A Jeffrey Fluid Model for a Porous-walled Channel: Application to Flat Plate Dialyzer

Creeping motion of a Jeffrey fluid in a small width porous-walled channel is presented with an application to flow in flat plate hemodialyzer. Darcy’s law is used to characterize the fluid leakage through channel walls. Using suitable physical approximations, approximate analytical solution of equations of motion is obtained by employing perturbation method. Expressions for velocity field and the hydrostatic pressure are obtained. Effects of filtration coefficient, the inlet pressure and Jeffrey fluid parameters on the flow characteristics are discussed graphically. The derived results are used to study the flow of filtrate in a flat plat hemodialyzer. Using the derived solutions, theoretical values of the filtration rate and the mean pressure difference in the hemodialyzer are calculated. On comparing the computed results with the available experimental data, a reasonable agreement between the two is found. It is concluded that the presented model can be used to study the hydrodynamical aspects of the fluid flow in a flat plate hemodialyzer.

The study of hydrodynamics of fluid flow in porous-walled channels and tubes have been remained a subject of interest for researchers since decades. This is because of the occurrence of such flows in many areas of science and engineering, particularly in those processes where mass transfer and filtration are encountered. Desalination process due to reverse osmosis, transpiration cooling, ultrafiltration in the tubules of glomerulus and fluid reabsorption through permeable walls of renal proximal tubule in kidneys, and blood filtration during hemodialysis in an artificial kidney are examples of such flows [1][2][3][4][5][6] . A common characteristic of such flows is the direct inapplicability of the usual Poiseuille law 7 in these situations because of the existence of non-zero velocity components of fluid in both axial and normal directions. This happens because of the fluid leakage through porous walls of the channel.
Literature survey reveals that various researchers have attempted the problem of fluid flow in porous-walled channels and tubes 5,[8][9][10][11][12] by assuming different forms of fluid reabsorption at the walls. Berman 8 , Sellars 9 , and Yuan 10 have studied the laminar flow of incompressible Newtonian fluid in a uniformly porous-walled channel. Solutions were obtained for a uniform suction or injection case using the regular perturbation method. Kozinski et al. 11 and Siddiqui et al. 12 also studied the creeping flow of a viscous fluid in a permeable channel. Exact solutions were found for the case in which the leakage has exponential and linear decaying rate along the channel length. Flow of a viscous fluid in a porous-walled channel was also studied by Marshall et al. 5 who assumed that the fluid leakage across the channel walls is proportional to the pressure differences across the walls. For further studies of fluid flows under different flow conditions authors would refer the reader to [13][14][15][16][17][18][19] .
In the literature presented above, all studies were performed for the Newtonian fluid model only. Despite the fact that most of the industrial and biological fluids are non-Newtonian 20 and the classical Newton's law of viscosity fails to describe the complex rheological properties of non-Newtonian fluids, a very little work has been done in order to study the non-Newtonian fluids flow in porous-walled channels and tubes. Amongst many non-Newtonian fluid models, one is the Jeffrey fluid model that has attracted many researchers due to its consideration as a better model for physiological fluids [21][22][23] . The Jeffrey model is recognized as a generalization of the frequently used Newtonian fluid model because of the fact tat its constitutive equation can be reduced to that of the Newtonian model's as a special case. The Jeffrey fluid model is capable of describing the stress relaxation property of non-Newtonian fluids, which the usual viscous fluid model cannot describe. Class of non-Newtonian fluids having the characteristic memory time scale, also known as the relaxation time, can be described well by the Jeffrey fluid model. Therefore, this article is devoted to the study of an incompressible Jeffrey fluids' flow in a porous-walled channel of small width. Fluid absorption at the channel wall is taken in accordance with the Darcy's law. The approach of the present analysis seems better from that of 2,3,11,24 . In the present approach, the fluid loss at the wall is taken as function of the wall permeability, the flow rate then determined can be found to decay linearly or exponentially along the channel wall as special case in our study. Whereas, in 2,3,11,24 a flow rate that decays linearly or exponentially, along the channel and tube was assumed in advance.

Problem Formulation
Let us consider the steady and creeping motion of an incompressible Jeffrey fluid in a porous-walled channel comprising of two equidistant parallel flat plates of length L distance 2a apart. A Cartesian coordinate system   x y ( , ) is chosen in a such that the axis of symmetry lies at y = 0 as shown in Fig. 1. It is assumed that the fluid moves only due to the hydrostatic pressure. Hydrostatic and osmotic pressures outside the channel are also assumed ot be constant. The flow is governed by the following equations xx xy where  p is the hydrostatic pressure of the fluid, ij are components of the extra stress tensor,  u and  v are components of the velocity vector ∼   x y V( , ) in the axial and transverse directions, respectively. The extra stress tensor  S for the Jeffrey fluid is given by the following equation [21][22][23] where μ is the dynamic viscosity of the fluid,  A is the shear rate, λ  2 is the retardation time, λ 1 is the ratio of relaxation time to the retardation time, and Ä represents differentiation of  A. The initial and boundary conditions corresponding to the prescribed flow are  the channel wall which is usually measured in the units of cm 2 (L p /μt is called the hydraulic permeability of the channel wall), t is the wall thickness, and p m can be visualized as the back pressure that opposes the fluid leakage and it is equal to the difference of hydrostatic and the osmotic pressures outside the channel wall. Equation (8) is the no slip condition at the wall, whereas Equations (9 and 10) are the inlet conditions. In these equations  p i is the mean pressure and  Q i is the flow rate at the inlet of the channel at =  x 0.

