Steady Creeping Slip Flow of Viscous Fluid through a Permeable Slit with Exponential Reabsorption 1

Transcription

1 Applied Mathematical Sciences, Vol., 7, no. 5, HIKARI Ltd, Stead Creeping Slip Flow of Viscous Fluid through a Permeable Slit with Exponential Reabsorption T. Haroon Department of Mathematics, Pennslvania State Universit, York Campus Edgecomb Avenue 743, USA A. M. Siddiqui Department of Mathematics, Pennslvania State Universit, York Campus Edgecomb Avenue 743, USA A. Shahzad COMSATS Institute of Information Technolog, Abbottabad, Pakistan J. H. Smeltzer Department of Mathematics, Pennslvania State Universit, York Campus Edgecomb Avenue 743, USA Copright c 7 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. Abstract This paper presents the hdrodnamics of creeping flow of viscous fluid through a permeable slit with exponential reabsorption b taking into consideration the influence of the slip velocit at the walls. A sstem of partial differential equations (PDEs) is converted into a single PDE using the stream function. Then, taking a particular form of the stream function, exact solutions are determined. Expressions for velocit components, axial flow rate, mass flow rate, pressure distribution, The authors are grateful to The Pennslvania State Universit for providing funding assistance for this research.

2 478 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer wall shear stress, fractional reabsorption and leakage flux are obtained explicitl. Using the data of a rat s kidne, variation of the exponential reabsorption parameter for various values of fractional reabsorption is tabulated. These values are calculated using Newton s method with the help of computer software MAPLE 5. The effect of exponential reabsorption and slip parameters on velocit components, axial flow rate, pressure distribution, wall shear stress and leakage flux are graphicall discussed. It is observed that these parameters pla a vital role in altering the flow properties, which are useful in analzing the reabsorption of glomerular filtrate through an abnormal proximal convoluted tubule. Kewords: Slip condition; Exact solution; Creeping flow; Porous slit; Proximal convoluted tubule Introduction The stud of creeping flow through a permeable duct has man applications in the field of engineering, such as flow through geothermal energ extraction, dring of food, insulation of buildings, transpiration cooling, reverse osmosis desalination, glomerular tubule ultrafiltration, proximal tubule reabsorption and artificial kidnes. Out of these man applications, flow through kidnes is perhaps the most significant. It has been reported in [] that about % of the up to 5, people who die each ear from kidne disease could benefit from artificial kidne treatment or kidne transplantation. Kidnes are natural filtration plants found in the bod of most living creatures. The main functions of kidnes are to remove waste materials from the blood and to balance bod fluid b the process of reabsorption. Each kidne contains over a million tin filtering units, known as nephrons, that are similar in structure and function. The two major parts of a kidne nephron are the renal corpuscle and the renal tubule. A renal corpuscle is the initial bloodfiltering component of a nephron, and consists of two structures: a glomerulus and a Bowman s capsule. Blood is forced from glomerular capillaries at higher pressure to allow filtration and the resulting filtrate known as glomerular filtrate enters the Bowman s capsule. The portion of the nephron following the Bowman s capsule is known as the renal tubule. This is where most of the materials obtained from glomerular filtrate such as water, glucose and electroltes are reabsorbed through the tubular walls if an of these substances are at low levels in the bod. Ever da, quarts of blood enter the nephrons for purification, and after reabsorption of valuable nutrients from the blood, about quarts of waste products and extra water creates urine []. If the kidnes of a patient no longer filters the blood and reabsorbs nutrients properl, artificial kidnes might be capable of postponing death from irreversible kidne failure for several ears.

