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Fluid Mechanics
Bernoulli's Equation & Flow
GATE 2019
Medium

Two immiscible, incompressible, viscous fluids with the same densities but different viscosities are contained between two infinite horizontal parallel plates 2 m apart, split into two equal 1 m layers. The bottom plate is fixed; the upper plate moves at a constant 3 m/s. Assuming steady, fully-developed laminar flow with zero pressure gradient, if the dynamic viscosity of the lower fluid μ2 is twice that of the upper fluid μ1, the velocity at the interface (round off to two decimal places) is ___ m/s.

1 m/s(± 0.02 accepted)

Explanation

With d²u/dy²=0 in each layer, velocity is linear in each. Shear stress is continuous at the interface: μ1(U−Vi)/h1 = μ2(Vi−0)/h2. With h1=h2=1m and μ2=2μ1: μ1(3−Vi) = 2μ1·Vi → 3−Vi = 2Vi → Vi = 1 m/s.

Source: GATE ME 2019 Session 1 (Set 1) (2019)

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