An Analytical Study of Creeping Flow of a Second-Order Fluid through a Small Diameter Leaky Tube with Linearly Diminishing Absorption

Bano, Zarqa; Siddiqui, Abdul Majeed; Bhatti, Kaleemullah

doi:https://doi.org/10.1155/2022/9907420

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Research Article | Open Access

Volume 2022 | Article ID 9907420 | https://doi.org/10.1155/2022/9907420

An Analytical Study of Creeping Flow of a Second-Order Fluid through a Small Diameter Leaky Tube with Linearly Diminishing Absorption

Zarqa Bano,¹Abdul Majeed Siddiqui,²and Kaleemullah Bhatti¹

Academic Editor: Abderrahim Wakif

Received19 Feb 2022

Revised28 Jul 2022

Accepted29 Jul 2022

Published01 Sept 2022

Abstract

This study is focused on investigating the effects of linear absorption parameter on creeping flow of a second-order fluid through a narrow leaky tube. These flows are experienced in several biological and industrial procedures, for instance, in proximal convoluted tubule of a human kidney, in hemodialysis devices, in filtration processes of food industry, journal bearing, and slide bearing. Inertial effects are neglected due to creeping motion assumption and low Reynolds number. Langlois recursive approach method is used to linearize the governing compatibility equation and to obtain an approximate analytical solution by reverse solution method. Analytical expressions for various physically important quantities like velocity profile, bulk flow, mean pressure drop in longitudinal direction, wall shear stress, leakage flux, fractional absorption, and stream function are obtained in dimensionless form. The obtained solution shows great similarity with the already available work in the literature. Variation in flow variables with linear absorption parameter is analysed in detail. The special case of uniform absorption in creeping motion of second-order fluid is presented in a separate section. It is noticed that velocity field, in case of uniform absorption, is independent of non-Newtonian parameters. Therefore, it is asserted that any two dimensional axisymmetric Newtonian velocity fields is also a solution for second-order fluids with identical boundary conditions.

1. Introduction

There are numerous physical situations where viscous fluid flow through a small diameter leaky tube has been observed, for instance, the flow of biological fluids through proximal convoluted tubule of a Nephron, blood flow, and solute transport in an artificial kidney (Hollow fiber dialyzer or Flat plate dialyzer), filtration processes in multi-component mixtures, in journal bearing, slide bearing, and most recently in cross flow microfiltration processes.

A nephron is a basic unit of kidney which is mainly responsible for blood filtration and removal of toxic substances like urea and creatinine. There are about 1 million nephrons in a human kidney. Glomerular filtrate flows through different parts of a nephron and contains water, , , , creatinine, and urea. Most of these substances are reabsorbed back into the blood through proximal convoluted tubule. Therefore there is a lot of effort devoted in mathematical modeling of flow in proximal convoluted tubule. Macy [1, 2] is believed to be one of the pioneers who modeled the flow of glomerular filtrate in human kidney. Kelman [3], on the basis of certain assumptions and experiments, established that the rate of absorption in proximal part diminishes exponentially as a function of longitudinal distance. There is a significant amount of contribution in modeling the flow of glomerular filtrate in a nephron under different circumstances [4–9]. Recently Siddiqui et al. [10] tackled the problem using Brinkman model [11] by considering a rectangular porous slit with linear absorption, axisymmetry, and no slip conditions at the walls.

Another situation where fluid flow within a membrane with absorbing walls is witnessed inside an artificial kidney (Hollow Fiber Dialyzer/Flat plate dialyzer). A hollow fiber dialyzer is a hemodialysis machine which consists of thousands of fibers of cylindrical shape having small diameter. These fibers are permeable to small solutes but not to large molecules like red blood cells (RBC). A counter current flow is initiated on each side of fiber membrane due to pressure gradient in each compartment. The quantity which estimates the efficiency of a dialyzer is the clearance of the solute which is defined as “Volume of the blood that is absolutely free from solute ”:where and are volume flow rates, respectively, at blood inlet and outlet, similarly and are, respectively, the concentrations of the solute at blood inlet and outlet. There is a prominent contribution by researchers in modeling the blood flow and solute transport in a hollow fiber dialyzer [12–14]. In most of these models, blood flow is modeled with Newtonian fluid. We believe that blood plasma behaves as a Newtonian fluid being consists of 90 percent of water, but the haematocrit (cell mater, 45percent of whole blood) behaves as a non-Newtonian fluid. Therefore, blood should be considered as a non-Newtonian fluid. There are rate type models (power law, oldroyd-B models) which are able to capture shear thinning behavior but not the normal stress effects. Rivlin and Ericksen [15] postulated a viscoelastic model which is capable of capturing normal stress effects. In this paper, we will be using second-order Rivlin Ericksen fluid as a model.

