If the velocity gradient is given by θ and dynamic viscosity of the fluid is given by μ. What is the shear stress on the wall of the boundary layer in the direction of motion?
Try it first: pick an option, then read the explanation.
Correct answer: A. μθ
Asked in SSC JE 24-01-2018 (Evening Shift)
Detailed Explanation
Wall shear stress governs skin friction drag at the boundary layer surface. Newton's law relates this wall shear stress to dynamic viscosity and the near-wall velocity gradient.
From Newton's law of viscosity:
\[ \tau = \mu \left(\frac{du}{dy}\right)_{\text{wall}} \]
Given \(\left(\frac{du}{dy}\right)_{\text{wall}} = \theta\):
\[ \tau = \mu\theta \]
Why “μθ” Is Correct
Shear stress is the product of dynamic viscosity \(\mu\) and the velocity gradient \(\theta\), which gives \(\tau = \mu\theta\).
Why the Other Options Are Wrong
- B. μ − θ: μ − θ subtracts quantities with different physical dimensions (Pa·s and s⁻¹).
- C. μ/θ: μ/θ divides viscosity by velocity gradient instead of multiplying.
- D. μ: μ has units of dynamic viscosity and represents stress only when the velocity gradient is unity.
Key Point
Wall shear stress τ_w = μ(du/dy)_w = μθ.
Key Concepts for Students
- Boundary layer no-slip condition: fluid velocity at the solid boundary equals wall velocity.
- Skin friction coefficient C_f = τ_w / (0.5 ρ U²).
- Velocity gradient is steepest immediately at the boundary wall.
Practice more: this is question 43 in Properties of Fluid MCQs – Part 3. Try the whole set and check every answer instantly.