Flow of a newtonian fluid through a permeable tube: the application to the proximal renal tubule

Geometrical Influence on Particle Transport in Cross-Flow Ultrafiltration: Cylindrical and Flat Sheet Membranes

Cross-flow membrane ultrafiltration (UF) is used for the enrichment and purification of small colloidal particles and proteins. We explore the influence of different membrane geometries on the particle transport in, and the efficiency of, inside-out cross-flow UF. For this purpose, we generalize the accurate and numerically efficient modified boundary layer approximation (mBLA) method, developed in recent work by us for a hollow cylindrical membrane, to parallel flat sheet geometries with one or two solvent-permeable membrane sheets. Considering a reference dispersion of Brownian hard spheres where accurate expressions for its transport properties are available, the generalized mBLA method is used to analyze how particle transport and global UF process indicators are affected by varying operating parameters and the membrane geometry. We show that global process indicators including the mean permeate flux, the solvent recovery indicator, and the concentration factor are strongly dependent on the membrane geometry. A key finding is that irrespective of the many input parameters characterizing an UF experiment and its membrane geometry, the process indicators are determined by three independent dimensionless variables only. This finding can be very useful in the design, optimization, and scale-up of UF processes.

A Jeffrey Fluid Model for a Porous-walled Channel: Application to Flat Plate Dialyzer

Creeping motion of a Jeffrey fluid in a small width porous-walled channel is presented with an application to flow in flat plate hemodialyzer. Darcy's law is used to characterize the fluid leakage through channel walls. Using suitable physical approximations, approximate analytical solution of equations of motion is obtained by employing perturbation method. Expressions for velocity field and the hydrostatic pressure are obtained. Effects of filtration coefficient, the inlet pressure and Jeffrey fluid parameters on the flow characteristics are discussed graphically. The derived results are used to study the flow of filtrate in a flat plat hemodialyzer. Using the derived solutions, theoretical values of the filtration rate and the mean pressure difference in the hemodialyzer are calculated. On comparing the computed results with the available experimental data, a reasonable agreement between the two is found. It is concluded that the presented model can be used to study the hydrodynamical aspects of the fluid flow in a flat plate hemodialyzer.

Hydrodynamical Study of Micropolar Fluid in a Porous-Walled Channel: Application to Flat Plate Dialyzer

This article investigates the two-dimensional creeping flow of a non-Newtonian micropolar fluid in a small width permeable channel. Fluid is absorbed through permeable walls at a variable rate. This situation arises in filtration and mass transfer phenomena in industrial and engineering processes. The exact solution of the equations of motion is obtained. Graphs of the velocity profiles and pressure drop reveal the significant impact of the non-Newtonian nature of the micropolar fluid on the flow. The obtained solutions are used to discuss the hydrodynamical aspects of the physiological phenomenon of blood filtration in an artificial kidney, the flat plate dialyzer (FPD). Expressions for finding the ultrafiltration rate and mean pressure drop in an FPD are derived. Ultrafiltration rate and the mean pressure difference in an FPD are computed using derived expressions. A comparison of these with the existing empirical and experimental results shows a good agreement. For certain values of parameters, the derived form of the flow rate reveals that the axial flow rate in an FPD decays exponentially along the membrane length. This is a well-established and admitted result used by several researchers for studying the hydrodynamics of blood flow in renal tubules of kidneys. It is concluded that the presented model can be used to study the hydrodynamical aspects of blood flow in an FPD.

Slip flow through a non-uniform channel under the influence of transverse magnetic field

This study deals with the steady laminar slip flow of an incompressible Newtonian fluid in a non-uniform permeable channel under the influence of transverse magnetic field. The reabsorption through the wall is accounted for by considering flux as a function of downstream distance. The non-linear coupled partial differential equations of motion are first transformed into a single fourth order partial differential equation and then solved analytically using Adomain decomposition method. Effects of pertinent parameters on different flow properties are discussed by plotting graphs. Results reveal that magnetic field considerably influences the behavior of flow.

Biophysics and biofluid dynamics of primary cilia: evidence for and against the flow-sensing function

Primary cilia have been called “the forgotten organelle” for over 20 yr. As cilia now have their own journal and several books devoted to their study, perhaps it is time to reconsider the moniker “forgotten organelle.” In fact, during the drafting of this review, 12 relevant publications have been issued; we therefore apologize in advance for any relevant work we inadvertently omitted. What purpose is yet another ciliary review? The primary goal of this review is to specifically examine the evidence for and against the hypothesized flow-sensing function of primary cilia expressed by differentiated epithelia within a kidney tubule, bringing together differing disciplines and their respective conceptual and experimental approaches. We will show that understanding the biophysics/biomechanics of primary cilia provides essential information for understanding any potential role of ciliary function in disease. We will summarize experimental and mathematical models used to characterize renal fluid flow and incident force on primary cilia and to characterize the mechanical response of cilia to an externally applied force and discuss possible ciliary-mediated cell signaling pathways triggered by flow. Throughout, we stress the importance of separating the effects of fluid shear and stretch from the action of hydrodynamic drag.

Performance of one- and two-dimensional models for a slow flow system in a long, permeable tubule

A radially symmetric slow flow system in a long cylinder with moderate wall leakage is considered, with physical parameters taken from the renal proximal tubule. Dimensional analysis yields a simplified system which, under certain assumptions, is well-posed, and possesses a unique solution if a solution exists. A lumped parameter analysis generates a one-dimensional model identical to a typical one-dimensional model in a special case. The one- and two-dimensional models agree well in the situations examined. When the wall fluxes are of a certain class, the concentration to which the solution tends as the fluid proceeds down the tubule can be computed algebraically.

Flow through brushborders and similar protuberant wall structures

Longitudinal flow through channels with protuberant wall structures such as brushborders were studied both experimentally and theoretically. The experiments were performed using tubes with synthetic internal brushborders, scaled to resemble the geometry and flow in proximal tubules of the kidney. Fractional flow rates in the brushborder were deduced from transit times of dye traces through the central core and total flow rate. The measured ratios of brushborder to core flow QB/QC were found to be independent of Reynolds number over the range Re = 0.01 - 0.2. They agreed reasonably well with theoretical predictions based on the Kozeny-Carman equation for flow through arrays of parallel cylinders. The predictions can be extended to arbitrary geometries and turbulent flow conditions by appropriate modifications. Extrapolation of the results to the proximal tubule of the kidney indicate that brushborder flowrates there vary from 0.003 to 0.09% of total flow. Any axial transport in these structures would be predominantly diffusive in nature, and would be highly unlikely to affect radial brushborder gradients and transmural flux.

Examination of transport equations pertaining to permeable elastic tubules such as Henle's loop

The transport equations applicable to loops of Henle and similar elastic permeable tubules were re-examined to assess the effect of radial transport resistance in the lumen and tubule geometry on solute transport. Active transport at the wall as well as external gradients equivalent to a 2--1,000-fold concentration increase per centimeter of distance were considered. Wall permeabilities and active transport constants were varied up to 2 . 10(-2) cm/s. It is shown that for conditions applicable to the loop of Henle, resistance to radial solute transfer in the lumen is negligible, both for passive and active transmural transport with concomitant water flux, and that axial dispersion further reduces that resistance. These conclusions apply equally to conical and elliptical geometries likely to arise in loop operation. The validity of Poiseuille's equation for these geometries is discussed. Ii is concluded that the one-dimensional transport equations are a valid representation of loop operation.