Question

Consider flow of an incompressible fluid of density ρ and viscosity μ through a long, horizontal section of round pipe of diameter

d. the velocity profile is shown below; v is the average speed across the pipe cross section, which by conservation of mass remains constant down the pipe. for a very long pipe, the flow eventually becomes hydrodynamically 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, as sketched. the shear stress is also constant down the pipe in the fully developed region. we assume some constant average roughness height ε 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 non- dimensional relationship between shear stress τw and the other parameters in the problem.

Answer 1

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