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

Authors: Zarqa Bano; Abdul Majeed Siddiqui; Kaleemullah Bhatti
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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,Na+, Cl−, K+, HCO3 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 clearanceClS of the solute S which is defined as “Volume of the blood that is absolutely free from solute S ”:(1)ClS=QbiCsbi−QboCsboCsbi,where Qbi and Qbo are volume flow rates, respectively, at blood inlet and outlet, similarly Csbi and Csbo are, respectively, the concentrations of the solute S 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. Section2 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 Figure1. 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 r,θ,z have been chosen and the corresponding velocity components are V⟶=ur,uθ,uz. Listed below are some additional assumptions on the flow:(1)
∂∗/∂t=0 for each flow variable i.e., the flow is steady state.(2)
uθ=0 and ∂∗/∂θ=0 for each flow variable i.e., the flow is axisymmetric.(3)
ur=ϵa1z+a0 at the walls i.e., absorption through the walls is linear in z.(4)
uz=0 at the walls i.e., the fluid in contact with the walls has zero slip velocity.(5)
Q=ϵQ0 at z=0 i.e., the inlet flow is assumed to be constant.Figure 1
Schematic diagram.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)∂ur∂r+urr+∂uz∂z=0.


#### 2.1.2. R-component

(3)∂p^∂r=−μ∂Ω∂z−α1uz∇2Ω−Ωr2+α1+α2Ω2r+2r∂∂zurΩ.


#### 2.1.3. Z-Component

(4)∂p^∂z=μ∂Ω∂r+Ωr+α1ur∇2Ω−Ωr2−α1+α22r∂∂rurΩ,with the following conditions:(5)uzR,z=0,urR,z=ϵa1z+a0,∂uz∂r0,z=0,ur0,z=0,ϵQ0=2π∫0Rruzr,0dr.


### 2.2. Model (Form 2)

By using the definition of stream functionur=1/r∂ψ/∂z and uz=−1/r∂ψ/∂r, the problem described above can be stated in more simplified form as follows:Compatibility equation:(6)E4ψ=−α1rμ∂ψ,E4ψ/r2∂z,r+α1+α2μ2E2∂ψ∂zE2ψr2−∂∂zE2ψr2,with the following conditions:(7)∂ψ∂rR,z=0,∂ψ∂zR,z=ϵRa1z+a0,∂∂r1r∂ψ∂rr=0=0,1r∂ψ∂zr=0=0,∫0R∂ψ∂rz=0dr=−ϵQ02π,where E4∗=E2E2∗.


## 2.1. Model (Form 1)

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


### 2.1.1. Continuity

(2)∂ur∂r+urr+∂uz∂z=0.


### 2.1.2. R-component

(3)∂p^∂r=−μ∂Ω∂z−α1uz∇2Ω−Ωr2+α1+α2Ω2r+2r∂∂zurΩ.


### 2.1.3. Z-Component

(4)∂p^∂z=μ∂Ω∂r+Ωr+α1ur∇2Ω−Ωr2−α1+α22r∂∂rurΩ,with the following conditions:(5)uzR,z=0,urR,z=ϵa1z+a0,∂uz∂r0,z=0,ur0,z=0,ϵQ0=2π∫0Rruzr,0dr.


## 2.1.1. Continuity

(2)∂ur∂r+urr+∂uz∂z=0.


## 2.1.2. R-component

(3)∂p^∂r=−μ∂Ω∂z−α1uz∇2Ω−Ωr2+α1+α2Ω2r+2r∂∂zurΩ.


## 2.1.3. Z-Component

(4)∂p^∂z=μ∂Ω∂r+Ωr+α1ur∇2Ω−Ωr2−α1+α22r∂∂rurΩ,with the following conditions:(5)uzR,z=0,urR,z=ϵa1z+a0,∂uz∂r0,z=0,ur0,z=0,ϵQ0=2π∫0Rruzr,0dr.