On the other hand, there is also a lot of research confined to only mathematical understanding of two-dimensional steady state laminar flow through porous tube [16–20], as understanding the hydrodynamics of such flows and the effects of permeable walls on flow variables is important in itself. In most of these works, flows are modeled with Newtonian model and the effects of wall porosity are discussed on velocity and pressure profiles. More recently, Shahzad and Khan [21] studied the effects of reabsorbing walls on heat and mass transfer of a viscous fluid in a narrow permeable tube, and Kashan et al. [22] investigated couple stress effects and wall porosity effects on the flow between two permeable membranes.

The purpose of the current study is to investigate the effects of linear absorption on creeping motion of a second-order Rivlin Ericksen fluid through a leaky tube having small diameter. This work is important to understand the blood flow and solute transport in a hollow fiber dialyzer. To the best knowledge of the authors, there is no work that combines the linear absorption with creeping motion of second-order fluid. The governing equation and boundary conditions are set in terms of the steam function and solved analytically for an approximate solution using Langlois recursive approach method [23, 24].

This paper is organized as follows. Section 2 gives formulation of the problem. Section 3 presents the detailed analytical solution of the problem. Sections 4 and 5 list the expressions of velocity profile and modified pressure, respectively. In Section 6, we list all the important physical quantities in dimensionless formulation. Section 7 is about a special case when linear absorption parameter vanishes. In Section 8, the results are presented graphically and finally in section 9, we give concluding remarks.

2. Problem Formulation

Consider the Stokes flow of a second-order incompressible fluid in a thin cylindrical tube having permeable walls or membrane as shown in Figure 1. It is further assumed that absorption through the walls is linearly decreasing with the longitudinal length of the tube and that inertial effects of the fluid flow are not taken into account due to Stokes flow consideration. Keeping in view the geometry of the problem, cylindrical coordinates have been chosen and the corresponding velocity components are . Listed below are some additional assumptions on the flow:(1) for each flow variable i.e., the flow is steady state.(2) and for each flow variable i.e., the flow is axisymmetric.(3) at the walls i.e., absorption through the walls is linear in .(4) at the walls i.e., the fluid in contact with the walls has zero slip velocity.(5) at i.e., the inlet flow is assumed to be constant.

With all these assumptions and neglecting thermal effects, inertial effects, and body forces, this problem is governed by the following set of partial differential equations [15,25–27].

2.1. Model (Form 1)

Mathematical model for the problem described above consists of the following set of PDEs.

2.1.1. Continuity

2.1.2. R-component

2.1.3. Z-Component

with the following conditions:

2.2. Model (Form 2)

By using the definition of stream function and , the problem described above can be stated in more simplified form as follows:

Compatibility equation:

with the following conditions:where .

3. Solution of the Problem

Both the model forms are equivalent. Langlois recursive approach [23, 24, 26, 27] is frequently applicable if the flow is slow enough, in which it is assumed that

andwhere is small dimensionless parameter. Making use of the assumption (8) in equations (6) and (7) breaks the problem (Model, form (2)) into three linear problems.

3.1. First-Order Solution

Comparing the coefficients of , we obtain the following homogeneous equation:

with the corresponding boundary conditions

Looking at condition given in (12) assume stream function to be of the form

and seek for a reverse solution of the (10) subject to the conditions (11)–(15). As so . Thus, equation (10) reduces to

Solving this is equivalent to solving following system of ordinary differential equations for and :

Following conditions are obtained using assumed form of stream function (8) in (11)–(15):where has been taken conventionally and therefore . Solution to ODEs (18) and (19) subject to conditions (20) and (21) has been obtained using Maple as

On substituting these expressions in (14), we have

3.2. Second-Order Solution

Comparing the coefficients of , we obtain the following nonhomogeneous equation:

By putting first-order solution given in (23) one can easily reach at

This nonhomogeneous nonlinear PDE need to be solved subject to following conditions:

Solution to (21) subject to conditions (26)–(30), is obtained similarly as in the first-order case.

3.3. Third Order Solution

Comparing the coefficients of , we obtain the following nonhomogeneous equation involving and :

By putting first and second-order solutions given in (23) and (31), respectively, we reach atwhere , , and are polynomials in .

