GATE 2019 CE (CE1) – Question 61
Laminar flow in the x direction occurs between infinite parallel plates. The lower plate is stationary; the upper moves at 1 cm/s in x. Gap is 5 mm and dynamic viscosity 0.01 N·s/m². If lower-plate shear stress is zero, find dp/dx (N/m² per m), to one decimal place.
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Correct answer: 7.9 to 8.1
Explanation
Steady parallel flow gives $\mu u^{\prime\prime}=dp/dx$. Zero lower-wall shear means $u^\prime(0)=0$; no slip gives u(0) = 0. Thus $u(y)=(dp/dx)y^2/(2\mu)$. At H = 0.005 m, u(H) = 0.01 m/s: $$\frac{dp}{dx}=\frac{2\mu U}{H^2}=\frac{2(0.01)(0.01)}{0.005^2}=8.0\ \mathrm{Pa/m}.$$ The positive adverse gradient cancels the wall-driven bottom shear.