Question

Consider the steady, incompressible flow of depth h of a liquid of known density ρ and unknown viscosity µ down a flat plate as shown in Figure 1. Air is the fluid above the liquid layer. The force of gravity is in the vertical direction with acceleration g, and the plate is at an angle θ with respect to the horizontal. Assuming the coordinate system as shown, with x aligned with the flow direction, and y normal to the plate, assume that the flow properties are independent of the z-direction. Furthermore, assume that the flow is fully-developed, so that it is independent of x.

ls air liquid се Figure 1: Flow of a liquid down a flat plate The exact solution to this problem is known to be: where U is the peak velocity, occurring at y- h. The boundary conditions for this problem are: #21(y) = 0 at y = 0 (no-slip) Tyz()0 at y (no-stress, assuming air imparts little viscous stress to the liquid) It is decided to use this simple setup to determine the viscosity of the liquid, μ, in terms of the other known parameters of the problem, ie., U, ρ, g, θ, and h. Ifp, g, θ and h are known, and then U is measured, then fu can be determined. (a) From the y-component of the momentum equation, show that the pressure in the liquid is given by: p(y)-Po + pg(h-y) cos independent of x. , where po is the constant ai r pressure above the liquid

Answer 1