## 2.2. Model (Form 2)

By using the definition of stream functionur=1/r∂ψ/∂z and uz=−1/r∂ψ/∂r, the problem described above can be stated in more simplified form as follows:Compatibility equation:(6)E4ψ=−α1rμ∂ψ,E4ψ/r2∂z,r+α1+α2μ2E2∂ψ∂zE2ψr2−∂∂zE2ψr2,with the following conditions:(7)∂ψ∂rR,z=0,∂ψ∂zR,z=ϵRa1z+a0,∂∂r1r∂ψ∂rr=0=0,1r∂ψ∂zr=0=0,∫0R∂ψ∂rz=0dr=−ϵQ02π,where E4∗=E2E2∗.


## 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(8)ψ=∑i=13ϵiψi=ϵψ1+ϵ2ψ2+ϵ3ψ3,and(9)p^=∑i=13ϵip^i=ϵp^1+ϵ2p^2+ϵ3p^3,where ϵ 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 ofϵ1, we obtain the following homogeneous equation:(10)E4ψ1=0,with the corresponding boundary conditions(11)∂ψ1∂rR,z=0,(12)∂ψ1∂zR,z=Ra1z+a0,(13)∂∂r1r∂ψ1∂rr=0=0,(14)1r∂ψ1∂zr=0=0,(15)∫0R∂ψ1∂rz=0dr=−Q02π.Looking at condition given in (12) assume stream function ψ1 to be of the form(16)ψ1r,z=a0z+12a1z2F1r+G1r,and seek for a reverse solution of the (10) subject to the conditions (11)–(15). As E2=∂2/∂r2−1/r∂/∂r+∂2/∂z2 so E4=∂2/∂r2−1/r∂/∂r2+2∂2/∂r2−1/r∂/∂r∂2/∂z2+∂4/∂z4. Thus, equation (10) reduces to(17)a0z+12a1z2d2dr2−1rddr2F1r+d2dr2−1rddr2G1r+2a1d2dr2−1rddrF1r=0.Solving this is equivalent to solving following system of ordinary differential equations forF1r and G1r:(18)d2dr2−1rddr2F1r=0,(19)d2dr2−1rd∂r2G1r+2a1d2dr2−1rd∂rF1r=0.Following conditions are obtained using assumed form of stream function (8) in (11)–(15):(20)F1′R=0,F1R=R,F10=0,ddr1rdF1dr0=0,(21)G1′R=0,G1R=−Q02π,G10=0,ddr1rdG1dr0=0,where ψ10,0=0 has been taken conventionally and therefore G10=0. Solution to ODEs (18) and (19) subject to conditions (20) and (21) has been obtained using Maple as(22)F1r=R2r2R2−r4R4,G1r=−Q02π2r2R2−r4R4+R3a112r2R2−2r4R4+r6R6.On substituting these expressions in (14), we have(23)ψ1r,z=a0z+12a1z2R−Q02π2r2R2−r4R4+R3a112r2R2−2r4R4+r6R6.


### 3.2. Second-Order Solution

Comparing the coefficients ofϵ2, we obtain the following nonhomogeneous equation:(24)E4ψ2=−α1rμ∂ψ1∂z∂∂rE4ψ1r2−∂ψ1∂r∂∂zE4ψ1r2+α1+α2μ2E2∂ψ1∂zE2ψ1r2−∂∂zE2ψ1r2.By putting first-order solution given in (23) one can easily reach at(25)E4ψ2=−16α1+α2a1z+a0r2a14R2−r2R6μ.This nonhomogeneous nonlinear PDE need to be solved subject to following conditions:(26)∂ψ2∂rR,z=0,(27)∂ψ2∂zR,z=0,(28)∂∂r1r∂ψ2∂rr=0=0,(29)1r∂ψ2∂zr=0=0,(30)∫0R∂ψ2∂rz=0dr=0.Solution to (21) subject to conditions (26)–(30), ψ2 is obtained similarly as in the first-order case.(31)ψ2r,z=−11a1a1z+a0α1+α2R−r2R2−r2/22r2R+r236R6μ.


