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 𝐷eff given by 𝐷eff=π‘Ž2𝑀2π‘š/48π·π‘š, where π·π‘š is the molecular diffusivity, π‘€π‘š is the normalized axial velocity, and π‘Ž 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 π‘€π‘š 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 β€œThe 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.

2.1. Flow in Circular Tube
2.1.1. Governing Equations

Cylindrical polar coordinate system (π‘Ÿ,πœ“,π‘₯) is used to analyze the flow through uniform circular tube, where π‘Ÿ and π‘₯ are the coordinates in the radial and axial directions, respectively, and πœ“ 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 𝑑𝑝1𝑑π‘₯=βˆ’π‘Ÿπ‘‘π‘‘π‘Ÿ(π‘Ÿπœ),(2.1) where 𝑑𝑝/𝑑π‘₯ 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 βˆ’π‘‘π‘’=ξƒ―1π‘‘π‘Ÿπœ‚ξ€·πœβˆ’πœπ‘¦ξ€Έπ‘›if𝜏β‰₯πœπ‘¦,0ifπœβ‰€πœπ‘¦,(2.2) where 𝑒 is the velocity in the axial direction; πœ‚ is the coefficient of viscosity of H-B fluid with dimension (π‘€πΏβˆ’1π‘‡βˆ’2)𝑛𝑇; πœπ‘¦ 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 𝑒, we utilize the following boundary conditions 𝜏isfiniteatπ‘Ÿ=0,(2.3)𝑒=0atπ‘Ÿ=π‘Ž,(2.4) where π‘Ž 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. 1π‘Ÿπœ•ξ‚΅π‘Ÿπœ•π‘Ÿπœ•πΆ1ξ‚Ά=πœ•π‘ŸΜ‚π‘’πΆπœ…πœ•πΆ1πœ•Μƒπ‘₯.(2.5) The species transport equation for the outer region simplifies to the following form: 1π‘Ÿπœ•ξ‚΅π‘Ÿπœ•π‘Ÿπœ•πΆ2ξ‚Ά=πœ•π‘ŸΜ‚π‘’πœ…πœ•πΆ2πœ•Μƒπ‘₯,(2.6) where 𝐢1 and 𝐢2 are the concentration of the solute in the plug core region and outer region; Μƒπ‘₯=π‘₯βˆ’π‘’π‘‘ is the new axial coordinate moving with the normalized velocity 𝑒;̂𝑒𝐢=π‘’πΆβˆ’π‘’;̂𝑒=π‘’βˆ’π‘’;̂𝑒 is the relative velocity in the outer region; ̂𝑒𝐢 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 πœ•πΆ1πΆπœ•π‘Ÿ=0atπ‘Ÿ=0,(2.7)1=0atπ‘Ÿ=0,(2.8)πœ•πΆ2πΆπœ•π‘Ÿ=0atπ‘Ÿ=π‘Ž,(2.9)2=𝐢1atπ‘Ÿ=π‘Ÿπ‘.(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 π‘Ÿ and then using (2.3), we get the expression for the shear stress as follows: π‘Ÿπœ=βˆ’2𝑑𝑝.𝑑𝑧(2.11) Using binomial series expansion in (2.2) and neglecting the terms involving (πœπ‘¦/𝜏)2 and higher powers of (πœπ‘¦/𝜏) (since (πœπ‘¦/𝜏)3β‰ͺ1), one can obtain the simplified form of the constitutive equation as follows: βˆ’π‘‘π‘’=1π‘‘π‘Ÿπœ‚ξ‚Έπœπ‘›βˆ’π‘›πœπ‘¦πœπ‘›βˆ’1+𝑛(π‘›βˆ’1)2πœπ‘¦πœπ‘›βˆ’2ξ‚Ή.