3 Stead creeping slip flow of viscous fluid through a permeable slit 479 The reabsorption of the glomerular filtrate is about 8% under normal conditions but is reduced in the presence of a tubular disorder. The structure of the renal tubule is composed of epithelial cells and nutrients from the blood are reabsorbed through these cells. Several diseases like tubular proteinuria, acute pelonephritis, allergic interstitial nephritis and tubular interstitial injur result in abnormal reabsorption, and this abnormal reabsorption has been studied b several researchers [3, 4, 5, 6]. Other conditions such as urinar tract infections, i.e., pelonephritis and obstructive uropath, can also cause abnormal reabsorption. In these tpes of diseases, microorganisms such as bacteria grow at the tubular walls and pore blockage phenomena results [7]. This pore blockage causes abnormal reabsorption, which then results in the reduction of the amount of useful fluid needed to fulfill the bod s needs. (see Figure 5) Tubular diseases have a major impact on kidne transplants and the usage of artificial kidnes; there is a need for mathematical models to help understand the phsiological process of flow through the renal tubule under both normal and diseased conditions. Mace [8, 9] studied the hdrodnamics of viscous fluid through a porous tube with linear reabsorption and exponential reabsorption at the walls. Kozinski et al. [] completed the solutions of Mace [9], and obtained the exact expressions for velocit and pressure difference. Several researchers [,, 3, 4, 5, 6, 7] later studied the flow behavior of glomerular filtrate through the renal tubule under various phsiological conditions. Recentl, Haroon et al. [8] studied the behavior of glomerular filtrate through a proximal convoluted tubule with uniform reabsorption at the walls. The determined the exact solution of the momentum equation and discussed the fractional reabsorption at the walls using the data from one of the rat kidnes. Siddiqui et al. [9] examined the hdrodnamics of a viscous fluid through a porous slit with linear absorption at the walls. The discovered that the adsorption of microorganisms at the tubular wall reduced the efficienc of the renal tubule. Later, Haroon et al. [] extended the work of Siddiqui et al. [9] and investigated the flow of glomerular filtrate through a renal tubule with periodic reabsorption at the walls. The concluded that the pore blockage phenomenon is random, and perhaps the periodic nature of pore blockage ma be one of the causes of the disease. The aim of this present stud is to investigate the behavior of glomerular filtrate through a renal tubule with exponential reabsorption b taking into consideration the influence of the slip velocit at the walls. This exponential reabsorption at the walls demonstrates that the pores are graduall becoming blocked from the opening to the end of the tubule. Because of the pores at the tubular walls, the glomerular filtrate ma slip at those walls, so the validit of the slip boundar condition cannot be ignored. Beavers et al. [] suggested that the slip velocit is proportional to the velocit gradient on the porous

4 48 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer boundar. Some theoretical support for the Beavers-Joseph slip condition was provided b Saffman []. He analzed the flow in a channel with one porous boundar as a particular case of flow through a non-uniform medium. He derived the form of the boundar condition correct up to order k as: k u u slip = + O(k), () γ where γ is a dimensionless constant and k is the specific permeabilit of the porous medium. Additional support for the Beavers-Joseph slip condition was provided b Talor [3] and Richardson [4]. Recentl, Siddiqui et al. [5] investigated the effects of slip on the flow of an incompressible fluid through a renal tubule. The considered the boundar condition () and discovered that the slip coefficient considerabl influenced the flow variables. This paper is arranged as follows: in section, the formulation of the problem is described b writing a mathematical equation that governs the flow of glomerular filtrate through a renal tubule. A rectangular coordinate sstem (x,, z) is chosen such that x ranges from at the Bowman s capsule to L at the loop of Henle and ranges from the axis to the tubular walls. The reabsorption velocit v at the walls is exponential with axial distance and longitudinal velocit u experiences a slip velocit due to smaller pores. In section 3, the exact solutions for velocit components, volume flow rate, mass flow rate, pressure distribution, wall shear stress, fractional reabsorption and leakage flux are determined. In section 4, a set of phsiological data from one of the rat kidnes is used to find the theoretical values of the exponential reabsorption parameter for various values of fractional reabsorption and is presented in the form of a table. The variations of flow properties are graphicall presented in section 5. Conclusions are then presented in the final section. Description of the problem The stead, laminar flow of an incompressible, homogeneous, Newtonian fluid of constant viscosit through a porous slit having width H is considered. A rectangular Cartesian coordinate sstem (x, ) is chosen with the x-axis aligned with the center line of the slit and the - axis normal to it, (Figure ). At the point x =, the slit becomes porous with the reabsorption velocit decaing exponentiall throughout the length L of the slit. The fluid ma slip on the surface of the porous walls and the Renolds number is assumed to be sufficientl small so that inertial forces can be neglected. The initial volume flow rate Q is assumed to be constant at x =. Assuming that the breadth of the slit W is large enough to neglect the third component of the velocit, the flow field can be considered to be two dimensional in the form given b: V = [u(x, ), v(x, )], ()

