The exact solutions for the viscous fluid through a porous slit with linear ab-sorption are obtained. The Stokes equation with non-homogeneous boundary conditions is solved to get the expressions for the velocity components, pressure distribution, wall shear stress, fractional absorption, and leakage flux. The volume flow rate and mean flow rate are found to be useful in obtaining a convenient form of the longitudinal velocity component and pressure difference. The points of the maximum velocity components for a fixed axial distance are identified. The value of the linear absorption parameter is ran-domly chosen, and the rest available data of the rat kidney to the tabulate pressure drop and fractional absorption are incorporated. The effects of the linear absorption, uniform absorption, and flow rate parameters on the flow properties are discussed by graphs. It is found that forward flow occurs only if the volume flux per unit width is greater than the absorption velocity throughout the length of the slit, otherwise back flow may occur. The leakage flux increases with the increase in the linear absorption parameter. Streamlines are drawn to help the analysis of the flow behaviors during the absorption of the fluid flow through the renal tubule and purification of blood through an artificial kidney.

Hydrodynamics of viscous fluid through porous slit with linear absorption

Abstract Creeping flow through a proximal tubule with a uniform form of wall reabsorption is modeled as a slit flow. Equations of motion along with appropriate boundary conditions are transformed in terms of stream function. Exact solutions are obtained for components of the velocity field , axial flow rate, pressure distribution, pressure drop, wall shear stress, fractional reabsorption and leakage flux. Using physiological data of rat kidney, fractional reabsorption and pressure drop are tabulated. It is observed that 80% fractional reabsorption from a proximal tubule through a single nephron can be achieved by setting the reabsorption velocity V0 = 1.6 × 10−6 cm/sec at the pressure drop of 8.4 dyn/cm2. The hydrodynamics of flow properties are graphically discussed.

Creeping flow of viscous fluid through a proximal tubule with uniform reabsorption: A mathematical study

Creeping flow of a viscous fluid in a uniformly porous slit with porous medium: An application to the diseased renal tubules

This paper is concerned with the Stokes flow of an incompressible viscous fluid through a slit with periodic reabsorption at the walls. The momentum equation for the two dimensional flow is exactly solved in terms of stream function for two different cases of boundary conditions. Dimensional forms of stream function, velocity components, axial flow rate, pressure distribution, mean pressure drop, wall shear stress, fractional reabsorption and leakage flux are obtained. The points of maximum velocity components are also identified for fixed axial distance. Using physiological data of rat kidney, the theoretical values of periodic reabsorption and pressure drop for various values of fractional reabsorption are tabulated. The graphs of flow properties for both the cases are compared with the case of uniform reabsorption. It is shown that the periodic reabsorption parameter for both the cases plays a vital role in altering the flow properties, which are useful in analyzing flow behavior during the reabsorption of glomerular filtrate through a renal tubule in normal and diseased conditions. It is found that 50% reabsorption of fluid from a single nephron can be achieved by setting α = 3.197500134 cm for one of the cases which indicates that there is a need of artificial kidney for survival. In case 2, a minor treatment is needed as the value of α for 80% reabsorption is not possible. Streamlines are also drawn to analyze the flow behavior through an abnormal renal tubule.

Stokes flow through a slit with periodic reabsorption: An application to renal tubule

An investigation of viscous incompressible fluid flow in a nonuniform rigid channel with permeable walls is presented by taking into consideration the influence of slip velocity at the walls. The effect of fluid absorption through permeable walls is accounted for by prescribing flux as a function of axial distance. The nonlinear governing equations of motion are linearized by the perturbation method by assuming ‐ (the ratio of inlet width to the length of the channel) as a small parameter to get an approximate analytical solution. The effects of reabsorption coefficient (a), slope parameter (k), and slip parameter (fl) on the velocity profiles, mean pressure drop, and wall shear stress are studied and presented graphically. The results indicate that the slip parameter influences the flow field considerably.

Flow through nonuniform channel with permeable wall and slip effect

In this paper, we investigate viscous, incompressible fluid flow in a channel with slowly varying cross-section with absorbing walls. The motion of fluid is assumed to be steady and laminar. The effect of fluid absorption through permeable wall is accounted for by prescribing flux as a function of axial distance. The nonlinear equations of motion are linearized by perturbation method by assuming δ (ratio of inlet half-width to wavelength) as a small parameter and are solved by numerical methods. The effects of slope parameter (k) on the velocity profiles, mean pressure drop and wall shear stress are presented graphically. It has been observed that the mean pressure drop and the magnitude of wall shear stress decreases in a diverging channel.

Mathematical model of flow in renal tubules

In this paper, a numerical method employing a finite difference technique is used for an investigation of viscous, incompressible fluid flow in a tube with absorbing wall and slowly varying cross-section. The effect of fluid absorption through permeable wall is accounted by prescribing flux as a function of axial distance. The method is not restricted by the parameters in the problem such as wave number, permeability parameter, amplitude ratio and Reynolds number. The effects of these parameters on the radial velocity and mean pressure drop is studied and the results are presented graphically. Comparison is also made between the results obtained by perturbation method of solution and present approach.

Flow of Newtonian Fluid in Non-Uniform Tubes with Application to Renal Flow: A Numerical Study

The problem of steady, viscous, incompressible fluid flow in a tube of slowly varying cross-section with absorbing wall is studied For the mathematical formulation of the problem the effect of fluid absorption through permeable wall is accounted by prescribing flux as a function of axial distance and the fluid is considered as a Newtonian fluid The nonlinear equations of motion are linearized by perturbation method by assuming δ (ratio of inlet radius to wavelength) as a small parameter and the resulting equations are solved by numerical methods The effects of reabsorption coefficient (α), slope parameter (k) and amplitude ratio (e) on the velocity profiles mean pressure drop and wall shear stress are presented graphically Results indicate that the variation of slope parameter and reabsorption coefficient influences the flow field considerably

Fluid flow in a rigid wavy non-uniform tube: application to flow in renal tubules

In this paper, we investigate steady, viscous, incompressible fluid flow in an asymmetric channel with absorbing walls which may have possible applications in flows in renal tubules. The effect of fluid absorption through permeable wall is accounted for by prescribing flux as a function of axial distance. The nonlinear governing equations of motion are linearized by perturbation method by assuming $\delta$ (ratio of inlet width to wavelength) as a small parameter and the resulting equations are solved by numerical methods. The effects of reabsorption coefficient ($ \alpha $) and phase difference ($ \phi $) on the velocity profiles, mean pressure drop and wall shear stress are studied and explained graphically.

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Tesfahun Berhane

Papers

Mathematical model of flow in renal tubules

Flow through nonuniform channel with permeable wall and slip effect

Flow of Newtonian Fluid in Non-Uniform Tubes with Application to Renal Flow: A Numerical Study

Fluid flow in a rigid wavy non-uniform tube: application to flow in renal tubules

Fluid flow in an asymmetric channel