Chapter 6, Problem 17P

A liquid layer separates two plane surfaces as shown. The lower surface is stationary; the upper surface moves downward at constant speed V. The moving surface has width w, perpendicular to the plane of the diagram, and w >> L. The incompressible liquid layer, of density ρ, is squeezed from between the surfaces. Assume the flow is uniform at any cross section and neglect viscosity as a first approximation. Use a suitably chosen control volume to show that u = Vx/b within the gap, where b = b₀ − Vt. Obtain an algebraic expression for the acceleration of a fluid particle located at x. Determine the pressure gradient, ∂p/∂x, in the liquid layer. Find the pressure distribution, p(x). Obtain an expression for the net pressure force that acts on the upper (moving) flat surface.

P6.17