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Journal of Applied Mathematics
Volume 2012, Article ID 812535, 24 pages
http://dx.doi.org/10.1155/2012/812535
Research Article

Nonlinear Analysis for Shear Augmented Dispersion of Solutes in Blood Flow through Narrow Arteries

1School of Advanced Sciences, VIT University, Chennai Campus, Chennai 48, India
2School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Received 23 May 2012; Accepted 2 July 2012

Academic Editor: Turgut Öziş

Copyright © 2012 D. S. Sankar 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.

Abstract

The shear augmented dispersion of solutes in blood flow (i) through circular tube and (ii) between parallel flat plates is analyzed mathematically, treating blood as Herschel-Bulkley fluid model. The resulting system of nonlinear differential equations are solved with the appropriate boundary conditions, and the expressions for normalized velocity, concentration of the fluid in the core region and outer region, flow rate, and effective axial diffusivity are obtained. It is found that the normalized velocity of blood, relative diffusivity, and axial diffusivity of solutes are higher when blood is modeled by Herschel-Bulkley fluid rather than by Casson fluid model. It is also noted that the normalized velocity, relative diffusivity, and axial diffusivity of solutes are higher when blood flows through circular tube than when it flows between parallel flat plates.

1. Introduction

The dispersion of a solute in a solvent flowing in a pipe/channel is an important physical phenomenon, which has wide applications in many fields of science and engineering and some potential application fields are chemical engineering, biomedical engineering, physiological fluid dynamics, and environmental sciences [1]. The physics involved in the dispersion theory is the spreading of a passive species (solute) in a flowing fluid (solvent) due to the combined action of molecular diffusion and nonuniform velocity distribution [2]. For better understanding of the concept of shear-augmented dispersion, let us consider a bolus of a solute in the fully developed laminar flow of an incompressible fluid in a conduit. The bolus is carried downstream by the Poiseuille flow and is subjected to the resulting transverse concentration gradient. At the leading edge of the bolus, the bolus diffuses from the high concentration region near the centre of the tube towards the low concentration region at the wall [3].

Taylor [4] initiated the study on the dispersion of solutes in fluid flow and reported that if a solute is injected into a solvent flowing steadily in a straight tube, the combined action of the lateral molecular diffusion and the variation of velocity over the cross-section would cause the solute ultimately to spread diffusively with the effective molecular diffusivity 𝐷e given by 𝐷e=𝑎2𝑤2𝑚/48𝐷𝑚, where 𝐷𝑚 is the molecular diffusivity, 𝑤𝑚 is the normalized axial velocity, and 𝑎 is the radius of the tube. He also pointed out that the spreading of the solute is symmetrical about a point moving with the average velocity 𝑤𝑚 of the fluid. Since many intravenous medications are therapeutic at low concentration, but toxic at high concentration, it is important to know the rate of dispersion of the material in the circulatory system [5]. The main objective of this study is to analyze the dispersion of solutes in blood flow. Sankarasubramanian and Gill [6] discussed the dispersion of solute undergoing first-order wall retention in Poiseuille flow through circular tube. Their generalized dispersion model gives rise to three effective transport coefficients, namely, the convection, the diffusion, and the exchange coefficients. Lungu and Moffatt [7] analyzed the effect of wall conductance on heat diffusion using Fourier transform with average function to obtain a series solution. Tsangaris and Athanassiadis [8] investigated the diffusion of solutes in an oscillatory flow in an annular pipe.

When blood flows through arteries and veins, it shows many fluid dynamic complexities such as pulsatility, curvature, branching and elasticity of walls, and thus, the dispersion of solutes in blood flow is affected by these factors as well as reactions and the multiphase character of fluid [9]. Hence, it is important to understand the modifications caused by non-Newtonian rheology to the dispersion of passive species. This analysis can also be applied to blood handling devises too. Rao and Deshikachar [10] studied the dispersion of solute in a steady flow of incompressible fluid in an annular pipe and showed that the axial dispersion of the normalized concentration decreases with the increase of the inner radius of the cylinder. They reported that the asymptotic solution, for large time, of effective diffusivity in the flow directions is a decreasing function of the wall conductance. Mazumdar and Das [11] investigated the effect of wall conductance on the axial dispersion in the pulsatile tube flow.

Sharp [12] investigated the shear-augmented dispersion of solutes in the steady flow of Casson fluid through a circular pipe and also flow between parallel plates using Taylor model [4]. Jiang and Grotberg [13] studied the dispersion of a bolus contaminant in a straight tube with oscillatory flow field and weak conductive walls and reported that the axial dispersion diminished by the wall conductance when the frequency parameter exceeds the critical value. Smith and Walton [14] discussed the dispersion of solutes in the fluid flow through inclined tube with an annulus. The dispersion of solutes in the flow of power law fluids was analyzed by Agarwal and Jayaraman [1]. They showed that the effective molecular diffusivity varies with yield stress for Casson and Bingham fluids and power law index in the case of power-law fluids. Dash et al. [15] studied the shear augmented dispersion of a solute in the Casson fluid flow in a conduit using the generalized dispersion model of Gill and Sankarasubramanian [16].

Herschel-Bulkley (H-B) fluid model and Casson fluid model are the non-Newtonian fluid models that are generally used in the studies of blood flow through narrow arteries [17, 18]. Tu and Deville [19] and Sankar et al. [20] mentioned that blood obeys Casson’s equation only for moderate shear rate, whereas the H-B equation can still be used at low shear rates and represent fairly closely what is occurring in blood. Several researchers proved that for tube diameter 0.095 mm blood behaves like H-B fluid rather than power law and Bingham fluids [21, 22]. Iida [23] reports “The velocity profile in the arterioles having diameter less than 0.1 mm are generally explained fairly by Casson and H-B fluid models. However, the velocity profiles in the arterioles whose diameters are less than 0.065 mm do not conform to the Casson model, but, can still be explained by H-B model.” Hence, it is appropriate to model blood as H-B fluid model rather than Casson fluid model when it flows through smaller diameter arteries. The theoretical analysis of shear-augmented dispersion in the steady flow of H-B fluid through circular tube has not been studied so far, to the knowledge of the authors. Hence, in this paper, we analyze the shear augmented dispersion of solutes in the steady flow of H-B fluid through a narrow cylindrical tube with possible application to blood flow. Since, some devices involve the flow between parallel flat plates or membranes rather than flow in tubes [12], the study on the dispersion of solutes in fluid flow between parallel flat plates is also important. Thus, it is also aimed to investigate the shear-augmented dispersion of solutes in the incompressible fluid flow between parallel flat plates. The layout of the paper is as follows.

Section 2 formulates the problem mathematically and then solves the resulting system of differential equations to obtain the expression for the flow quantities such as normalized velocity, concentration of the fluid in the core region and outer region, flow rate, and the effective axial diffusivity. The effects of various parameters such as power law index and yield stress on these flow quantities are discussed through appropriate graphs in the numerical simulation of the results and discussion Section 3. Also, some possible physiological application of this study to blood flow is given in Section 3. The main results are summarized in the concluding Section 4.

