## Abstract

The theory of miscible dispersion in a straight circular pipe with interphase mass transfer that was investigated by Sankarasubramanian & Gill (1973 *Proc. R. Soc. Lond. A* 333, 115–132. (doi:10.1098/rspa.1973.0051); 1974 *Proc. R. Soc. Lond. A* 341, 407–408. (doi:10.1098/rspa.1974.0195)) in Newtonian fluid flow is extended by considering various non-Newtonian fluid models, such as the Casson (Rana & Murthy 2016 *J. Fluid Mech.* 793, 877–914. (doi:10.1017/jfm.2016.155)), Carreau and Carreau–Yasuda models. These models are useful to investigate the solute dispersion in blood flow. The three effective transport coefficients, i.e. exchange, convection and dispersion coefficients, are evaluated to analyse the dispersion process of solute. The convection and dispersion coefficients are determined asymptotically at large time which is sufficient to understand the nature of the solute dispersion process in a tube. The axial mean concentration is analysed, using the asymptotic expressions for these three coefficients. The effect of the wall absorption parameter, Weissenberg number, power-law index, Yasuda parameter and Peclet number on the dispersion process is discussed clearly in this study. A comparative study of the solute dispersion among the Newtonian and all other non-Newtonian models is presented. At low shear rate, it is observed that Carreau fluid behaves like Newtonian fluid, whereas the other fluids exhibit significant differences during the solute dispersion. This study may be applicable to understand the dispersion process of drugs in the blood stream.

## 1. Introduction

Nowadays, the solute dispersion in a fluid flow is the most fascinating topic of research and it has broad applications, such as the chromatographic separations in chemical engineering, pollutant transport in the environment, the mixing and the transportation of drugs or toxins in physiological systems. There are many applications on the dispersion process in blood flow through the vessels. In the indicator dilution technique, a quantity of solute is introduced into the blood stream to measure its concentration at some downstream point as it flows through blood. Because many intravenous medications are therapeutic at low concentration but toxic at high concentration, it is important to know the rate of dispersion of drugs in the blood flow of the circulatory system.

Analytically and experimentally, the theory on the dispersion process of solute was first invented by Taylor [1] in a steady laminar viscous fluid flow through a straight tube, and it was discussed that solute in the fluid is dispersed due to both radial molecular diffusion and axial convection with the effective Taylor dispersion coefficient, *D*_{eff}, defined by *D*_{m} is the molecular diffusivity, *w*_{m} is the mean axial velocity and *R* is the radius of the tube. Aris [2] introduced a new idea to study the Taylor dispersion in a steady flow using the method of moments and discussed the asymptotic behaviour of the second moment about the mean by modifying the effective molecular diffusion coefficient as

The Taylor−Aris dispersion theory and the generalized dispersion model have been applied by many researchers to study the solute dispersion process in Newtonian and non-Newtonian fluid flows. Chatwin [9] analysed the longitudinal dispersion of a passive contaminant in oscillatory flow through a tube with an effective molecular diffusion coefficient. Joshi *et al.* [10] investigated the axial transport of contaminant gas in oscillatory flow in a tube experimentally to determine the effective molecular diffusion coefficient. By following the method of moments, Mazumder & Das [11] studied the axial dispersion of a passive contaminant in pulsatile flow of a viscous incompressible fluid with the effect of first-order irreversible catalytic reaction at the tube wall. The theory of solute dispersion in a steady or unsteady flow of non-Newtonian fluids has wide applications in polymers, biochemical processing and cardiovascular flows. Despite a lot of demand to understand the solute dispersion process in non-Newtonian fluids, limited literature exists. Sharp [12] determined the effective axial diffusion coefficient in a fully developed steady flow of non-Newtonian fluids, such as the Casson, power-law and Bingham fluids through a straight tube, by following the theory of Taylor's dispersion. Nagarani *et al.* [13] presented work on solute dispersion in a Casson fluid flow in a pipe and channel with the combined effect of yield stress and irreversible wall reaction which discussed the solute dispersion at large time. Recently, Rana & Murthy [14] analysed the axial dispersion of solute in pulsatile flow of a Casson fluid through a tube which clearly demonstrates the dispersion phenomenon owing to the effect of boundary reaction catalysed by the tube wall, yield stress of the fluid, amplitude of fluctuating pressure component and small and large Womersley frequency parameter.

In the recent past, a lot of research activity took place by treating the non-Newtonian fluid as Carreau and Carreau–Yasuda models. The Carreau fluid model is relatively easy for mathematical treatment. The greatest advantage of the Carreau–Yasuda fluid model over the other non-Newtonian models is that it contains five parameters to interpret the rheology of the fluid, and this five-parameter model has sufficient flexibility to fit a wide variety of experimental apparent viscosity curves and it has proven to be useful in haemodynamics [15]. This model explains the behaviour of the fluid viscosity at low shear rate to high shear rate regions and also describes the shear thinning nature of the fluid; that is why this model is gaining much popularity in recent times. Under the consideration of zero Reynolds number and infinitely long wavelength, Naby & Misiery [16] aimed to obtain velocity distribution in a peristaltic flow of Carreau fluid with the effect of an inserted endoscope. Boyd *et al.* [17] analysed the velocity distribution and shear rate for Casson and Carreau–Yasuda fluids in steady and oscillatory flows through a straight and curved pipe using the lattice Boltzmann method. From this study, it is noted that both the Casson and Carreau–Yasuda fluids exhibit significant differences in the steady and oscillatory flow situations. The dispersion process of solute was not investigated in Carreau and Carreau–Yasuda fluids through a tube with wall absorption.