5 Stead creeping slip flow of viscous fluid through a permeable slit 48 where u and v are velocit components in the x and directions. Using profile (, the continuit equation becomes: u x + v =, (3) and for creeping flow the momentum equation takes the form: The boundar conditions are: u u + φ u = p x + µ u, (4) = p + µ v. (5) =, v =, at =, (6) =, v = V e αx, at = H, (7) Q = W H u(, )d, (8) k where φ =. In equation (7, the reabsorption velocit v depends upon α γ showing that the pores at the tubular walls near the exit of the slit are becoming blocked. If α, no pore blockage phenomenon occurs; additionall, if φ, the problem of uniform reabsorption that Haroon et al. [8] studied, is recovered. Introducing the stream function ψ(x, ) of the following form: u = ψ, v = ψ x, (9) it is observed that the equation of continuit (3 is identicall satisfied and equations (4-5) reduce to the following forms: p x = µ p ( ) ψ = µ ( ψ x, () ). () Eliminating the pressure gradient from the above equations, we obtain the following partial differential equation: 4 ψ =, ()

6 48 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer where 4 = ( ) and = x + is a Laplacian operator. Further, the boundar conditions (6-8) in term of ψ(x, ) become: and ψ =, ψ x =, at =, (3) ψ + ψ φ =, ψ x = V e αx, at = H, (4) ψ(, ) =, (5) ψ(, H) = Q W. (6) Equation ( along with boundar conditions (3-4) and (5-6) is a boundar value problem (BVP), describing the two dimensional slip flow of Newtonian fluid through a porous slit with exponential reabsorption at the walls. The solution of this problem will be illustrated in the following section. 3 Method of solution To obtain exact solutions of the above two dimensional BVP, we analze the boundar condition and choose the stream function ψ(x, ) of the form: ψ = V e αx F () + K(), (7) where F () and K() are arbitrar functions that need to be determined. Using equation (7 in equation (, we obtain the following equation: [ d V e αx 4 ] F d + d F 4 α d + α4 F + d4 K =, (8) d4 which can take the form: [ d V e αx 4 ] F d + d F 4 α d + α4 F and =, (9) d 4 K d 4 =. () Since V e αx, therefore, [ d 4 ] F d + d F 4 α d + α4 F =. ()

7 Stead creeping slip flow of viscous fluid through a permeable slit 483 With the help of equation (7, the boundar conditions (3-6) reduce to: and F () =, F (H) = α, df (H) d K() =, K(H) = αq V W αw d F () d =, () + φ d F (H) d =, (3) d K() d =, (4), dk(h) d + φ d K(H) d =. (5) The solution of equation ( using boundar conditions (-3) is found to be: F () = αh sin (αh) cos (αh) + φα H cos (αh) + φα sin (αh) (αh cos (αh) sin (αh) + αφ sin sin (α) αh)α (αφ sin (αh) cos (αh)) αh cos (αh) sin (αh) + αφ sin cos (α), (6) (αh) and the solution of equation ( along with the boundar conditions (4-5) is: K() = αq V W [ ] 3H(H + φ) 3. (7) 4αW H (H + 3φ) The expression for the stream function ψ(x, ) is obtained b combining equations (6 and (7 in equation (7 and writing as: [ αh sin (αh) cos (αh) + φα ψ(x, ) = V e αx H cos (αh) + φα sin (αh) (αh cos (αh) sin (αh) + αφ sin sin (α) αh)α ] (αφ sin (αh) cos (αh)) αh cos (αh) sin (αh) + αφ sin cos (α) (αh) αq V W [ + ] 3H(H + φ) 3, (8) 4αW H (H + 3φ) which strongl depends upon slip parameter φ, uniform reabsorption velocit V and exponential reabsorption parameter α. 3. Components of velocit The velocit components are obtained using relation (9: u(x, ) = V e αx [ αh sin (αh) cos (αh) + φα H cos (αh) + φα sin (αh) (αh cos (αh) sin (αh) + αφ sin αh) cos (α)