2. Mathematical Formulation

Consider the dispersion of a solute in the axi-symmetric, steady, laminar, and fully developed unidirectional flow (in the axialdirection) of Herschel-Bulkley (H-B) fluid (viscous incompressiblenon-Newtonian fluid) through (i) circular tube and (ii) between parallel flat plates. The geometry of the flow fields in circular tube and between parallel flat plates are shown in Figures 1(a) and 1(b), respectively.

fig1
Figure 1: The geometry of the fluid flow.
2.1. Flow in Circular Tube
2.1.1. Governing Equations

Cylindrical polar coordinate system (𝑟,𝜓,𝑥) is used to analyze the flow through uniform circular tube, where 𝑟 and 𝑥 are the coordinates in the radial and axial directions, respectively, and 𝜓 is the azimuthal angle. It has been reported that the radial velocity is negligibly small and can be neglected for a low Reynolds number flow in a narrow artery with mild stenosis [23]. For the steady flow of incompressible viscous fluid, the axial component of the momentum equation simplifies to 𝑑𝑝1𝑑𝑥=𝑟𝑑𝑑𝑟(𝑟𝜏),(2.1) where 𝑑𝑝/𝑑𝑥 is the axial pressure gradient, 𝑝 is the pressure and 𝜏 is the shear stress. The constitutive equation of the H-B fluid is given by 𝑑𝑢=1𝑑𝑟𝜂𝜏𝜏𝑦𝑛if𝜏𝜏𝑦,0if𝜏𝜏𝑦,(2.2) where 𝑢 is the velocity in the axial direction; 𝜂 is the coefficient of viscosity of H-B fluid with dimension (𝑀𝐿1𝑇2)𝑛𝑇; 𝜏𝑦 is the yield stress and 𝑛 the is power law index of H-B fluid. To solve (2.1) and (2.2) for the unknowns shear stress 𝜏 and velocity 𝑢, we utilize the following boundary conditions 𝜏isniteat𝑟=0,(2.3)𝑢=0at𝑟=𝑎,(2.4) where 𝑎 is the radius of the tube. For steady flow, the simplified form of the species transport equation in the plug core region and outer (nonplug core) region are given below in (2.5) and (2.6), respectively. 1𝑟𝜕𝑟𝜕𝑟𝜕𝐶1=𝜕𝑟̂𝑢𝐶𝜅𝜕𝐶1𝜕̃𝑥.(2.5) The species transport equation for the outer region simplifies to the following form: 1𝑟𝜕𝑟𝜕𝑟𝜕𝐶2=𝜕𝑟̂𝑢𝜅𝜕𝐶2𝜕̃𝑥,(2.6) where 𝐶1 and 𝐶2 are the concentration of the solute in the plug core region and outer region; ̃𝑥=𝑥𝑢𝑡 is the new axial coordinate moving with the normalized velocity 𝑢;̂𝑢𝐶=𝑢𝐶𝑢;̂𝑢=𝑢𝑢;̂𝑢 is the relative velocity in the outer region; ̂𝑢𝐶 is the relative velocity in the plug core region; t is the time. The boundary conditions for the concentration of the fluid in the core region are 𝜕𝐶1𝐶𝜕𝑟=0at𝑟=0,(2.7)1=0at𝑟=0,(2.8)𝜕𝐶2𝐶𝜕𝑟=0at𝑟=𝑎,(2.9)2=𝐶1at𝑟=𝑟𝑐.(2.10) Equations (2.5) and (2.6) can be solved with the help of the boundary conditions (2.7) to get the expressions for the concentrations in the plug core region and outer region.