### 3.3. Third Order Solution

Comparing the coefficients ofϵ3, we obtain the following nonhomogeneous equation involving ψ1 and ψ2:(32)E4ψ3=−α1rμ∂ψ1∂z∂∂rE4ψ2r2−∂ψ1∂r∂∂zE4ψ2r2+∂ψ2∂z∂∂rE4ψ1r2−∂ψ2∂r∂∂zE4ψ1r2+α1+α2μ2E2∂ψ1∂zE2ψ2r2+∂ψ2∂zE2ψ1r2−2∂∂zE2ψ1rE2ψ2r.By putting first and second-order solutions given in (23) and (31), respectively, we reach at(33)E4ψ3=Arz2+Brz+Cr,where Ar, Br, and Cr are polynomials in r.(34)Ar=−384a13α1+α2r2μ2R9α1+2α23R4−13r2R212α1+10α213+5r4α1+α236,Br=−768U0a02r2α1+α2μ2R9α1+23α2R4−13r2R212α1+10α213+5r4α1+α236,Cr=88a1α1+α2r23πR10μ2πa12α1+15α211R7−43πR544a12α1+111α243r2+384a02α1+α243+48R4a1Q0α111−6πa12α1−5α2/3r2−16a02α1+5α2/4r2R311−60R2r2a1Q0α111+9πr4R44a12α1−α23r2−32a02α1+5α2/29+12r4a1Q0α111.PDE (33) need to be solved with following set of conditions:(35)∂ψ3∂rR,z=0,(36)∂ψ3∂zR,z=0,(37)∂∂r1r∂ψ3∂rr=0=0,(38)1r∂ψ3∂zr=0=0,(39)∫0R∂ψ3∂rz=0dr=0.Looking at equation (33) and the associated conditions given in equations (35)–(39), ψ3 is set to be quadratic in z.(40)ψ3r,z=z2F3r+zG3r+H3r.Following the similar procedure as in first- and second-order case, we get(41)F3r=−95R+r2R−r2α1+α2r2a1372R9μ2Lr,G3r=−95R+r2a12R−r2α1+α2r2a036R9μ2Lr,with Lr=α1+59α2/95R4−24α1+18α2r2R2/95+r4α1+α2/95 and(42)H3r==−2621−r+R2R+r2α1+α2r2a021600πR10μ2πa02α1+703α22621R7−2173πR52621a02α1+729α22173r2−19560U022173α1+295α2326−8940R4a0Q0α12621+433πr2R32621a02α1+139α2433r2−4560U02433α1+45α238+2640R2r2a0Q0α12621−47πr4R5242a02α1+11α247r2−240U02α1+5/2α247−180r4a0Q0α12621,and hence third order solution is obtained by (40).


## 3.1. First-Order Solution

Comparing the coefficients ofϵ1, we obtain the following homogeneous equation:(10)E4ψ1=0,with the corresponding boundary conditions(11)∂ψ1∂rR,z=0,(12)∂ψ1∂zR,z=Ra1z+a0,(13)∂∂r1r∂ψ1∂rr=0=0,(14)1r∂ψ1∂zr=0=0,(15)∫0R∂ψ1∂rz=0dr=−Q02π.Looking at condition given in (12) assume stream function ψ1 to be of the form(16)ψ1r,z=a0z+12a1z2F1r+G1r,and seek for a reverse solution of the (10) subject to the conditions (11)–(15). As E2=∂2/∂r2−1/r∂/∂r+∂2/∂z2 so E4=∂2/∂r2−1/r∂/∂r2+2∂2/∂r2−1/r∂/∂r∂2/∂z2+∂4/∂z4. Thus, equation (10) reduces to(17)a0z+12a1z2d2dr2−1rddr2F1r+d2dr2−1rddr2G1r+2a1d2dr2−1rddrF1r=0.Solving this is equivalent to solving following system of ordinary differential equations forF1r and G1r:(18)d2dr2−1rddr2F1r=0,(19)d2dr2−1rd∂r2G1r+2a1d2dr2−1rd∂rF1r=0.Following conditions are obtained using assumed form of stream function (8) in (11)–(15):(20)F1′R=0,F1R=R,F10=0,ddr1rdF1dr0=0,(21)G1′R=0,G1R=−Q02π,G10=0,ddr1rdG1dr0=0,where ψ10,0=0 has been taken conventionally and therefore G10=0. Solution to ODEs (18) and (19) subject to conditions (20) and (21) has been obtained using Maple as(22)F1r=R2r2R2−r4R4,G1r=−Q02π2r2R2−r4R4+R3a112r2R2−2r4R4+r6R6.On substituting these expressions in (14), we have(23)ψ1r,z=a0z+12a1z2R−Q02π2r2R2−r4R4+R3a112r2R2−2r4R4+r6R6.