In this study, the axial solute dispersion in Carreau and Carreau–Yasuda fluid flows through a circular tube is analysed by considering boundary absorption/reaction at the tube wall. The generalized dispersion model proposed by Sankarasubramanian & Gill [7] is considered along with the constitutive relation for these models. An asymptotic analysis of convection and dispersion coefficients is made for large time. A comparative study of the solute dispersion among the Newtonian, Casson [14], Carreau and Carreau–Yasuda models is presented. The study of solute dispersion may be applied to understand the transport of drug/proteins in plasma in blood flow through the vessels. The boundary absorption or interphase mass transfer plays an important role in cardiovascular flow. Interphase mass transfer occurs at the wall of permeable blood vessels. The motivation of this study is to understand the different physiological processes that are involved in the dispersion of solute in blood flow by considering non-Newtonian fluid models.

## 2. Mathematical formulation

Consider an axisymmetric, fully developed one-dimensional flow of Carreau–Yasuda fluid in a straight circular tube of radius *R* to study the dispersion process of solute with the effect of wall absorption. At the tube wall, much of the solute is required to conduct the irreversible first-order catalytic reaction, and, for this reason, solute is continuously depleted from the system and this rate of depletion is proportional to the solute concentration at the tube wall. By ignoring free convection effects in this study, a dilute solution is assumed, so that the product of reaction does not cause any coupling of flux [7].

### (a) Constitutive equation for Carreau–Yasuda fluid and momentum equation

In the present investigation, the rheology of blood is characterized by non-Newtonian Carreau–Yasuda fluid owing to the shear thinning nature of blood. The rheological properties of blood are influenced by red blood cells. The shear thinning behaviour of blood is due to the aggregation of red blood cells at the condition of rest or at low shear rates, deformation and alignment of the erythrocytes at high shear rates [18]. The constitutive equation [15] for Carreau–Yasuda fluid is given by
*τ*′ and *μ*_{0} is the zero shear rate viscosity, *n* is the power-law exponent, *λ* is a time constant and *a* is the Yasuda parameter that represents the transition region between the zero shear rate region and the power-law region. The parameters *a* and *n* are dimensionless. However, the parameter *λ* has units of seconds. These parameters determine the behaviour of the non-Newtonian fluid with the viscosities *μ*_{0} and *n*<1 and *n*>1, respectively. For *n*<1, *μ*_{0} at low shear rates and *μ*_{0} at low shear rate, next a shear thinning region and, finally, a Newtonian region with viscosity *n*=1 or *λ*=0, the fluid represents a Newtonian fluid with viscosity *μ*_{0} and for the case of *a*=2, the fluid is known as Carreau fluid. The non-Newtonian behaviour of blood is very significant in small arteries at low shear rates. Owing to the presence of fibrinogen and globulins in blood plasma, red blood cells aggregate and form a three-dimensional microstructure at low shear rates. This microstructure of RBCs is mainly responsible for the shear thinning behaviour of blood at low shear rate and it exists significantly for normal blood with shear rates below 1 s^{−1} in required time intervals [20]. The data from low shear rate 10^{−3} s^{−1} to high shear rate 10 s^{−1} were considered by Picart *et al.* [21] for the measurement of shear stress of the blood. In this study, low shear rate is considered, so that *a*=2, equation (2.2) becomes the constitutive equation for Carreau fluid. The momentum equation for Carreau–Yasuda fluid along with the appropriate boundary conditions is written as
*a* that can take physically realistic real values. But for certain integral values of *a*, the time-dependent velocity distribution can be obtained. We have obtained the steady-state solution which is valid for all values of *a* and it is presented in §2b.

### (b) Governing equation and initial and boundary conditions

The solute concentration is governed by the unsteady convective diffusion equation
*C*′ is the local concentration of the solute, *w*′ is the axial velocity of the fluid in the tube, *D*_{m} is the molecular diffusivity, *t*′ is the time and, *r*′ and *z*′ represent the radial and axial directions, respectively. The solute of mass *M* at the plane *z*′=0 is considered to be uniformly distributed initially at *t*′=0 in a circular tube over the cross section of a circle of radius *R* [7]. The initial distribution for concentration is represented as
*C*_{0}=*M*/*πR*^{3}, *ψ*_{1}(*z*′)=*Rδ*(*z*′), *Y* _{1}(*r*′)=1 and *δ*(*z*′) is the Dirac delta function. Boundary conditions that are due to geometrical symmetry, an irreversible first-order heterogeneous reaction at the tube wall and the finite quantity of the solute in the system, are mathematically written as
*k* is the reaction rate constant. The non-dimensional variables are given by
*w*_{0}=−(*R*^{2}/4*μ*_{0})(d*p*′/d*z*′) is the characteristic velocity (centreline velocity in a Poiseuille flow) and d*p*′/d*z*′ is the applied pressure gradient along axis of the tube. The constitutive equation (2.2) reduces to a non-dimensional form as
*We*=*λw*_{0}/*R* is known as the Weissenberg number, which is the product of shear rate and characteristic time of the fluid. The non-dimensional form of momentum equation in steady state can be written as
*We*^{a} as the perturbation parameter, and we could obtain a second-order approximate solution. Thus, the velocity distribution for Carreau–Yasuda fluid is obtained as
*Pe*=*Rw*_{0}/*D*_{m} is the Peclet number. The initial condition (2.7) and boundary conditions (2.8)–(2.10) can be written in non-dimensional form as
*β*=*kR*/*D*_{m} is the wall absorption parameter. Also the initial concentration *ψ*(*z*)=*δ*(*z*)/*Pe*, *Y* (*r*)=1; it is worth mentioning that the initial solute distribution is governed by the Peclet number. For a blood vessel of fixed radius, the greater the value of *Pe*, the lower is the initial solute concentration.

