- Consider A Layer Of Fluid Flowing Down A Solid Plane Inclined At An Angle From The Horizontal As Shown In Figure 6 H 1 (173.71 KiB) Viewed 34 times
Consider a layer of fluid flowing down a solid plane inclined at an angle ( from the horizontal, as shown in figure 6. Here, ê, is place along the flow direction (wall tangent) and è, is the wall normal. Gravity acts normal to the horizontal. The density of the incompressible fluid is p and y depicts its viscosity. Assume the flow has reached a steady state so that the layer has a constant thickness h. The atmosphere exerts constant pressure, Poo, and negligible shear on the top layer surface at y = h. y=h h ! Pop ey 01 ex Figure 6: Flow down an inclined plate. 1. Assume two-dimensional flow (velocity/pressure fields can vary in t and y only) and there is no z-velocity. Start with the full incompressible Navier-Stokes equations. Simplify the governing equations by dropping out zero terms. For any terms you cross out, give explicit reasons for why they are zero. For each term dropped out with wrong/no reason, one point will be deducted. (4 pts) 2. Find an expression for the wall-normal velocity u, (z,y); (4 pts) 3. Find an expression for the wall-tangential velocity field u.(z,y): (8 pts) 4. Find an expression for the pressure field p(r, y): (8 pts) (Not necessarily do these field variables depend on both s and y.) 5. Now you tilt the inclined plane so the angle 0 varies. Using the expression you found in 3, report the pressure fields for 0 = 0 and 0 = 90°. (4 pts) 6. For the pressure fields you found for 0 = 0 and 8 = 90°, what do you observe (beyond the mathe- matical expressions themselves?) (2 pts)