## 3.2. Second-Order Solution

Comparing the coefficients ofϵ2, we obtain the following nonhomogeneous equation:(24)E4ψ2=−α1rμ∂ψ1∂z∂∂rE4ψ1r2−∂ψ1∂r∂∂zE4ψ1r2+α1+α2μ2E2∂ψ1∂zE2ψ1r2−∂∂zE2ψ1r2.By putting first-order solution given in (23) one can easily reach at(25)E4ψ2=−16α1+α2a1z+a0r2a14R2−r2R6μ.This nonhomogeneous nonlinear PDE need to be solved subject to following conditions:(26)∂ψ2∂rR,z=0,(27)∂ψ2∂zR,z=0,(28)∂∂r1r∂ψ2∂rr=0=0,(29)1r∂ψ2∂zr=0=0,(30)∫0R∂ψ2∂rz=0dr=0.Solution to (21) subject to conditions (26)–(30), ψ2 is obtained similarly as in the first-order case.(31)ψ2r,z=−11a1a1z+a0α1+α2R−r2R2−r2/22r2R+r236R6μ.


## 3.3. Third Order Solution

Comparing the coefficients ofϵ3, we obtain the following nonhomogeneous equation involving ψ1 and ψ2:(32)E4ψ3=−α1rμ∂ψ1∂z∂∂rE4ψ2r2−∂ψ1∂r∂∂zE4ψ2r2+∂ψ2∂z∂∂rE4ψ1r2−∂ψ2∂r∂∂zE4ψ1r2+α1+α2μ2E2∂ψ1∂zE2ψ2r2+∂ψ2∂zE2ψ1r2−2∂∂zE2ψ1rE2ψ2r.By putting first and second-order solutions given in (23) and (31), respectively, we reach at(33)E4ψ3=Arz2+Brz+Cr,where Ar, Br, and Cr are polynomials in r.(34)Ar=−384a13α1+α2r2μ2R9α1+2α23R4−13r2R212α1+10α213+5r4α1+α236,Br=−768U0a02r2α1+α2μ2R9α1+23α2R4−13r2R212α1+10α213+5r4α1+α236,Cr=88a1α1+α2r23πR10μ2πa12α1+15α211R7−43πR544a12α1+111α243r2+384a02α1+α243+48R4a1Q0α111−6πa12α1−5α2/3r2−16a02α1+5α2/4r2R311−60R2r2a1Q0α111+9πr4R44a12α1−α23r2−32a02α1+5α2/29+12r4a1Q0α111.PDE (33) need to be solved with following set of conditions:(35)∂ψ3∂rR,z=0,(36)∂ψ3∂zR,z=0,(37)∂∂r1r∂ψ3∂rr=0=0,(38)1r∂ψ3∂zr=0=0,(39)∫0R∂ψ3∂rz=0dr=0.Looking at equation (33) and the associated conditions given in equations (35)–(39), ψ3 is set to be quadratic in z.(40)ψ3r,z=z2F3r+zG3r+H3r.Following the similar procedure as in first- and second-order case, we get(41)F3r=−95R+r2R−r2α1+α2r2a1372R9μ2Lr,G3r=−95R+r2a12R−r2α1+α2r2a036R9μ2Lr,with Lr=α1+59α2/95R4−24α1+18α2r2R2/95+r4α1+α2/95 and(42)H3r==−2621−r+R2R+r2α1+α2r2a021600πR10μ2πa02α1+703α22621R7−2173πR52621a02α1+729α22173r2−19560U022173α1+295α2326−8940R4a0Q0α12621+433πr2R32621a02α1+139α2433r2−4560U02433α1+45α238+2640R2r2a0Q0α12621−47πr4R5242a02α1+11α247r2−240U02α1+5/2α247−180r4a0Q0α12621,and hence third order solution is obtained by (40).