2.1.2. Solution Method

Integrating (2.1) with respect to 𝑟 and then using (2.3), we get the expression for the shear stress as follows: 𝑟𝜏=2𝑑𝑝.𝑑𝑧(2.11) Using binomial series expansion in (2.2) and neglecting the terms involving (𝜏𝑦/𝜏)2 and higher powers of (𝜏𝑦/𝜏) (since (𝜏𝑦/𝜏)31), one can obtain the simplified form of the constitutive equation as follows: 𝑑𝑢=1𝑑𝑟𝜂𝜏𝑛𝑛𝜏𝑦𝜏𝑛1+𝑛(𝑛1)2𝜏𝑦𝜏𝑛2.(2.12) Using (2.11) in (2.12) and integrating the resulting differential equation with respect to 𝑟 and then using the boundary condition (2.4), we get the expression for the velocity in the outer (non-plug core) region as 1𝑢(𝑟)=1(𝑛+1)𝜂2𝑑𝑝𝑑𝑧𝑛𝑎𝑛+1𝑟𝑛+1(𝑛+1)𝑟𝑐(𝑎𝑛𝑟𝑛)+𝑛(𝑛+1)2𝑟2𝑐𝑎𝑛1𝑟𝑛1,(2.13) where 𝑟𝑐 is the plug core radius, which is defined as follows: 𝑟𝑐=2𝜏𝑦.(𝑑𝑝/𝑑𝑧)(2.14) The expression for the plug flow velocity is obtained by replacing 𝑟 by 𝑟𝑐 in (2.13) as given in the following: 𝑢𝑐1(𝑟)=1(𝑛+1)𝜂2𝑑𝑝𝑑𝑧𝑛𝑎𝑛+1(𝑛+1)𝑟𝑐𝑎𝑛+𝑛(𝑛+1)2𝑟2𝑐𝑎𝑛1𝑛(𝑛1)2𝑟𝑐𝑛+1.(2.15) Using (2.14) and (2.15), one can obtain the following expression for the bulk velocity or normalized velocity: 𝑢=𝑢𝐻𝐴𝑧𝑐,(2.16) where 𝑢𝐻𝑎=𝑛+11𝜂(𝑛+3)2𝑑𝑝𝑑𝑧𝑛,𝐴𝑧(2.17)𝑐𝑛=1(𝑛+3)𝑧(𝑛+2)𝑐+𝑛(𝑛+3)(𝑛1)𝑧2(𝑛+1)𝑐2𝑛4+2𝑛35𝑛26𝑛+4𝑧2(𝑛+1)(𝑛+2)𝑐𝑛+3,(2.18) where 𝑧𝑐=𝑟𝑐/𝑎. When 𝑛=1, one can get the bulk velocity of Newtonian fluid from (2.17). Solving (2.5) with the help of the boundary conditions (2.7) and (2.8), one can get the expression for the concentration of the solute in the plug core region as follows: 𝐶1=𝑢𝐻𝑟22𝜅(𝑛+1)𝜕𝐶1𝐵𝑧𝜕̃𝑥𝑐,(2.19) where 𝐵𝑧𝑐=1(𝑛+1)(𝑛+3)(𝑧𝑛+2)𝑐+𝑛(𝑛+3)2𝑧𝑐2𝑛(𝑛1)(𝑛+3)4𝑧𝑐𝑛+1𝑛4+2𝑛35𝑛26𝑛+4𝑧2(𝑛+1)(𝑛+2)𝑐𝑛+3.(2.20) One can get the concentration of the solute in the core region by integrating (2.6) and then using the boundary conditions (2.9) and (2.10) which is given as follows: 𝐶2=𝑢𝐻𝑎2𝜅𝜕𝐶21𝜕̃𝑥𝑧(𝑛+2)(𝑛+3)𝑛+3+(𝑛+3)(𝑛+2)2𝑧𝑐𝑧𝑛+2𝑛(𝑛+3)2(𝑛+1)2𝑧2𝑐𝑧𝑛+1+𝑧212(𝑛+1)(𝑛+3)𝑧2(𝑛+2)𝑐+𝑛(𝑛+3)𝑧4(𝑛+1)2𝑐+𝑛4+2𝑛35𝑛26𝑛+4𝑧8(𝑛+1)(𝑛+2)𝑐𝑛+3𝑛4+2𝑛35𝑛26𝑛+4𝑧4(𝑛+1)(𝑛+2)𝑐𝑛+3𝑧log𝑧𝑐𝑛7+10𝑛6+32𝑛5+18𝑛493𝑛3164𝑛252𝑛+408(𝑛+1)2(𝑛+2)2𝑧(𝑛+3)𝑐𝑛+3.(2.21) The flux of solute across a cross section at constant ̃𝑥 is defined as follows [12] 1𝑞=𝜋𝑎2𝑟𝑐0̂𝑢𝑐𝐶1𝜅𝜕𝐶𝜕̃𝑥2𝜋𝑟𝑑𝑟+𝑎𝑟𝑐̂𝑢𝐶2𝜅𝜕𝐶.𝜕̃𝑥2𝜋𝑟𝑑𝑟(2.22) For our convenience, (2.22) is rewritten as follows: 𝑞=𝜅𝜕𝐶+2𝜕̃𝑥𝑎2𝐼1+𝐼2,(2.23) where 𝐼1=𝑟𝑐0̂𝑢𝑐𝐶1𝑟𝑑𝑟=𝑢2𝐻𝐵2𝑧𝑐𝑟4𝑐4𝜅(𝑛+1)2𝜕𝐶,𝐼𝜕̃𝑥(2.24)2=𝑎𝑟𝑐(̂𝑢𝑟)𝐶2𝑑𝑟=𝑢2𝐻𝑎24𝜅𝜕𝐶𝜕̃𝑥𝑎𝑟𝑐𝑇1(𝑟)𝑇2(𝑟)𝑑𝑟.(2.25) The functions 𝑇1(𝑟) and 𝑇2(𝑟) appearing in (2.25) are given as follows: 𝑇1=(𝑛+3)(𝑛+1)𝑟(𝑛+3)𝑟(𝑛+1)𝑛+2𝑎𝑛+1(𝑛+3)𝑧𝑐𝑟+(𝑛+3)𝑧𝑐𝑟𝑛+1𝑎𝑛+𝑛(𝑛+3)2𝑧2𝑐𝑟𝑛(𝑛+3)2𝑧2𝑐𝑟𝑛𝑎𝑛1𝑧𝐴𝑐𝑟=𝑇11+𝑇12+𝑇13+𝑇14+𝑇15+𝑇16+𝑇17,𝑇21=𝑟(𝑛+1)(𝑛+3)𝑎𝑛+3+(𝑛+3)𝑧𝑐(𝑛+2)2𝑟𝑎𝑛+2𝑛(𝑛+3)2(𝑛+1)2𝑧2𝑐𝑟𝑎𝑛+1𝑧+𝐺𝑐𝑟𝑎2𝑛4+2𝑛35𝑛26𝑛+4𝑧4(𝑛+1)(𝑛+2)𝑐𝑛+3𝑟log𝑟𝑐𝑧+𝐻𝑐,(2.26) where 𝐺𝑧𝑐=12(𝑛+1)(𝑛+3)𝑧2(𝑛+2)𝑐+𝑛(𝑛+3)𝑧4(𝑛+1)2𝑐+𝑛4+2𝑛35𝑛26𝑛+4𝑧8(𝑛+1)(𝑛+2)𝑐𝑛+3,𝐻𝑧𝑐=𝑛7+10𝑛6+32𝑛5+18𝑛493𝑛3164𝑛252𝑛+408(𝑛+1)2𝑧(𝑛+2)(𝑛+3)𝑐𝑛+3.(2.27) For the easy evaluation of the integral in (2.25), it is rewritten as follows: 𝐼2=𝑢2𝐻𝑎24𝜅𝜕𝐶𝜕̃𝑥𝑎𝑟𝑐𝑇1𝑇2=𝑑𝑟𝑢2𝐻𝑎24𝜅𝜕𝐶𝜕̃𝑥𝑎𝑟𝑐𝑇11+𝑇12+𝑇13+𝑇14+𝑇15+𝑇16+𝑇17𝑇2=𝑑𝑟𝑢2𝐻𝑎24𝜅𝜕𝐶𝑆𝜕̃𝑥1+𝑆2+𝑆3+𝑆4+𝑆5+𝑆6+𝑆7,(2.28) where 𝑆1=𝑎𝑟𝐶𝑇11𝑇2𝑑𝑟,𝑆2=𝑎𝑟𝐶𝑇12𝑇2𝑑𝑟,𝑆3=𝑎𝑟𝐶𝑇13𝑇2𝑆𝑑𝑟,4=𝑎𝑟𝐶𝑇14𝑇2𝑑𝑟,𝑆5=𝑎𝑟𝐶𝑇15𝑇2𝑑𝑟,𝑆6=𝑎𝑟𝐶𝑇16𝑇2𝑑𝑟,𝑆7=𝑎𝑟𝐶𝑇17𝑇2𝑑𝑟.(2.29) The details of obtaining the expressions for 𝑆1,𝑆2,𝑆3,𝑆4,𝑆5,𝑆6 and 𝑆7 are given in Appendix A. The effective axial diffusivity is defined as 𝐷e𝑞=𝜕𝐶/𝜕̃𝑥=𝜅1+Pec2𝐸𝑧48𝑐𝐴2𝑧𝑐.(2.30) From the simplified form of the expression obtained for the flux of solute (defined in (2.23)), the expression for 𝐸(𝑧𝑐) is obtained as 𝐸𝑧𝑐=24(𝑛+3)(𝑛+5)48𝑛(𝑛+3)(2𝑛+9)𝑧𝑐𝑛(𝑛+2)(𝑛+4)(𝑛+5)(2𝑛+5)12𝑛6+13𝑛5+58𝑛4+82𝑛391𝑛2𝑧305𝑛1422𝑐(1+𝑛)(2+𝑛)3(4+𝑛)(5+𝑛)24𝑛2(𝑛1)(𝑛+3)(2𝑛+7)𝑧3𝑐𝑛(𝑛+1)(𝑛+2)(𝑛+4)(2𝑛+3)3(𝑛+3)8+12𝑛7+36𝑛646𝑛5257𝑛4+74𝑛3+404𝑛2𝑧112𝑛64𝑐5+𝑛(𝑛+1)2+(𝑛+2)(𝑛+4)(𝑛+5)24(𝑛+3)2𝑛2𝑧+4𝑛3𝑐𝑛+6+𝑛(𝑛+1)(𝑛+2)(𝑛+4)(𝑛+5)6𝑛(𝑛+3)4+6𝑛33𝑛2𝑧36𝑛+24𝑐𝑛+73(𝑛+1)(𝑛+4)(𝑛+5)4𝑛13+48𝑛12+195𝑛11+160𝑛10913𝑛91878𝑛8+1865𝑛7+6772𝑛62535𝑛518918𝑛412512𝑛3+5656𝑛2+4872𝑛1152𝑧𝑐2𝑛+6(𝑛+1)3(𝑛+2)3(+3𝑛𝑛+3)(2𝑛+3)(2𝑛+5)4+2𝑛35𝑛2𝑛6𝑛+44+6𝑛33𝑛2𝑧36𝑛+24𝑐2𝑛+8(+𝑛𝑛+1)(𝑛+2)(𝑛+4)(𝑛+5)6(𝑛1)(𝑛+3)4+2𝑛35𝑛2𝑧6𝑛+4𝑐𝑛+5𝑧log𝑐6𝑛(𝑛+1)(𝑛+2)4+2𝑛35𝑛26𝑛+42𝑧𝑐2𝑛+6𝑧log𝑐(𝑛+1)2(𝑛+2)2.(2.31)

