Question

In plane Couette flow, what determines the shear stress between the plates?

• A. The fluid's density
• B. The distance between the plates
• C. The velocity of the moving plate
• D. The temperature of the fluid

Hint:

Couette Flow Shear Stress. For laminar flow between parallel plates (distance h, relative velocity U), \(\tau = \mu (U/h)\). Stress depends on viscosity, relative velocity, and plate separation.

Answer 1

The Correct Option is C

Plane Couette flow is the flow between two parallel plates, one stationary and the other moving with a constant velocity \(U\), separated by a distance \(h\).
Assuming laminar flow of a Newtonian fluid, the velocity profile is linear: \(u(y) = U \frac{y}{h}\), where y is the distance from the stationary plate.
The shear stress (\(\tau\)) in a Newtonian fluid is given by Newton's law of viscosity: $$ \tau = \mu \frac{du}{dy} $$ where \(\mu\) is the dynamic viscosity.
Differentiating the velocity profile: $$ \frac{du}{dy} = \frac{d}{dy} \left( U \frac{y}{h} \right) = \frac{U}{h} $$ Substituting this into the shear stress formula: $$ \tau = \mu \frac{U}{h} $$ This shows that the shear stress depends on the fluid's viscosity (\(\mu\)), the velocity of the moving plate (\(U\)), and the distance between the plates (\(h\)).
Comparing the options: - Density (1) doesn't directly appear in the shear stress formula for incompressible Couette flow.
- Distance between plates (2) affects the shear stress (\(\tau \propto 1/h\)).
- Velocity of the moving plate (3) directly affects the shear stress (\(\tau \propto U\)).
This is the driving factor for the flow and stress.
- Temperature (4) affects viscosity (\(\mu\)), which in turn affects shear stress, but it's an indirect effect.
Given the options, the velocity of the moving plate (U) is a primary determinant of the velocity gradient and thus the shear stress.
While viscosity and distance are also crucial, the velocity is the direct cause of the shearing motion.