Incompressible laminar flow through hollow fibers: a general study by means of a two-scale approach

Abstract

We study the laminar flow of an incompressible Newtonian fluid in a hollow fiber, whose walls are porous. We write the Navier-Stokes equations for the flow in the inner channel and Darcy's law for the flow in the fiber, coupling them by means of the Beavers-Joseph condition which accounts for the (possible) slip at the membrane surface. Then, we introduce a small parameter ε 1 (the ratio between the radius and the length of the fiber) and expand all relevant quantities in powers of ε. Averaging over the fiber cross section, we find the velocity profiles for the longitudinal flow and for the cross-flow, and eventually, we determine the explicit expression of the permeability of the system. This work is also preliminary to the study of more complex systems comprising a large number of identical fibers (e.g., ultrafiltration modules and dialysis). (2010). Primary 35Q35; Secondary 76S05. Mathematics Subject Classification

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.…”

Designing irrigation pipes

“…Equation (4) is mass balance, expressed in the form of flux continuity, equation (6) is peculiar to the dead end configuration, equation (5) is the classical no-slip condition, and equations (7), (8) are consequence of the symmetry of the flow field around the pipe axis. In [1] interface conditions more general than no-slip have been examined, but (5) is adequate for the present case. Indeed an estimate of the Beavers-Joseph coefficient based on the specific values of porosity and permeability (see [1], for more details) shows that the slip effect plays no role at the zero order in ε.…”

“…In [1] interface conditions more general than no-slip have been examined, but (5) is adequate for the present case. Indeed an estimate of the Beavers-Joseph coefficient based on the specific values of porosity and permeability (see [1], for more details) shows that the slip effect plays no role at the zero order in ε.…”

“…In all cases the end of the pipe is sealed so that the flow takes place in the so called dead end configuration. In the paper [1] we have studied in detail the dead-end flow in hollow fibers filtration modules. That study was promoted by the European Life + Project PURIFAST, and opened the way to a number of further investigations in different areas.…”

“…The technique we will use to obtain a reliable mathematical model of an irrigation pipe, inspired by [1], is the one of upscaling, that requires the following steps:…”