PDE (33) need to be solved with following set of conditions:

Looking at equation (33) and the associated conditions given in equations (35)–(39), is set to be quadratic in z.

Following the similar procedure as in first- and second-order case, we getwith and

and hence third order solution is obtained by (40).

4. Expressions for Velocity Components

As and ,the following expressions for first-order velocity components are obtained:

It should be noted that this is the solution corresponding to the viscous case. Therefore, the above expressions refer exactly to the expression obtained by Macey [1]. These expressions are also reduced to those that Bhatti et al. [26] obtained when uniform suction was considered. Also as and , the following expressions for second-order velocity components are obtained as follows:

It should be observed that the second-order radial velocity component does not depend on z, and if the absorption is assumed to be uniform, all of these components will vanish. These terms are in full accordance with our previous paper [26] in which the flow with inertial effects was considered. Similarly as and , the following expressions for third order velocity components are obtained:

Now we are able to write the complete expressions of velocity components correct to third-order by using equations (43)–(48) in the following:

5. Modified Pressure

The analytical expression for modified pressure will be obtained here, which will be correct to order . (3) and (4) can be written as

and

In these equations, assumptions (8) and (9) will lead to the following equations governing modified pressure of order :

On substituting expression of from (19) in (29) and simple integration gives the following expression of modified pressure of order :

Similarly and are obtained.

6. Nondimensionalization of Various Important Physical Quantities

The following transformations are used to simplify the obtained expressions. Most of these transformations are the same which were used in earlier works [26, 27].

The description of all the physical variables and parameters is given in Nomenclature. This work primarily focuses on linearly decreasing absorption at the walls of the tube and its consequences. Therefore, it is important to comment on parameter . This parameter and its dimensionless version are named in the literature as slope-parameter of absorption or linear absorption parameter since . In the following, we firstly present the analytical expressions for velocity components, bulk flow, wall shear stress, fractional absorption at the walls, leakage flux, mean pressure drop in axial direction, and stream function in dimensionless form and then see the dependence of all these flow related physical quantities upon the linear absorption parameter .

6.1. Components of Velocity

Transformations given in equations (54)–(57) help us to write the axial velocity component as follows:where

And the radial velocity component is written aswhere

6.2. Bulk Flow

Bulk flow or volume flow rate is defined , or , where second definition is written after applying transformations (54)–(57). Using these definitions, we get

This is quite comparable to the expression of bulk flow obtained in [1, 26] because naturally here we have an additional term involving linear absorption parameter . If we ignore reverse flow, then which implies that .

6.3. Wall Shear Stress

It is defined as or in terms of dimensionless variables this can be redefined as . Using this definition, the expression of wall shear stress is obtained to be

6.4. Fractional Absorption and Leakage Flux

Fractional absorption is defined as and leakage flux is defined as in dimensionless formulation. Expressions for FR and leakage flux are obtained to be

6.5. Mean Pressure Drop in Axial Direction and Stream Function

Using transformations (54)–(57), detailed expressions of dimensionless stream function and mean pressure drop along axial direction have been obtained from the fundamental solution of the problem . Again the graphical approach is suitable in order to determine influence of slip parameter and other non-Newtonian parameters on and stream lines. In order to see complete method of computation of such expressions, one can go through our previous work [26, 27].

7. Special Case: Creeping Flow with Uniform Absorption

Assumption implies uniform absorption through the walls of the tube. Under such assumption, velocity components, bulk flow, and mean pressure drop along z-axis, respectively, will reduce to the following:

It has been noticed here that the velocity field is independent of non-Newtonian parameters . This means that in case of axisymmetric creeping motion under uniform absorption or suction, no contribution is coming from the non-Newtonian terms of the governing equations. At this stage, we make an important assertion. In 1966, Tanner [28] shown that any plane creeping Newtonian velocity field is also a solution for second-order fluids under identical velocity boundary conditions. Therefore, we assert that any 2D axisymmetric Newtonian velocity field is also a solution for second-order fluids under the same velocity boundary conditions. This claim is very evident since velocity field given in equations (65) and (66) match exactly with the axisymmetric Newtonian velocity field [1].

8. Results and Discussions

The expressions obtained in Section 6 will be analysed graphically in this section, and the dependence of flow variables and other physically important quantities upon linear absorption parameter will be discussed in detail. The tube dimensions are assumed to be and . Inlet volume flow rate and dynamic viscosity are also assumed [26]. With these assumptions, we have and . Inequality ensures no reverse flow occurs, and the values of and are chosen accordingly. If , we may observe some reverse flow.