## 3. Method of solution

By adopting the method given in Sankarasubramanian & Gill [7], the solution of unsteady convective diffusion equation (2.17) can be assumed as
*C*_{m}(*t*,*z*) is defined by
*r* and integrating with respect to *r* from 0 to 1, the governing equation (2.17) can be rewritten as
*C*_{m} is derived as
*δ*_{ij} denotes Kronecker delta
*K*_{i}(*t*) (*i*=3,4,5,…) in equation (3.4) can be truncated as their contribution is insignificant [4]. Then, the resulting generalized dispersion model for mean concentration *C*_{m}(*t*,*z*) is represented by
*K*_{0}(*t*), *K*_{1}(*t*) and *K*_{2}(*t*) signify the exchange coefficient owing to the non-zero solute flux at the tube wall, the convection coefficient owing to the velocity of solute and the dispersion coefficient owing to the molecular diffusion and velocity of the fluid, respectively. Now, the solute concentration *C*(*t*,*z*,*r*) can be written as
*f*_{0}, *K*_{0}, *f*_{1}, *K*_{1} and *K*_{2} in this specific order for obtaining mean concentration *C*_{m} from equation (3.7). So, we substitute equations (3.7) and (3.9) into equation (2.17) and equate the coefficients of (∂^{l}*C*_{m}/∂*z*^{l}) (*l*=0,1,2) to obtain the following set of differential equations for *f*_{l}:
*l*=0,1,2 and *f*_{−1}=*f*_{−2}=0. From equations (2.18)–(2.21), the initial and boundary conditions in terms of *C*_{m} and *f*_{l} are expressed as
*δ*_{l0} denotes as Kronecker delta defined by equation (3.6). By solving the coupled equations (3.8) and (3.10), the unknown functions *f*_{0}, *K*_{0}, *f*_{1}, *K*_{1} and *K*_{2} are determined, using the method of eigenfunction expansion. In this study, we are interested to see the behaviour of the effective transport coefficients *K*_{0}, *K*_{1} and *K*_{2} at large time by taking

### (a) Solution for *f*_{0}(*t*,*r*) and *K*_{0}(*t*)

The function *f*_{0}(*t*,*r*) and exchange coefficient *K*_{0}(*t*) are given below, which are obtained as in Sankarasubramanian & Gill [7] and Rana & Murthy [14] (the derivation is not repeated here for brevity)
*J*_{0} and *J*_{1} are the Bessel functions of the first kind of order zero and one, respectively, *l*=0,1,2,…, and *μ*_{l} are the roots of the transcendental equation

### (b) Asymptotic representation of *K*_{l}(*t*) (*l*=0,1 and 2)

We study the dispersion phenomenon of solute at large time by taking *f*_{l}(*t*,*r*) (*l*=0,1) and *K*_{l}(*t*) (*l*=0,1,2). As noted earlier, the function *f*_{0}(*t*,*r*) and exchange coefficient *K*_{0}(*t*) are independent of the velocity field, but the function *f*_{l}(*t*,*r*) and coefficient *K*_{l}(*t*) (*l*=1,2) depend on the velocity of the fluid. So, the steady flow is assumed to obtain the asymptotic value of *f*_{l}(*t*,*r*) and *K*_{l}(*t*) (*l*=1,2). This asymptotic study provides an understanding of the solute dispersion process in blood flow with non-Newtonian models in a circular tube. The asymptotic representation of *f*_{0}(*t*,*r*) and *K*_{0}(*t*) at large time is obtained from equations (3.18) and (3.19); these are given by
*μ*_{0} is the first (lowest in magnitude) root of the transcendental equation (3.20). The asymptotic expressions for *f*_{l}(*r*)′*s* are obtained from equation (3.10)
*l*=1,2, *f*_{−1}=0 and *K*_{l} (*l*=1,2) are given by
*K*_{l} (*l*=1,2) are represented in terms of functions *f*_{0} and *f*_{1} as
*K*_{1} is obtained from equation (3.28) using the known function *f*_{0} and it is presented by
*K*_{1} is obtained for non-zero *β*. To obtain the dispersion coefficient *K*_{2} from equation (3.28), the function *f*_{1} must be known. For *l*=1, using the method of eigenfunction expansion, the solution for *f*_{1} is obtained from equation (3.23) satisfying the boundary conditions (3.25) and (3.26)
*μ*_{l} are the roots of the transcendental equation *μ*_{l}*J*_{1}(*μ*_{l})=*βJ*_{0}(*μ*_{l}), *l*=0,1,2,…. In equation (3.30), *a*_{0} is unknown and using the additional condition (3.27) for *l*=1, *a*_{0} is obtained as
*f*_{1}(*r*) is represented as
*K*_{2} is given by
*K*_{2} is obtained for non-zero *β*. When *β*=0, solute in the system is not depleted at the tube wall, and the solute flux is zero. Therefore, the exchange coefficient *K*_{0}, which is due to the presence of non-zero solute flux at the tube wall, is absent. However, the convection coefficient *K*_{1} and dispersion coefficient *K*_{2} control the solute dispersion in the system. Now, the asymptotic expressions of *K*_{1} and *K*_{2} are obtained as

### (c) Solution for mean concentration *C*_{m}(*t*,*z*)

The solution of equation (3.7) for mean concentration *C*_{m}(*t*,*z*) in the presence of solute absorption at the wall with initial and boundary conditions (3.11) and (3.16) is given by
*K*_{0}(*t*), *K*_{1}(*t*) and *K*_{2}(*t*). Therefore, for large time, the mean concentration of solute with the effect of wall absorption is obtained from equations (3.39) and (3.41a)–(3.41c). For no absorption of solute at the wall (*β*=0), the solution for axial mean concentration *C*_{m}(*t*,*z*) becomes
*T** are those values of *z*_{1} and *T* computed for *β*=0 at large time.

