Mathematical Model of Fluid Flow and Solute Transport in ‎Converging-diverging Permeable Tubes

Document Type : Research Paper

Authors

1 Department of Mathematics, School of Advance Sciences, VIT-AP University, Amaravati-522237, Andhra Pradesh, India

2 Department of Mathematics, National Institute of Technology, Warangal-506004, Telangana State, India

Abstract

This article presents a mathematical model for fluid and solute transport in an ultra-filtered glomerular capillary. Capillaries ‎are assumed to be converging-diverging tubes with permeable boundaries. According to Starling's hypothesis, ultrafiltration is ‎related to the variations in hydrostatic and osmotic pressures in the capillary and Bowman's space and takes place along the ‎length of the capillary. The governing equations of fluid flow for the case of axisymmetric motion of viscous incompressible ‎Newtonian fluid have been considered along with the solute transfer equation. The non-uniform geometry has been mapped ‎into a finite regular computational domain via a coordinate transformation. Correspondingly, the governing equations are ‎transformed to the computational space and solved to get the velocity and pressure values. The solute transfer equation is ‎also solved numerically using a finite difference scheme. The solutions provide the predictions of the axial distribution of ‎hydrostatic pressure and osmotic pressure, velocities, and concentration profiles at various points along the axis and solute ‎clearance quantities along the capillary. The current results are in good agreement with the earlier findings in limiting cases of ‎cylindrical tubes with a constant radius. It is found that there is a significant effect of osmotic pressure on the solute ‎concentration. Using a set of data, the influence of various physiological parameters on the velocity components and solute ‎concentration are presented and discussed through graphs to correlate with physiological situations. The generic structure of ‎the current model also provides an acceptable approach to exploring fluid exchange in organs apart from the glomerular ‎capillary.‎