Figure 2 depicts the influences of linear absorption parameter on dimensionless axial velocity component when is fixed. It is observed that the magnitude of axial velocity increases by increasing the magnitude of linear absorption parameter. It is also seen that the axial velocity decreases in first half of the tube while it increases after almost midway from the inlet for .

(a)

(b)

The effect of on dimensionless radial velocity component is observed in Figure 3. First of all, radial velocity component decreases along axial distance; see Figures 3(a) and 3(b). Secondly attains its maximum within the tube. Radial velocity component reduces with the increasing strength of linear absorption parameter ; see Figure 3(c). Fixing for small suction is observed while for small injection is witnessed. Thus, while applying this model to study blood flow and solute transport in a hollow fiber dialyzer, the value of may be adjusted to have zero net ultrafiltration, i.e., zero net transfer of fluid between blood and dialysate [14].

(a)

(b)

(c)

The effect of on bulk flow and wall shear stress is observed in Figures 4(a) and 4(b), respectively, both of which decrease from entrance to exit of the tube for if we fix . With the increasing value of magnitude of , both and also increase. The magnitude of leakage flux decreases while the mean pressure drop along axial direction decreases with the increase in magnitude of . These relationships can be visualized in Figures 5(a) and 5(b).

(a)

(b)

(a)

(b)

In Figures 6 and 7, stream lines are plotted for fixed and different values of linear absorption parameter . There is no reverse flow as evident from these figures for given values of and . However, small suction can be seen in first half of the tube and small injection in second half in we increase the magnitude of .

(a)

(b)

(a)

(b)

9. Concluding Remarks

The steady creeping flow of second-order fluid in a leaky tube with linear absorption has been studied in this work. The Langlois recursive approach method has been remarkable in solving highly linear PDEs. Analytical expressions for velocity components, bulk flow, wall shear stress, fractional absorption, leakage flux, mean pressure drop along the axis of the tube, and the stream function have been obtained in dimensionless formulation and analysed for different values of linear absorption parameter . We believe that this model will be further helpful in understanding the flow of blood in hollow fiber dialyzer and solute transfer through its membrane, where advection-diffusion equation may be considered governing the concentration of the solutes. Following conclusions are drawn from this study:(1)The magnitude of radial velocity decreases with the increasing strength of linear absorption parameter.(2)The magnitude of axial velocity increases with the increasing strength of linear absorption parameter.(3)If absorption is observed to be negative (i.e., small injection). For this reason, current model may be more appropriate to deal with the blood flow and solute transport in a hollow fiber dialyzer.(4)Bulk flow and wall shear stress both increase with the increasing strength of linear absorption parameter.(5)Mean pressure drop in longitudinal direction and leakage flux decrease with the increasing strength of linear absorption parameter.(6)The velocity profile when presented in Section 7 is in agreement with the corresponding Newtonian problem under identical velocity boundary conditions [1]. Therefore, it is asserted not only in plane creeping flow but also in 2D axisymmetric flow, the Newtonian velocity field is also a solution to the second-order fluid equations under same boundary conditions.

Nomenclature

:	Elastic parameter
:	Cross-viscosity parameter
:	Ratio of radius to length
:	Dimensionless coordinate of tube’s transversal axis
:	Dimensionless elastic parameter
:	Dimensionless cross-viscosity parameter
:	Dynamic viscosity
:	Dimensionless stream function
:	Stream function
:	Constant fluid density
:	Wall shear stress
:	Dimensionless wall shear stress
:	Coordinate of tube’s azimuthal axis
:	Dimensionless coordinate of tube’s longitudinal axis,

Other Symbols

:	Mean pressure drop in major flow direction in the tube at point
:	Characteristic pressure in the tube at point
:	Fractional reabsorption
:	Length of the tube
:	Inlet flow Reynolds number
:	Dimensionless pressure in the tube at point
:	Pressure in the tube at point
:	Volume flow rate at any point
:	Leakage flux
:	Dimensionless volume flow rate at any point
:	Dimensionless leakage flux
:	Inlet volume flow rate
:	Radius of the tube
:	Coordinate of tube’s transversal axis
:	Linear absorption parameter
:	Dimensionless fluid velocity along r-direction
:	Fluid velocity along r-direction
:	Dimensionless fluid velocity along z-direction
:	Fluid velocity along z-direction
:	Cross flow radial velocity at wall of the pipe
:	Wall Reynolds number
:	Coordinate of tube’s longitudinal axis

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Zarqa Bano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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