## 4. Expressions for Velocity Components

Asur1=1/r∂ψ1/∂z and uz1=−1/r∂ψ1/∂r,the following expressions for first-order velocity components are obtained:(43)ur1=a0+a1z2rR−r3R3,(44)uz1=−16a1R2−3Rr2R2−4z2R2a1−8a0zR2−12Q0R21−r2R2.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 ur2=1/r∂ψ2/∂z and uz2=−1/r∂ψ2/∂r, the following expressions for second-order velocity components are obtained as follows:(45)ur2=−a12α1+α2r1−r/R21+r/R222−r2/R272μ,(46)uz2=−a1α1+α2a0+a1z11−34r2/R2+2r4/R41−r2/R218μ.It should be observed that the second-order radial velocity component does not depend onz, 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 ur3=1/r∂ψ3/∂z and uz3=−1/r∂ψ3/∂r, the following expressions for third order velocity components are obtained:(47)ur3=−a12α1+α2a0+a1zr2/R2−12r/Rr4/R4α1+r4/R4α2−24r2/R2α1−18r2/R2α2+95α1+59α236μ2,(48)uz3=−11r/R−11r/R+1α1+α2a13600R2πμ2α2+47α111rR8+−239α211−753α111rR6+101α2+9991α133rR4+−1189α211−12209α133rR2+703α233+2621α133a12R2+−500z2α1+α2a1211−1000a0zα1+α2a111−500α2+200α1a0211rR6+7500z2a1211α2+33α125+15000a0za111α2+33α125+7500a0211α2+62α175rR4+−21300z2a1211α2+111α171−42600a0za111α2+111α171−21300a0211α2+226α1213rR2+5900z2a1211α2+95α159+11800a0za111α2+95α159+5900a0211α2+326α1295Rπ+300Q0a1r/R6−37r/R4/3+107r/R2/3−149/15α111.Now we are able to write the complete expressions of velocity components correct to third-order by using equations (43)–(48) in the following:(49)ur=ϵur1+ϵ2ur2+ϵ3ur3,uz=ϵuz1+ϵ2uz2+ϵ3uz3.


## 5. Modified Pressure

The analytical expression for modified pressurep^ will be obtained here, which will be correct to order ϵ3. (3) and (4) can be written as(50)∂p^∂r=μr∂∂zE2ψ−α1r2∂ψ∂rE4ψ−1rα1+α22∂∂z∂ψ∂zE2ψr2−E2ψr2,and(51)∂p^∂z=−μr∂∂rE2ψ−α1r2∂ψ∂zE4ψ+2rα1+α2∂∂r∂ψ∂zE2ψr2.In these equations, assumptions (8) and (9) will lead to the following equations governing modified pressure of order ϵ:(52)∂p1^∂r=μr∂∂zE2ψ1,∂p1^∂z=−μr∂∂rE2ψ1.On substituting expression ofψ1 from (19) in (29) and simple integration gives the following expression of modified pressure of order ϵ:(53)p1^=−4μR3a0+a1zr2+2Q0zπR+13R2a1z−6a0z2−2a1z3.Similarlyp2^ and p3^ 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].(54)ζ=zL,γ=rR,Uγ=πR2Q0ur,Uζ=πR2Q0uz,(55)τwζ=πR3μQ0τw,Qζ=QzQ0,Ψγ,ζ=πQ0ψr,z,Pγ,ζ=ρL2μ2pr,z,(56)W0=ρRa0μ,Nre=2ρQ0πμR,λ1=α1ρR2,λ2=α2ρR2,δ=RL,(57)S=ρRLa1μ.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 parametera1<0. This parameter a1 and its dimensionless version S are named in the literature as slope-parameter of absorption or linear absorption parameter since a1=d/dzUr1,z. 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 S.


