Cylinder Oil film, h

A thin film of oil of thickness Y and dynamic viscosity u flows uniformly down an inclined plane as shown in Figure Q2 (in which p is the pressure and A is the cross-sectional area of the element of fluid shown) under laminar conditions. Given that the velocity gradient is zero at the free surface and that the pressure gradient must be zero (i.e. the flow is uniform and atmospheric conditions prevail) at the surface, show that the velocity profile is pg u = (Yy- y/2) sine where u is the local velocity at an elevation y above the bed, 0 is the inclination of the plane to the horizontal and pis the fluid density. Also, du dy where t is the shear stress and du/dy is the velocity gradient. A reservoir discharges through a sluice gate that is 1.2 m wide by 1.4 m deep. The top of the opening is 0.8 m below the water level in the reservoir. The water level just downstream of the sluice gate is below the bottom of the opening. Calculate the theoretical discharge. The theoretical discharge…

Oil Coating: A long, continuous belt is pulled upwards through a chemical oil bath at velocity V0. The belt has rectangular cross-section and has length (L), width into the paper (W). The belt picks up a film of oil of thickness h, density ρ, and dynamic viscosity μ. Gravity g tends to make the oil drain down, but the movement of the belt keeps the fluid from running off completely. Assume fully developed, steady, laminar, incompressible and two-dimensional flow of oil to answer the following questions. Assume that no pressure gradient is needed in the vertical direction to drive the film flow. Also assume that the shear stress at the air-oil interface is zero (free shear condition). Assume no-slip condition for the fluid in contact with the moving belt. Justify any other assumptions you may make. Show all steps. (a) Derive an expression for the two-dimensional velocity field inside the oil film in terms of the known parameters. Clearly indicate your co-ordinates and origin. You must…

Find the force required to pull a shaft with diameter D, steadily into a sleeve with diameter D, and length L at a velocity of V. The clearance between the shaft and the sleeve is filled with an oil of viscosity μ.