(2.12) Using (2.11) in (2.12) and integrating the resulting differential equation with respect to π‘Ÿ and then using the boundary condition (2.4), we get the expression for the velocity in the outer (non-plug core) region as 1𝑒(π‘Ÿ)=ξ‚΅βˆ’1(𝑛+1)πœ‚2π‘‘π‘ξ‚Άπ‘‘π‘§π‘›ξ‚Έπ‘Žπ‘›+1βˆ’π‘Ÿπ‘›+1βˆ’(𝑛+1)π‘Ÿπ‘(π‘Žπ‘›βˆ’π‘Ÿπ‘›)+𝑛(𝑛+1)2π‘Ÿ2π‘ξ€·π‘Žπ‘›βˆ’1βˆ’π‘Ÿπ‘›βˆ’1ξ€Έξ‚Ή,(2.13) where π‘Ÿπ‘ is the plug core radius, which is defined as follows: π‘Ÿπ‘=2πœπ‘¦βˆ’.(𝑑𝑝/𝑑𝑧)(2.14) The expression for the plug flow velocity is obtained by replacing π‘Ÿ by π‘Ÿπ‘ in (2.13) as given in the following: 𝑒𝑐1(π‘Ÿ)=ξ‚΅βˆ’1(𝑛+1)πœ‚2π‘‘π‘ξ‚Άπ‘‘π‘§π‘›ξ‚Έπ‘Žπ‘›+1βˆ’(𝑛+1)π‘Ÿπ‘π‘Žπ‘›+𝑛(𝑛+1)2π‘Ÿ2π‘π‘Žπ‘›βˆ’1βˆ’π‘›(π‘›βˆ’1)2π‘Ÿπ‘π‘›+1ξ‚Ή.(2.15) Using (2.14) and (2.15), one can obtain the following expression for the bulk velocity or normalized velocity: 𝑒=𝑒𝐻𝐴𝑧𝑐,(2.16) where π‘’π»π‘Ž=βˆ’π‘›+1ξ‚΅βˆ’1πœ‚(𝑛+3)2𝑑𝑝𝑑𝑧𝑛,𝐴𝑧(2.17)𝑐𝑛=1βˆ’(𝑛+3)𝑧(𝑛+2)𝑐+𝑛(𝑛+3)(π‘›βˆ’1)𝑧2(𝑛+1)𝑐2βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧2(𝑛+1)(𝑛+2)𝑐𝑛+3,(2.18) where 𝑧𝑐=π‘Ÿπ‘/π‘Ž. 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: 𝐢1=π‘’π»π‘Ÿ2ξ‚΅2πœ…(𝑛+1)πœ•πΆ1ξ‚Άπ΅ξ€·π‘§πœ•Μƒπ‘₯𝑐,(2.19) where 𝐡𝑧𝑐=1βˆ’(𝑛+1)(𝑛+3)(𝑧𝑛+2)𝑐+𝑛(𝑛+3)2𝑧𝑐2βˆ’π‘›(π‘›βˆ’1)(𝑛+3)4𝑧𝑐𝑛+1βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+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: 𝐢2=π‘’π»π‘Ž2πœ…ξ‚΅πœ•πΆ2βˆ’1πœ•Μƒπ‘₯𝑧(𝑛+2)(𝑛+3)𝑛+3+(𝑛+3)(𝑛+2)2𝑧𝑐𝑧𝑛+2βˆ’π‘›(𝑛+3)2(𝑛+1)2𝑧2𝑐𝑧𝑛+1+𝑧2ξ‚΅1βˆ’2(𝑛+1)(𝑛+3)𝑧2(𝑛+2)𝑐+𝑛(𝑛+3)𝑧4(𝑛+1)2𝑐+𝑛4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧8(𝑛+1)(𝑛+2)𝑐𝑛+3ξƒͺβˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧4(𝑛+1)(𝑛+2)𝑐𝑛+3𝑧logπ‘§π‘ξ‚Άβˆ’ξ€·π‘›7+10𝑛6+32𝑛5+18𝑛4βˆ’93𝑛3βˆ’164𝑛2ξ€Έβˆ’52𝑛+408(𝑛+1)2(𝑛+2)2𝑧(𝑛+3)𝑐𝑛+3ξƒ­.(2.21) The flux of solute across a cross section at constant Μƒπ‘₯ is defined as follows [12] 1π‘ž=πœ‹π‘Ž2ξ‚Έξ€œπ‘Ÿπ‘0̂𝑒𝑐𝐢1βˆ’πœ…πœ•πΆξ‚ξ€œπœ•Μƒπ‘₯2πœ‹π‘Ÿπ‘‘π‘Ÿ+π‘Žπ‘Ÿπ‘ξ‚€Μ‚π‘’πΆ2βˆ’πœ…πœ•πΆξ‚ξ‚Ή.πœ•Μƒπ‘₯2πœ‹π‘Ÿπ‘‘π‘Ÿ(2.22) For our convenience, (2.22) is rewritten as follows: ξ‚€π‘ž=βˆ’πœ…πœ•πΆξ‚+2πœ•Μƒπ‘₯π‘Ž2𝐼1+𝐼2ξ€»,(2.23) where 𝐼1=ξ€œπ‘Ÿπ‘0̂𝑒𝑐𝐢1π‘Ÿπ‘‘π‘Ÿ=𝑒2𝐻𝐡2ξ€·π‘§π‘ξ€Έπ‘Ÿ4𝑐4πœ…(𝑛+1)2ξ‚€πœ•πΆξ‚,πΌπœ•Μƒπ‘₯(2.24)2=ξ€œπ‘Žπ‘Ÿπ‘(Μ‚π‘’π‘Ÿ)𝐢2π‘‘π‘Ÿ=𝑒2π»π‘Ž2ξ‚€4πœ…πœ•πΆξ‚ξ€œπœ•Μƒπ‘₯π‘Žπ‘Ÿπ‘π‘‡1(π‘Ÿ)𝑇2(π‘Ÿ)π‘‘π‘Ÿ.