2.2. Flow between Parallel Flat Plates
2.2.1. Governing Equations

Cartesian coordinate system (𝑥,𝑦) is used to analyze the flow between parallel flat plates. The width of the flow region is taken as 2 h (h is half of the spacing between the flat plates). Since, the flow is assumed as steady, laminar, and fully developed, the velocity of the fluid in the 𝑦 direction is negligibly small and can be neglected for low Reynolds number flow [12]. Thus, for the steady flow of viscous incompressible fluid between the parallel flat plates, the axial component of the momentum equation simplifies to 𝑑𝑝𝑑𝑑𝑥=𝑑𝑦(𝜏),(2.32) where 𝜏 is the shear stress and 𝑝 is the pressure. The constitutive equation of the H-B fluid model in Cartesian coordinate system is defined by 𝑑𝑢=1𝑑𝑦𝜂𝜏𝜏𝑦𝑛if𝜏𝜏𝑦,0if𝜏𝜏𝑦,(2.33) where 𝑢 is the velocity in the 𝑥 direction, 𝜂 is the coefficient of viscosity of H-B fluid, 𝜏𝑦 is the yield stress, and 𝑛 the is power law index of H-B fluid. The following boundary conditions are used to solve (2.32) and (2.33) for the unknowns shear stress 𝜏 and velocity u𝜏isniteat𝑦=0,(2.34)𝑢=0at𝑦=.(2.35) The simplified form of the species transport equation in the plug core region and outer region for the flow between flat plates are 𝜕2𝐶1𝜕𝑦2=̂𝑢𝑐𝜅𝜕𝐶1,𝜕𝜕̃𝑥(2.36)2𝐶2𝜕𝑦2=̂𝑢𝜅𝜕𝐶2,𝜕̃𝑥(2.37) where 𝐶1 and 𝐶2 are the concentration of the species in the plug core region and outer region, respectively, and ̃𝑥=𝑥𝑢𝑡 is the coordinate moving in the 𝑥 direction with the normalized velocity 𝑢,̂𝑢𝑐=𝑢𝑐𝑢,̂𝑢=𝑢𝑢,̂𝑢 is the relative velocity in the outer region, and ̂𝑢𝑐 is the relative velocity in the plug core region, 𝑡 is the time. The boundary conditions of the concentration of the species in the plug core region and outer region are 𝜕𝐶1𝐶𝜕𝑟=0at𝑦=0,(2.38)1=0at𝑦=0,(2.39)𝜕𝐶2𝐶𝜕𝑟=0at𝑦=,(2.40)1=𝐶2at𝑦=𝑦𝐶,(2.41) where 𝑦𝐶 is half the width of the plug core region. Equations (2.36) and (2.37) can be solved by utilizing the boundary conditions (2.38)–(2.41) to get the expressions for the concentrations of the solute in the plug core region and outer region.

