This research explores the creeping flow of a Jeffrey fluid through a narrow permeable slit with an application of blood flow through a planer hemodialyzer. The fluid motion of Jeffrey fluid in a two-dimensional conduit with nonhomogeneous boundary conditions due to constant reabsorption on the wall is a complicated problem. The viscous effect of Jeffrey fluid in a cross-sectional area of a narrow slit is computed with the help of continuity and momentum equation. The stress component, velocity profile, stream function, and pressure gradient show the behavior of creeping flow of Jeffrey fluid in a narrow slit. To find the explicit expression of velocity, pressure, stream function, and flux, recursive (Langlois) approach is adopted. Maximum velocity, shear stress, leakage flux, and fractional absorption on the wall are also calculated in this research. The mathematical results of this research are very helpful to study the blood flow through planer hemodialyzer; therefore, this theoretical model has significant importance in the field of renal physiology.
The small conduit (mini/microchannel) has gained an important role in biofluid mechanics. The main problem in the development of mini and microslits are wall properties (friction on the surface and permeability of the wall). Through experimental results, it is proved that flow of non-Newtonian fluids [1–5] through slit is beneficial which improves the lubrication performance in the hydrodynamics system.
The microtribology of different types of complex fluids [6–8] has gained a lot of attention in these days due to its frequent use in industry and physics. These fluids are characterized as viscoelastic, time-dependent, and time-independent fluids. Richard Bird  proposed a theory of viscoelastic fluids that explains the viscous and elastic effects. These viscoelastic fluids have complex mathematical structures as well. Different researchers [10–15] have discussed many interesting and challenging issues of viscoelastic fluids. An important viscoelastic fluid is a Jeffrey fluid which has a simple constitutive relation and explains the tribology of viscoelastic fluid. The Jeffrey model uses the local and convective derivatives of first Rivlin Erickson tensor. Few researchers like Nadeem et al.  discussed the similarity solution for a stagnation point flow of a Jeffrey fluid over a shrinking sheet. In his research, he has simplified the two-dimensional momentum equation under the boundary layer assumption, and less complex partial differential equations are reduced into the nonlinear ordinary differential equation by the suitable transformation and he solved the resulting problem by homotopy analysis method. Partial slip effect on Jeffrey fluid in a porous medium is investigated by Khan  and flow of Jeffrey fluid in a rotating frame is studied by Hayat et al. . These researchers have considered the boundary layer approach and discussed the approximate series solution instead of full two-dimensional and two directional flow in the closed duct. The two-dimensional flow of Jeffrey fluid has been discussed by the lubrication and boundary layer approach but no one has addressed the Jeffrey model in two-dimensional slit without lubrication approach.
A lot of attention has been given to the flows of rectangular conduit because of its application in design and technology. Some researchers like Fetecau and Dimitrue [19–21] examined the viscoelastic fluid flow in a rectangular conduit and found the exact solutions of unidirectional and two-dimensional flow by the analytic methods. The slow motion of a viscous fluid through permeable rectangular slit has been investigated by Siddiqui and co-workers [22–25] and found exact solutions by the Inverse method. The two-dimensional flow of a viscoelastic fluid in a permeable rectangular conduit (two directional) is a big challenge of the present era. A wide range of applications for viscous fluid flow through permeable rectangular slit exist in different mechanisms like oil filtration, blood circulation through kidney, and urine flow through renal tubule; therefore, different scientists [26–29] have thrown the light on creeping flow of viscous fluid through permeable slits and tube with uniform and nonuniform reabsorption on the wall. Due to dire need of biofluid engineering, blood flow through a two-dimensional channel has been discussed in Ref.  and perturbation solution was obtained for the pressure, velocity, and volume flow rate. In these days, biofluid engineering requires the study of microorganisms through narrow tubes; therefore, to fulfill this need, Bhatti  proposed the flow of microorganism in nanofluid via artery and attained the series solution through perturbation method.
Besides the above studies, Nadeem with his co-authors [31–37] explained different microbiological phenomenon with the help of analytical and numerical techniques, but all these studies were limited under the lubrication approach. As per our knowledge, no attention has been paid to study the creeping flow of a Jeffrey fluid in a rectangular slit with uniform reabsorption on the wall that can be used in the process of desalination due to osmosis acting in the backward direction, reabsorption of the fluid on the boundary wall of the proximal tube present in the kidneys, and in artificial kidneys when blood is filtered during hemodialysis.
