Water at 60°C flows between two large flat plates. The lower plate moves to the left at a speed of 0·3 m/s. The plate spacing is 3 mm and the flow is laminar. The pressure gradient required to produce zero net flow at a cross-section is : (Take μ = 5 × 10⁻⁴ Ns/m² at 60°C)
- A– 100 N/m².mCorrect
- B– 10 N/m².m
- C– 1000 N/m².m
- D– 50 N/m².m
Solution
Concept
Combined Couette and Poiseuille flow
Formula
q = U h / 2 − (h³ / 12 μ) (dp/dx)
Step-by-step solution
1. Zero net flow: U h/2 = (h³/12μ)(dp/dx). 2. dp/dx = 6 μ U / h² = 6 × 5×10⁻⁴ × (−0.3) / (0.003)². 3. = −9×10⁻⁴ / 9×10⁻⁶ = −100 N/m² per m (the sign is negative since the plate moves in the negative direction).
Explanation
Flow caused by the moving plate (Couette flow) is exactly cancelled by the pressure-driven flow when dp/dx = 6μU/h². The pressure falls in the direction opposite to the plate motion.
Why the other options are wrong
−10, −50 and −1000 N/m²·m are 0.1, 0.5 and 10 times the correct value.
Shortcut
Zero net flow: dp/dx = 6μU/h².
Common mistake
Using h in mm.
Correct answer: A) – 100 N/m².m