8 484 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer ] (αφ sin (αh) cos (αh)) αh cos (αh) sin (αh) + αφ sin {cos α α sin (α)} (αh) + 3(αQ V W ) {H(H + φ) }, (9) 4αW H (H + 3φ) [ αh sin (αh) cos (αh) + φα v(x, ) = V e αx H cos (αh) + φα sin (αh) (αh cos (αh) sin (αh) + αφ sin sin (α) αh)α α(αφ sin (αh) cos (αh)) αh cos (αh) sin (αh) + αφ sin cos (α) (αh) It is found that if α and φ, the velocit components of Haroon et al. [8] are recovered. Equations (9 and (3 give a complete description of the fluid velocities at all the points inside the slit. From equation (9, we obtain the expression for maximum longitudinal velocit as follows: [ αh sin (αh) cos (αh) + φα u max = V e αx H cos (αh) + φα sin (αh) (αh cos (αh) sin (αh) + αφ sin αh) ] (αφ sin (αh) cos (αh)) αh cos (αh) sin (αh) + αφ sin + 3(αQ V W )(H + φ), (αh) 4αW H(H + 3φ) at = for fixed axial distance x. From equation (3, we get the maximum transverse velocit at the wall, which is due to reabsorption at the walls, i.e., v max = V e αx for fixed x. The axial volume flow rate Q(x) can be obtained b using the relation: Q(x) = W H Substituting equation (9, the above equation becomes: Q(x) = W V e αx + αq V W α u(x, )d. (3) ]., (3) which is independent of slip parameter but strongl dependent upon reabsorption velocit. The mass flow rate q(x) can be obtained b using the following formula: q(x) = ρ Q(x), (33) where ρ is the densit of the fluid. With the help of equation (3 in (33, we get the expression for mass flow rate inside the slit as: q(x) = ρ [ Q + W V (e αx ) α ]. (34) (3)

9 Stead creeping slip flow of viscous fluid through a permeable slit 485 It is noted that at the end of the porous slit (x = L), the expression for mass flow rate becomes: [ q(l) = ρ Q + W V (e αl ] ). (35) α To find the expression for the amount of fluid mass reabsorbed through the walls of the slit, we can write: which depends upon α. 3. Pressure distribution q() q(l) = ρw V ( e αl ), (36) α To get the expression for pressure distribution, we will use equation (7 in equations (-) to get: { p [V x = µ e αx α df } ] d + d3 F + d3 K, (37) d 3 d 3 { }] p = µ [αv e αx α F + d F. (38) d Upon integrating equation (3 with respect to x, we get: [ p(x, ) = µ V e αx { α df } ] α d + d3 F + x d3 K + R(), (39) d 3 d 3 where R() is an unknown function that needs to be determined. B differentiating equation (39 with respect to and comparing this with equation (38 along with using of equations (-), we find that: dr d = R() = C, (4) where C is an unknown constant of integration. Equation (39 can be now written as: [ p(x, ) = µ V e αx { α df } ] α d + d3 F + x d3 K + C. (4) d 3 d 3 Inserting equations (6 and (7 into equation (4, we have: p(x, ) p(, ) = µv αe αx {cos (αh) φα sin (αh)}(cos (α) ) αh cos (αh) sin (αh) + αφ sin (αh) 3µ(αQ V W )x H W α(h + 3φ), (4)

10 486 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer where p(, ) is the value of the pressure at the entrance of the slit at =. The mean pressure p(x) is calculated b using the following relation: p(x) = H H [p(x, ) p(, )]d. (43) With the help of equation (4, we get: p(x) = µv αe αx {cos (αh) φα sin (αh)}(sin (αh) H) H αh cos (αh) sin (αh) + αφ sin (αh) 3µ(αQ V W )x H W α(h + 3φ). (44) The mean pressure drop is calculated b using the following relation: p(l) = p() p(l). (45) After substituting equation (44 in equation (45, we get: p(l) = p() p(l), (46) p(l) = µv α{cos (αh) φα sin (αh)} sin (αh) ( e αx ) H αh cos (αh) sin (αh) + αφ sin (αh) + 3µ(αQ V W )L H W α(h + 3φ). (47) 3.3 Wall shear stress The wall shear stress ma be calculated using the following expression: τ w =H = µ ( u + v ). (48) x =H If we substitute equations (9-3) in the above equation, we obtain: =H µα V e αx (H + φ + φ sin αh) τ w = αh cos (αh) sin (αh) + αφ sin (αh) + 3µ(αQ W V ) αw H(H + 3φ), (49) which shows that wall shear stress decas from entrance to exit of the slit.