Therefore, in this problem two-dimensional creeping flow of non-Newtonian fluid through rectangular slit has been presented with uniform reabsorption on the wall, which leads to a highly nonlinear set of partial differential equations (PDEs) with nonhomogeneous boundary conditions in a finite domain. This type of system can be handled by numerical and analytical technique which requires high speed computers, but for such problems Langlois  introduced a recursive approach which can be used to solve the complicated system of PDEs. In this research, two-dimensional momentum and continuity equations with the nonhomogeneous boundary conditions are linearized into steady slow flow of Jeffrey fluid in a narrow conduit with Langlois approach.
This research is organized in six sections. In Sec. 1, introduction and literature review with shortcoming is presented, and mathematical formulation of two-dimensional creeping flow of a Jeffrey fluid in a rectangular slit with uniform reabsorption has been made in Sec. 2. Recursive approach is used to solve the nonlinear problem in Sec. 3, and expression for stream function, velocity profile, pressure distribution, shear stress on the wall, flow rate, and leakage flux are calculated by the Inverse method. The graphical results for pressure difference, velocity profile (on the entrance, middle and exit region), and stream function are presented in Sec. 4, application of the proposed model is included in the Sec. 5, and conclusions are added in Sec. 6.
2 Mathematical Modeling
Consider an incompressible, steady, and two-dimensional Jeffrey fluid flow (blood flow) across a rectangular cross section of the rectangular slit (planer hemodialyzer) with -axis locating at center of the rectangular slit and -axis in the perpendicular direction of centerline. A constant reabsorption rate at the permeable walls of rectangular slit is uniformly distributed. The walls of the slit are separated by the distance and width of the slit is . The volume flow rate at the entrance of the slit is .
The geometry of the slit in Fig. 1 shows that flow is symmetric about the centerline of the slit; therefore, for computational purpose we will consider only the upper half of a slit.
where and show velocity components in the horizontal and transverse direction, is small parameter due to creeping flow, represents axial flow rate, and shows reabsorption velocity.
where and are Jeffrey fluid parameters, and
where are defined in the nomenclature.
where and are the Jeffrey parameter and reabsorption velocity, respectively.
In this research, creeping (slow) flow of Jeffrey fluid suggests that the velocity, pressure, and stresses are slowly varying; therefore, we expand velocity, pressure, and stresses in the power of which is a small nondimensional parameter. This recursive approach  is pioneered by Langlois to solve the momentum and continuity equation for slow viscoelastic fluid flow.
3.1 First-Order Problem
3.2 Second-Order Problem.
3.3 Third-Order Problem.
To solve the first-, second-, and third-order system, we use the symbolic software mathematica. The expressions of first-, second-, and third-order solutions are included in the Appendix and their analyses are constructed through the graphs.
Note that fractional reabsorption and leakage flux both depend upon porosity parameter but pressure distribution, velocity, and stress components depend upon porosity parameter and Jeffrey parameter also the negative pressure indicates that the Jeffrey fluid is reabsorbed when it flows through the slit (planer hemodialyzer).
4 Graphical Results and Discussion
In this section, graphical behavior of horizontal and vertical component of velocity, flow rate, pressure difference, stream function, and wall shear stress are observed for distinct values of porosity parameter and Jeffrey parameter . In this study, show the entrance, middle, and exit points of the rectangular cross section, respectively. Here we have chosen three different positions on the axis of the slit to observe the forward and backward flow of Jeffrey fluid for the reabsorption analysis.
Figures 2(a)–2(c) indicates that horizontal velocity decelerate by mounting the porosity parameter at entrance, middle, and exit region of rectangular cross section, and decline is faster in the middle region but at the exit region reverse flow has been observed. It is also observed that near the center of rectangular cross section, flow is maximum due to pressure gradient and near the walls of rectangular cross section the fluid flow become stationary due to wall friction.
Figures 3(a)–3(c) shows the axial velocity for distinct values of Jeffrey parameter at entrance, middle and exit region of the slit for as given in Ref. . As it can be seen that increasing values of Jeffrey parameter accelerate the axial velocity near the walls and decelerate near the center of the slit, and also in the middle and exit region of the rectangular cross section, reverse flow has been observed.