## 4. Results and discussion

In this study, the dispersion of solute is analysed in a steady flow of Carreau and Carreau–Yasuda fluids through a tube with the effect of solute absorption at the tube wall owing to an irreversible first-order reaction. The asymptotic values of three effective transport coefficients, i.e. the exchange coefficient *K*_{0}(*t*), convection coefficient *K*_{1}(*t*) and dispersion coefficient *K*_{2}(*t*) at large time, are determined to study the dispersion phenomenon of solute using the generalized dispersion model suggested by Sankarasubramanian & Gill [7]. It is observed that the parameters *a*, *n* and *λ* which are due to the non-Newtonian Carreau–Yasuda fluid have a significant effect on the dispersion process of solute. The effect of wall absorption parameter *β* on the three transport coefficients is investigated. In addition, the effect of the Yasuda parameter *a*, power-law index *n* and Weissenberg number *We* on both *K*_{1} and *K*_{2} has been analysed at large time in the present investigation. The significance of the flow driving Peclet number *Pe* on *K*_{2} is studied at large time. Subsequently, the significance of these parameters *β*, *a*, *n*, *We* and *Pe* on the axial mean concentration *C*_{m} at large time is presented. In addition, a comparison of the solute dispersion has been made considering various non-Newtonian fluid models, such as Carreau fluid, Carreau–Yasuda fluid and Casson fluid [14] along with the Newtonian counterpart.

The present investigation is suitable for blood flow in the arterioles and medium-sized arteries. The Carreau and Carreau–Yasuda models represent the non-Newtonian nature of the blood. The data used for non-Newtonian fluid models throughout this investigation are physically realistic and relevant to blood flow. On the basis of significant contributions made in the direction of Carreau–Yasuda and Carreau fluid models by Bird *et al.* [15], Boyd *et al.* [17], Bernabeu *et al.* [19], Abraham *et al.* [22], Cho & Kensey [23] and Biasetti *et al.* [24], in this study, we have considered four different sets of parameter values for Carreau–Yasuda and Carreau fluids, which are as given in table 1. In addition, four different yield stress values, 0.001,0.005,0.01 and 0.05 are considered to represent the Casson fluid 1, Casson fluid 2, Casson fluid 3 and Casson fluid 4 models following the recent study of Rana & Murthy [14] to compare the results with other non-Newtonian and Newtonian models. Newtonian fluid is discussed by taking *n*=1 in this study. In this analysis, the non-Newtonian parameter *n* is given values 0.265, 0.441, 0.538 and 0.728 for Carreau–Yasuda fluid (C–Y fluid 4) and Carreau fluid [15]. The effect of the Weissenberg number *We* is seen to be more significant for normal blood with red cell aggregation at low shear rate in small vessels [25]. At very low shear rate, the parameter *We* is very small (≪1) [25]. In this investigation, *We* is given values 0.03, 0.04, 0.05, 0.08, 0.1 and 0.18 [16,26]. Following Bird *et al.* [15] and Abraham *et al.* [22], the Yausda parameter *a* is assigned values 0.64, 0.98 and 1.25. The value of *β* is taken in the range from 0 to 10^{4} (it is observed that beyond this number, the variation in all the relevant physical quantities is insignificant, this value is treated as *β* are 0.7 for carbon dioxide and *n*=1 (Newtonian fluid) for all the three transport coefficients are in the best agreement with the results given in Sankarasubramanian & Gill [7,8]. For brevity, we have shown this comparison for the mean concentration (which is derived using these three transport coefficients) in figure 4*a*.

### (a) Exchange coefficient *K*_{0}(*t*) and its asymptotic behaviour

The exchange coefficient *K*_{0}(*t*) is present in the system owing to non-zero solute flux at the tube wall with the irreversible first-order reaction. Initially, at time *t*=0, no amount of solute is present for the reaction to take place, and the exchange coefficient is zero. After the injection of solute, it is transported by convection of the non-Newtonian fluid along the tube and also diffuses to the wall region, thereby the solute concentration increases with time at the tube wall. The depletion of solute starts rapidly owing to wall absorption and *K*_{0}(*t*) becomes significant in the system. Owing to the reduction of the solute in the tube, *K*_{0}(*t*) attains a negative value. The expressions for *f*_{0}(*t*,*r*) and *K*_{0}(*t*) given in equations (3.18) and (3.19) clearly indicate that these are dependent only on *β*, *μ*_{l} and *A*_{l} (while *A*_{l} is dependent on the initial distribution of solute) and are independent of the parameters *We*, *a*, *Pe* and *n* which govern the non-Newtonian nature of the fluid and its velocity, as also reported in Rana & Murthy [14].

From figure 1*a*, it is evident that for small values of *β* (say 0.01 and 1), negative exchange coefficient −*K*_{0}(*t*) reaches its asymptotic value at very small time, and there is no significant change in −*K*_{0}(*t*) with time. So, for small *β*, asymptotic value of *K*_{0}(*t*) at large time is the same as the value at small time. However, the magnitude of −*K*_{0}(*t*) decreases sharply with time for large values of *β* (say 100) and reaches its asymptotic value after a certain time. Physical explanation is that in steady state, an equilibrium between the diffusive transport of solute from the centre of the tube to the wall region and the reaction at the wall occurs and, as a result, *K*_{0}(*t*) attains its asymptotic value. Because the reaction rate at the wall is small for small values of *β*, a small amount of solute diffuses to the wall region and steady state occurs quickly. So, *K*_{0}(*t*) reaches its asymptotic value within small time for small values of *β*. However, owing to the large reaction rate at the wall for large values of *β*, solute near the wall region is depleted more rapidly and much of the solute is transported by diffusion to the tube wall and large time is required to reach a steady state. So, for this reason, *K*_{0}(*t*) reaches its asymptotic value at large time for large *β*. It is also noted that the magnitude of −*K*_{0}(*t*) increases with *β* at very small time after the injection of solute. The similar phenomenon for −*K*_{0}(*t*) was described for *β*=0.01, 1 and 100 by Sankarasubramanian & Gill [7] and Rana & Murthy [14]. The asymptotic behaviour of *K*_{0}(*t*) at large time (taking *β*. From figure 1*b*, it is observed that as *β* increases, the magnitude of −*K*_{0} increases gradually for small *β* and increases sharply for large *β* and it reaches a constant value at very large values of *β*. At large time, the magnitude of −*K*_{0} becomes 5.669 for *β*=100 and further, for very large values of *β*, it approaches a constant value 5.78, as discussed in Sankarasubramanian & Gill [7] and Rana & Murthy [14]. The physical explanation for this is that owing to the increase of *β*, the first-order reaction rate at the wall increases, and solute is consumed rapidly at the wall. Therefore, more solute is required to maintain the reaction rate at the wall, and the transport of solute to the wall region is controlled by molecular diffusion. As a result, the diffusion process is stabilized in the system for rapid rate of reaction at very large values of *β*.

