hw11.pdf - (Advanced Transport Phenomena(I Problem Set 11(due Friday December 2 Problems on viscous flows exact solutions stream function and vorticity

hw11.pdf - (Advanced Transport Phenomena(I Problem Set...

(Advanced) Transport Phenomena (I)Problem Set 11(due Friday, December 2)Problems on viscous flows: exact solutions, stream function,and vorticityProblem 1.This problem is motivated by flow in a membrane separator. You are asked toanswer the following questions as well as derive all equations, justify every claim made, andcorrect any errors in the classic paper by A. S. Berman (1953), a copy of which is attachedherewith. You should provide detailed derivations and answer questions and justify claimsinplain English.Consider the steady, laminar flow of an incompressible Newtonian fluid of constant densityρand constant viscosityμin a two-dimensional channel with two equally porous walls thatare separated by a distance 2has shown in the accompanying figure (see the last page ofthe problem set). The fluid flowing in the channel permeates or leaves the channel wallswith constant velocityvwthat is independent of position. Choose a Cartesian coordinatesystem (x,y) with origin at the center of the channel. They-axis is perpendicular to thechannel walls and thex-axis is in a plane parallel to the channel walls.(a) The flow in the channel is governed by the Navier-Stokes and continuity equations,which in vector notation read:v· ∇v=−1ρ∇p+ν∇2v,∇ ·v= 0(1), (2)wherevis the velocity vector with componentsuandvin thex- andy-directions,pis thepressure, andν≡μ/ρis the kinematic viscosity. The effect of external forces, includinggravity, is to be neglected in the analysis. Using the dimensionless variableλ≡y/h, write(1) and (2) in component form. Since the flow in the channel should be symmetric aboutλ= 0, we only need to solve the problem over half the channel, viz. 0≤λ≤1. State theboundary conditions along the symmetry planeλ= 0 and the upper porous wallλ= 1.