The variation of porosity parameter and Jeffrey parameter on the magnitude of vertical component of velocity is shown in Figs. 4(a) and 4(b), which indicates that flow in transverse direction is symmetric about the centerline and also it grows by increasing , i.e., porosity parameter along vertical direction helps to accelerate the flow near the walls of rectangular cross section but reverse behavior is observed in horizontal direction. It is also observed that magnitude of vertical velocity decreases by increasing Jeffrey fluid parameter
Figures 5(a)–5(c) displays the pressure difference for distinct values of , and ; it is noted that more pressure is required to flow the fluid in a slit when the reabsorption rate rises but the amount of pressure from one point to another point falls with the extending amount of viscosities and of the Jeffrey fluid. Figures 6(a)–6(c) illustrates that by increasing porosity parameter , Jeffrey fluid parameters and result in decrease of the wall shear stress. Figure 6(a) indicates that porosity parameter reduces the shear stress on the wall because uniform reabsorption helps to accelerate the flow due to decline in resistive force on the surface of the slit; therefore, fluid moves with the less amount of tangential force along the wall.
It can be noticed that by increasing the values of porosity parameter the contour size increases, which shows that reabsorption causes thinning of fluid, and increasing value of Jeffrey fluid parameter shows that the contour size decreases which causes thickening of the Jeffrey fluid; these contour plots also show that flow is in the backward direction which is due to the presence of reabsorption velocity on the wall.
5 Application to Planer Hemodialyzer
In this section, we will make the computational analysis of fractional reabsorption for the different values of velocity on the wall and mean pressure drop in a planer hemodialyzer by using the results calculated in this research. An artificial kidney (planer dialyzer) has several blood compartments, and each compartment consists of two planer surfaces that are composed of recovered cellulose. The blood streams between the cellulose surfaces go in a cross-current stream along the sections in hemodialyzer plane. The filtration of blood through cellulose in a given time is due to the process of reabsorption on the boundary. To check the accuracy of the formula, experimental data [40,41] have been mentioned in Table 1.
These data correspond to RP kidney and also correspond to a disposable planer artificial kidney. By incorporating the values of parameters mentioned in Table 1 with and the reabsorption velocity and non-Newtonian parameter are evaluated, which are involved in the formulation of pressure rise and fractional reabsorption.
Table 1 shows the values of experimental data mentioned in Refs. [40,41] that are used to find the values of reabsorption velocity and fluid viscosity in this study. The numerical values mentioned in Tables 2 and 3 are evaluated with the formulas derived in Eqs. (68) and (76); the calculated values of pressure and wall velocity show that fractional reabsorption increases with the increase in permeability on the boundary and for high amount of fractional reabsorption less amount of the pressure is required. Table 3 shows the calculated values of pressure drop is 15 mmHg against the reabsorption velocity 0.3237 cm/s which is also mentioned through experiments given in Refs.  and  that the pressure drop in a planer hemodialyzer is approximately 15 mm Hg. Thus, a good agreement between the present and earlier obtained empirical values of pressure drop and fractional reabsorption is found which can build a confidence in stating that the present model can be used to obtain theoretical results in advance to study the hydrodynamics aspects of the flow in a planer hemodialyzer.
In this study, slow flow of a Jeffrey fluid through a permeable rectangular slit of cross-sectional area is discussed. The mathematical model of creeping flow of a Jeffrey fluid is presented by the set of complicated nonlinear partial differential equation which is solved by the Langlois approach. The result of this study can be compared with the results of Siddiqui  for Newtonian fluid if and or This research analyses different features for velocity, pressure and stream functions as it can be seen through the graphical results that the axial velocity diminishes with the extending values of porosity parameter in the rectangular cross section of the slit also decline in axial velocity is dominant in the middle region of the slit and reverse behavior is observed at the exit of slit. It is predicted that the mounted values of Jeffrey parameter shrink the magnitude of axial velocity at the center of entrance, middle, and exit region.
Moreover, it can be scrutinized that the magnitude of transverse velocity grows with the extending values of and lows with the improving values of but at the center of slit, the fluid comes at rest and then starts to move in opposite direction. In this study, it is also noticed that more pressure is required to flow the fluid in a slit when the reabsorption rate rises but the amount of pressure from one point to another point fall off with the extending amount of viscosities and This research indicates that porosity parameter reduces the shear stress on the wall because uniform reabsorption helps to accelerate the flow therefore fluid moves with the less amount of tangential force along the wall. The contour size increases by increasing the values of porosity parameter wich shows that reabsorption causes thinning of the Jeffrey fluid and increase of value of Jeffrey fluid parameter shows that the contour size decreases which causes thickening of the Jeffrey fluid. The theoretical results of the proposed problem can be used for the hydrodynamic study of the blood through artificial kidney that is analyzed with the help of tabular data.
We thank the reviewers for their constructive suggestions which led to improvement in this research.