2.2.2. Solution Method

Integrating (2.32) with respect to 𝑦 and then using (2.34), one can easily get the following expression for the shear stress 𝜏: 𝜏=𝑑𝑝𝑑𝑥𝑦.(2.42) Using (2.42) in (2.33) and then utilizing the boundary condition (2.35), the expression for the velocity of H-B fluid in the outer region is obtained as follows: 1𝑢(𝑦)=(𝑛+1)𝜂𝑑𝑝𝑑𝑥𝑛𝑛+1𝑦𝑛+1(𝑛+1)𝑦𝑐𝑛𝑦𝑛+𝑛(𝑛+1)2𝑦2𝑐𝑛1𝑦𝑛1,(2.43) where 𝑦𝑐=𝜏𝑦.(𝑑𝑝/𝑑𝑥)(2.44) One can obtain the expression for the velocity of H-B fluid in the plug core region as below by replacing 𝑦 by 𝑦𝑐 in (2.43) and then simplifying the resulting expression 𝑢𝑐1(𝑦)=(𝑛+1)𝜂𝑑𝑝𝑑𝑥𝑛𝑛+1(𝑛+1)𝑦𝑐𝑛+𝑛(𝑛+1)2𝑦2𝑐𝑛1𝑛(𝑛1)2𝑦𝑐𝑛+1.(2.45) The normalized velocity or bulk velocity of the H-B fluid at a cross section is obtained as 𝑢=𝑢𝐻𝐹𝑧𝑐,(2.46) where 𝑢𝐻=𝑛+1(𝑛+2)𝜂𝑑𝑝𝑑𝑥𝑛,𝐹𝑧𝑐𝑛=1(𝑛+2)𝑧(𝑛+1)𝑐+(𝑛1)(𝑛+2)2𝑧𝑐2𝑛𝑛23𝑧2(𝑛+1)𝑐𝑛+2,(2.47) where 𝑧𝑐=𝑦𝑐/. Solving (2.36) with the help of the boundary conditions (2.38) and (2.39), one can get the expression for the concentration of the species in the plug core region as follows: 𝐶1=𝑢𝐻𝑦22𝜅(𝑛+1)𝜕𝐶1𝑀𝑧𝜕̃𝑥𝑐,(2.48) where 𝑀𝑧𝑐=1(𝑛+2)𝑧𝑐+(𝑛+1)(𝑛+2)2𝑧𝑐2𝑛(𝑛1)(𝑛+2)2𝑧𝑐𝑛+1+𝑛𝑛232𝑧𝑐𝑛+2.(2.49) The following expression is obtained for the concentration of the species in the outer region by solving (2.37) with the help of the boundary conditions (2.40) and (2.41) 𝐶2=(𝑛+2)𝑢𝐻2𝜅𝜕𝐶21𝜕̃𝑥(𝑧𝑛+1)(𝑛+2)(𝑛+3)𝑛+3+1(𝑛+1)(𝑛+2)2𝑧𝑐𝑧𝑛+212(𝑛+1)2𝑧2𝑐𝑧𝑛+1+𝑧2112(𝑛+1)(𝑛+2)𝑧2(𝑛+1)𝑐+14𝑧2𝑐+𝑛𝑛23𝑧4(𝑛+1)(𝑛+2)𝑐𝑛+2𝑛𝑛23𝑧2(𝑛+1)(𝑛+2)𝑐𝑛+2𝑛𝑧+4+2𝑛35𝑛26𝑛+4𝑧4(𝑛+1)(𝑛+2)(𝑛+3)𝑐𝑛+3,(2.50) where 𝑧=𝑦/. The flux of the solute across a cross section at constant ̃𝑥 is defined as 1𝑞=𝑦𝑐0̂𝑢𝑐𝐶1𝜅𝜕𝐶𝜕̃𝑥𝑑𝑦+𝑦𝑐̂𝑢𝑐𝐶2𝜅𝜕𝐶𝜕̃𝑥𝑑𝑦.(2.51) For our convenience, (2.51) is rewritten as 𝑞=𝜅𝜕𝐶+1𝜕̃𝑥𝐼3+𝐼4,(2.52) where 𝐼3=𝑦𝐶0̂𝑢𝑐𝐶1𝑑𝑦=𝑢2𝐻𝑀2𝑧𝑐𝑦3𝐶6𝜅(𝑛+1)2𝜕𝐶,𝐼𝜕̃𝑥(2.53)4=𝑦𝐶̂𝑢𝐶2𝑑𝑦=𝑢2𝐻(𝑛+2)2(𝑛+1)2𝜅𝜕𝐶𝜕̃𝑥𝑦𝐶𝑊1(𝑦)𝑊2(𝑦)𝑑𝑦.(2.54) The functions 𝑊1(𝑦) and 𝑊2(𝑦) appearing in (2.54) are 𝑊11(𝑦)=(𝑛+1)(𝑛+2)𝑦(𝑛+1)𝑛+1𝑛+1(𝑛+2)𝑧(𝑛+1)𝑐+(𝑛+2)𝑧𝑐𝑦𝑛𝑛+(𝑛+2)2𝑧2𝑐𝑛(𝑛+2)2𝑧2𝑐𝑦𝑛1𝑛1𝑛𝑛232𝑧(𝑛+1)𝑐𝑛+2=𝑊11+𝑊12+𝑊13+𝑊14+𝑊15+𝑊16+𝑊17,𝑊(2.55)21(𝑦)=𝑦(𝑛+1)(𝑛+2)(𝑛+3)𝑛+3+𝑧𝑐𝑦(𝑛+1)(𝑛+2)𝑛+21𝑧2(𝑛+1)2𝑐𝑦𝑛+1𝑧+𝐽𝑐𝑦2𝑛𝑛23𝑧2(𝑛+1)(𝑛+2)𝑐𝑛+2𝑦𝑧+𝐾𝑐,(2.56) where 𝐽𝑧𝑐=112(𝑛+1)(𝑛+2)𝑧2(𝑛+1)𝑐+14𝑧2𝑐+𝑛𝑛23𝑧4(𝑛+1)(𝑛+2)𝑐𝑛+2,𝐾𝑧𝑐=𝑛4+2𝑛35𝑛26𝑛+4𝑧4(𝑛+1)(𝑛+2)(𝑛+3)𝑐𝑛+3.(2.57) Equation (2.54) is rewritten as below for the easy evaluation of the integral appearing in it 𝐼4=(𝑛+2)𝑢2𝐻2𝜅𝜕𝐶𝜕̃𝑥𝑦𝐶𝑊1(𝑦)𝑊2=(𝑦)𝑑𝑦(𝑛+2)𝑢2𝐻2𝜅𝜕𝐶𝜕̃𝑥𝑦𝐶𝑊11+𝑊12+𝑊13+𝑊14+𝑊15+𝑊16+𝑊17𝑊2=𝑑𝑦𝑢2𝐻𝑎24𝜅𝜕𝐶𝐷𝜕̃𝑥1+𝐷2+𝐷3+𝐷4+𝐷5+𝐷6+𝐷7,(2.58) where 𝐷1=𝑦𝐶𝑊11𝑊2𝑑𝑦,𝐷2=𝑦𝐶𝑊12𝑊2𝑑𝑦,𝐷3=𝑦𝐶𝑊13𝑊2𝐷𝑑𝑦,4=𝑦𝐶𝑊14𝑊2𝑑𝑦,𝐷5=𝑦𝐶𝑊15𝑊2𝐷𝑑𝑦,6=𝑦𝐶𝑊16𝑊2𝑑𝑦,𝐷7=𝑦𝐶𝑊17𝑊2𝑑𝑦.(2.59) The details of obtaining the expressions for 𝐷1,𝐷2,𝐷3,𝐷4,𝐷5,𝐷6 and 𝐷7 are given in Appendix B. The effective axial diffusivity is defined as 𝐷e𝑞=𝜕𝐶/𝜕̃𝑥=𝜅1+2Pec2𝑁𝑧105𝑐𝐹2𝑧𝑐,(2.60) where 𝑁𝑧𝑐=35(𝑛+4)(2𝑛+5)35𝑛(2𝑛+7)𝑧𝑐+𝑛2(𝑛+1)(𝑛+3)(𝑛+4)354+6𝑛3+8𝑛2𝑧3𝑛3𝑐2(𝑛+1)2(𝑛+3)(𝑛+4)35𝑛(𝑛1)(𝑛+2)𝑧𝑐3+8(𝑛+1)(𝑛+3)35(𝑛1)2(𝑛+2)𝑧4𝑐𝑛4(2𝑛+1)35𝑛(𝑛+6)2𝑧3𝑐𝑛+2+4(𝑛+1)(𝑛+3)(𝑛+4)35𝑛2𝑛(𝑛+5)2𝑧3𝑐𝑛+34(𝑛+1)2𝑛(𝑛+3)357+10𝑛6+24𝑛532𝑛4133𝑛3+22𝑛2𝑧+156𝑛24𝑐𝑛+48(𝑛+1)2𝑛(𝑛+3)(𝑛+4)35(𝑛+2)4+4𝑛35𝑛2𝑧18𝑛+12𝑐𝑛+5+𝑛4(𝑛+1)(𝑛+3)(𝑛+4)35(𝑛+2)4+4𝑛35𝑛2𝑧18𝑛+12𝑐𝑛+6+8(𝑛+3)(𝑛+4)35𝑛2𝑛232𝑧𝑐2𝑛+48(𝑛+1)235957𝑛10+16𝑛9+80𝑛84𝑛7492𝑛6380𝑛5113𝑛4+740𝑛3396𝑛2𝑧192𝑛+72𝑐2𝑛+516(𝑛+1)2+𝑛(𝑛+3)(2𝑛+1)(2𝑛+5)35𝑛2𝑛34+4𝑛35𝑛2𝑧18𝑛+12𝑐2𝑛+6.8(𝑛+1)(𝑛+3)(𝑛+4)(2.61)

3. Results and Discussion

The objective of this study is to analyze the blood flow characteristics due to the shear augmented dispersion of solutes when blood flows (i) through circular tubes and (ii) between parallel flat plates, modeling blood as H-B fluid. It is also aimed to discuss the effects of various physical parameters on the velocity distribution of blood, relative diffusivity, and effective axial diffusivity of the solute [19].

3.1. Normalized Velocity Distribution

The normalized velocity profiles of H-B and Casson fluids (for different values of the power law index 𝑧𝑐 and yield stress n) flow (i) through circular tube and (ii) between parallel flat plates are sketched in Figure 2. It is observed that the normalized velocity decreases marginally with the increase of power law index and yield stress. It is also seen that the normalized velocity of Newtonian fluid model is marginally higher than those of the H-B and Casson fluid models, and it is slightly higher than that of Power law fluid model. It is clear that the normalized velocity of H-B fluid model is considerably higher than that of the Casson fluid model. One can notice that the normalized velocity of any fluid model when it flows between parallel flat plates is very similar to its normalized velocity when it flows through a circular tube. It is of interest to note that the normalized velocity profile of the Newtonian fluid model is in good agreement with the corresponding normalized velocity profile in Figure 3 of Sharp [12].

