SUMMARY The effect of wall porosity on the two dimensional steady state

# Summary the effect of wall porosity on the two

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SUMMARY The effect of wall porosity on the two-dimensional steady-state incompressible laminar flow of a fluid in a channel having rectangular cross section has been investigated in detail by the solution of the Navier-Stokes equations. . With the assumption of uniform suction at the walls an exact solution of the flow equations is obtained leading to a third-order nonlinear differential equation with appropriate boundary conditions. For small flows through the porous walls, this latter equation is solved by a perturbation method. First-order expressions for the dependence of the velocity components and the pressure on positional coordinates, channel dimensions and fluid properties are obtained. ' The velocity "profile" in the major flow direction is found to deviate from the Poiseuille parabola by being flatter at the center of the channel and steeper in the region close to the walls, the degree of deviation de-pending on a Reynolds number for the flow through the channel walls. The pressure drop in the direction of the main flow is found to be appreciably less in the porous wall channel 40 80 120 160 200 FIG. 2. Flow lines in the upper half of a channel with porous walls. R= 1; N R.= 1000. 240 than that in a solid wall channel having the same dimensions and the same entrance Reynolds number. NOMENCLATURE 2h= distance between the channel walls. k=an integration constant. p(x, X)=the pressure in the channel at the point (x, X). u(x, X) = the velocity component in the x direction at the point (x, X) in the channel. u(x) = the x component of velocity averaged over the channel cross section at position x along the channel. vex, X) = the X component of velocity at the point (x, X) in the channel. Vw= the velocity of the fluid leaving the channel walls. R = a Reynolds number for the flow through the channel wall, R=hvw/ll. N Re= a Reynolds number for the flow entering the channel, N Re=4hil(0)/II. X=dimensionless distance parameter, X= (y/h). ~ = the fluid viscosity. 11= the kinematic viscosity ~/p). 1f= the stream function. 'Ir = the dimensionless stream function. ACKNOWLEDGMENT Dr. R. F. Fledderman'st contributions, in the form of suggestions concerning various methods of attack on the problem, are gratefully acknowledged. Appreciation and thanks are extended to Dr. L. Geller, Dr. J. D. Justice, Dr. F. A. Ficken, and Mr. H. L. Weissberg for many stimulating discussions during the course of this work. The author wishes particularly to express his in-debtedness to Dr. L. Geller for his suggestions leading to the clarification and simplification of the mathe-matical arguments. t Under U. S. Atomic Energy Commission contract with the State Engineerin& Experiment Station at the Georgia Institute of T~chnology. Project No. 202-146. Dr. M. J. Goglia, Project Dlreetor. Downloaded 09 Dec 2011 to 128.210.126.199. Redistribution subject to AIP license or copyright; see