Question

Consider steady fully-developed incompressible flow of a Newtonian fluid between two infinite parallel flat plates. The plates move in the opposite directions, as shown in the figure. In the absence of body force and pressure gradient, the ratio of shear stress at the top surface ($y=H$) to that at the bottom surface ($y=0$) is

• A. $1$
• B. $\dfrac{U_1}{U_2}$
• C. $\dfrac{U_1-U_2}{U_2}$
• D. $\dfrac{U_1+U_2}{U_2}$

Hint:

With no pressure gradient, Couette flow has a {linear} profile and a {constant} shear rate across the gap. Hence the wall shear magnitudes at both plates are equal.

Answer 1

The Correct Option is A, C

Step 1: Governing equation for Couette flow (no $dp/dx$).
Steady, fully developed, incompressible, Newtonian $\Rightarrow$ \[ \mu \frac{d^2 u}{dy^2} = \frac{dp}{dx}=0 \ \Rightarrow\ \frac{d^2 u}{dy^2}=0. \] Hence $u(y)$ is linear: \[ u(y)=Ay+B. \]
Step 2: Apply boundary conditions from plate motions.
Let $u(0)=-U_2$ (bottom plate moving left) and $u(H)=+U_1$ (top moving right). Then \[ u(y)=-U_2+\frac{U_1+U_2}{H}\,y. \] Therefore the velocity gradient is constant: \[ \frac{du}{dy}=\frac{U_1+U_2}{H}. \]
Step 3: Wall shear stresses.
For a Newtonian fluid, $\tau=\mu \dfrac{du}{dy}$ (shear within the fluid). At $y=H$: $\tau_H=\mu\dfrac{U_1+U_2}{H}$. At $y=0$: $\tau_0=\mu\dfrac{U_1+U_2}{H}$. Thus the magnitudes (in the fluid) are {equal}; the directions on the two plates are opposite, but the requested ratio of values is \[ \frac{\tau_H}{\tau_0}=1. \] Final Answer: \[ \boxed{1} \]