Problem 4. Water flows down an inclined river bed as shown in the figure. The river bed has an inclined angle of , the depth of the water is h, the width of the bed (in z-direction) is W, the fluid is Newtonian and incompressible, with dynamic viscosity u and density p. y = 0 at the river bottom. a) Simplify the Navier-Stokes equation in in x-direction by eliminating all the trivial terms, and solve the velocity profile by using proper boundary conditions. b) Derive the viscous forces in x- and y- directions, respectively, acted on the cross-section control surface indicated by the blue dashed line, the surface normal of which is (-1, 0, 0). (hint: use an integral of viscous stress over the cross-section area) c) Derive the volumetric flow rate, Q, of the river by integrating velocity over a cross-section area. d) What is mean velocity () as a function of the maximum velocity (Vmax )? n= (-1,0,0) 00