Dimensionless Formulation
Following parameters are used to transform equations into dimensionless form: i i 2 Using the above quantities, Eqs (1−3) take the following form xx xy where δ = a L is the ratio of channel height to its length, = = u x ( , 1) 0, is the dimensionless wall filtration parameter and λ = is the Jeffrey fluid parameter in dimensionless form, = A w a is the ratio of channel width to height and Using Eq. (4) and the dimensionless parameters, the components of the stress tensor S can be readily obtained in the following dimensionless form

Solution to the Problem
The set of Eqs (12−14) contains nonlinear coupled partial differential equations in three unknowns u, v and p and it is not possible to determine an exact solution of it. Since the channel is narrow and has small width as compared with its length, therefore the dimensionless parameter δ is very small. Thus it seems reasonable to ignore terms of the order δ 2 and higher order. Note that this is also true for the flat plate hemodialyzer, for which the parameter δ 2 is of the order 10 −8 (see the data given in Table 1 and [25][26][27] ). Now expanding u, v and p in the power series of the dimensionless Jeffrey fluid parameter as we obtain the following systems of initial-boundary-value problem up-to the Order (λ 2 ): Zeroth order system.
Zeroth order solution. The exact solution of the zeroth order system is given in the following First order solution. The exact solution of the first order system is given in the following where the constants C 1 , C 2 , C 3 and C 4 are respectively given by the following expressions It is remarkable to point out that in the above velocity and pressure profiles, we get contribution of both parameters λ 1 and λ 2 of the Jeffrey fluid. For the limiting case when λ 1 → 0 and λ 2 → 0, the presented solutions reduce to the solution for Newtonian fluid flow in a permeable channel.

Expressions for Various Quantities of Interest
The dimensionless mean pressure p x ( ) taken over any cross-section of the channel is defined as Hence the difference of mean pressure Δp(x) can be obtained as The non-dimensional wall shear stress may be obtained as The dimensionless volume flow rate at any cross-section of the channel can be computed as The fractional re-absorption (FR) is the amount of fluid that has been reabsorbed through the channel walls. It can be computed using the following expression The streamlines can be determined by using the following relation Thus streamlines can be obtained up to the order λ 2 by the following equation where C 5 is an arbitrary constant.

Results and Discussion
This section describes effects of the inlet pressure p i , wall permeability parameter K and the Jeffrey fluid parameters λ 1 and λ 2 on the axial and normal velocity components u(x, y), v(x, y), the mean pressure difference Δp(x) and the streamline patterns. Figures 2−5 present the variation of u(x, y) with y at the cross-section x = 0.3 of the channel for different values of K, p i , λ 1 and λ 2 , respectively. A parabolic axial velocity is formed that has the maximum value at the center line of channel and is zero at the boundary. These figures reveal that the magnitude of axial velocity decreases rapidly as the inlet pressure and the wall permeability are increased. Similar effects are also noticed when the magnitudes of Jeffery fluid parameters is increased. However relatively small variations happen due to increase of λ 1 and λ 2 as compared to those due to p i and K.