812535.fig.002
Figure 2: Normalized velocity profiles of some non-Newtonian fluids flowing in a tube and between parallel flat plates.
812535.fig.003
Figure 3: Variation of the relative diffusivity with yield stress for some non-Newtonian fluids when they flow in a tube and between parallel flat plates.
3.2. Relative Diffusivity

The variation of relative diffusivity with yield stress of H-B and Casson fluids when flowing (i) through circular tube and (ii) between parallel flat plates is shown in Figure 3. It is observed that the relative diffusivity decreases slowly with the increase of the yield stress of H-B and Bingham fluid models, but it decreases rapidly (nonlinearly) with the increase of the yield stress for Casson fluid model. It is also noted that the relative diffusivity decreases considerably with the increase of the power law index of the H-B fluid model. It is also found that the relative diffusivity is significantly higher for H-B fluid model than that of the Casson fluid model. It is seen that for any fluid model, the relative diffusivity is higher when it flows through circular pipe than when it flows between parallel flat plates.

Figure 4 sketches the variation of relative diffusivity with the reciprocal of the power law index for H-B and power law fluids when they flow (i) through a circular tube and (ii) between flat plates. It is clear that for power law fluid, the relative diffusivity increases rapidly with the increase of the reciprocal of the power law index from 0 to 0.5, and then it increases slowly with the increase of the reciprocal of the power law index from 0.5 to 1. The same behavior is also noticed for H-B fluid, but some nonlinearity is found at lower values of the reciprocal of the power law index. It is noted that for both H-B and power law fluid models, the relative diffusivity increases almost linearly with the increase of the reciprocal of the power law index. It is found that the relative diffusivity is marginally higher for power law fluid model than that of the H-B fluid model.

812535.fig.004
Figure 4: Comparison of relative diffusivity of non-Newtonian fluids when flowing in a tube and between parallel flat plates.
3.3. Some Physiological Applications

The estimates of yield stress 𝑧𝑐 and relative diffusivity in the canine vascular system are useful to understand the dispersion of solutes in blood flow through arterioles, venules, inferior vena cava, and ascending aorta. Using the expressions obtained for flow in tubes, the estimates of yield stress 𝑧𝑐 and relative diffusivity in the canine vascular system [12, 24] (in arteries of different diameters) are computed in Table 1. It is observed that the estimates of the relative diffusivity decreases slowly with the increase of the yield stress. It is also noted that the relative diffusivity decreases gradually with the increase of the power law index. It is found that the solute disperses rapidly in arterioles than in ascending aorta. It is also noticed that the solute dispersion is faster when blood is modeled by H-B fluid or Bingham fluid than when it is modeled by Casson fluid.

tab1
Table 1: Estimates of yield stress 𝑧𝑐 and relative diffusivity (𝐸/𝐴2) in the canine vascular system for flow in tube.

From the expressions obtained for flow between parallel flat plates, the estimates of yield stress 𝑧𝑐 and relative diffusivity in the same canine vascular system are computed in Table 2. It is noted that the variation in the relative diffusivity with the yield stress/diameter of the canine artery is the similar to the one that was observed in the case of flow in tube flow in Table 1. From Tables 1 and 2, it is observed that for any fluid model, the relative diffusivity of the solute is slightly higher when it disperses in circular tube than when it disperses between parallel flat plates.

tab2
Table 2: Estimates of yield stress 𝑧𝑐 and relative diffusivity (𝑁/𝐹2) in the canine vascular system for flow between parallel flat plates.

4. Conclusion

This mathematical analysis exhibits many interesting results on the dispersion of the solutes in blood flow when blood is modeled by H-B fluid model and compares the results of the present study with the results of Sharp [12]. The main findings of this theoretical study are summarized as follows.(i)The normalized velocity of blood flow is considerably higher when it is modeled by H-B fluid rather than Casson fluid model.(ii)The normalized velocity of blood is significantly higher when it flows through circular tube than when it flows between parallel plates.(iii)The relative diffusivity and axial diffusivity of the solute are marginally higher when blood is modeled by H-B fluid rather than by Casson fluid.(iv)The relative diffusivity and axial diffusivity of the solute are slightly higher when blood flows in circular tubes than when it flows between parallel flat plates.(v)The normalized velocity of blood, relative diffusivity, and axial diffusivity of solute decrease with the increase of the yield stress of the blood.

Based on these results, one can note that there is a substantial difference between the flow quantities of H-B fluid model (present results) and Casson fluid model (results of Sharp [12]), and thus, it is expected that the present H-B model may be useful to predict physiologically important flow quantities. Hence, it is concluded that the present study can be treated as an improvement in the mathematical modeling of dispersion of solutes in blood flow through narrow diameter arteries. Since the solutes may disperse unsteadily, the study on the unsteady diffusion of solutes in blood flow with effects on boundary absorption would be more realistic, and this will be done in the near future.

Appendices

A.

