1) Consider steady state flow of an incompressible Newtonian fluid between parallel plates of length L...

Transcribed Image Text:

1) Consider steady state flow of an incompressible Newtonian fluid between parallel plates of length L and width has shown. P. U h P P 1 L The flow is driven by the pressure difference between the entrance and the exit of the channel. The Navier-Stokes equations that govern the flow are Continuity: V.V=0 Momentum: p(V VV) = + where the effect of gravity is neglected. Here, V is the velocity vector, p is the density, p is the pressure and is the viscosity. Assume fully developed flow. a) Briefly explain the physics of the developing flow in the entrance region and the fully developed flow regime that govern the velocity field, i.e., qualitatively describe the behavior of the flow in terms of the cross section and along the x- direction relating this to the terms in the Navier-Stokes equation. b) Non-dimensionalize the full Navier-Stokes equation using suitable variables and discuss the flow physics of resulting non-dimensional parameters. Simplify this equation, justifying all simplifications, and arrive at the equations that govern the flow in the fully-developed region. Solve for the velocity profile. c) Now consider the entrance region, the flow is developing from the uniform flow at the channel entrance to the fully developed velocity profile found in (b). Consider that the fully developed flow starts at the location, Le, where the two boundary layers (one over the upper plate and one over the lower plate) merge. Show from the order of magnitude analysis that the entrance length Le is a function of the Reynolds number. 1) Consider steady state flow of an incompressible Newtonian fluid between parallel plates of length L and width has shown. P. U h P P 1 L The flow is driven by the pressure difference between the entrance and the exit of the channel. The Navier-Stokes equations that govern the flow are Continuity: V.V=0 Momentum: p(V VV) = + where the effect of gravity is neglected. Here, V is the velocity vector, p is the density, p is the pressure and is the viscosity. Assume fully developed flow. a) Briefly explain the physics of the developing flow in the entrance region and the fully developed flow regime that govern the velocity field, i.e., qualitatively describe the behavior of the flow in terms of the cross section and along the x- direction relating this to the terms in the Navier-Stokes equation. b) Non-dimensionalize the full Navier-Stokes equation using suitable variables and discuss the flow physics of resulting non-dimensional parameters. Simplify this equation, justifying all simplifications, and arrive at the equations that govern the flow in the fully-developed region. Solve for the velocity profile. c) Now consider the entrance region, the flow is developing from the uniform flow at the channel entrance to the fully developed velocity profile found in (b). Consider that the fully developed flow starts at the location, Le, where the two boundary layers (one over the upper plate and one over the lower plate) merge. Show from the order of magnitude analysis that the entrance length Le is a function of the Reynolds number.