11 Stead creeping slip flow of viscous fluid through a permeable slit Fractional reabsorption The fractional reabsorption (F A) is defined as: F A = Q() Q(L). (5) Q() B using equation (9 in the above equation, the expression for F A reduces to: F A = W V ( e αl ) αq. (5) The above equation depends upon the exponential reabsorption parameter, the reabsorption velocit and the axial flow rate. 3.5 Leakage flux The leakage flux q(x) is defined as: Using equation (3 in the above equation, we get: q(x) = dq(x) dx. (5) q(x) = W V e αx, (53) which demonstrates that the leakage flux is directl proportional to the reabsorption velocit. 4 Application to the flow through the proximal convoluted tubule To investigate the hdrodnamic contribution to the proximal convoluted tubular reabsorption, a set of data available in the literature [8, 8, 9, ] is utilized. This data was recentl used b Haroon et al. [8, ] and Siddiqui et al. [9] to determine the theoretical values of reabsorption velocit and the linear and periodic reabsorption parameters for different values of fractional reabsorption. In a recent work, Haroon et al. [8] observed that for normal nephron function, the tubular pores remain opened and the reabsorption velocit remains uniform throughout the axial distance of the renal tubule. Theoretical values of reabsorption velocit at the wall for different values of fractional reabsorption were obtained; for example, when normal nephron function of approximatel 8% of the glomerular filtrate was transferred through

12 488 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer the tubular epithelium, the theoretical values of reabsorption velocit had to be.6 6 cm/sec. Therefore, the end of the renal tubule Q(L) represents the filtrate entering the descending limb of Henle s loop, and when Q(L) Q =., the expression takes the form: Q(L) Q = W V (e αl ) αq. (54) In order to find the theoretical values of the exponential reabsorption parameter α, the data is given in [8, 8, 9, ]: H = 3 cm, L = cm, µ = 7 3 dn sec/cm, Q = 4 7 cm 3 /sec and W = cm. The theoretical value of the reabsorption velocit V is chosen as V =.6 6 cm/sec, recentl calculated b Haroon et al. [8], and the value of α is obtained for various values of fractional reabsorption that can be calculated from the following nonlinear equation: F A% W V (e αl ) αq =. (55) To determine the values of α, the computer software MAPLE 5 is used with an initial guess of.. After appling five iterations to approximate the root α, the obtained values are listed in the following table: Table : Values of exponential reabsorption parameter α for different values of fractional reabsorption. F A 8% 7 % 6 % 5% α cm It is observed from the above table that as α increases, F A decreases. It can be concluded that, α is a controlling parameter to regulate the pore blockage at the tubular walls. 5 Results and Discussion The creeping slip flow of Newtonian fluid through a permeable slit with exponential reabsorption at the walls is analzed. Exact solutions are determined for velocit components, axial flow rate, mass flow rate, pressure distribution, mean pressure drop, wall shear stress, fractional reabsorption and leakage flux.