(2.25) The functions 𝑇1(π‘Ÿ) and 𝑇2(π‘Ÿ) appearing in (2.25) are given as follows: 𝑇1=(𝑛+3)(𝑛+1)π‘Ÿβˆ’(𝑛+3)π‘Ÿ(𝑛+1)𝑛+2π‘Žπ‘›+1βˆ’(𝑛+3)π‘§π‘π‘Ÿ+(𝑛+3)π‘§π‘π‘Ÿπ‘›+1π‘Žπ‘›+𝑛(𝑛+3)2𝑧2π‘π‘Ÿβˆ’π‘›(𝑛+3)2𝑧2π‘π‘Ÿπ‘›π‘Žπ‘›βˆ’1ξ€·π‘§βˆ’π΄π‘ξ€Έπ‘Ÿ=𝑇11+𝑇12+𝑇13+𝑇14+𝑇15+𝑇16+𝑇17,𝑇21=βˆ’ξ‚€π‘Ÿ(𝑛+1)(𝑛+3)π‘Žξ‚π‘›+3+(𝑛+3)𝑧𝑐(𝑛+2)2ξ‚€π‘Ÿπ‘Žξ‚π‘›+2βˆ’π‘›(𝑛+3)2(𝑛+1)2𝑧2π‘ξ‚€π‘Ÿπ‘Žξ‚π‘›+1𝑧+πΊπ‘ξ€Έξ‚€π‘Ÿπ‘Žξ‚2βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧4(𝑛+1)(𝑛+2)𝑐𝑛+3ξ‚΅π‘Ÿlogπ‘Ÿπ‘ξ‚Άξ€·π‘§+𝐻𝑐,(2.26) where 𝐺𝑧𝑐=1βˆ’2(𝑛+1)(𝑛+3)𝑧2(𝑛+2)𝑐+𝑛(𝑛+3)𝑧4(𝑛+1)2𝑐+𝑛4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧8(𝑛+1)(𝑛+2)𝑐𝑛+3,𝐻𝑧𝑐=𝑛7+10𝑛6+32𝑛5+18𝑛4βˆ’93𝑛3βˆ’164𝑛2ξ€Έβˆ’52𝑛+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=𝑒2π»π‘Ž2ξ‚€4πœ…πœ•πΆξ‚ξ€œπœ•Μƒπ‘₯π‘Žπ‘Ÿπ‘π‘‡1𝑇2=π‘‘π‘Ÿπ‘’2π»π‘Ž2ξ‚€4πœ…πœ•πΆξ‚ξ€œπœ•Μƒπ‘₯π‘Žπ‘Ÿπ‘ξ€·π‘‡11+𝑇12+𝑇13+𝑇14+𝑇15+𝑇16+𝑇17𝑇2=π‘‘π‘Ÿπ‘’2π»π‘Ž2ξ‚€4πœ…πœ•πΆξ‚ξ€Ίπ‘†πœ•Μƒπ‘₯1+𝑆2+𝑆3+𝑆4+𝑆5+𝑆6+𝑆7ξ€»,(2.28) where 𝑆1=ξ€œπ‘Žπ‘ŸπΆπ‘‡11𝑇2π‘‘π‘Ÿ,𝑆2=ξ€œπ‘Žπ‘ŸπΆπ‘‡12𝑇2π‘‘π‘Ÿ,𝑆3=ξ€œπ‘Žπ‘ŸπΆπ‘‡13𝑇2π‘†π‘‘π‘Ÿ,4=ξ€œπ‘Žπ‘ŸπΆπ‘‡14𝑇2π‘‘π‘Ÿ,𝑆5=ξ€œπ‘Žπ‘ŸπΆπ‘‡15𝑇2π‘‘π‘Ÿ,𝑆6=ξ€œπ‘Žπ‘ŸπΆπ‘‡16𝑇2π‘‘π‘Ÿ,𝑆7=ξ€œπ‘Žπ‘ŸπΆπ‘‡17𝑇2π‘‘π‘Ÿ.(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 𝐷eο¬€π‘ž=βˆ’ξƒ¬πœ•πΆ/πœ•Μƒπ‘₯=πœ…1+Pec2𝐸𝑧48𝑐𝐴2𝑧𝑐.(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 𝐸𝑧𝑐=24βˆ’(𝑛+3)(𝑛+5)48𝑛(𝑛+3)(2𝑛+9)π‘§π‘βˆ’ξ€·π‘›(𝑛+2)(𝑛+4)(𝑛+5)(2𝑛+5)12𝑛6+13𝑛5+58𝑛4+82𝑛3βˆ’91𝑛2ξ€Έπ‘§βˆ’305π‘›βˆ’1422𝑐(1+𝑛)(2+𝑛)3βˆ’(4+𝑛)(5+𝑛)24𝑛2(π‘›βˆ’1)(𝑛+3)(2𝑛+7)𝑧3π‘βˆ’ξ€·π‘›(𝑛+1)(𝑛+2)(𝑛+4)(2𝑛+3)3(𝑛+3)8+12𝑛7+36𝑛6βˆ’46𝑛5βˆ’257𝑛4+74𝑛3+404𝑛2ξ€Έπ‘§βˆ’112π‘›βˆ’64𝑐5+𝑛(𝑛+1)2+(𝑛+2)(𝑛+4)(𝑛+5)24(𝑛+3)2𝑛2𝑧+4π‘›βˆ’3𝑐𝑛+6+𝑛(𝑛+1)(𝑛+2)(𝑛+4)(𝑛+5)6𝑛(𝑛+3)4+6𝑛3βˆ’3𝑛2ξ€Έπ‘§βˆ’36𝑛+24𝑐𝑛+7βˆ’3ξ‚€(𝑛+1)(𝑛+4)(𝑛+5)4𝑛13+48𝑛12+195𝑛11+160𝑛10βˆ’913𝑛9βˆ’1878𝑛8+1865𝑛7+6772𝑛6βˆ’2535𝑛5βˆ’18918𝑛4βˆ’12512𝑛3+5656𝑛2+4872π‘›βˆ’1152𝑧𝑐2𝑛+6(𝑛+1)3(𝑛+2)3(+3𝑛𝑛+3)(2𝑛+3)(2𝑛+5)4+2𝑛3βˆ’5𝑛2π‘›βˆ’6𝑛+4ξ€Έξ€·4+6𝑛3βˆ’3𝑛2ξ€Έπ‘§βˆ’36𝑛+24𝑐2𝑛+8(+𝑛𝑛+1)(𝑛+2)(𝑛+4)(𝑛+5)6(π‘›βˆ’1)(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έπ‘§βˆ’6𝑛+4𝑐𝑛+5𝑧logπ‘ξ€Έβˆ’6𝑛(𝑛+1)(𝑛+2)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+42𝑧𝑐2𝑛+6𝑧log𝑐(𝑛+1)2(𝑛+2)2.(2.31)

2.2. Flow between Parallel Flat Plates
2.2.1. Governing Equations

