A continuous belt, passing upward through a chemical bath at speed U 0 , picks up a liquid film of thickness h , density ρ , and viscosity μ . Gravity tends to make the liquid drain down, but the movement of the belt keeps the liquid from running off completely. Assume that the flow is fully developed and laminar with zero pressure gradient, and that the atmosphere produces no shear stress at the outer surface of the film. State clearly the boundary conditions to be satisfied by the velocity at y = 0 and y = h . Obtain an expression for the velocity profile. P8.31
A continuous belt, passing upward through a chemical bath at speed U 0 , picks up a liquid film of thickness h , density ρ , and viscosity μ . Gravity tends to make the liquid drain down, but the movement of the belt keeps the liquid from running off completely. Assume that the flow is fully developed and laminar with zero pressure gradient, and that the atmosphere produces no shear stress at the outer surface of the film. State clearly the boundary conditions to be satisfied by the velocity at y = 0 and y = h . Obtain an expression for the velocity profile. P8.31
A continuous belt, passing upward through a chemical bath at speed U0, picks up a liquid film of thickness h, density ρ, and viscosity μ. Gravity tends to make the liquid drain down, but the movement of the belt keeps the liquid from running off completely. Assume that the flow is fully developed and laminar with zero pressure gradient, and that the atmosphere produces no shear stress at the outer surface of the film. State clearly the boundary conditions to be satisfied by the velocity at y = 0 and y = h. Obtain an expression for the velocity profile.
A viscous fluid of viscosity 2.48 Pa-s and
density 884 kg/m³ is dragged by a rigid flat surface that moves
upward with speed V, as shown. The velocity profile in the fluid layer
is of the form,
pg
v(x) = (x? – 2hx) + V
Find the minimum speed V for which the entire fluid layer moves
upward. What are the minimum and maximum values of the shear
stress in the layer? Assume the flow to be incompressible and fully
developed.
5 mm
Consider flow of an incompressible fluid of density ρ and viscosity µ through a long, horizontal round pipe of diameter D. V is the average speed remains constant down the pipe. For a very long pipe, the flow eventually becomes fully developed, which means that the velocity profile also remains uniform down the pipe. Because of frictional forces between the fluid and the pipe wall, there exists a shear stress τw on the inside pipe wall. We assume some constant average roughness height, along the inside wall of the pipe. In fact, the only parameter that is not constant down the length of pipe is the pressure, which must decrease (linearly) down the pipe in order to “push” the fluid through the pipe to overcome friction. Develop a nondimensional relationship between shear stress τw and the other parameters in the problem.
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…
Chapter 8 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
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