Creeping Flow of Viscous Fluid through a Proximal Tubule with Uniform Reabsorption: A Mathematical Study

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1 Applied Mathematical Sciences, Vol., 26, no. 6, HIKARI Ltd, Creeping Flow of Viscous Fluid through a Proximal Tubule with Uniform Reabsorption: A Mathematical Study T. Haroon, A. M. Siddiqui 2 and A. Shahzad Department of Mathematics, COMSATS Institute of Information Technology Park Road, Chak Shahzad, Islamabad, 44, Pakistan 2 Department of Mathematics,Pennsylvania State University York Campus Edgecomb 73, USA Copyright c 26 T. Haroon, A. M. Siddiqui and A. Shahzad. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 8% fractional reabsorption from a proximal tubule through a single nephron can be achieved by setting the reabsorption velocity V =.6 6 cm/sec at the pressure drop of 8.4 dyn/cm 2. The hydrodynamics of flow properties are graphically discussed. Mathematics Subject Classification: 76A5 Keywords: Proximal tubule, Uniform reabsorption, Slit, Exact solutions Introduction Kidneys are like a bean shaped organ, which perform two critical functions for the body. The first is the removal of metabolic waste products from the

2 796 T. Haroon, A. M. Siddiqui and A.Shahzad body and the second is the maintenance of the water volume within the body. Each kidney contains over a million tiny filtering units, known as nephrons, all similar in structure and function. Each nephron functions independently and in most instances, it is sufficient to study the function of a single nephron to understand the mechanism of kidney in terms of mathematical models. In nephrons, the portion after the Bowman s capsule is called proximal convoluted tubule. It is the place where most of the material like water, glucose and electrolytes from glomerular filtrate are reabsorbed whenever any of these substances run low in the body. Every day, kidneys process about 2 liters of blood to sift out about 2 liters of waste products and extra water. The waste and extra water become urine, which flows to our bladder through tubes called ureters. The reabsorption in the proximal convoluted tubule is carried out through the small pores of uniform size, where the velocity and pressure fields differ from simple Poiseuille s flow since the fluid on the wall has a transverse velocity. Flow in the proximal convoluted tubule has been studied by various authors. Wesson [] and Burgen [2] theoretically discussed renal model assuming a constant rate of reabsorption. Macey [3] was the first who studied the mathematical modeling of the flow of an incompressible viscous fluid through a renal tubule. He obtained exact solution using a circular tube with linear rate of reabsorption. Kelman [4] proposed that the bulk flow in the proximal tubule decays exponentially with the axial distance. Later, Macey [5] used the condition of Kelman [4] and solved the transport equations to find velocity components and pressure drop. Kozinski et al. [6] not only modified Macey s work in tube [5], but also extended it for porous slit. They obtained analytic solutions for velocity and pressure difference. Later, several researchers [7-2] studied the hydrodynamics of viscous fluid through the proximal tubule with various physical assumptions. This work is motivated by the problem associated with understanding the fluid reabsorption through the proximal tubule with uniform wall reabsorption. To study the hydrodynamics of such flow, the geometry of the proximal tubule is approximated with a porous slit of length L, width 2H and breadth W. A rectangular Coordinate system (x, y, z) is chosen such that x ranges from at Bowman s capsule to L at the end, which empties into the loop of henle and y ranges from the axis to the tubular walls. The flow of glomerular filtrate in renal tubule is assumed Newtonian which becomes two dimensional due to permeability of the walls and creeping because the smaller tubular diameter which employs the viscous forces is much larger than the inertial forces. According to the authors knowledge no work has been cited so far. This study is not only important because of its biological significance, but also in the view of the interesting mathematical features presented by the equations governing the flow.

3 Creeping flow of viscous fluid through a proximal tubule Governing equations and problem formulation We consider a steady, laminar, isothermal creeping flow of an incompressible Newtonian fluid through a porous slit with the x-axis aligned with the center line of the slit and y normal to it, Fig. (). A constant axial flow rate Q enters the slit which is reabsorbed from the walls with uniform velocity V throughout the length of the slit. As this physical model is symmetric, therefore we will take upper half of the given configuration. The velocity profile for the stated Figure : Geometry of the problem with uniform reabsorption. problem is of the form V = [u(x, y), v(x, y)], () where u and v are the velocity components in x and y directions. Using the above profile, the continuity equation in two dimensions becomes u x + v y =, (2) and for creeping flow momentum equation takes the form [ p 2 ] x = µ u x + 2 u, (3) 2 y 2 [ p 2 ] y = µ v x + 2 v, (4) 2 y 2 where p is the hydrodynamic pressure and µ is the viscosity of the fluid. The boundary conditions are u y =, v =, at y =, (5) u =, v = V, at y = H, (6) Q = 2W H u(, y)dy, (7)