Cartesian coordinate system (π‘₯,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 𝑑𝑝𝑑𝑑π‘₯=βˆ’π‘‘π‘¦(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 βˆ’π‘‘π‘’=ξƒ―1π‘‘π‘¦πœ‚ξ€·πœβˆ’πœπ‘¦ξ€Έπ‘›if𝜏β‰₯πœπ‘¦,0ifπœβ‰€πœπ‘¦,(2.33) where 𝑒 is the velocity in the π‘₯ direction, πœ‚ is the coefficient of viscosity of H-B fluid, πœπ‘¦ 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 u𝜏isfiniteat𝑦=0,(2.34)𝑒=0at𝑦=β„Ž.(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 πœ•2𝐢1πœ•π‘¦2=Μ‚π‘’π‘πœ…πœ•πΆ1,πœ•πœ•Μƒπ‘₯(2.36)2𝐢2πœ•π‘¦2=Μ‚π‘’πœ…πœ•πΆ2,πœ•Μƒπ‘₯(2.37) where 𝐢1 and 𝐢2 are the concentration of the species in the plug core region and outer region, respectively, and Μƒπ‘₯=π‘₯βˆ’π‘’π‘‘ is the coordinate moving in the π‘₯ direction with the normalized velocity 𝑒,̂𝑒𝑐=π‘’π‘βˆ’π‘’,̂𝑒=π‘’βˆ’π‘’,̂𝑒 is the relative velocity in the outer region, and ̂𝑒𝑐 is the relative velocity in the plug core region, 𝑑 is the time. The boundary conditions of the concentration of the species in the plug core region and outer region are πœ•πΆ1πΆπœ•π‘Ÿ=0at𝑦=0,(2.38)1=0at𝑦=0,(2.39)πœ•πΆ2πΆπœ•π‘Ÿ=0at𝑦=β„Ž,(2.40)1=𝐢2at𝑦=𝑦𝐢,(2.41) where 𝑦𝐢 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: ξ‚΅πœ=βˆ’π‘‘π‘ξ‚Άπ‘‘π‘₯𝑦.(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: 1𝑒(𝑦)=ξ‚΅βˆ’(𝑛+1)πœ‚π‘‘π‘ξ‚Άπ‘‘π‘₯π‘›ξ‚Έβ„Žπ‘›+1βˆ’π‘¦π‘›+1βˆ’(𝑛+1)π‘¦π‘ξ€·β„Žπ‘›βˆ’π‘¦π‘›ξ€Έ+𝑛(𝑛+1)2𝑦2π‘ξ€·β„Žπ‘›βˆ’1βˆ’π‘¦π‘›βˆ’1ξ€Έξ‚Ή,(2.43) where 𝑦𝑐=πœπ‘¦.βˆ’(𝑑𝑝/𝑑π‘₯)(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 𝑒𝑐1(𝑦)=ξ‚΅βˆ’(𝑛+1)πœ‚π‘‘π‘ξ‚Άπ‘‘π‘₯π‘›ξ‚Έβ„Žπ‘›+1βˆ’(𝑛+1)π‘¦π‘β„Žπ‘›+𝑛(𝑛+1)2𝑦2π‘β„Žπ‘›βˆ’1βˆ’π‘›(π‘›βˆ’1)2𝑦𝑐𝑛+1ξ‚Ή.(2.45) The normalized velocity or bulk velocity of the H-B fluid at a cross section is obtained as 𝑒=𝑒𝐻𝐹𝑧𝑐,(2.46) where π‘’π»β„Ž=βˆ’π‘›+1ξ‚΅βˆ’(𝑛+2)πœ‚π‘‘π‘ξ‚Άπ‘‘π‘₯𝑛,𝐹𝑧𝑐𝑛=1βˆ’(𝑛+2)𝑧(𝑛+1)𝑐+(π‘›βˆ’1)(𝑛+2)2𝑧𝑐2βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’3𝑧2(𝑛+1)𝑐𝑛+2,(2.47) where 𝑧𝑐=𝑦𝑐/β„Ž. 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: 𝐢1=𝑒𝐻𝑦2ξ‚΅2πœ…(𝑛+1)πœ•πΆ1ξ‚Άπ‘€ξ€·π‘§πœ•Μƒπ‘₯𝑐,(2.48) where 𝑀𝑧𝑐=1βˆ’(𝑛+2)𝑧𝑐+(𝑛+1)(𝑛+2)2𝑧𝑐2βˆ’π‘›(π‘›βˆ’1)(𝑛+2)2𝑧𝑐𝑛+1+𝑛𝑛2ξ€Έβˆ’32𝑧𝑐𝑛+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) 𝐢2=(𝑛+2)π‘’π»β„Ž2πœ…ξ‚΅πœ•πΆ2βˆ’1πœ•Μƒπ‘₯ξ‚Άξ‚Έ(𝑧𝑛+1)(𝑛+2)(𝑛+3)𝑛+3+1(𝑛+1)(𝑛+2)2𝑧𝑐𝑧𝑛+2βˆ’12(𝑛+1)2𝑧2𝑐𝑧𝑛+1+𝑧21βˆ’12(𝑛+1)(𝑛+2)𝑧2(𝑛+1)𝑐+14𝑧2𝑐+𝑛𝑛2ξ€Έβˆ’3𝑧4(𝑛+1)(𝑛+2)𝑐𝑛+2ξƒͺβˆ’π‘›ξ€·π‘›2ξ€Έβˆ’3𝑧2(𝑛+1)(𝑛+2)𝑐𝑛+2𝑛𝑧+4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧4(𝑛+1)(𝑛+2)(𝑛+3)𝑐𝑛+3ξƒ­,(2.50) where 𝑧=𝑦/β„Ž. The flux of the solute across a cross section at constant Μƒπ‘₯ is defined as 1π‘ž=β„Žξ‚Έξ€œπ‘¦π‘0̂𝑒𝑐𝐢1βˆ’πœ…πœ•πΆξ‚ξ€œπœ•Μƒπ‘₯𝑑𝑦+β„Žπ‘¦π‘ξ‚€Μ‚π‘’π‘πΆ2βˆ’πœ…πœ•πΆξ‚ξ‚Ήπœ•Μƒπ‘₯𝑑𝑦.