{V}{2} \left( \frac{y}{H} - 1 \right)$