4 798 T. Haroon, A. M. Siddiqui and A.Shahzad where W is breadth of the slit. Introducing the stream function ψ(x, y) in the following form u = ψ y, v = ψ x, (8) we find that equation (2) is identically satisfied and equations (3-4) take the form [ p 3 ] x = µ ψ x 2 y + 3 ψ, (9) y 3 [ p 3 ] ψ = µ y x + 3 ψ. () 3 x y 2 To eliminate pressure gradient, we differentiate equation (9) with respect to y and equation () with respect to x, then subtract the resulting equations and get 4 ψ x ψ 4 x 2 y + 4 ψ =. () 2 y4 Further, the boundary conditions (5-7) are transformed into 2 ψ y 2 =, ψ y =, ψ =, x at y =, (2) ψ x = V, at y = H, (3) ψ(, ) = and ψ(, H) = Q 2W. (4) Equation () along with (2-4) is a boundary value problem (BVP), describing the two dimensional flow of an incompressible Newtonian fluid through a slit with uniform reabsorption at the walls. 3 Solution of the problem To obtain solution of the above BVP, inverse method [3-7] is used and we choose stream function ψ(x, y) of the form ψ(x, y) = V xr(y) + T (y), (5) where R(y) and T (y) are unknown functions to be determined. Using equation (5) in equations (- 4 ), we get d 4 R =, (6) dy4

5 Creeping flow of viscous fluid through a proximal tubule 799 with boundary conditions and with boundary conditions R() =, d 2 R() =, dy 2 (7) R(H) =, dr(h) =, dy (8) d 4 T dy 4 =, (9) d 2 T () T () =, =, (2) dy 2 T (H) = Q 2W, dt (H) =. (2) dy The solution of equation (6) along with the boundary conditions (7-8) is obtained as R(y) = V [ ( ) ( ) ] y y 3 3, (22) 2 H H and the solution of equation (9) with corresponding boundary conditions is transformed to T (y) = Q [ ( ) ( ) ] y y 3 3. (23) 4W H H Using solutions (22) and (23) in equation (5), the expression for stream function ψ(x, y) becomes ψ(x, y) = { } [ ( ) ( ) ] Q y y 3 2 2W V x 3. (24) H H It is observed that forward flow is possible only if volume flux per unit breadth is greater than the reabsorption velocity throughout the slit. Otherwise the possibility of back flow is there. 4 Velocity components The velocity components are obtained using relation (8): u(x, y) = 3 (Q [ ( ) ] 2W V x) y 2, (25) 4W H H v(x, y) = V [ ( ) ( ) ] y y 3 3. (26) 2 H H

6 8 T. Haroon, A. M. Siddiqui and A.Shahzad From equation (26), it is noted that the transverse component of velocity is independent of x coordinate. From equation (25), parabolic profile is observed for longitudinal velocity, which is maximum at the center for fixed axial distance x. Expression for the maximum longitudinal velocity is u max = 3(Q 2W V x). (27) 4W H The maximum transverse velocity is observed at the wall surface, i.e. v max = V. (28) The axial flow rate Q(x) inside the slit can be obtained by using Q(x) = 2W H u(x, y)dy. (29) Substituting equation (25) in equation (29), the axial flow rate becomes Q(x) = Q 2W V x, (3) which shows that axial flow rate decreases from entrance to exit of the slit. 5 Pressure distribution To determine the expression of pressure distribution, we substitute equation (5) in equations (9-) to get { } p x = µ x d3 R dy + d3 T, (3) 3 dy 3 p y Integration of equation (3) with respect to x yields p(x, y) = µ = µ d2 R dy 2. (32) { } d3 R 2 x2 dy + d3 T 3 dy x + H(y), (33) 3 where H(y) is some unknown function of y. In order to determine H(y), we differentiate equation (33) with respect to y, and comparing with equation (32), along with the use of equation (6-9) and get dh dy = R µd2, H(y) = µdr + C. (34) dy2 dy

