Question

Water is in steady fully developed laminar flow between two horizontal, very wide (W) and long (L) parallel surfaces separated by a distance b. The bottom surface at y 0 moves in the negative x-direction at a speed vo while the top surface at y b is stationary. In addition, a constant pressure gradient dP/dx is acting on the liquid in the x-direction. (a) Write the simplified form of the Navier-Stokes equation and the appropriate boundary conditions. (b) Derive an equation for the velocity profile in terms of the parameters, dPds, b, Vo and μ . (c) Obtain an equation for the volumetric flow rate. (d) Assume that water has a density ρ-1000 kg/m 3 and viscosity μ = 1.0 x 103 kg/ms. If the pressure gradient is dP/dx =-1.5 Pam, b = 6 mm, W = 1.0 m, and the speed of the bottom surface is 1.0 cm/s, calculate the volumetric flow rate of water and specify the direction of flow (positive or negative x-direction).

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