(2.51) For our convenience, (2.51) is rewritten as ξ‚€π‘ž=βˆ’πœ…πœ•πΆξ‚+1πœ•Μƒπ‘₯β„Žξ€ΊπΌ3+𝐼4ξ€»,(2.52) where 𝐼3=ξ€œπ‘¦πΆ0Μ‚π‘’π‘πΆξ…ž1𝑑𝑦=𝑒2𝐻𝑀2𝑧𝑐𝑦3𝐢6πœ…(𝑛+1)2ξ‚€πœ•πΆξ‚,πΌπœ•Μƒπ‘₯(2.53)4=ξ€œβ„Žπ‘¦πΆΜ‚π‘’πΆ2𝑑𝑦=𝑒2𝐻(𝑛+2)β„Ž2(𝑛+1)2πœ…ξ‚€πœ•πΆξ‚ξ€œπœ•Μƒπ‘₯β„Žπ‘¦πΆπ‘Š1(𝑦)π‘Š2(𝑦)𝑑𝑦.(2.54) The functions π‘Š1(𝑦) and π‘Š2(𝑦) appearing in (2.54) are π‘Š11(𝑦)=βˆ’(𝑛+1)(𝑛+2)𝑦(𝑛+1)𝑛+1β„Žπ‘›+1βˆ’(𝑛+2)𝑧(𝑛+1)𝑐+(𝑛+2)π‘§π‘π‘¦π‘›β„Žπ‘›+(𝑛+2)2𝑧2π‘βˆ’π‘›(𝑛+2)2𝑧2π‘π‘¦π‘›βˆ’1β„Žπ‘›βˆ’1βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’32𝑧(𝑛+1)𝑐𝑛+2=π‘Š11+π‘Š12+π‘Š13+π‘Š14+π‘Š15+π‘Š16+π‘Š17,π‘Š(2.55)21(𝑦)=βˆ’ξ‚€π‘¦(𝑛+1)(𝑛+2)(𝑛+3)β„Žξ‚π‘›+3+𝑧𝑐𝑦(𝑛+1)(𝑛+2)β„Žξ‚π‘›+2βˆ’1𝑧2(𝑛+1)2π‘ξ‚€π‘¦β„Žξ‚π‘›+1𝑧+π½π‘ξ€Έξ‚€π‘¦β„Žξ‚2βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’3𝑧2(𝑛+1)(𝑛+2)𝑐𝑛+2ξ‚€π‘¦β„Žξ‚ξ€·π‘§+𝐾𝑐,(2.56) where 𝐽𝑧𝑐=1βˆ’12(𝑛+1)(𝑛+2)𝑧2(𝑛+1)𝑐+14𝑧2𝑐+𝑛𝑛2ξ€Έβˆ’3𝑧4(𝑛+1)(𝑛+2)𝑐𝑛+2,𝐾𝑧𝑐=𝑛4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+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)𝑒2π»β„Ž2πœ…ξ‚€πœ•πΆξ‚ξ€œπœ•Μƒπ‘₯β„Žπ‘¦πΆπ‘Š1(𝑦)π‘Š2=(𝑦)𝑑𝑦(𝑛+2)𝑒2π»β„Ž2πœ…ξ‚€πœ•πΆξ‚ξ€œπœ•Μƒπ‘₯β„Žπ‘¦πΆξ€·π‘Š11+π‘Š12+π‘Š13+π‘Š14+π‘Š15+π‘Š16+π‘Š17ξ€Έπ‘Š2=𝑑𝑦𝑒2π»π‘Ž2ξ‚€4πœ…πœ•πΆξ‚ξ€Ίπ·πœ•Μƒπ‘₯1+𝐷2+𝐷3+𝐷4+𝐷5+𝐷6+𝐷7ξ€»,(2.58) where 𝐷1=ξ€œβ„Žπ‘¦πΆπ‘Š11π‘Š2𝑑𝑦,𝐷2=ξ€œβ„Žπ‘¦πΆπ‘Š12π‘Š2𝑑𝑦,𝐷3=ξ€œβ„Žπ‘¦πΆπ‘Š13π‘Š2𝐷𝑑𝑦,4=ξ€œβ„Žπ‘¦πΆπ‘Š14π‘Š2𝑑𝑦,𝐷5=ξ€œβ„Žπ‘¦πΆπ‘Š15π‘Š2𝐷𝑑𝑦,6=ξ€œβ„Žπ‘¦πΆπ‘Š16π‘Š2𝑑𝑦,𝐷7=ξ€œβ„Žπ‘¦πΆπ‘Š17π‘Š2𝑑𝑦.(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 𝐷eο¬€π‘ž=βˆ’ξƒ¬πœ•πΆ/πœ•Μƒπ‘₯=πœ…1+2Pec2𝑁𝑧105𝑐𝐹2𝑧𝑐,(2.60) where 𝑁𝑧𝑐=35(βˆ’π‘›+4)(2𝑛+5)35𝑛(2𝑛+7)𝑧𝑐+𝑛2(𝑛+1)(𝑛+3)(𝑛+4)354+6𝑛3+8𝑛2ξ€Έπ‘§βˆ’3π‘›βˆ’3𝑐2(𝑛+1)2βˆ’(𝑛+3)(𝑛+4)35𝑛(π‘›βˆ’1)(𝑛+2)𝑧𝑐3+8(𝑛+1)(𝑛+3)35(π‘›βˆ’1)2(𝑛+2)𝑧4π‘βˆ’ξ€·π‘›4(2𝑛+1)35𝑛(𝑛+6)2ξ€Έπ‘§βˆ’3𝑐𝑛+2+4(𝑛+1)(𝑛+3)(𝑛+4)35𝑛2𝑛(𝑛+5)2ξ€Έπ‘§βˆ’3𝑐𝑛+34(𝑛+1)2βˆ’ξ€·π‘›(𝑛+3)357+10𝑛6+24𝑛5βˆ’32𝑛4βˆ’133𝑛3+22𝑛2𝑧+156π‘›βˆ’24𝑐𝑛+48(𝑛+1)2βˆ’ξ€·π‘›(𝑛+3)(𝑛+4)35(𝑛+2)4+4𝑛3βˆ’5𝑛2ξ€Έπ‘§βˆ’18𝑛+12𝑐𝑛+5+𝑛4(𝑛+1)(𝑛+3)(𝑛+4)35(𝑛+2)4+4𝑛3βˆ’5𝑛2ξ€Έπ‘§βˆ’18𝑛+12𝑐𝑛+6+8(𝑛+3)(𝑛+4)35𝑛2𝑛2ξ€Έβˆ’32𝑧𝑐2𝑛+48(𝑛+1)2βˆ’ξ€·35957𝑛10+16𝑛9+80𝑛8βˆ’4𝑛7βˆ’492𝑛6βˆ’380𝑛5βˆ’113𝑛4+740𝑛3βˆ’396𝑛2ξ€Έπ‘§βˆ’192𝑛+72𝑐2𝑛+516(𝑛+1)2+𝑛(𝑛+3)(2𝑛+1)(2𝑛+5)35𝑛2π‘›βˆ’3ξ€Έξ€·4+4𝑛3βˆ’5𝑛2ξ€Έπ‘§βˆ’18𝑛+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].

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.

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.

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.