### (b) Asymptotic behaviour of convection coefficient *K*_{1}(*t*)

The asymptotic behaviour of convection coefficient *K*_{1}(*t*) is obtained to analyse the dispersion phenomenon of solute with and without the effect of wall absorption at the tube wall. The convection coefficient *K*_{1}, which arises because of the fluid movement, depends on *β*, and the non-Newtonian parameters *a*, *n* and *We*. In this study, the effect of these parameters *β*, *a*, *n* and *We* on the negative asymptotic convection coefficient −*K*_{1} has been discussed for large time. The present results of *K*_{1} for Newtonian fluid with *n*=1 at large time which are shown in table 2 agree well with those published in Sankarasubramanian & Gill [7] (figure 3) to check the accuracy of the present investigation.

Figure 2*a* shows the variation of −*K*_{1} at large time with *β* for different fluid models, such as Carreau–Yasuda, Carreau, Newtonian and Casson models. These different fluid models are considered to study the asymptotic convection process of the solute. From this figure, it is noted that Carreau–Yasuda, Carreau, Newtonian and Casson fluids exhibit significant variations of −*K*_{1} with varying values of *β* in the steady flow situation. The behaviour of −*K*_{1} as *β* varies with the chosen set of parameter values for Carreau fluids (Carreau fluid 1, 2, 3 and 4) and Newtonian fluid is not significantly different even in magnitude, as both these fluid models exhibit similarity at low shear rate [28,29]. The deviation of Carreau–Yasuda models (C–Y fluid 1, 2, 3 and 4) from the Newtonian fluid behaviour is seen for low shear rates, and the significant difference in Casson models (Casson fluid 1, 2, 3 and 4) is also observed from the other non-Newtonian and Newtonian fluid behaviour. As *β* increases, the magnitude of −*K*_{1} increases slowly for small values of *β* and increases rapidly for large values of *β* and, further, for very large values of *β*, it approaches a constant value. The physical significance for the enhancement of −*K*_{1} at a large value of *β* is due to the consumption of solute in the presence of boundary absorption at the wall. As *β* increases, the reaction rate at the tube wall increases and higher values of *β* satisfy more absorption at the wall region and in this region, a very low amount of solute is available for convection compared with that at the central region of the tube. So, the solute distribution is weighted in favour of the central region, and solute is convected with faster velocity near the central region than that at the wall region. In table 2, the magnitude of −*K*_{1} for all fluids with different values of *β* is reported. Among the discussed Newtonian and non-Newtonian fluids, lower and higher magnitudes of −*K*_{1} with various values of *β* are observed for Casson fluid and Carreau–Yasuda fluid, respectively. From table 2, it is seen that solute is convected along the tube with a higher velocity in the case of Carreau–Yasuda fluid for every value of *β* when compared with Carreau, Newtonian and Casson models.

The effect of *We* on −*K*_{1} has been studied for Carreau–Yasuda and Carreau fluids. Figure 2*b* shows that as *We* increases, the magnitude of −*K*_{1} increases for every value of *β* and also it increases with increasing *β*. From figure 2*b*, it is noted that solute is convected with a higher velocity of the fluid for higher values of *We*. The solute is transported with a higher fluid velocity for *We*=0.18 in comparison with other values of *We*. The effect of the shear thinning nature of the non-Newtonian Carreau–Yasuda and Carreau fluids on −*K*_{1} is investigated for *n*=0.265, 0.441, 0.538 and 0.728 [15] by fixing other parameter values. The asymptotic variation of −*K*_{1} with *β* reveals that as *n* decreases, the magnitude of −*K*_{1} increases. It also increases with the rate of wall reaction. The reason is that as *n* decreases, shear thinning nature of the fluid becomes more predominant and because of this solute is convected with a higher velocity of the fluid. For *n*=0.265, solute is convected with a higher velocity for all values of *β* in comparison with other values of *n*. These results are reported in table 3 for C–Y fluid 4. A similar investigation for different values of parameter *a* reveals that the magnitude of −*K*_{1} decreases as the parameter *a* increases for every value of *β*. For small and large values of *β*, −*K*_{1} attains a larger magnitude for *a*=0.64 than the other values of *a*, such as 0.98 and 1.25. So, solute is convected with a higher velocity for *a*=0.64; these details are reported in table 3 for C–Y fluid 4. Similar behaviour is noted for all other fluid models with different values of *n*, *We* and *a*.

