Consider a steady, fully-developed, uni-directional laminar flow of an incompressible Newtonian fluid (viscosity μ) between two infinitely long horizontal plates separated by a distance 2H, as shown in the figure. The flow is driven by the combined action of a pressure gradient and the motion of the bottom plate at y = −H in the negative x direction with speed V, relative to the stationary top plate at y = H. Given (P1−P2)/L 0, where P1 and P2 are the pressures at two x locations separated by distance L. Which one of the following represents the x-component of the fluid velocity vector?

Question

Consider a steady, fully-developed, uni-directional laminar flow of an incompressible Newtonian fluid (viscosity μ) between two infinitely long horizontal plates separated by a distance 2H, as shown in the figure. The flow is driven by the combined action of a pressure gradient and the motion of the bottom plate at y = −H in the negative x direction with speed V, relative to the stationary top plate at y = H. Given (P1−P2)/L > 0, where P1 and P2 are the pressures at two x locations separated by distance L. Which one of the following represents the x-component of the fluid velocity vector?

Accepted Answer

The correct answer is A, [(P1−P2)H²/(2μL)]·(1−(y/H)²) + (V/2)·((y/H)−1). GATE 2024 official key: (A). Solving μ·d²u/dy² = −(P1−P2)/L with u(H)=0 and u(−H)=−V gives u(y) = [(P1−P2)H²/(2μL)]·(1−(y/H)²) + (V/2)·((y/H)−1).