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[1] Apelblat, A., Katchasky, A.K., Silberberg, A., A mathematical analysis of capillary tissue fluid exchange, Biorheology, 11, 1974, 1-49.
[2] Palatt, J.P., Henry S., Roger, I.T., A hydrodynamical model of a permeable tubule, Journal of Theoretical Biology, 44, 1974, 287-303.
[3] Brenner, B.M., Troy, J.L., Daugharty, T.M., Deen, W.M., Dynamics of glomerular ultrafiltration in the rat. II. Plasma flow dependence of GFR, American Journal of Physiology, 223, 1972, 1184-1190.
[4] Keener, J., Sneyd, J., Mathematical Physiology, 2nd Edition, Springer, 2009.
[5] Guyton, A.C., John, E.H., Text Book of Medical Physiology, 11th Edition, Elsevier Saunders Publishing, 2011.
[6] Brenner, B.M., Troy, J.L., Daugharty, T.M., Deen, W.M., Dynamics of glomerular ultrafiltration in the rat. II. Plasma flow dependence of GFR, American Journal of Physiology 223, 1972, 1184-1190.
[7] Brenner, B.M., Baylis, C., Deen, W.M., Transport of molecules across renal glomerular capillaries, Physiological Reviews, 56, 1978, 502-534.
[8] Martin, R.P., Susan, E.Q., Melanie, P.H., Lance, D.D., The glomerulus: The sphere of influence, Clinical Journal of the American Society of Nephrology, 9, 2014, 1461-1469.
[9] Misra, J.C., Ghosh, S.K., A mathematical model for the study of blood flow through a channel with permeable walls, Acta Mechanica, 122, 1997, 137-153.
[10] Moustafa, E., Blood flow in capillary under starling hypothesis, Applied Mathematics and Computation, 149, 2004, 431-439.
[11] Yuan, S.W., Finkelstein, A.B., Laminar pipe flow with injection and suction through a porous wall, Transactions of the American Society of Mechanical Engineers, 78, 1956, 719-724.
[12] Kedem, O., Katchalsky, A., A physical interpretation of the phenomenological coefficients of membrane permeability, Journal of General Physiology, 45, 1961, 143-179.
[13] Deen, W.M., Robertson, C.R., Brenner, B.M., A model of glomerular ultrafiltration in the rat. American Journal of Physiology, 223, 1972, 1178-1183.
[14] Marshall, E.A., Trowbridge, E.A., A mathematical model of the ultrafiltration process in a single glomerular capillary, Journal of Theoretical Biology, 48, 1974, 89-412.
[15] Papenfuss, H.D., Gross, J.F., Analytical study of the influence of capillary pressure drop and permeability on glomerular ultrafiltration, Microvascular Research, 16, 1978, 59-72.
[16] Papenfuss, H.D., Gross, J.F., Transcapillary exchange of fluid and plasma proteins, Biorheology, 24, 1987, 319-335.
[17] Salathe, E.P., Mathematical studies of capillary tissue exchange, Bulletin of Mathematical Biology, 50, 1988, 289-311.
[18] Deen, W.M., Robertson, C.R., Brenner, B.M., Concentration polarization in an ultrafiltering capillary, BioPhysical Journal, 14, 1974, 412- 431.
[19] Ross, M.S., A mathematical model of mass transport in a long permeable tube with radial convection, Journal of Fluid Mechanics, 63, 1974, 157-175.
[20] Tyagi, V.P., Abbas, M., An exact analysis for a solute transport, due to simultaneous dialysis and ultrafiltration, in a hollow-fibre artificial kidney, Bulletin of Mathematical Biology, 49, 1987, 697-717.
[21] Chaturani, P., Ranganatha, T.R., Flow of Newtonian fluid in non-uniform tubes with variable wall permeability with application to flow in renal tubules, Acta Mechanica, 88, 1991, 11-26.
[22] Layton, A.T., Layton, H.E., A computational model of epithelial solute and water transport along a human nephron, PLoS Computational Biology, 15(2), 2019, e1006108.
[23] Pstras, L., Waniewski, J., Lindholm, B., Transcapillary transport of water, small solutes and proteins during hemodialysis, Scientific Reports, 10, 2020, 18736.
[24] Waniewski, J., Pietribiasi, M., Pstras, L., Calculation of the Gibbs–Donnan factors for multi-ion solutions with non-permeating charge on both sides of a permselective membrane, Scientific Reports, 11, 2021, 22150.
[25] Radhakrisnamacharya, G., Peeyush, C., Kaimal, M.R., A hydrodynamical study of the flow in renal tubules, Bulletin of Mathematical Biology, 43, 1981, 151-163.
[26] Varunkumar, M., Muthu, P., Fluid flow and solute transfer in a tube with variable wall permeability, Zeitschrift für Naturforschung A, 74, 2019, 1057-1067.
[27] Varunkumar, M., Muthu, P., Fluid flow and solute transfer in a permeable tube with influence of slip velocity, An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity, 9, 2020, 153-166.
[28] Radhakrishnamacharya, G., Muthu, P., Varunkumar, M., Mathematical Model of Fluid Flow and Solute Transfer in a Permeable Channel with Slip Velocity at the Boundaries, Proceedings of National Academy Sciences, India, Section-A Physical Sciences, 91, 2021, 611-622.
[29] Krishnaprasad, J.S.V.R., Chandra, P., Low Reynolds number flow in a channel with varying cross section and permeable boundaries, In: Bio Mechanics, New Delhi, WillEastern Ltd, 1988.
[30] Chaturani, P., Ranganatha, T.R., Solute transfer in fluid in permeable tubes with application to flow in glomerular capillaries, Acta Mechanica, 96, 1993, 139-154.
[31] Regirer, S.A., Quasi one-dimensional model of transcapillary filtration, Journal of Fluid Dynamics, 10, 1975, 442-446.
[32] Shettinger, U.R., Prabhu, H.J., Ghista. D.N., Blood Ultrafiltration: A Design Analysis, Medical and Biological Engineering and Computation, 15, 1977, 32-38.
[33] John, D.A.Jr., Computational Fluid Dynamics - The Basics with Applications, McGraw-Hill, 1995.
[34] Chung, T.J., Computational Fluid Dynamics, 2nd Edition, Cambridge University Press, 2010.