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=π‘Ž2(𝑛+7)βˆ’ξ€·π‘›8(𝑛+1)(𝑛+5)4+12𝑛3+45𝑛2𝑧+54𝑛𝑐8(𝑛+1)(𝑛+2)2+𝑛(𝑛+4)4+7𝑛3+7𝑛2ξ€Έπ‘§βˆ’15𝑛2𝑐16(𝑛+1)3βˆ’(𝑛+3)𝑧4𝑐8(𝑛+1)2+(𝑛+3)28𝑛2𝑧+3𝑛+25π‘βˆ’π‘›(𝑛+3)216(𝑛+1)2𝑧6𝑐+𝑛7+4𝑛6βˆ’10𝑛5βˆ’60𝑛4βˆ’39𝑛3+112𝑛2ξ€Έ+128π‘›βˆ’832(𝑛+1)3(𝑛+2)2𝑧𝑐𝑛+3βˆ’ξ€·π‘›5+11𝑛4+37𝑛3+27𝑛2ξ€Έβˆ’42π‘›βˆ’188(𝑛+1)2𝑧(𝑛+4)(𝑛+5)𝑐𝑛+5βˆ’ξ€·π‘›(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+432(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+7+𝑛(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+48(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+3log𝑧𝑐,𝑆2=π‘Ž2ξƒ¬βˆ’(𝑛+4)+𝑛2(𝑛+1)(𝑛+3)(𝑛+5)2𝑛3+19𝑛2ξ€Έ+60𝑛+632(𝑛+1)(𝑛+2)2𝑧(𝑛+5)(2𝑛+5)𝑐+𝑛𝑛4+8𝑛3+18𝑛2ξ€Έβˆ’274(𝑛+1)3𝑧(𝑛+2)(𝑛+5)2𝑐+ξ€·n6+7𝑛5+17𝑛4βˆ’π‘›3βˆ’54𝑛2ξ€Έπ‘§βˆ’50𝑛+8𝑐𝑛+34(𝑛+1)3(𝑛+2)2+((𝑛+5)𝑛+3)𝑧𝑐𝑛+52(𝑛+1)2βˆ’((𝑛+5)𝑛+3)2𝑧𝑐𝑛+62𝑛3+8𝑛2ξ€Έ+𝑛+17𝑛+10(𝑛+3)24(𝑛+1)2𝑧(𝑛+5)𝑐𝑛+7βˆ’ξ€·2𝑛6+15𝑛5+39𝑛4+33𝑛3βˆ’25𝑛2ξ€Έβˆ’52π‘›βˆ’88(𝑛+1)2𝑧(𝑛+2)(𝑛+3)(2𝑛+5)𝑐2𝑛+6+𝑛(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+48(𝑛+1)2𝑧(𝑛+2)(𝑛+5)𝑐2𝑛+8βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+44(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+3log𝑧𝑐,𝑆3=π‘Ž2ξ‚Έβˆ’π‘›(π‘›βˆ’1)(𝑛+3)(𝑛+5)16(𝑛+1)2βˆ’(𝑛+7)𝑧8(𝑛+5)𝑐+𝑛(𝑛+3)2(𝑛+6)8(𝑛+2)2𝑧(𝑛+4)2𝑐+(𝑛+3)𝑧8(𝑛+1)5π‘βˆ’(𝑛+3)2𝑧8(𝑛+2)6𝑐+𝑛(𝑛+3)2𝑧16(𝑛+1)7π‘βˆ’ξ€·π‘›7+4𝑛6βˆ’10𝑛5βˆ’60𝑛4βˆ’39𝑛3+112𝑛2ξ€Έ+128π‘›βˆ’832(𝑛+1)2(𝑛+2)2𝑧𝑐𝑛+4+𝑛(𝑛+3)4+8𝑛3+13𝑛2ξ€Έβˆ’12π‘›βˆ’6𝑧8(𝑛+1)(𝑛+4)(𝑛+5)𝑐𝑛+6+𝑛(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧32(𝑛+1)(𝑛+2)𝑐𝑛+8βˆ’ξ€·π‘›(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧8(𝑛+1)(𝑛+2)𝑐𝑛+4log𝑧𝑐,𝑆4=π‘Ž2ξ‚Έ(2𝑛+7)𝑧2(𝑛+4)(2𝑛+5)π‘βˆ’π‘›(𝑛+3)32(𝑛+2)3(𝑧4+𝑛)2𝑐+𝑛(π‘›βˆ’1)(𝑛+3)2(2𝑛+5)4(𝑛+1)2(𝑧𝑛+4)(2𝑛+3)3π‘βˆ’ξ€·π‘›6+7𝑛5+15𝑛4+𝑛3βˆ’44𝑛2ξ€Έβˆ’56𝑛+44(𝑛+1)2(𝑛+2)2𝑧(𝑛+4)𝑐𝑛+4βˆ’(𝑛+3)2𝑧(𝑛+1)(𝑛+4)𝑐𝑛+5+(𝑛+3)4(𝑛+2)2(𝑧𝑛+4)𝑐𝑛+6βˆ’π‘›(𝑛+3)2𝑧2(𝑛+2)𝑐𝑛+7+(𝑛+3)2ξ€·4𝑛6+28𝑛5+51𝑛4βˆ’15𝑛3βˆ’97𝑛2ξ€Έβˆ’39𝑛+248(𝑛+1)(𝑛+2)3𝑧(2𝑛+3)(2𝑛+5)𝑐2𝑛+6βˆ’(𝑛𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧8(𝑛+1)(𝑛+2)(𝑛+4)𝑐2𝑛+8+𝑛(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+44(𝑛+1)(𝑛+2)2𝑧𝑐𝑛+4log𝑧𝑐,𝑆5=π‘Ž2𝑛(𝑛+7)𝑧16(𝑛+5)2π‘βˆ’π‘›2(𝑛+3)2(𝑛+6)16(𝑛+2)2𝑧(𝑛+4)3𝑐+𝑛2(π‘›βˆ’1)(𝑛+3)(𝑛+5)32(𝑛+1)2𝑧4π‘βˆ’π‘›(𝑛+3)𝑧16(𝑛+1)6𝑐+𝑛(𝑛+3)2𝑧16(𝑛+2)7π‘βˆ’π‘›2(𝑛+3)2𝑧32(𝑛+1)8π‘βˆ’π‘›ξ€·π‘›7+4𝑛6βˆ’10𝑛5βˆ’60𝑛4βˆ’39𝑛3+112𝑛2ξ€Έ+128π‘›βˆ’864(𝑛+1)2(𝑛+2)2𝑧𝑐𝑛+5+𝑛𝑛(𝑛+3)4+8𝑛3+13𝑛2ξ€Έβˆ’12π‘›βˆ’6𝑧16(𝑛+1)(𝑛+4)(𝑛+5)𝑐𝑛+7βˆ’ξ€·π‘›π‘›(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧64(𝑛+1)(𝑛+2)𝑐𝑛+9+𝑛(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧16(𝑛+1)(𝑛+2)𝑐𝑛+5log𝑧𝑐,𝑆6=π‘Ž2ξ‚Έβˆ’π‘›π‘§4(𝑛+2)2𝑐+𝑛2(𝑛+3)(2𝑛+5)4(𝑛+2)2𝑧(2𝑛+3)3π‘βˆ’π‘›2(π‘›βˆ’1)(𝑛+2)(𝑛+3)8(𝑛+1)3𝑧4𝑐+𝑛2𝑛4+4𝑛3+3𝑛2ξ€Έ+2𝑛+108(𝑛+1)2(𝑛+2)2𝑧𝑐𝑛+5βˆ’π‘›(𝑛+3)𝑧4(𝑛+2)𝑐𝑛+6+𝑛2(𝑛+3)𝑧8(𝑛+1)𝑐𝑛+7βˆ’π‘›(𝑛+3)2ξ€·2𝑛5+3𝑛4βˆ’2𝑛3+𝑛2ξ€Έ+2π‘›βˆ’416(𝑛+1)3𝑧(𝑛+2)(2𝑛+3)𝑐2𝑛+6+𝑛𝑛4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧16(𝑛+1)(𝑛+2)𝑐2𝑛+8βˆ’ξ€·π‘›π‘›(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+48(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+5log𝑧𝑐,𝑆7=βˆ’π‘Ž2ξƒ¬βˆ’(𝑛+7)+𝑛8(𝑛+3)(𝑛+5)2𝑛3+27𝑛2ξ€Έ+113𝑛+1468(𝑛+2)2(𝑛+4)2𝑧(𝑛+5)π‘βˆ’π‘›ξ€·π‘›7+19𝑛6+142𝑛5+538𝑛4+1078𝑛3+1005𝑛2ξ€Έ+147π‘›βˆ’2428(𝑛+1)2(𝑛+2)3𝑧(𝑛+4)(𝑛+5)2𝑐+𝑛6+6𝑛5βˆ’22𝑛3+11𝑛2ξ€Έβˆ’8π‘›βˆ’432(𝑛+1)3𝑧4π‘βˆ’(𝑛+3)(2𝑛+1)𝑧8(𝑛+1)(𝑛+2)5𝑐+𝑛𝑛(𝑛+3)(2𝑛+3)2ξ€Έ+3𝑛+18(𝑛+1)2(𝑛+2)2𝑧6𝑐+𝑛(𝑛+3)(2π‘›βˆ’1)𝑧16(𝑛+1)(𝑛+2)7π‘βˆ’ξ€·π‘›8+7𝑛7βˆ’14𝑛6βˆ’186𝑛5βˆ’347𝑛4+203𝑛3+1024𝑛2ξ€Έ+616π‘›βˆ’15232(𝑛+1)2(𝑛+2)2𝑧(𝑛+3)(𝑛+5)𝑐𝑛+3+𝑛𝑛8+6𝑛7βˆ’18𝑛6βˆ’184𝑛5βˆ’311𝑛4+266𝑛3+1000𝑛2ξ€Έ+504π‘›βˆ’17632(𝑛+1)2(𝑛+2)3(𝑧𝑛+4)𝑐𝑛+4+𝑛𝑛10+21𝑛9+150𝑛8+378𝑛7βˆ’235𝑛6βˆ’2391𝑛5βˆ’2388𝑛4+1288𝑛3+1800𝑛2ξ€Έ+608𝑛+19264(𝑛+1)2(𝑛+2)2𝑧(𝑛+4)(𝑛+5)𝑐𝑛+5+(𝑛𝑛+3)4+8𝑛3+13𝑛2ξ€Έβˆ’12π‘›βˆ’6𝑧8(𝑛+1)(𝑛+2)(𝑛+4)(𝑛+5)𝑐𝑛+6βˆ’ξ€·(π‘›βˆ’1)2𝑛7+25𝑛6+107𝑛5+169𝑛4+35𝑛3βˆ’54𝑛2ξ€Έ+12π‘›βˆ’8032(𝑛+1)2𝑧(𝑛+2)(𝑛+4)(𝑛+5)𝑐𝑛+7βˆ’(ξ€·π‘›π‘›βˆ’2)(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+432(𝑛+1)(𝑛+2)2𝑧𝑐𝑛+8+𝑛𝑛(π‘›βˆ’3)(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+464(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+9+𝑛4+2𝑛3βˆ’5𝑛2π‘›βˆ’6𝑛+4ξ€Έξ€·7+4𝑛6βˆ’10𝑛5βˆ’60𝑛4βˆ’39𝑛3+112𝑛2ξ€Έ+128π‘›βˆ’864(𝑛+1)2(𝑛+2)3𝑧(𝑛+3)𝑐2𝑛+6βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2π‘›βˆ’6𝑛+4ξ€Έξ€·4+8𝑛3+13𝑛2ξ€Έβˆ’12π‘›βˆ’616(𝑛+1)2𝑧(𝑛+2)(𝑛+4)(𝑛+5)𝑐2𝑛+8βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4264(𝑛+1)2(𝑛+2)2𝑧𝑐2𝑛+10βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧8(𝑛+1)(𝑛+2)𝑐𝑛+3𝑧log𝑐+𝑛𝑛(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+48(𝑛+1)(𝑛+2)2𝑧𝑐𝑛+4𝑧logπ‘ξ€Έβˆ’ξ€·π‘›π‘›(π‘›βˆ’1)(𝑛+3)4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+416(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+5𝑧log𝑐+𝑛4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+416(𝑛+1)2(𝑛+2)2𝑧𝑐2𝑛+6𝑧log𝑐.(A.1)

B.