### 6.1. Components of Velocity

Transformations given in equations (54)–(57) help us to write the axial velocity component as follows:(58)Uζ=A1γ,ζS3+B1γ,ζS2+C1γ,ζS+D1γ,ζ,where(59)A1γ,ζ=5γ2−1δλ2+λ118Nre−47δ2γ8500λ1+11λ247+753λ1500+239λ2500δ2+ζ2λ2+λ1γ6+−9991λ11500−1111λ2500δ2−99ζ25λ1+25λ233γ4+12209λ11500+1189λ2500δ2+333ζ25λ1+71λ2111γ2+−2621λ11500−703λ21500δ2−19λ1+59λ295ζ2,B1γ,ζ=−γ2−1λ2+λ1δ12Nre−20W0λ2+λ1ζ3+λ1Nreδγ6+132W0λ1+100W0λ2+83ζ−37λ1Nreδ3γ4+−444W0λ1−284W0λ2−1363ζ+107λ1Nreδ3γ2+380W0λ13+236W0λ23+443ζ−149λ1Nreδ15C1γ,ζ=γ2−19δNreλ1+5λ22γ6+−31λ1−75λ22γ4+113λ1+213λ22γ2−163λ15−59λ22λ2+λ1W02−2γ4−17γ2+112λ2+λ1W0−9γ2+3δ2+36ζ2,D1γ,ζ=−2γ2−11−4δW0Nreζ.And the radial velocity component is written as(60)Uγ=A2γ,ζS3+B2γ,ζS2+C2γ,ζS+D2γ,ζ,where(61)A2γ,ζ=−λ2+λ118Nreδ2ζγλ2+λ1γ4+−24λ1−18λ2γ2+95λ1+59λ2γ2−12,B2γ,ζ=−λ2+λ118Nreδ2γW0λ2+λ1γ4+−12+−24λ1−18λ2W0γ2+11+95λ1+59λ2W0γ2−12,C2γ,ζ=−2Nreζγγ2−2,D2γ,ζ=−2NreγW0γ2−2.


### 6.2. Bulk Flow

Bulk flow or volume flow rate is definedQz=2π∫0Rruzr,zdr, or Qζ=2∫01γUζdγ, where second definition is written after applying transformations (54)–(57). Using these definitions, we get(62)Qζ=1−4δW0Nreζ−2δSNreζ2.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 S. If we ignore reverse flow, then Qζ1≥0 which implies that W0+S/2≤δNre/4.


### 6.3. Wall Shear Stress

It is defined asτw=−μ∂uz/∂r|r=R or in terms of dimensionless variables this can be redefined as τwζ=−∂Uζ/∂γγ=1. Using this definition, the expression of wall shear stress is obtained to be(63)τwζ=−16δλ2+λ1Nre343λ18640+43λ28640δ2+ζ2λ1+7λ212S3+215δ18λ1δNre−240λ1W0ζ−140W0ζλ2−35ζλ2+λ1NreS2+11520−168λ2+λ1λ1+5λ2/6W02−70λ1+λ2W0δ2−120ζ2δNreS+4−16W0ζδNre.


### 6.4. Fractional Absorption and Leakage Flux

Fractional absorption is defined asFR=Qζ0−Qζ1/Qζ0 and leakage flux is defined as q=−∂Qζ/∂ζ in dimensionless formulation. Expressions for FR and leakage flux are obtained to be(64)FR=2S+2W0δNre,qζ=4W0+SζδNre.


