Effect of slip on existence, uniqueness, and behavior of similarity solutions for steady incompressible laminar flow in porous tubes and channels

Abstract

The existence and multiplicity of similarity solutions for steady, fully developed, incompressible laminar flow in uniformly porous tubes and channels with one or two permeable walls is investigated from first principles. A fourth-order ordinary differential equation obtained by simplifying the Navier-Stokes equations by introducing Berman’s stream function [A. S. Berman, J. Appl. Phys. 24, 1232 (1953)] and Terrill’s transformation [R. M. Terrill, Aeronaut. Q. 15, 299 (1964)] is probed analytically. In this work that considers only symmetric flows for symmetric ducts; the no-slip boundary condition at porous walls is relaxed to account for momentum transfer within the porous walls. By employing the Saffman [P. G. Saffman, Stud. Appl. Math. 50, 93 (1971)] form of the slip boundary condition, the uniqueness of similarity solutions is investigated theoretically in terms of the signs of the guesses for the missing initial conditions. Solutions were obtained for all wall Reynolds numbers for channel flows whereas no solutions existed for intermediate values for tube flows. Introducing slip did not fundamentally change the number or the character of solutions corresponding to different sections. However, the range of wall Reynolds numbers for which similarity solutions are theoretically impossible in tube flows was found to be a weak function of the slip coefficient. Slip also weakly influenced the transition wall Reynolds number corresponding to flow in the direction of a favorable axial pressure gradient to one in the direction of an adverse pressure gradient. Momentum transfer from the longitudinal axis to the walls appears to occur more efficiently in porous channels compared to porous tubes even in the presence of slip.

Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

“…The classic Beavers-Joseph slip condition [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] was originally phrased in terms of the normal derivative of the tangential velocity (i.e., only the first term in square brackets, above), whereas we add the transposed gradient term to make the slip velocity proportional to the wall shear stress-as appears in [27] and [30]. The difference is negligible at macroscopic lengthscales, but becomes significant when the problem is scaled to resolve the (weakly singular) fine structure at the origin.…”

Laminar flow with injection through a long dead‐end cylindrical porous tube: Application to a hollow fiber membrane

“…26 indicates the outsidein direction of the permeate flux. The (positive) ratio of u to Àv linearly increases from the dead end to the open end: (27) which is one at the distance equal to the tube half-radius (n ¼ 1/2), and is much larger than one near the open end (n ) 1).…”

“…Terrill and Thomas 26 transformed the two-point boundary value problem into an initial value problem, provided fundamentally more rigorous analysis (than Berman's 25 ) of numerical and theoretical solutions of YF's problem 24 with an arbitrary wall Reynolds number, and showed the existence of multiple numerical solutions based on the initial guesses using the no-slip boundary condition at the porous wall. Recent theoretical achievements on the fluid flow in permeable tubes include Chellam and Liu's work regarding slip effects on existence and multiplicity of the similarity solutions, 27 which concluded that the slip boundary condition on the permeable wall only weakly influences the transition wall Reynolds number of the flow re-versal 28 with minimal changes in the similarity solutions. In parallel with the further theoretical development from YF's original work on laminar flow through porous wall, noticeable applications of YF's research include a plethora of engineering processes: crossflow ultrfiltration of particulate materials [29][30][31][32] ; computational fluid mechanics modeling in membrane channels and module design [33][34][35] ; fouling behavior of reverse osmosis membranes 36 ; constant properties of duct flow, 37 and laminar elasticoviscous, 38 pulsatile, 39 compressible, 40 and oscillatory 41 flows in permeable channels; laminar heat transfer, 42,43 combustion, 44 and gas/vapor separation 45 ; food science 46 ; and medical applications such as the uptake of tritium-cholesterol on the arterial wall, 47 convective flow and solute concentration in the (bioartificial) kidney, [48][49][50] and the role of the ''resting'' eccrine sweat gland in thermoregulation.…”

Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities

“…The similar method was used by Cox [4] to analyze the symmetric solutions when the two walls are accelerating equally and when one wall is accelerating and the other is stationary. The uniqueness of similarity solution was investigated theoretically by Chenllam and Liu [3] and their work mainly considered the symmetric flow in a channel with slip boundary conditions. All studies mentioned above are for symmetrical flows.…”

“…Existence of multiple solutions. Skalak [22], Cox [4] and Chenllam [3] considered symmetric flow in a channel with porous walls, accelerating walls and slip boundary conditions, respectively. In this section, we extend previous analysis [22,4,3] to investigate asymmetric flow in the channel with porous walls and to discuss the existence of multiple solutions.…”