### (c) Asymptotic behaviour of dispersion coefficient *K*_{2}(*t*)

The asymptotic analysis of dispersion coefficient *K*_{2}(*t*) at large time is made to investigate the dispersion process of the solute in the absence and presence of absorption at the tube wall. The asymptotic dispersion coefficient *K*_{2} depends on *β*, *a*, *n*, *We* and *Pe*. To check the accuracy of the present investigation, our results for *K*_{2}−1/*Pe*^{2} with *n*=1 (Newtonian fluid) at large time are compared with the results published in Sankarasubramanian & Gill [7,8] (figure 4), and these results are in the best agreement. From this analysis, it is also observed that the magnitude of *K*_{2}−1/*Pe*^{2} decreases rapidly for large values of *β*. For *β*=0 and *β*=0.01, the magnitude of *K*_{2}−1/*Pe*^{2} is 5.2083×10^{−3} and 5.21×10^{−3}, respectively, and as *K*_{2}−1/*Pe*^{2} reaches its constant value 1.2487×10^{−3}, which is consistent with the data given in Sankarasubramanian & Gill [7,8].

Mass transport mechanisms in large and medium-sized arteries are usually advection dominated, which are associated with rather large values of the Peclet number [30]. This investigation deals with low shear rate which is seen in small and medium-range blood vessels; the value of Peclet number, *Pe*, is considered in the range of 10 to 10^{4}, such as 10,100,200,500,1000 and 10 000. From this investigation, it is seen that the effect of *Pe* is significant on *K*_{2} for all Carreau and Carreau–Yasuda fluids. It is noted that as *Pe* increases, the magnitude of *K*_{2} decreases for each value of *β*. It is also noted that for very large values of *Pe*, the effect of *Pe* on *K*_{2} is insignificant. The magnitude of *K*_{2} with different values of *Pe* is tabulated for Carreau–Yasuda fluid (C–Y fluid 4) in table 4. For all other calculations in this analysis, the value of *Pe* is fixed as 1000 [14]. In figure 3*a*, the variation of *K*_{2} at large time is shown with *β* for various fluids, such as Carreau–Yasuda, Carreau, Newtonian and Casson fluids. The Carreau–Yasuda fluid with four sets of parameter values, Carreau fluid with four sets of parameter values, Newtonian fluid for *n*=1 and Casson fluid with four values of yield stress of the fluid [14] are incorporated in this study, and the comparison of these fluids on *K*_{2} has been made with the effect of reaction rate at the tube wall. Figure 3*a* explains that the differences of magnitude of *K*_{2} occur between these Newtonian and non-Newtonian fluids in the steady flow situation. The behaviour of *K*_{2} for all Carreau fluid models (Carreau fluid 1, 2, 3 and 4) is close to the Newtonian behaviour with the chosen set of parameters at low shear rate [28,29]. However, the nature of the Carreau fluid depends on the parameter *n* and Weissenberg number *We*. The difference in the value of *K*_{2} between the Carreau and Newtonian fluids can be observed with the change of parameters *n* and *We*. It is noted from the literature that Carreau–Yasuda fluid models (C–Y fluid 1, 2, 3 and 4) and Casson fluid models (Casson fluid 1, 2, 3 and 4) exhibit significant differences from the Newtonian fluid. From figure 3*a*, it is also noted that as *β* increases, the coefficient *K*_{2} decreases sharply for large values of *β* and reaches a constant value as *β* reaches a very large value. Physically, the reason for the decrement of *K*_{2} with the increase of *β* is that the reaction rate at the wall increases owing to the increment of the number of moles of reactive material at the tube wall, and more solute near the wall is continuously reduced in the system. Therefore, the solute distribution is weighted in favour of the region near the centre of the tube and in this region the solute is convected with faster velocity than that near the wall region. So, the velocity gradients are smaller near the central region than near the wall and smaller gradients cause smaller dispersion. In addition, transverse diffusion of solute is increased by the larger transverse concentration gradients in the system with the presence of wall absorption, and this causes the decrement in axial dispersion. The magnitude of *K*_{2} for different fluids and various values of *β* is described in table 5. For Carreau–Yasuda fluid (C–Y fluid 1), the magnitude of *K*_{2} is 7.9775×10^{−3} and 1.9034×10^{−3} for small *β*=0.01 and large *β*=100, respectively, and it reaches a constant value 1.7882×10^{−3} as *β* gets very large. It is seen from table 5 that in the case of Carreau–Yasuda fluid model, the magnitude of *K*_{2} is larger for every value of *β* in comparison with the other discussed Carreau, Newtonian and Casson fluid models, and the reason is the convection of solute with a higher velocity of the fluid for Carreau–Yasuda fluid. It is also noted that for the Casson model, *K*_{2} attains lower magnitude for every value of *β* in comparison with the other discussed fluid models.