𝑆1=𝑎2(𝑛+7)𝑛8(𝑛+1)(𝑛+5)4+12𝑛3+45𝑛2𝑧+54𝑛𝑐8(𝑛+1)(𝑛+2)2+𝑛(𝑛+4)4+7𝑛3+7𝑛2𝑧15𝑛2𝑐16(𝑛+1)3(𝑛+3)𝑧4𝑐8(𝑛+1)2+(𝑛+3)28𝑛2𝑧+3𝑛+25𝑐𝑛(𝑛+3)216(𝑛+1)2𝑧6𝑐+𝑛7+4𝑛610𝑛560𝑛439𝑛3+112𝑛2+128𝑛832(𝑛+1)3(𝑛+2)2𝑧𝑐𝑛+3𝑛5+11𝑛4+37𝑛3+27𝑛242𝑛188(𝑛+1)2𝑧(𝑛+4)(𝑛+5)𝑐𝑛+5𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+432(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+7+𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+48(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+3log𝑧𝑐,𝑆2=𝑎2(𝑛+4)+𝑛2(𝑛+1)(𝑛+3)(𝑛+5)2𝑛3+19𝑛2+60𝑛+632(𝑛+1)(𝑛+2)2𝑧(𝑛+5)(2𝑛+5)𝑐+𝑛𝑛4+8𝑛3+18𝑛2274(𝑛+1)3𝑧(𝑛+2)(𝑛+5)2𝑐+n6+7𝑛5+17𝑛4𝑛354𝑛2𝑧50𝑛+8𝑐𝑛+34(𝑛+1)3(𝑛+2)2+((𝑛+5)𝑛+3)𝑧𝑐𝑛+52(𝑛+1)2((𝑛+5)𝑛+3)2𝑧𝑐𝑛+62𝑛3+8𝑛2+𝑛+17𝑛+10(𝑛+3)24(𝑛+1)2𝑧(𝑛+5)𝑐𝑛+72𝑛6+15𝑛5+39𝑛4+33𝑛325𝑛252𝑛88(𝑛+1)2𝑧(𝑛+2)(𝑛+3)(2𝑛+5)𝑐2𝑛+6+𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+48(𝑛+1)2𝑧(𝑛+2)(𝑛+5)𝑐2𝑛+8𝑛4+2𝑛35𝑛26𝑛+44(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+3log𝑧𝑐,𝑆3=𝑎2𝑛(𝑛1)(𝑛+3)(𝑛+5)16(𝑛+1)2(𝑛+7)𝑧8(𝑛+5)𝑐+𝑛(𝑛+3)2(𝑛+6)8(𝑛+2)2𝑧(𝑛+4)2𝑐+(𝑛+3)𝑧8(𝑛+1)5𝑐(𝑛+3)2𝑧8(𝑛+2)6𝑐+𝑛(𝑛+3)2𝑧16(𝑛+1)7𝑐𝑛7+4𝑛610𝑛560𝑛439𝑛3+112𝑛2+128𝑛832(𝑛+1)2(𝑛+2)2𝑧𝑐𝑛+4+𝑛(𝑛+3)4+8𝑛3+13𝑛212𝑛6𝑧8(𝑛+1)(𝑛+4)(𝑛+5)𝑐𝑛+6+𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+4𝑧32(𝑛+1)(𝑛+2)𝑐𝑛+8𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+4𝑧8(𝑛+1)(𝑛+2)𝑐𝑛+4log𝑧𝑐,𝑆4=𝑎2(2𝑛+7)𝑧2(𝑛+4)(2𝑛+5)𝑐𝑛(𝑛+3)32(𝑛+2)3(𝑧4+𝑛)2𝑐+𝑛(𝑛1)(𝑛+3)2(2𝑛+5)4(𝑛+1)2(𝑧𝑛+4)(2𝑛+3)3𝑐𝑛6+7𝑛5+15𝑛4+𝑛344𝑛256𝑛+44(𝑛+1)2(𝑛+2)2𝑧(𝑛+4)𝑐𝑛+4(𝑛+3)2𝑧(𝑛+1)(𝑛+4)𝑐𝑛+5+(𝑛+3)4(𝑛+2)2(𝑧𝑛+4)𝑐𝑛+6𝑛(𝑛+3)2𝑧2(𝑛+2)𝑐𝑛+7+(𝑛+3)24𝑛6+28𝑛5+51𝑛415𝑛397𝑛239𝑛+248(𝑛+1)(𝑛+2)3𝑧(2𝑛+3)(2𝑛+5)𝑐2𝑛+6(𝑛𝑛+3)4+2𝑛35𝑛26𝑛+4𝑧8(𝑛+1)(𝑛+2)(𝑛+4)𝑐2𝑛+8+𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+44(𝑛+1)(𝑛+2)2𝑧𝑐𝑛+4log𝑧𝑐,𝑆5=𝑎2𝑛(𝑛+7)𝑧16(𝑛+5)2𝑐𝑛2(𝑛+3)2(𝑛+6)16(𝑛+2)2𝑧(𝑛+4)3𝑐+𝑛2(𝑛1)(𝑛+3)(𝑛+5)32(𝑛+1)2𝑧4𝑐𝑛(𝑛+3)𝑧16(𝑛+1)6𝑐+𝑛(𝑛+3)2𝑧16(𝑛+2)7𝑐𝑛2(𝑛+3)2𝑧32(𝑛+1)8𝑐𝑛𝑛7+4𝑛610𝑛560𝑛439𝑛3+112𝑛2+128𝑛864(𝑛+1)2(𝑛+2)2𝑧𝑐𝑛+5+𝑛𝑛(𝑛+3)4+8𝑛3+13𝑛212𝑛6𝑧16(𝑛+1)(𝑛+4)(𝑛+5)𝑐𝑛+7𝑛𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+4𝑧64(𝑛+1)(𝑛+2)𝑐𝑛+9+𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+4𝑧16(𝑛+1)(𝑛+2)𝑐𝑛+5log𝑧𝑐,𝑆6=𝑎2𝑛𝑧4(𝑛+2)2𝑐+𝑛2(𝑛+3)(2𝑛+5)4(𝑛+2)2𝑧(2𝑛+3)3𝑐𝑛2(𝑛1)(𝑛+2)(𝑛+3)8(𝑛+1)3𝑧4𝑐+𝑛2𝑛4+4𝑛3+3𝑛2+2𝑛+108(𝑛+1)2(𝑛+2)2𝑧𝑐𝑛+5𝑛(𝑛+3)𝑧4(𝑛+2)𝑐𝑛+6+𝑛2(𝑛+3)𝑧8(𝑛+1)𝑐𝑛+7𝑛(𝑛+3)22𝑛5+3𝑛42𝑛3+𝑛2+2𝑛416(𝑛+1)3𝑧(𝑛+2)(2𝑛+3)𝑐2𝑛+6+𝑛𝑛4+2𝑛35𝑛26𝑛+4𝑧16(𝑛+1)(𝑛+2)𝑐2𝑛+8𝑛𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+48(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+5log𝑧𝑐,𝑆7=𝑎2(𝑛+7)+𝑛8(𝑛+3)(𝑛+5)2𝑛3+27𝑛2+113𝑛+1468(𝑛+2)2(𝑛+4)2𝑧(𝑛+5)𝑐𝑛𝑛7+19𝑛6+142𝑛5+538𝑛4+1078𝑛3+1005𝑛2+147𝑛2428(𝑛+1)2(𝑛+2)3𝑧(𝑛+4)(𝑛+5)2𝑐+𝑛6+6𝑛522𝑛3+11𝑛28𝑛432(𝑛+1)3𝑧4𝑐(𝑛+3)(2𝑛+1)𝑧8(𝑛+1)(𝑛+2)5𝑐+𝑛𝑛(𝑛+3)(2𝑛+3)2+3𝑛+18(𝑛+1)2(𝑛+2)2𝑧6𝑐+𝑛(𝑛+3)(2𝑛1)𝑧16(𝑛+1)(𝑛+2)7𝑐𝑛8+7𝑛714𝑛6186𝑛5347𝑛4+203𝑛3+1024𝑛2+616𝑛15232(𝑛+1)2(𝑛+2)2𝑧(𝑛+3)(𝑛+5)𝑐𝑛+3+𝑛𝑛8+6𝑛718𝑛6184𝑛5311𝑛4+266𝑛3+1000𝑛2+504𝑛17632(𝑛+1)2(𝑛+2)3(𝑧𝑛+4)𝑐𝑛+4+𝑛𝑛10+21𝑛9+150𝑛8+378𝑛7235𝑛62391𝑛52388𝑛4+1288𝑛3+1800𝑛2+608𝑛+19264(𝑛+1)2(𝑛+2)2𝑧(𝑛+4)(𝑛+5)𝑐𝑛+5+(𝑛𝑛+3)4+8𝑛3+13𝑛212𝑛6𝑧8(𝑛+1)(𝑛+2)(𝑛+4)(𝑛+5)𝑐𝑛+6(𝑛1)2𝑛7+25𝑛6+107𝑛5+169𝑛4+35𝑛354𝑛2+12𝑛8032(𝑛+1)2𝑧(𝑛+2)(𝑛+4)(𝑛+5)𝑐𝑛+7(𝑛𝑛2)(𝑛+3)4+2𝑛35𝑛26𝑛+432(𝑛+1)(𝑛+2)2𝑧𝑐𝑛+8+𝑛𝑛(𝑛3)(𝑛+3)4+2𝑛35𝑛26𝑛+464(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+9+𝑛4+2𝑛35𝑛2𝑛6𝑛+47+4𝑛610𝑛560𝑛439𝑛3+112𝑛2+128𝑛864(𝑛+1)2(𝑛+2)3𝑧(𝑛+3)𝑐2𝑛+6𝑛4+2𝑛35𝑛2𝑛6𝑛+44+8𝑛3+13𝑛212𝑛616(𝑛+1)2𝑧(𝑛+2)(𝑛+4)(𝑛+5)𝑐2𝑛+8𝑛4+2𝑛35𝑛26𝑛+4264(𝑛+1)2(𝑛+2)2𝑧𝑐2𝑛+10𝑛4+2𝑛35𝑛26𝑛+4𝑧8(𝑛+1)(𝑛+2)𝑐𝑛+3𝑧log𝑐+𝑛𝑛(𝑛+3)4+2𝑛35𝑛26𝑛+48(𝑛+1)(𝑛+2)2𝑧𝑐𝑛+4𝑧log𝑐𝑛𝑛(𝑛1)(𝑛+3)4+2𝑛35𝑛26𝑛+416(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+5𝑧log𝑐+𝑛4+2𝑛35𝑛26𝑛+416(𝑛+1)2(𝑛+2)2𝑧𝑐2𝑛+6𝑧log𝑐.(A.1)

B.

