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

The Correct Option is A

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