7 Creeping flow of viscous fluid through a proximal tubule 8 where C is unknown constant to be determined. Using equation (34) in equation (33), we arrive at { } p(x, y) = µ d3 R 2 x2 dy + d3 T 3 dy x µ dr + C. (35) 3 dy Substituting equations (22) and (23) in equation (35), after simplification the final expression for pressure distribution is obtained as p(x, y) p(, ) = 3µ (Q W V x) x 2W H 3 3µV 2H 3 y2, (36) where p(, ) is a constant equal to the value of the pressure at the entrance of the slit at y =. 5. Mean pressure drop The mean pressure p(x) at any section of the slit can be obtained by using the following formula p(x) = H H [p(x, y) p(, )]dy. (37) By substituting equation (36) in above equation, we get [ 3 (Q 2W V x) x p(x) = µ + V ]. (38) 2W H 3 2H Since the pressure drop over the length L of the slit is which reduces to p(l) = p() p(l), (39) p(l) = 3µ (Q 2W V L) L 2W H 3. (4) It is observed that above result strongly depends upon the fluid viscosity. 5.2 Wall shear stress The wall shear stress is defined as τ w y=h = µ Using (25) and (26) in above formula, we get ( u y + v ). (4) x y=h τ w y=h = 3µ(Q 2W V x) 2W H 2, (42) which shows that the walls shear stress decays from entrance to exit of the slit.

8 82 T. Haroon, A. M. Siddiqui and A.Shahzad 5.3 Fractional Reabsorption The fractional reabsorption is the amount of fluid that has been reabsorbed through the walls of the slit and is defined as [6]: F A = Q() Q(L). (43) Q() By using equation (3) in above formula we get F A = 2W V L Q, (44) which shows that F A varies with the change of reabsorption velocity and axial flow rate. 5.4 Leakage Flux The leakage flux q(x) is defined as [6]: Using equation (3) in above equation, we get q(x) = dq(x) dx. (45) q(x) = 2W V, (46) which shows the linear relationship of leakage flux with reabsorption velocity. 6 Application to the Flow Through Proximal Tubule The proximal tubule is a major site for reabsorption. 8% of the filtrate is reabsorbed and the amount of waste at the end is Q() =.2. To obtain the Q(L) theoretical value of V, we have used the physiological data of rat kidney given in [3, 8]. The data according to our problem is H = 3 cm, L = cm, µ = 7 3 dynsec/cm 2, Q = 4 7 cm 3 /sec. Since H << W, let we take W = cm. Using the data in expression (44) for fractional reabsorption and expression (4) for pressure drop, the corresponding reabsorption velocity V and pressure drop p() are obtained as given in the following table. From this table it is noted that 8% fractional reabsorption through a proximal tubule of single nephron can be achieved by setting the reabsorption velocity V =.6 6 cm/sec at the pressure drop of 8.4 dyn/cm 2. Further, the pressure drop increases in case of low fractional reabsorption.

9 Creeping flow of viscous fluid through a proximal tubule 83 Table : Values of reabsorption velocity and pressure drop for different values fractional reabsorption. F A 8% 7 % 6 % 5% V cm/sec p() dyn/cm Results and discussion To study the behavior of velocity components, volume flow rate, pressure difference, wall shear stress and leakage flux through graphs, we used the following dimensionless parameters: x = x H, y = y H, V = V U, ψ = ψ U H, p = Q = Q U W H, τ w = p µu /H, τ w µu /H, (47) where U is the fluid velocity at the entrance of porous slit. The effects of reabsorption velocity V on velocity components, axial flow rate, pressure difference and wall shear stress at different positions x =. (entrance), x =.5 (middle place) and x =.9 (exit) of the slit are shown in Figures. (2-8). Figs.(2-4) indicate the effect of V on longitudinal velocity component u at different positions, i.e., entrance, middle place and exit of the slit, keeping the volume flow rate Q = 3. Parabolicity at the entrance of the slit is higher than at the middle position and exit of the slit. It is observed that with the increase in V, u decreases in both at the entrance and middle position of the slit. Reverse flow phenomenon can be observed at higher values of V near the exit of the slit. In Fig. (5), effect of V on transverse component of velocity v is depicted. It is observed that with increasing V, the magnitude of the v increases and remain uniform at all the position downstream due to uniform reabsorption. Axial flow rate decreases, with increasing V, Fig. (6). It is also noted that axial flow rate and wall shear stress are reduced with the increase in V, while pressure difference increases at the center line of the slit, with V, see Fig. (7). 8 Concluding remarks In this paper, creeping flow of Newtonian fluid through a proximal tubule is modeled as slit flow with uniform reabsorption at the walls. Exact solutions are obtained by using inverse method. The hydrodynamics of flow properties are studied graphically. The major findings of the present study are summarised as follows:

