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 \( \frac{\Delta P}{L} = \frac{P_1 - P_2}{L}>0 \), 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. \( \frac{\Delta P H^2}{2\mu L} \left( 1 - \frac{y^2}{H^2} \right) + \frac{V}{2} \left( \frac{y}{H} - 1 \right) \)
• B. \( \frac{\Delta P H^2}{2\mu L} \left( \frac{y^2}{H^2} - 1 \right) + \frac{V}{2} \left( \frac{y}{H} - 1 \right) \)
• C. \( \frac{\Delta P H^2}{2\mu L} \left( \frac{y^2}{H^2} - 1 \right) - \frac{V}{2} \left( \frac{y}{H} - 1 \right) \)
• D. \( \frac{\Delta P H^2}{2\mu L} \left( 1 - \frac{y^2}{H^2} \right) - \frac{V}{2} \left( \frac{y}{H} - 1 \right) \)
  \includegraphics[width=0.5\linewidth]{q36 CE.PNG} \vspace{0.5cm}

Hint:

For combined Couette and Poiseuille flow, the velocity profile is the sum of the contributions from the pressure gradient (parabolic) and the relative plate motion (linear).

Answer 1

The Correct Option is A

Step 1: Velocity profile for combined Couette and Poiseuille flow. The velocity profile for a combined Couette (due to plate motion) and Poiseuille (due to pressure gradient) flow is given by: \[ u_x(y) = u_{\text{Poiseuille}}(y) + u_{\text{Couette}}(y). \] 1. Poiseuille flow contribution: The parabolic velocity profile due to the pressure gradient is: \[ u_{\text{Poiseuille}}(y) = \frac{\Delta P H^2}{2\mu L} \left( 1 - \frac{y^2}{H^2} \right). \] 2. Couette flow contribution: The linear velocity profile due to the relative motion of the plates is: \[ u_{\text{Couette}}(y) = \frac{V}{2} \left( \frac{y}{H} - 1 \right). \] Step 2: Combine the contributions. The total velocity profile is: \[ u_x(y) = \frac{\Delta P H^2}{2\mu L} \left( 1 - \frac{y^2}{H^2} \right) + \frac{V}{2} \left( \frac{y}{H} - 1 \right). \] Step 3: Conclusion. The \( x \)-component of the fluid velocity vector is: \[ u_x(y) = \frac{\Delta P H^2}{2\mu L} \left( 1 - \frac{y^2}{H^2} \right) + \frac{V}{2} \left( \frac{y}{H} - 1 \right). \] This corresponds to option (1).