13 Stead creeping slip flow of viscous fluid through a permeable slit 489 To stud the variation of flow properties, the following dimensionless quantities are introduced: x = x L, = L, ψ = ψ V L, Q = Q V W L, α = αl, p p = µv /L, τ w = τ w µv /L, (56) and the variations have been explained in figures (3-3), after skipping. In figures (3-), a comparison of the exponential reabsorption (v = V e αx ) with the uniform reabsorption (v = V ) is provided. Fig. (3) demonstrates that the longitudinal velocit profiles u at the entrance (x =.) of the slit for both cases are parabolic and approximatel the same. In Figs. (4-5), it is noted that the velocit profile of u for the exponential reabsorption is higher in magnitude than the uniform reabsorption at the middle position of the center of the slit and the exit of the slit. Using these three figures, we observe that the u at the entrance is higher than at the middle position and the exit of the slit. Figs. (6-8) show the comparison of reabsorption velocit v for different positions of the slit. Using these figures, it is noted that the profile of the exponential reabsorption as compared to uniform reabsorption is low at all the positions due to the pore blockage phenomena. It is also noticed that for exponential reabsorption, v is higher at the entrance than at the middle position and exit of the slit. In Fig. (9), the volume flow rate Q(x) is depicted for the exponential reabsorption and the uniform reabsorption from entrance to exit, which decreases from entrance to exit of the slit. The volume flow rate due to exponential reabsorption is higher than due to uniform reabsorption because of abnormal reabsorption of fluid across the walls of the slit. Fig. () shows the pressure difference from entrance to exit of the slit. It is observed that the magnitude of the pressure difference at the center line p(x, ) p(, ) of the exponential reabsorption is less than the uniform reabsorption at the exit of the slit. The shear stress τ w at the walls is shown in Fig. (). Due to pore blockage, the profile of τ w for the exponential reabsorption has large magnitude when compared with the uniform reabsorption. The comparison of wall leakage can be seen in Fig. (). The effects of the exponential reabsorption parameter α on u at different positions of the slit are shown in Figs. (3-5). From Fig. (3), it is observed that with increasing α, u increases near the walls and decreases at the center of the slit. At the middle position of the slit as shown in Fig. (4), it is observed that as α increases, u increases. Similarl, at the exit of the slit, u has a parabolic profile for all values of α and increases with increasing α (see Fig. (5). The pore blockage phenomenon can be controlled b varing α and the effect of α on reabsorption velocit v is shown in Figs. (6-8). It is also observed that v decreases with increasing α at different positions of the slit. The effect of α on Q(x) can be seen in Fig. (9). With increasing

14 49 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer α, Q(x) increases and reaches highest magnitude at the exit of the slit. In Fig. (), the effect of α on p(x, ) p(, ) is seen. It is observed that p(x, ) p(, ) decreases downstream for α = and α = ; otherwise, as α increases, p(x, ) p(, ) also increases. For α = 3, p(x, ) p(, ) decreases downstream after the middle position of the slit. The effect of α on wall shear stress in shown in Fig. (). Maximum shear stress at the walls is recorded at the entrance of the slit and the magnitude increases as α increases. Fig. () depicts the effect of α on leakage flux. The effect of slip parameter φ on u at different positions of the slit is shown in Figs. (3-5). It is observed that with increasing φ, u increases near the walls due to the permeabilit of the walls. Due to this permeabilit, φ also affects the reabsorption velocit v. (see Fig. (6-8). It is observed that φ has some effect on v between the center and the walls of the slit. The effect of φ on p(x, ) p(, ) and τ w are shown in Figs. (9-3). It is observed that p(x, ) p(, ) increases and wall shear stress decreases with increasing φ. 6 Conclusion The problem of creeping slip flow of Newtonian fluid through a porous slit with exponential reabsorption was considered. Exact solutions were obtained for the expressions of the components of velocit, volume flow rate, mass flow rate, pressure difference, pressure drop, wall shear stress, fractional reabsorption and leakage flux. From this work, we made the following observations:. Due to pore blockage, the profile of longitudinal velocit for exponential reabsorption at the exit is higher than the profile of uniform reabsorption, but it is lower for transverse velocit.. Blockage phenomenon can be controlled b varing the values of α, i.e., with decreasing α the longitudinal velocit increases while the transverse velocit decreases. 3. Volume flow rate, pressure difference, wall shear stress and leakage flux decrease downstream. 4. 8% fractional reabsorption can be achieved if α is set at.56cm. 5. Slip parameter φ controls the longitudinal velocit at the slit walls, since it increases with increasing φ. 6. φ does not affect the profile of transverse velocit. 7. Pressure difference and wall shear stress are affected with slip parameter φ.