Figure 3*b* shows the plot of *K*_{2} against *β* with different values of *We* for Carreau–Yasuda fluid (C–Y fluid 4). Figure 3*b* reveals that the magnitude of *K*_{2} increases as *We* increases in the presence of the rate of wall absorption. It is also observed that *K*_{2} decreases with the increase of *β* for each value of *We*. The difference in the value of *K*_{2} with varying *We* is seen to be more significant for small values of *β*. But for large values of *β*, this difference in the magnitude of *K*_{2} diminishes and for very large values of *β*, it becomes very small. For higher values of *We*, solute is convected with a higher velocity of the fluid and owing to which the value of *K*_{2} increases for each value of *β*. For this reason, *K*_{2} increases at a small value of *β*, and the larger difference in the magnitude of *K*_{2} with the various of values of *We* is seen for small values of *β*. As *β* increases, more solute is absorbed at the wall of the tube, and solute dispersion decreases rapidly as discussed earlier in this study. This decrement of *K*_{2} for large *β* is seen to be greater with increasing *We*. Hence, the difference in the values of *K*_{2} with varying parameter *We* becomes small for very large values of *β*. The variation of *K*_{2} at large time for different values of parameter *n* with shear thinning nature (*n*<1) of the non-Newtonian Carreau–Yasuda and Carreau fluids has been studied with the reaction of solute at the tube wall. It is observed that as the parameter *n* decreases (for more shear thinning nature), the magnitude of *K*_{2} increases for every value of *β*. In addition, the magnitude of *K*_{2} decreases with the increment of *β* for each value of *n*. The difference in the magnitude of *K*_{2} with the various values of *n* is more significant for small values of *β* but for large values of *β*, this difference in magnitude becomes small as described for the case of parameter *We*, because the decrement of *K*_{2} at large values of *β* becomes more rapid with the decreasing values of *n*. The magnitude of *K*_{2} with different values of *β* is given for various values of *n* in table 6 (tabulated data are for the C–Y fluid 4). The reason for the enhancement of *K*_{2} as the shear thinning nature of the fluid increases is that the shear thinning behaviour enhances the fluid velocity and owing to which the dispersion of solute increases. The magnitude of *K*_{2} with different values of *β* is presented in table 6 (tabulated data are for the C–Y fluid 4) for various values of parameter *a*. The data reveal that as *a* increases, the magnitude of *K*_{2} decreases with the effect of *β*. Owing to the increase of *a*, flow velocity decreases and hence decreases the solute dispersion for each value of *β*. For small values of *β*, the difference in the value of *K*_{2} with parameter *a* is larger than that of large *β*. For large values of *β*, *K*_{2} decreases rapidly with smaller values of *a*. Similar behaviour is seen for all other fluid models with different values of *n*, *We* and *a*.

### (d) Mean concentration *C*_{m}(*t*,*z*)

The expression (3.39) for the axial mean concentration of solute *C*_{m}(*t*,*z*) is obtained by considering the influence of the initial condition (2.18). This expression for *C*_{m}(*t*,*z*) is seen to depend on the asymptotic values of exchange coefficient *K*_{0}(*t*), convection coefficient *K*_{1}(*t*) and dispersion coefficient *K*_{2}(*t*) at large time. Thus, the mean solute concentration changes with the parameters *β*, *n*, *a*, *We* and *Pe*. What follows is the discussion on the significant effect of these parameters on the axial distribution of mean concentration of solute. The present results for mean concentration have been validated by comparing with the results given for a Newtonian fluid in Sankarasubramanian & Gill [7,8]. Figure 4*a* shows the comparison of our results for *n*=1 with the results of Sankarasubramanian & Gill [7,8] (figure 5, neglected the effect of *Pe* on *K*_{2}) on the distribution of mean concentration *C*_{m} with time *t* at axial distance *z*=0.5, and this comparison shows good agreement. From figure 4*a*, it is seen that as *β* increases, the peak of *C*_{m} decreases, and the concentration profile becomes more blunt. At a fixed axial distance, as *β* increases, the mean concentration of solute becomes maximum at a shorter time. The reason is that for large values of *β*, the rate of reaction at the tube wall becomes more rapid and solute is transported quickly to the wall region by convection, dispersion and molecular diffusion. At very large values of *β*, the process of solute transport becomes more diffusion controlled. So, solute distribution starts quickly in time for larger values of *β*. The magnitude of peak concentration (*C*_{m}×*Pe*) at *z*=0.5 is 3.8559, 1.143 and 0.2671 for *β*=0.01, 1 and 100, respectively.

Figure 4*b* displays the distribution of *C*_{m} with time *t* at *z*=0.5 for different values of *β* for Carreau–Yasuda fluid (C–Y fluid 4). It is noted from figure 4*b* that as *β* increases, the peak of *C*_{m} decreases. For small values of *β*, the peak of mean concentration is maximum and it decreases more for large *β*. A very small difference in the peak of the mean concentration is seen at large time for *β*=0 and *β*=0.01. The solute in the system is continuously absorbed at the tube wall owing to the presence of the boundary absorption. As *β* increases, the rate of wall absorption increases. Therefore, more solute moves to the wall region and it is depleted rapidly in the system. The magnitude of peak concentration *C*_{m} at *z*=0.5 is 3.8616×10^{−3},3.7935×10^{−3},1.2045×10^{−3} and 0.3155×10^{−3} for *β*=0, 0.01, 1 and 100, respectively. When we attempt to compare the mean concentration distribution profiles at a fixed cross section as a function of time and also at a fixed time along various axial positions, we note the following trend. In figure 5*a*, the axial distribution of *C*_{m} is presented with *t* at *z*=0.5 for different Newtonian and non-Newtonian fluid models. The comparison of these fluid models on the axial mean concentration of solute has been discussed by considering the Carreau–Yasuda fluid with four sets of parameter values, Carreau fluid with four sets of parameter values, Newtonian fluid for *n*=1 and Casson fluid with four values of yield stress of the fluid [14]. From these figures, it is noted that the maximum mean concentration is for the Casson model and the minimum mean concentration is seen for the Carreau–Yasuda model. With the Casson model for the fluid, the velocity is less compared with other described fluid models and hence, it decreases the solute dispersion. Therefore, the mean concentration of solute increases. But for Carreau–Yasuda fluid, solute is convected with a higher velocity of the fluid in comparison with other discussed fluid models and solute dispersion increases. Hence, it decreases the mean concentration of solute. The magnitude of the peak of *C*_{m} at *z*=0.5 is 4.84×10^{−3} and 3.5399×10^{−3} for Casson fluid (Casson fluid 4) and Carreau–Yasuda fluid (C–Y fluid 1), respectively. Figure 5*b* depicts the plot of *C*_{m} along the axial distance *z* with these discussed fluid models at large time *t*=0.5. Figure 5*b* also reports the similar nature of the fluid models on axial mean concentration as discussed earlier in this paragraph. It is observed that for Casson fluid, the distribution profile of the mean concentration is more peaky, and for the Carreau–Yasuda fluid, the mean concentration profile becomes more blunt and dispersed axially. For the Carreau–Yasuda fluid model, solute is convected with higher velocity of the fluid as well as the molecular diffusion is quite effective at large time after the injection of solute in the blood stream. So, solute is dispersed by convection and molecular diffusion and thereby results in more axial spread of the solute distribution. The magnitude of the peak of mean concentration at *z*=0.5 and *t*=0.5 is given in table 7.