  π‘Š1ξ‚Έ=β„Ž(𝑛+6)6βˆ’(𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)𝑛(𝑛+5)𝑧𝑐6(𝑛+1)2+(𝑛+2)(𝑛+3)(π‘›βˆ’1)(𝑛+4)𝑧2𝑐12(𝑛+1)2βˆ’π‘§(𝑛+2)3𝑐6(𝑛+1)2(+𝑧𝑛+2)4𝑐6(𝑛+1)2+1𝑧12(𝑛+1)5π‘βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’326(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+2+𝑛4+2𝑛3+112𝑛2ξ€Έβˆ’π‘›βˆ’84(𝑛+1)2𝑧(𝑛+2)(𝑛+3)𝑐𝑛+3+𝑛4+6𝑛3+7𝑛2ξ€Έβˆ’6π‘›βˆ’44(𝑛+1)2𝑧(𝑛+2)(𝑛+3)(𝑛+4)𝑐𝑛+4βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’312(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+5ξƒ­,π‘Š2ξ‚Έβˆ’=β„Ž(2𝑛+7)+2(𝑛+1)(𝑛+3)(𝑛+4)(2𝑛+5)𝑛(𝑛+3)2(𝑛+1)2𝑧(𝑛+2)(𝑛+4)𝑐+(π‘›βˆ’1)(𝑛+2)(2𝑛+5)4(𝑛+1)2𝑧(𝑛+4)(2𝑛+3)2𝑐+𝑛𝑛2ξ€Έπ‘§βˆ’3𝑐2𝑛+64(𝑛+1)2ξƒ­,π‘Š(𝑛+4)3ξ‚Έβˆ’=β„Ž(𝑛+6)𝑧6(𝑛+1)(𝑛+3)(𝑛+4)𝑐+𝑛(𝑛+5)6(𝑛+1)2𝑧(𝑛+3)2π‘βˆ’(π‘›βˆ’1)(𝑛+4)12(𝑛+1)2𝑧3𝑐+16(𝑛+1)2𝑧4𝑐+(𝑛+2)𝑧12(𝑛+1)6𝑐+𝑛𝑛2ξ€Έβˆ’36(𝑛+1)2𝑧𝑐𝑛+3βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+44(𝑛+1)2𝑧(𝑛+3)𝑐𝑛+4βˆ’ξ€·π‘›4+6𝑛3+7𝑛2ξ€Έβˆ’6π‘›βˆ’44(𝑛+1)2𝑧(𝑛+3)(𝑛+4)𝑐𝑛+5+𝑛𝑛2ξ€Έβˆ’312(𝑛+1)2𝑧𝑐𝑛+6ξƒ­,π‘Š4ξ‚Έ1=β„Žπ‘§2(𝑛+2)(𝑛+3)π‘βˆ’π‘›(2𝑛+5)𝑧2(𝑛+1)(𝑛+3)(2𝑛+3)2𝑐+(π‘›βˆ’1)(𝑛+2)24(𝑛+1)2𝑧(𝑛+3)3π‘βˆ’ξ€·π‘›π‘›(𝑛+4)2ξ€Έβˆ’3𝑧4(𝑛+1)(𝑛+2)(𝑛+3)𝑐𝑛+3+𝑛4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’8𝑛+24(𝑛+1)2𝑧(𝑛+3)𝑐𝑛+4+(𝑛+2)𝑧2(𝑛+1)(𝑛+3)𝑐𝑛+5βˆ’(𝑛+2)𝑧4(𝑛+3)𝑐𝑛+6+ξ€·2𝑛5+5𝑛4+𝑛3βˆ’3𝑛2ξ€Έβˆ’2π‘›βˆ’24(𝑛+1)3𝑧(𝑛+2)(2𝑛+3)𝑐2𝑛+5𝑛𝑛2ξ€Έβˆ’3𝑧4(𝑛+1)(𝑛+3)𝑐2𝑛+6ξƒ­,π‘Š5ξ‚Έ=β„Ž(𝑛+6)𝑧12(𝑛+3)(𝑛+4)2π‘βˆ’π‘›(𝑛+5)𝑧12(𝑛+1)(𝑛+3)3𝑐+(π‘›βˆ’1)(𝑛+4)𝑧24(𝑛+1)4π‘βˆ’(𝑛+3)𝑧12(𝑛+1)6π‘βˆ’(𝑛+2)𝑧247π‘βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’3𝑧12(𝑛+1)𝑐𝑛+4βˆ’ξ€·π‘›4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+4𝑧8(𝑛+1)(𝑛+3)𝑐𝑛+5βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’3𝑧4(𝑛+1)𝑐𝑛+7+𝑛𝑛(𝑛+3)4+8𝑛3+13𝑛2ξ€Έβˆ’12π‘›βˆ’6𝑧16(𝑛+1)(𝑛+4)(𝑛+5)𝑐𝑛+7ξƒ­,π‘Š6ξ‚Έβˆ’=β„Žπ‘›(2𝑛+5)𝑧4(𝑛+2)(𝑛+3)(2𝑛+3)2𝑐+𝑛24(𝑛+1)2𝑧3π‘βˆ’π‘›(π‘›βˆ’1)(2𝑛+3)𝑧8(𝑛+1)(2𝑛+1)4𝑐+𝑛𝑛4+3𝑛3βˆ’3𝑛2ξ€Έβˆ’7𝑛+28(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+4βˆ’ξ€·π‘›3+𝑛2ξ€Έβˆ’4𝑛+4𝑧8(𝑛+3)𝑐𝑛+5+𝑛8𝑧𝑐𝑛+6βˆ’ξ€·(𝑛+2)2𝑛2ξ€Έ+π‘›βˆ’28(𝑛+1)2𝑧(2𝑛+1)(2𝑛+3)𝑐2𝑛+5+𝑛2𝑛2ξ€Έβˆ’3𝑧8(𝑛+1)(𝑛+2)𝑐2𝑛+6ξƒ­,π‘Š7𝑛=β„Žπ‘›(𝑛+6)2ξ€Έβˆ’3𝑧12(𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)𝑐𝑛+2+𝑛2(𝑛𝑛+5)2ξ€Έβˆ’312(𝑛+1)2𝑧(𝑛+2)(𝑛+3)𝑐𝑛+3+𝑛𝑛(π‘›βˆ’1)(𝑛+4)2ξ€Έβˆ’324(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+4βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’312(𝑛+1)2𝑧(𝑛+2)𝑐𝑛+5+𝑛𝑛2ξ€Έβˆ’312(𝑛+1)2𝑧𝑐𝑛+6βˆ’π‘›ξ€·π‘›2ξ€Έβˆ’3𝑧24(𝑛+1)𝑐𝑛+7βˆ’π‘›2𝑛2ξ€Έβˆ’312(𝑛+1)2𝑧(𝑛+2)𝑐2𝑛+4+𝑛𝑛2π‘›βˆ’3ξ€Έξ€·4+2𝑛3βˆ’5𝑛2ξ€Έβˆ’6𝑛+48(𝑛+1)2𝑧(𝑛+2)(𝑛+3)𝑐2𝑛+5+𝑛𝑛2π‘›βˆ’3ξ€Έξ€·4+6𝑛3+7𝑛2ξ€Έβˆ’6π‘›βˆ’48(𝑛+1)2𝑧(𝑛+2)(𝑛+3)(𝑛+4)𝑐2𝑛+6βˆ’π‘›2𝑛2ξ€Έβˆ’3224(𝑛+1)2𝑧(𝑛+2)𝑐2𝑛+7ξƒ­.(B.1)

Acknowledgment

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