Assume the liquid film in Example 5.9 is horizontal (i.e., \(\theta=0^{\circ}\) ) and that the flow is driven by a constant shear stress on the top surface \((y=h), \tau_{y x}=C\). Assume that the liquid film is thin enough and flat and that the flow is fully developed with zero net flow rate (flow rate \(Q=0\) ). Determine the velocity profile \(u(y)\) and the pressure gradient \(d p / d x\).

Data From Example 5.9

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Example 5.9 ANALYSIS OF FULLY DEVELOPED LAMINAR FLOW DOWN AN INCLINED PLANE SURFACE A liquid flows down an inclined plane surface in a steady, fully developed laminar film of thickness h. Simplify the continuity and Navier-Stokes equations to model this flow field. Obtain expressions for the liquid velocity profile, the shear stress distribution, the volume flow rate, and the average velocity. Relate the liquid film thickness to the volume flow rate per unit depth of surface normal to the flow. Calculate the volume flow rate in a film of water h=1 mm thick, flowing on a surface b = 1 m wide, inclined at 0 15 to the horizontal.