The effect of *Pe* on *C*_{m} is shown in figure 6*a*. It is worth noting that the mean concentration depends on the initial source concentration (i.e. the initial condition) and from this figure it is clear that the peak of *C*_{m} decreases as *Pe* increases and the concentration profiles become more flat with increasing values of *Pe*. The magnitude of peak concentration *C*_{m} is 2.3089×10^{−1}, 3.7613×10^{−2}, 1.8928×10^{−2}, 7.585×10^{−3}, 3.7935×10^{−3} and 3.7938×10^{−4} for *Pe*=10,100,200,500,1000 and 10 000, respectively. In figure 6*b*, *C*_{m} is plotted versus *t* at *z*=0.5 with different values of *n* for Carreau–Yasuda fluid (C–Y fluid 4). From this figure, it is noted that the peak of *C*_{m} increases with the increase in the values of the Carreau–Yasuda parameter *n*. As *n* increases, the shear thinning nature of the fluid decreases and, for that, both the flow velocity and the dispersion coefficient decrease. Therefore, the mean concentration of the solute increases. The magnitude of peak of *C*_{m} at *z*=0.5 is 3.7935×10^{−3}, 3.8171×10^{−3} and 3.8332×10^{−3} for *n*=0.265, 0.538 and 0.728, respectively. The variation of *C*_{m} with *t* is shown for various values of Weissenberg number *We* for Carreau–Yasuda fluid (C–Y fluid 4) in figure 6*c*. From this figure, it is observed that as *We* increases, the peak of *C*_{m} decreases and it decreases more for higher values of *We*. Owing to an increase of *We*, the solute is convected with a higher velocity of the fluid and the solute dispersion increases and, hence, it decreases the solute concentration in the system. The magnitude of peak concentration *C*_{m} at *z*=0.5 is 3.8092×10^{−3}, 3.7776×10^{−3}, 3.6952×10^{−3} and 3.5544×10^{−3} for *We*=0.03, 0.05, 0.1 and 0.18, respectively. Figure 6*d* for Carreau–Yasuda fluid (C–Y fluid 4) explains that the peak of *C*_{m} increases as the Yasuda parameter *a* increases. Owing to the increase in *a*, both flow velocity and solute dispersion decreases. As a result, the concentration of solute increases. The magnitude of the peak of *C*_{m} at *z*=0.5 is 3.5552×10^{−3}, 3.7935×10^{−3} and 3.8356×10^{−3} for *a*=0.64, 0.98 and 1.25, respectively.

## 5. Conclusion

The solute dispersion in a non-Newtonian fluid flow through a straight tube is studied in the absence and presence of wall absorption/reaction at the tube wall using the generalized dispersion model. The different non-Newtonian fluid models such as Carreau fluid (four sets of parameter values), Carreau–Yasuda fluid (four sets of parameter values) and Casson fluid (four sets of yield stress parameter values) are considered to investigate the convection, dispersion and mean concentration distribution of the solute at large time after the injection of solute in the blood flow. The exchange coefficient *K*_{0} is obtained explicitly for all time and is shown to be independent of the nature of the fluid. But owing to difficulty in obtaining the time-dependent velocity distribution for the Carreau and Carreau–Yasuda fluid models, the asymptotic analysis for the convection coefficient *K*_{1} and dispersion coefficient *K*_{2} is made at large time. Physically, the large time behaviour is sufficient to justify the dispersion process of solute. By considering *n*=1 in the present investigation, our results reduce to the Newtonian fluid case and these results are compared with those presented in Sankarasubramanian & Gill [7,8] and it is observed that the comparison with three transport coefficients *K*_{0}, *K*_{1}, *K*_{2} and mean concentration *C*_{m} are in best agreement. It is noted that *K*_{0} depends on wall absorption parameter *β* only while *K*_{1}, *K*_{2} and *C*_{m} depend on parameter *β*, Weissenberg number *We*, Yasuda parameter *a* and power-law exponent *n*. As the Peclet number *Pe* increases, the magnitude of *K*_{2} and the peak of *C*_{m} decrease. It is reported in the literature that the behaviour of a Carreau fluid is close to a Newtonian fluid at low shear rate whereas the other fluids exhibit a significant difference in the steady flow situation. It is observed that as *β* increases, the asymptotic value of −*K*_{1} increases and *K*_{2} decreases. Therefore, the peak of *C*_{m} also decreases and the distribution profile of *C*_{m} becomes more blunt and dispersed. At fixed time for large values of *β*, the peak of mean concentration of solute occurs at a larger axial distance in comparison with small values of *β*. As the non-Newtonian parameter *n* increases, the asymptotic value of both −*K*_{1} and *K*_{2} decreases. Therefore, the peak of *C*_{m} increases with *n*. In this study, it is seen that as *We* increases, both the flow velocity and solute dispersion increases and it decreases the solute concentration. The asymptotic value of both coefficients −*K*_{1} and *K*_{2} decreases with the increment of the parameter *a*. Hence, the peak of mean concentration increases with *a*.

## Authors' contributions

All authors contributed for the development and improvement of the present study. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This research received no specific grant from any funding agency, commercial or not-for-profit sectors.

## Acknowledgements

The authors are thankful to all the reviewers for their valuable comments and constructive criticism for the improvement of the technical content of the manuscript.

- Received May 10, 2016.
- Accepted August 25, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.