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GATE 2024 Chemical Engineering, Q16 · Fluid Mechanics

The correct answer is B, −αy²/2.

Question

The velocity field in an incompressible flow is v =αxy= αxy î +vy+ v_{y} ĵ +β+ β k̂, where î, ĵ, k̂ are unit vectors in the (x,y,z) Cartesian coordinate system. Given that αα and ββ are constants, and vy=0v_{y} = 0 at y = 0, the correct expression for vyv_{y} is

  • A. −αxy/2−αxy/2
  • B. −αy2/2−αy²/2
  • C. αy2/2αy²/2
  • D. αxy/2αxy/2

Official answer: −αy2/2−αy²/2

Why

GATE 2024 official key: (B). Continuity: ∂(αxy)/∂x+∂vy/∂y+∂β/∂z=0⇒αy+∂vy/∂y=0.\partial (αxy)/\partial x + \partial v_{y}/\partial y + \partial β/\partial z = 0 \Rightarrow αy + \partial v_{y}/\partial y = 0. Integrating: vy=−αy2/2+v_{y} = −αy²/2 + C; at y=0, C=0, so vy=−αy2/2.v_{y} = −αy²/2.

Explanation cross-checked for consistency. Final answer matches the official IIT answer key.

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