Incorporating Darcy's law for pure solvent flow through porous tubes: Asymptotic solution and numerical simulations

Abstract

International audienceA generalized solution for pressure-driven, incompressible, Newtonian flow in a porous tubular membrane is challenging due to the coupling between the transmembrane pressure and velocity. To date, all analytical solutions require simplifications such as neglecting the coupling between the transmembrane pressure and velocity, assuming the form of the velocity fields, or expanding in powers of parameters involving the tube length. Moreover, previous solutions have not been validated with comparison to direct numerical simulation (DNS). We comprehensively revisit the problem to present a robust analytical solution incorporating Darcy's law on the membrane. We make no assumptions about the tube length or form of the velocity fields. The analytic solution is validated with detailed comparison to DNSs, including cases of axial flow exhaustion and cross flow reversal. We explore the validity of typical assumptions used in modeling porous tube flow and present a solution for porous channels in Supporting Information

An Exact Solution for Stokes Flow in a Channel with Arbitrarily Large Wall Permeability

“…We note that in the limit of small permeability, λ → 0, and thus the pressure profile appears to be constant in y as the leading order of cos(λy) is 1. This agrees with the results ofHaldenwang 7 , Karode 10 , Tilton et al 12 , Regirer 16 to leading order. Each of these works shows a similar profile in x to the pressure profile we have presented, namely a linear combination of a hyperbolic sine and cosine.…”

“…Similarly we may find that the leading order profile in v depends on y as (y 3 − 3y) which may be found by noting that the two terms (the sum and sin(λy)) may be found in the leading order to be v(x, y) ≈ G(x)λ(y − y 3 ) + 2G(x)λy = λ(3y − y 3 ), (60) where G(x) is some function of x. We further remark that this is precisely the leading order expression found in Berman 1 and Tilton et al 12 .…”

“…Each of these works shows a similar profile in x to the pressure profile we have presented, namely a linear combination of a hyperbolic sine and cosine. The scale of the arguments of these hyperbolic trig functions presented in Karode 10 and Tilton et al 12 is √ 3A 1 which we have shown is roughly λ in the limit of small permeability (equation 55). The nondimensional approximation for the pressure profile is in agreement in Karode 10 and Tilton et al 12 and is given as…”

“…Although the latter is a more standard choice (see for example Tilton et al 12 ) we elect to use the former for mathematical convenience. Fluid flow between the interior and exterior of the channel is assumed to be driven by the pressure difference across the two regions.…”

“…Also similar to much of the existing work in this field, we assume a symmetric flow profile in the transverse direction (see for example Refs. [7][8][9]12 ). These assumptions are consistent with previous problem statements in this field, and for biological flows the porosity of the channels will be due to orthogonal water channels making the no slip condition in the axial velocity more robust.…”