### 6.5. Mean Pressure Drop in Axial Direction and Stream Function

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


## 6.1. Components of Velocity

Transformations given in equations (54)–(57) help us to write the axial velocity component as follows:(58)Uζ=A1γ,ζS3+B1γ,ζS2+C1γ,ζS+D1γ,ζ,where(59)A1γ,ζ=5γ2−1δλ2+λ118Nre−47δ2γ8500λ1+11λ247+753λ1500+239λ2500δ2+ζ2λ2+λ1γ6+−9991λ11500−1111λ2500δ2−99ζ25λ1+25λ233γ4+12209λ11500+1189λ2500δ2+333ζ25λ1+71λ2111γ2+−2621λ11500−703λ21500δ2−19λ1+59λ295ζ2,B1γ,ζ=−γ2−1λ2+λ1δ12Nre−20W0λ2+λ1ζ3+λ1Nreδγ6+132W0λ1+100W0λ2+83ζ−37λ1Nreδ3γ4+−444W0λ1−284W0λ2−1363ζ+107λ1Nreδ3γ2+380W0λ13+236W0λ23+443ζ−149λ1Nreδ15C1γ,ζ=γ2−19δNreλ1+5λ22γ6+−31λ1−75λ22γ4+113λ1+213λ22γ2−163λ15−59λ22λ2+λ1W02−2γ4−17γ2+112λ2+λ1W0−9γ2+3δ2+36ζ2,D1γ,ζ=−2γ2−11−4δW0Nreζ.And the radial velocity component is written as(60)Uγ=A2γ,ζS3+B2γ,ζS2+C2γ,ζS+D2γ,ζ,where(61)A2γ,ζ=−λ2+λ118Nreδ2ζγλ2+λ1γ4+−24λ1−18λ2γ2+95λ1+59λ2γ2−12,B2γ,ζ=−λ2+λ118Nreδ2γW0λ2+λ1γ4+−12+−24λ1−18λ2W0γ2+11+95λ1+59λ2W0γ2−12,C2γ,ζ=−2Nreζγγ2−2,D2γ,ζ=−2NreγW0γ2−2.


## 6.2. Bulk Flow

Bulk flow or volume flow rate is definedQz=2π∫0Rruzr,zdr, or Qζ=2∫01γUζdγ, where second definition is written after applying transformations (54)–(57). Using these definitions, we get(62)Qζ=1−4δW0Nreζ−2δSNreζ2.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 S. If we ignore reverse flow, then Qζ1≥0 which implies that W0+S/2≤δNre/4.


## 6.3. Wall Shear Stress

It is defined asτw=−μ∂uz/∂r|r=R or in terms of dimensionless variables this can be redefined as τwζ=−∂Uζ/∂γγ=1. Using this definition, the expression of wall shear stress is obtained to be(63)τwζ=−16δλ2+λ1Nre343λ18640+43λ28640δ2+ζ2λ1+7λ212S3+215δ18λ1δNre−240λ1W0ζ−140W0ζλ2−35ζλ2+λ1NreS2+11520−168λ2+λ1λ1+5λ2/6W02−70λ1+λ2W0δ2−120ζ2δNreS+4−16W0ζδNre.


## 6.4. Fractional Absorption and Leakage Flux

Fractional absorption is defined asFR=Qζ0−Qζ1/Qζ0 and leakage flux is defined as q=−∂Qζ/∂ζ in dimensionless formulation. Expressions for FR and leakage flux are obtained to be(64)FR=2S+2W0δNre,qζ=4W0+SζδNre.


