Question

A steady incompressible laminar flow exists between two long parallel plates separated by \(h = 1\ \text{m}\). The bottom plate is fixed and the upper plate moves with velocity \(U = 10\ \text{m/s}\). The kinematic viscosity of the fluid is \(10^{-6}\ \text{m}^2/\text{s}\) and density is \(10^3\ \text{kg/m}^3\). No pressure gradient acts. Then the shear stress at the bottom plate is ____________ N/m\(^2\) (correct to two decimal places).

Hint:

In Couette flow, shear stress is uniform across the gap and depends only on viscosity and plate velocity.

Answer 1

Correct Answer: 0.01

For Couette flow, velocity profile is linear: \[ u(y) = U \frac{y}{h} \] Shear stress: \[ \tau = \mu \frac{du}{dy} \] Dynamic viscosity: \[ \mu = \rho \nu = 10^3 \times 10^{-6} = 10^{-3}\ \text{Pa·s} \] Velocity gradient: \[ \frac{du}{dy} = \frac{U}{h} = \frac{10}{1} = 10\ \text{s}^{-1} \] Thus: \[ \tau = 10^{-3} \times 10 = 0.01\ \text{N/m}^2 \] Thus the value lies in: \[ \boxed{0.01\ \text{to}\ 0.01} \]
Final Answer: 0.01 N/m\(^2\)