15 Stead creeping slip flow of viscous fluid through a permeable slit 49 It should be mentioned that this stud is theoretical in nature and significant additional experimental and phsiological work is needed to have complete insight in to the transport of glomerular filtrate through the renal tubule in a normal and abnormal kidne nephron. This present stud would have significant impact from a biomedical point of view, as there is little information on this topic. References [] E. F. Leonard and R. L. Dedrick, Artificial Kidne: Problems and Approaches for the Chemical Engineer, Chem. Eng. Progr. Smp. Series, 84 (968), 5-3. [] S. I. Fox, Human Phsiolog, 9th Editon, 996. [3] E. A. Butler and F. V. Flnn, The proteinuria of renal tubular disorders, The Lancet, 7 (958), no. 754, [4] C. R. Kleeman, W. Hewitt, and L. B. Guze, Pelonephritis, Journal of the American Medical Association, 73 (96), no. 3, [5] D. H. Lehman, C. B. Wilson and F. J. Dixon, Interstitial nephritis in rats immunized with heterologous tubular basement membrane, Kidne International, 5 (974), no. 3, [6] A. A. Edd, Proteinuria and interstitial injur, Nephrolog Dialsis Transplantation, 9 (4), no., [7] S. Dürr and J. C. Thomason, Biofouling, John Wile & Sons, 9. [8] R. I. Mace, Pressure flow patterns in a clinder with reabsorbing walls, The Bulletin of Mathematical Biophsics, 5 (963), no., [9] R. I. Mace, Hdrodnamics in the renal tubule, The Bulletin of Mathematical Biophsics, 7 (965), no., [] A. A. Kozinski, F. P. Schmidt and E. N. Lightfoot, Velocit profiles in porous-walled ducts, Industrial & Engineering Chemistr Fundamentals, 9 (97), no. 3,

16 49 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer [] P. J. Palatt, H. Sackin and R. I. Tanner, A hdrodnamic model of a permeable tubule, Journal of Theoretical Biolog, 44 (974), no., [] 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 Biolog, 36 (974), [3] G. Radhakrishnamachara, P. Chandra and M. R. Kaimal, A hdrodnamical stud of the flow in renal tubules, Bulletin of Mathematical Biolog, 43 (98), no., [4] P. Chaturani and T. R. Ranganatha, Flow of Newtonian fluid in nonuniform tubes with variable wall permeabilit with application to flow in renal tubules, Acta Mechanica, 88 (99), no. -, [5] P. Muthu and T. Berhane, Mathematical model of flow in renal tubules, Inter. J. App. Maths. Mechanics, 6 (), [6] P. Muthu and T. Berhane, Fluid flow in asmmetric channel, Tamkang Journal of Mathematics, 4 (), no., [7] P. Muthu and T. Berhane, Flow of Newtonian fluid in non-uniform tubes with application to renal flow: A numerical stud, Advances in Applied Mathematics and Mechanics, 3 (), no. 5, [8] T. Haroon, A. M. Siddiqui and A. Shahzad, Creeping flow of viscous fluid through a proximal tubule with uniform reabsorption: A mathematical stud, Applied Mathematical Sciences, (6), no. 6, [9] A. M. Siddiqui, T. Haroon and A. Shahzad, Hdrodnamics of viscous fluid through porous slit with linear absorption, Applied Mathematics and Mechanics, 37 (6), no. 3, [] T. Haroon, A. M. Siddiqui and A. Shahzad, Stokes flow through a slit with periodic reabsorption: An application to renal tubule, Alexandria Engineering Journal, 55 (6), no.,

17 Stead creeping slip flow of viscous fluid through a permeable slit 493 [] G. S. Beavers and D. D. Joseph, Boundar conditions at a naturall permeable wall, Journal of Fluid Mechanics, 3 (967), no., [] P. G. Saffman, On the boundar condition at the surface of a porous medium, Studies in Applied Mathematics, 5 (97), no., [3] G. I. Talor, A model for the boundar condition of a porous material. Part, Journal of Fluid Mechanics, 49 (97), no., [4] S. Richardson, A model for the boundar condition of a porous material. Part, Journal of Fluid Mechanics, 49 (97), no., [5] A. M. Siddiqui, T. Haroon, M. Kahshan and M. Z. Iqbal, Slip effects on the flow of Newtonian fluid in renal tubule, Journal of Computational and Theoretical Nanoscience, (5), no., Received: September, 7; Published: September 8, 7

18 494 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer Partial Pore Blockage Epithelial Cell Basement Membrane Complete Pore Blockage Figure : Tubular walls without/with pore blockage phenomenon. Figure : Geometr of the problem with exponential reabsorption at the walls. 3.5 V V e αx u(.,) Figure 3: Longitudinal velocit profile u for exponential reabsorption and uniform reabsorption at the entrance of the slit, when Q = 3, V = and α =

19 Stead creeping slip flow of viscous fluid through a permeable slit V V e α x u(.5,) Figure 4: Longitudinal velocit profile u for exponential reabsorption and uniform reabsorption at the middle position of the slit, when Q = 3, V = and α = Figure 5: Longitudinal velocit profile u for exponential reabsorption and uniform reabsorption at the exit of the slit, when Q = 3, V = and α =

