Application of creeping flow through a linearly absorbing slit filled with porous medium to diseased renal tubules
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ORIGINAL PAPER
Application of creeping flow through a linearly absorbing slit filled with porous medium to diseased renal tubules A. M. Siddiqui1 · Q. A. Azim2 · D. A. Sunny3 Received: 5 December 2019 / Accepted: 18 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we have discussed the flow of a Newtonian fluid through a slit filled with porous medium and linearly reabsorbing porous walls. The study is motivated by fluid flow in diseased renal tubules in a kidney. Due to diseases, some fibrous material, fatty substances and solid waste particles, etc., may get suspended in tubule channel as well as on the pores of the wall, resulting in the porous filling in the slit and biofouling, respectively. In this study, the absorption at the wall is assumed to follow a linear pattern and the fluid is assumed to be entering the channel at a prescribed initial flow rate. The problem of the two-dimensional fluid flow is formulated using stream function, and inverse solution method is applied to obtain an exact solution of the fourth-order compatibility equation. Some special cases are also deduced from the obtained results and compared with available results from literature. Expressions for various physically relevant quantities like fluid velocities, volume flow rate, fractional reabsorption, leakage flux and pressure distribution are obtained. The results are used to demonstrate how medium porosity and biofouling parameter may affect average pressure differences in the renal tubules of a rat kidney. Finally, the results are presented graphically and effects of changing various parameters on the flow are analysed. We have also deduced some special cases when the wall reabsorption is uniform, and when there is no medium porosity. We have shown these special cases match with the already present results in the literature. Keywords Renal tubule · Creeping flow · Newtonian fluid · Linear reabsorption · Biofouling · Porous medium
1 Introduction Kidneys are vital organs in living beings. They perform various important functions in a body including the removal of toxic elements from the system through urination. They also perform blood filtration and send the purified blood back to the heart. Every day, about 200 litres of blood enters the kidneys, while only about two litres of waste products and excess water is passed out in the form of glomerular filtrate (urine). After entering the kidneys, blood is filtered by tiny * Q. A. Azim [email protected] 1
Department of Mathematics, Pennsylvania State University, York Campus, York, PA 17403, USA
2
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Defence Road, Lahore 54000, Pakistan
3
Department of Applied Mathematics and Statistics, Institute of Space Technology, 1, Islamabad Highway, Islamabad 44000, Pakistan
functional units called nephrons. Each kidney has a million of these nephrons. Nephrons consist of two parts, Bowman’s capsule and renal tubules. Renal tubules are a collecti
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