Laminar flow analysis in a pipe with locally pressure-dependent leakage through the wall

Bernales, B.; Haldenwang, Pierre

doi:10.1016/j.euromechflu.2013.07.006

Cited by 15 publications

(14 citation statements)

References 28 publications

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“…Indeed, much effort has been devoted toward this topic. More recently, Haldenwang has studied the problem in more detail; nonetheless, this work still considers a channel or pipe of finite length with prescribed inflow and outflow conditions [7][8][9] ; the inflow condition in these last set of papers is matched with the analytic solution by Berman 1 in which the normal velocities at the wall are matched. In the work of Tilton et al 12 , the authors relaxed the assumption of a finite length pipe and found a self similar profile in a pipe of arbitrary length.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Herschlag¹,

Liu²

2015

SIAM J. Appl. Math.

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We derive an exact solution for Stokes flow in a channel with permeable walls. At the channel walls, the normal component of the fluid velocity is described by Darcy's law and the tangential component of the fluid velocity is described by the no slip condition. The pressure exterior to the channel is assumed to be constant. Although this problem has been well studied, typical studies assume that the permeability of the wall is small relative to other non-dimensional parameters; this work relaxes this assumption and explores a regime in parameter space that has not yet been well studied. A consequence of this relaxation is that transverse velocity is no longer necessarily small when compared with the axial velocity. We use our result to explore how existing asymptotic theories break down in the limit of large permeability for channels of small length.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Herschlag¹,

Liu²

2015

SIAM J. Appl. Math.

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“…We will therefore restrict our focus to Equations (11)- (14). We begin by nondimensionalizing this system at a fixed time with nondimensional variables,…”

Section: Problem Statement and Solutionmentioning

confidence: 99%

“…Subsequently, the problem of flow across channels driven by Darcy's law has been expanded in a number of ways. [8][9][10][11][12][13][14][15] A related problem considering prescribed cross-channel velocities in pipes and channels has also been the subject of significant investigation, beginning in 1953 with Berman, 16 and this problem has also been extended in a variety of ways. [17][18][19] Whereas the above works assume static channels, other investigations have examined channels and cylinders with deforming walls as a function of time.…”

Section: Introductionmentioning

confidence: 99%

Fluid extraction across pumping and permeable walls in the viscous limit

Herschlag

Liu

2016

Physics of Fluids

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In biological transport mechanisms such as insect respiration and renal filtration, fluid travels along a leaky channel allowing material exchange with systems exterior to the channel. The channels in these systems may undergo peristaltic pumping which is thought to enhance the material exchange. To date, little analytic work has been done to study the effect of pumping on material extraction across the channel walls. In this paper, we examine a fluid extraction model in which fluid flowing through a leaky channel is exchanged with fluid in a reservoir. The channel walls are allowed to contract and expand uniformly, simulating a pumping mechanism. In order to efficiently determine solutions of the model, we derive a formal power series solution for the Stokes equations in a finite channel with uniformly contracting/expanding permeable walls. This flow has been well studied in the case in which the normal velocity at the channel walls is proportional to the wall velocity. In contrast we do not assume flow that is proportional to the wall velocity, but flow that is driven by hydrostatic pressure, and we use Darcy's law to close our system for normal wall velocity. We incorporate our flow solution into a model that tracks the material pressure exterior to the channel. We use this model to examine flux across the channel-reservoir barrier and demonstrate that pumping can either enhance or impede fluid extraction across channel walls. We find that associated with each set of physical flow and pumping parameters, there are optimal reservoir conditions that maximize the amount of material flowing from the channel into the reservoir.

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“…On the one hand, the basic theoretical description of the cross-flow filtration refers to the seminal contribution by Berman (Berman (1953)) that treats of the channel flow driven by a uniform leakage at the wall. Other contributions on the case of the "pure solvent" flow proposed several analytical solutions that accounts for the pressure dependence on permeate flux (see Regirer (1960), Haldenwang (2007), Tilton et al (2012), Bernales and Haldenwang (2014)). On the other hand, an exact analytical solution for the solute transfer in Berman flow has been derived in Haldenwang et al (2010) and accounts for concentration polarization and the subsequent hindrance to permeation in the limit of certain RO/NF configurations.…”

Section: Introductionmentioning

confidence: 99%

Prandtl model for concentration polarization and osmotic counter-effects in a 2-D membrane channel

et al. 2017

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Laminar flow analysis in a pipe with locally pressure-dependent leakage through the wall

Cited by 15 publications

References 28 publications

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

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

Fluid extraction across pumping and permeable walls in the viscous limit

Prandtl model for concentration polarization and osmotic counter-effects in a 2-D membrane channel

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