20 496 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer.5 V α V e x v(.,) Figure 6: Transverse velocit profile v for exponential reabsorption and uniform reabsorption at the exit of the slit, when Q = 3, V = and α =.5 V V e α x v(.5,) Figure 7: Transverse velocit profile v for exponential reabsorption and uniform reabsorption at the middle position of the slit, when Q = 3, V = and α =

21 Stead creeping slip flow of viscous fluid through a permeable slit V V e α x v(.9,) Figure 8: Transverse velocit profile v for exponential reabsorption and uniform reabsorption at the exit of the slit, when Q = 3, V = and α = V V e α x.5 Q(x) x Figure 9: Axial flow rate Q(x) for exponential reabsorption and uniform reabsorption inside the slit, when Q = 3, V = and α =

22 498 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer V Ve α x p(x,) p(,) x Figure : Pressure difference p(x, ) p(, ) for exponential reabsorption and uniform reabsorption inside the slit, when Q = 3, V = and α = 5 V V e αx 4 τ w x Figure : Wall shear stress τ w for exponential reabsorption and uniform reabsorption inside the slit, when Q = 3, V = and α =

23 Stead creeping slip flow of viscous fluid through a permeable slit 499 V V e α x q(x) x Figure : Leakage flux q(x) for exponential reabsorption and uniform reabsorption inside the slit, when Q = 3, V = and α =.5 α = α = α = 3 u(.,) Figure 3: Effect of α on u at the entrance of the slit, when Q = 3, φ = and V =

24 5 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer.5 α = α = α = 3 u(.5,) Figure 4: Effect of α on u at the middle position of the slit, when Q = 3, φ = and V =.5 α = α = α = 3 u(.9,) Figure 5: Effect of α on u at the exit of the slit, when Q = 3, φ = and V =.5 α = α = α = 3 v(.,) Figure 6: Effect of α on v at the entrance of the slit, when Q = 3, φ = and V =.5 α = α = α = 3 v(.5,) Figure 7: Effect of α on v at the middle position of the slit, when Q = 3, φ = and V =

25 Stead creeping slip flow of viscous fluid through a permeable slit 5.5 α = α = α = 3 v(.9,) Figure 8: Effect of α on v at the exit of the slit, when Q = 3, φ = and V = 3 α = α = α = 3.5 Q(x) x Figure 9: Effect of α on Q(x) inside the slit, when Q = 3, φ = and V =.5 α = α = α = 3 p(x,) p(,) Figure : Effect of α on p(x, ) p(, ) inside the slit, when Q = 3, φ = and V = x 8 α = α = α = 3 τ w x Figure : Effect of α on τ w inside the slit, when Q = 3, φ = and V =

26 5 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer.75.5 α = α = α = 3.5 q(x) x Figure : Effect of α on q(x) inside the slit, when Q = 3, φ = and V =.5 u(.,) φ=.5 φ= φ= Figure 3: Effect of φ on u at the entrance of the slit, when Q = 3, α = and V =.5 φ =.5 φ = φ =.5 u(.5,) Figure 4: Effect of φ on u at the middle position of the slit, when Q = 3, α = and V = u(.9,).5 φ =.5 φ = φ = Figure 5: Effect of φ on u at the exit of the slit, when Q = 3, α = and V =

27 Stead creeping slip flow of viscous fluid through a permeable slit 53.5 φ=.5 φ= φ=.5 v(.,) Figure 6: Effect of φ on v at the entrance of the slit, when Q = 3, α = and V =.5 φ =.5 φ = φ =.5 v(.5,) Figure 7: Effect of φ on v at the middle position of the slit, when Q = 3, α = and V =.5 φ =.5 φ = φ =.5 v(.9,) Figure 8: Effect of φ on v at the exit of the slit, when Q = 3, α = and V = φ =.5 φ = φ =.5 p(x,) p(,) x Figure 9: Effect of φ on p(x, ) p(, ) inside the slit, when Q = 3, α = and V =

28 54 T. Haroon, A. M. Siddiqui, A. Shahzad and J. H. Smeltzer φ =.5 φ = φ =.5 3 τw x Figure 3: Effect of φ on τ w inside the slit, when Q = 3, α = and V =