𝑊1=(𝑛+6)6(𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)𝑛(𝑛+5)𝑧𝑐6(𝑛+1)2+(𝑛+2)(𝑛+3)(𝑛1)(𝑛+4)𝑧2𝑐12(𝑛+1)2𝑧(𝑛+2)3𝑐6(𝑛+1)2(+𝑧𝑛+2)4𝑐6(𝑛+1)2+1𝑧12(𝑛+1)5𝑐𝑛𝑛2326(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+2+𝑛4+2𝑛3+112𝑛2𝑛84(𝑛+1)2𝑧(𝑛+2)(𝑛+3)𝑐𝑛+3+𝑛4+6𝑛3+7𝑛26𝑛44(𝑛+1)2𝑧(𝑛+2)(𝑛+3)(𝑛+4)𝑐𝑛+4𝑛𝑛2312(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+5,𝑊2=(2𝑛+7)+2(𝑛+1)(𝑛+3)(𝑛+4)(2𝑛+5)𝑛(𝑛+3)2(𝑛+1)2𝑧(𝑛+2)(𝑛+4)𝑐+(𝑛1)(𝑛+2)(2𝑛+5)4(𝑛+1)2𝑧(𝑛+4)(2𝑛+3)2𝑐+𝑛𝑛2𝑧3𝑐2𝑛+64(𝑛+1)2,𝑊(𝑛+4)3=(𝑛+6)𝑧6(𝑛+1)(𝑛+3)(𝑛+4)𝑐+𝑛(𝑛+5)6(𝑛+1)2𝑧(𝑛+3)2𝑐(𝑛1)(𝑛+4)12(𝑛+1)2𝑧3𝑐+16(𝑛+1)2𝑧4𝑐+(𝑛+2)𝑧12(𝑛+1)6𝑐+𝑛𝑛236(𝑛+1)2𝑧𝑐𝑛+3𝑛4+2𝑛35𝑛26𝑛+44(𝑛+1)2𝑧(𝑛+3)𝑐𝑛+4𝑛4+6𝑛3+7𝑛26𝑛44(𝑛+1)2𝑧(𝑛+3)(𝑛+4)𝑐𝑛+5+𝑛𝑛2312(𝑛+1)2𝑧𝑐𝑛+6,𝑊41=𝑧2(𝑛+2)(𝑛+3)𝑐𝑛(2𝑛+5)𝑧2(𝑛+1)(𝑛+3)(2𝑛+3)2𝑐+(𝑛1)(𝑛+2)24(𝑛+1)2𝑧(𝑛+3)3𝑐𝑛𝑛(𝑛+4)23𝑧4(𝑛+1)(𝑛+2)(𝑛+3)𝑐𝑛+3+𝑛4+2𝑛35𝑛28𝑛+24(𝑛+1)2𝑧(𝑛+3)𝑐𝑛+4+(𝑛+2)𝑧2(𝑛+1)(𝑛+3)𝑐𝑛+5(𝑛+2)𝑧4(𝑛+3)𝑐𝑛+6+2𝑛5+5𝑛4+𝑛33𝑛22𝑛24(𝑛+1)3𝑧(𝑛+2)(2𝑛+3)𝑐2𝑛+5𝑛𝑛23𝑧4(𝑛+1)(𝑛+3)𝑐2𝑛+6,𝑊5=(𝑛+6)𝑧12(𝑛+3)(𝑛+4)2𝑐𝑛(𝑛+5)𝑧12(𝑛+1)(𝑛+3)3𝑐+(𝑛1)(𝑛+4)𝑧24(𝑛+1)4𝑐(𝑛+3)𝑧12(𝑛+1)6𝑐(𝑛+2)𝑧247𝑐𝑛𝑛23𝑧12(𝑛+1)𝑐𝑛+4𝑛4+2𝑛35𝑛26𝑛+4𝑧8(𝑛+1)(𝑛+3)𝑐𝑛+5𝑛𝑛23𝑧4(𝑛+1)𝑐𝑛+7+𝑛𝑛(𝑛+3)4+8𝑛3+13𝑛212𝑛6𝑧16(𝑛+1)(𝑛+4)(𝑛+5)𝑐𝑛+7,𝑊6=𝑛(2𝑛+5)𝑧4(𝑛+2)(𝑛+3)(2𝑛+3)2𝑐+𝑛24(𝑛+1)2𝑧3𝑐𝑛(𝑛1)(2𝑛+3)𝑧8(𝑛+1)(2𝑛+1)4𝑐+𝑛𝑛4+3𝑛33𝑛27𝑛+28(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+4𝑛3+𝑛24𝑛+4𝑧8(𝑛+3)𝑐𝑛+5+𝑛8𝑧𝑐𝑛+6(𝑛+2)2𝑛2+𝑛28(𝑛+1)2𝑧(2𝑛+1)(2𝑛+3)𝑐2𝑛+5+𝑛2𝑛23𝑧8(𝑛+1)(𝑛+2)𝑐2𝑛+6,𝑊7𝑛=𝑛(𝑛+6)23𝑧12(𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)𝑐𝑛+2+𝑛2(𝑛𝑛+5)2312(𝑛+1)2𝑧(𝑛+2)(𝑛+3)𝑐𝑛+3+𝑛𝑛(𝑛1)(𝑛+4)2324(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+4𝑛𝑛2312(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+5+𝑛𝑛2312(𝑛+1)2𝑧𝑐𝑛+6𝑛𝑛23𝑧24(𝑛+1)𝑐𝑛+7𝑛2𝑛2312(𝑛+1)2𝑧(𝑛+2)𝑐2𝑛+4+𝑛𝑛2𝑛34+2𝑛35𝑛26𝑛+48(𝑛+1)2𝑧(𝑛+2)(𝑛+3)𝑐2𝑛+5+𝑛𝑛2𝑛34+6𝑛3+7𝑛26𝑛48(𝑛+1)2𝑧(𝑛+2)(𝑛+3)(𝑛+4)𝑐2𝑛+6𝑛2𝑛23224(𝑛+1)2𝑧(𝑛+2)𝑐2𝑛+7.(B.1)

Acknowledgment

The present work is supported by the research university grant of Universiti Sains Malaysia, Malaysia (Grant no.: 1001/PMATHS/811177).

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