Application to Flat Plate Hemodialyzer
In this section, we aim to derive equations for the computation of ultrafiltration rate Q A and the mean pressure drop ΔP in an artificial kidney (FPD) by using results of the former section. An FPD comprises of several compartments of blood. Every compartment involves two rectangular sheets that are composed of recovered cellulose. The two sheets are clipped at their edges by a couple of rectangular notched plastic sheets. The blood streams between the cellulose sheets whereas the dializing liquid goes in a counter-current or a cross-current stream along the sections in hemodialyzer board 6,27-29 . The volume of blood lost by the leakage through cellulose in a given time, from a known recycling volume is taken to be the ultrafiltration rate.  www.nature.com/scientificreports www.nature.com/scientificreports/ Let L be the cellulose length, then we can compute the ultrafiltration rate for the presented model as following where sign denotes the dimensional quantity, Q 0 = Q(0) and Q L = Q(L). The non-dimensional expression for  Q A can be obtained using the dimensionless parameters defined earlier. Making use of (46), (49) and (60) in (65), the following expression for the ultrafiltration rate is determined     www.nature.com/scientificreports www.nature.com/scientificreports/ A similar approach can be adopted to evaluate the expression for calculation of mean pressure drop in FPD between x = 0 and x = L. By using Eq. (58) the following expression can be obtained for the mean pressure drop in a flat plate hemodialyzer   www.nature.com/scientificreports www.nature.com/scientificreports/ For checking the accuracy of these expressions, we are employing the experimental data provided in 5,29 as shown in Table 1. This data corresponds to a disposable flat-plate artificial kidney and is also referred to as the data for an RP kidney. By substituting these parameters in Eq. (66) along with λ 1 = 0.1 and λ 2 = 0.03 [21][22][23]  3 . This results in L p = 6.66 × 10 −16 cm 2 (see Table 2). In a similar way by substituting K = 6.23 × 10 −4 and the parameters given in Table 1 along with λ 1 = 0.1 and λ 2 = 0.03 in Eq. (67), mean pressure drop in an FPD can be computed. This results in the mean pressure drop − p p L (0) ( ) = 11.5 mm Hg (see Table 2). In the data for membranes of hemodialyzer, filtration coefficient L p value is usually not given. Results of experiments performed by Kaufmann et al. 30 show that at the normal body temperature regenerated cellulose  www.nature.com/scientificreports www.nature.com/scientificreports/ has the hydraulic permeability equal to 2.41 × 10 −11 cm 3 /dynes-sec when the membrane has a thickness equal to 7.5 × 10 −3 cm. Taking the fluid viscosity equal to 6.9 × 10 −3 dynes-sec/cm 2 from Table 1, this results in L p = 1.25 × 10 −15 cm 2 . The ultrafiltration coefficient computed by the empirical results of Marshall et al. 5 shows that Lp = 6.36 × 10 −16 cm 2 . It is also revealed in the experiments discussed in 5,29 that the mean pressure drop in an FPD is approximately 15 mm Hg. Thus a good agreement between the presented and earlier obtained experimental and empirical values of the ultrafiltration coefficient and mean pressure drop is found. This builds a confidence in stating that the presented model can be used to obtain theoretical results in advance to study the hydrodynamical aspects of the flow in a flat plate hemodialyzer.
Graphs of the ultrafiltration rate Q A versus p i , K and A are plotted in Figs 18−20, for fixed values of other parameters. These graphs state that Q A depends linearly on p i , or alternatively the ultrafiltration rate is linearly dependent on the trans-membrane pressure difference − p p (0) m . It is also evident from these graphs that Q A is  www.nature.com/scientificreports www.nature.com/scientificreports/ directly proportional to the dimensionless filtration coefficient K. Expressions for the dimensionless parameters and Fig. 19 also reveal that the ultrafiltration rate is directly proportional to the mechanical filtration coefficient L p and membrane area, and is inversely proportional to the membrane thickness and channel half width. Dependence of the ultrafiltration rate on the dimensionless filtration coefficient is also presented in Table 3. The linear dependence of Q A on p(0)−p m have been found experimentally by Malino et al. 31 and Mcdonald 32 . A series of experiments performed by Brown et al. 33 also highlights the dependence of Q A on the mechanical filtration coefficient, membrane thickness and the membrane area.
For a given or desired mechanical filtration coefficient L p , Eq. (66) can also be used to determine the magnitude of membrane thickness. This fact is explained in Fig. 21 which shows the behavior of dimensionless ultrafiltration rate Q A with the trans-membrane pressure difference for different values of membrane thickness. The experimental curve in this graph is plotted by admitting the experimental result of Kauffmann et al. 30 which     www.nature.com/scientificreports www.nature.com/scientificreports/ states that L p is 1.25 × 10 −15 cm 2 . From this figure it can be seen that for experimental and theoretical curves to be in good agreement, the membrane thickness t should be approximately 1.5.0 × 10 −3 cm. This thickness is approximately half of the estimated thickness obtained from the data of Funck-Bretano et al. 29 . The importance of membrane thickness in determining the ultrafiltration rate can also be seen from this graph. For example, it   www.nature.com/scientificreports www.nature.com/scientificreports/ can be seen that magnitude of the ultrafiltration rate can be doubled if the membrane thickness is halved. In order to determine a desired ultrafiltration rate for variations of the dimensionless filtration coefficient K, theoretical values of the ultrafiltration rate are tabulated in Table 3. It can be seen that increase in K causes the higher ultrafiltration rate.

Conclusions
The Jeffrey fluid model is a simpler model that describes well the physiological flows of non-Newtonian nature. In the study of fluid flow problems in porous-walled channels of small width, the presented model can serve as generalization of the usual Newtonian fluid model, since the obtained results can be reduced to the later one's by substituting parameters λ 1 and λ 2 equal to zero. The derived equations for the ultrafiltration rate and the mean pressure difference can be confidently used in studying the flow in a flat plate hemodialyzer. In applying the current results to study the problem of flow in flat plat hemodialyzer, one should not overlook the physical aspects of the flow phenomenon. It is also concluded the presented results are theoretical in their soul, therefore one should perform more experimental and theoretical investigation in order to have a complete understanding of the flow in a flat plat hemodialyzer.