## 6.5. Mean Pressure Drop in Axial Direction and Stream Function

Using transformations (54)–(57), detailed expressions of dimensionless stream function Ψγ,ζ and mean pressure drop along axial direction ΔP have been obtained from the fundamental solution of the problem ψr,z. Again the graphical approach is suitable in order to determine influence of slip parameter and other non-Newtonian parameters on ΔP 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

AssumptionS=0 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:(65)Uζ=21−γ21−4δW0Nreζ,(66)Uγ=2W0Nre2γ−γ3,(67)Qζ=1−4δW0Nreζ,(68)ΔP=8W0δ4λ1+2λ2Nreδ−2W0ζζ.It has been noticed here that the velocity field is independent of non-Newtonian parametersλ1,λ2. 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 Section6 will be analysed graphically in this section, and the dependence of flow variables and other physically important quantities upon linear absorption parameter S will be discussed in detail. The tube dimensions are assumed to be L=0.76cm and R=1.08×10−3cm. Inlet volume flow rate Q0=4.02×10−7cm3/s and dynamic viscosity μ=7.37×10−3dyn−s/cm2 are also assumed [26]. With these assumptions, we have δ=1.6×10−3 and Nre=3.2×10−2. Inequality W0+S/2≤δNre/4 ensures no reverse flow occurs, and the values of S∈−1.3,−0.3×10−5 and W0∈0.5,1.5×10−5 are chosen accordingly. If W0+S/2≥1.28×10−5, we may observe some reverse flow.Figure2 depicts the influences of linear absorption parameter S on dimensionless axial velocity component Uζ when W0=0.5×10−5 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 S>0.5×10−5.Figure 2
Behavior of axial velocityUζ for W0=0.5×10−5 and for different values of linear absorption parameter S. (a) γ=0. (b) γ=0.5.
(a)(b)The effect ofS 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 Uγ attains its maximum within the tube. Radial velocity component reduces with the increasing strength of linear absorption parameter S; see Figure 3(c). Fixing W0=0.5×10−5 for S≤0.5×10−5 small suction Uγ>0 is observed while for S>0.5×10−5 small injection Uγ<0 is witnessed. Thus, while applying this model to study blood flow and solute transport in a hollow fiber dialyzer, the value of S may be adjusted to have zero net ultrafiltration, i.e., zero net transfer of fluid between blood and dialysate [14].Figure 3
Behavior of radial velocityUγ for W0=0.5×10−5 (a) for S=−0.5×10−5, (b) for S=−0.5×10−5, and (c) for different values of S, S=−0.3×10−5 (Orange), S=−0.5×10−5 (Blue), S=−0.7×10−5 (Red), S=−0.9×10−5 (Purple), S=−1.1×10−5 (Green), and S=−1.3×10−5 (Navy). (a) At different cross-sections. (b) At different γ. (c) 3D plot for different values of S.
(a)(b)(c)The effect ofS on bulk flow Qζ and wall shear stress τwζ is observed in Figures 4(a) and 4(b), respectively, both of which decrease from entrance to exit of the tube for S≤0.5×10−5 if we fix W0=0.5×10−5. With the increasing value of magnitude of S, both Qζ and τwζ also increase. The magnitude of leakage flux q decreases while the mean pressure drop along axial direction ΔP decreases with the increase in magnitude of S. These relationships can be visualized in Figures 5(a) and 5(b).Figure 4
(a) Volume flow rate againstζ and (b) Wall shear stress against ζ for different values of linear absorption parameter S.
(a)(b)Figure 5
(a) Leakage flux againstζ and (b) mean pressure drop in axial direction against ζ for different values of linear absorption parameter S.
(a)(b)In Figures6 and 7, stream lines are plotted for fixed W0=0.5×10−5 and different values of linear absorption parameter S. There is no reverse flow as evident from these figures for given values of S and W0. However, small suction can be seen in first half of the tube and small injection in second half in we increase the magnitude of S.Figure 6
Stream lines forW0=0.5×10−5 and different values of S. (a)S=−0.3×10−5. (b)S=−0.5×10−5.
(a)(b)Figure 7
Stream lines forW0=0.5×10−5 and different values of S. (a)S=−0.7×10−5. (b)S=−0.9×10−5.
(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 parameterS. 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)
IfS>W0 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 whenS=0 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.



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