10 84 T. Haroon, A. M. Siddiqui and A.Shahzad V =.5 V = V =2 2 V =.5 V = V = 2 u(., y).5 u(.5, y) y y Figure 2: Effect of V on u at entrance of the slit. Figure 3: Effect of V on u at the middle place of the slit V =.5 V = V = V =.5 V = V = 2.5 u(.9, y).5 v(y) Reverse Flow y y Figure 4: Effect of V on u at the exit of the slit. Figure 5: Effect of V on v at all the positions of the slit V =.5 V = V = 2 2 V =.5 V = V = 2 Q(x) p(x,)-p(,) x x Figure 6: Effect of V on axial flow rate. Figure 7: Effect of V on pressure difference.. Forward flow in slit is possible only if the axial flow rate per unit width is greater than reabsorption velocity otherwise reverse flow phenomenon occurs. 2. Maximum longitudinal and transverse velocities are observed at the center and at the walls of the slit, respectively. 3. Axial flow rate, pressure and wall shear stress decrease downstream. 4. Reabsorption velocity parameter provides a mechanism to control the fractional reabsorption and mean pressure drop. 8% fractional reab-

11 Creeping flow of viscous fluid through a proximal tubule 85 Figure 8: Effect of V on wall shear stress. sorption from a proximal tubule through a single nephron can be achieved by setting V =.6 6 cm/sec at pressure drop of 8.4 dyn/cm Wall shear stress decreases with the increase in reabsorption velocity. 6. % reabsorption at the end of the proximal tubule can be achieved if Q /LW = 2V. Further, we would like to point out here that our study is of theoretical nature and much more experimental and physiological works are needed to have a complete insight of the flow phenomenon through proximal tubule. References [] Laurence G. Wesson, A theoretical analysis of urea excretion by the mammalian kidney, American Journal of Physiology, Legacy Content, 79 (954), no. 2, [2] A. S. V. Burgen, A theoretical treatment of glucose reabsorption in the kidney, Canadian Journal of Biochemistry and Physiology, 34 (956), no. 3, [3] Robert I. Macey, Pressure flow patterns in a cylinder with reabsorbing walls, The Bulletin of Mathematical Biophysics, 25 (963), no., [4] R. B. Kelman, A theoretical note on exponential flow in the proximal part of the mammalian nephron, The Bulletin of Mathematical Biophysics, 24 (962), no. 3, [5] Robert I. Macey, Hydrodynamics in the renal tubule, The Bulletin of Mathematical Biophysics, 27 (965), no. 2,

12 86 T. Haroon, A. M. Siddiqui and A.Shahzad [6] A. A. Kozinski, F. P. Schmidt, and E. N. Lightfoot, Velocity profiles in porous-walled ducts, Industrial and Engineering Chemistry Fundamentals, 9 (97), no. 3, [7] E. A. Marshall, and E. A. Trowbridge, Flow of a Newtonian fluid through a permeable tube: The application to the proximal renal tubule, Bulletin of Mathematical Biology, 36 (974), no. 5, [8] Paul J. Palatt, Henry Sackin, and Roger I. Tanner, A hydrodynamic model of a permeable tubule, Journal of Theoretical Biology, 44 (974), no. 2, [9] G. Radhakrishnamacharya, Peeyush Chandra, and M. R. Kaimal, A hydrodynamical study of the flow in renal tubules, Bulletin of Mathematical Biology, 43 (98), no. 2, [] P. Chaturani, and T. R. Ranganatha, Flow of Newtonian fluid in nonuniform tubes with variable wall permeability with application to flow in renal tubules, Acta Mechanica, 88 (99), no. -2, [] P. Muthu, and Tesfahun Berhane, Mathematical model of flow in renal tubules, Inter. Jr. App. Maths. Mechanics, 6 (2), [2] Sultan Ahmad, and Naseem Ahmad, On Flow through Renal Tubule in case of Periodic Radial Velocity Component, International Journal of Emerging Multidisciplinary Fluid Sciences, 3 (2), no. 4, [3] Ratip Berker, Intégration des Équations du Mouvement d un Fluide Visqueux Incompressible, Handbuch der physik 2, (963), [4] Dryden, Hugh Latimer, Francis Dominic Murnaghan, and Harry Bateman, Report of the Committee on hydrodynamics, Division of physical sciences, National research council. No. 73, Published by the National research council of the National academy of sciences, 932. [5] C. Y. Wang, Fluid flow due to a stretching cylinder, Physics of Fluids, 3 (988), no. 3, [6] A. M. Siddiqui, T. Haroon, and A. Shahzad, Hydrodynamics of viscous fluid through porous slit with linear absorption, Applied Mathematics and Mechanics, 37 (26), no. 3,

13 Creeping flow of viscous fluid through a proximal tubule 87 [7] A. M. Siddiqui, Some more inverse solutions of a non-newtonian fluid, Mechanics Research Communications, 7 (99), no. 3, [8] Jagat Narain Kapur, Mathematical Models in Biology and Medicine, Affiliated East-West Press, 985. Received: January 7, 26; Published: March 9, 26