2.1 Governing equation

The concentration of the solute is analysed by the unsteady convective diffusion equation

where $C^{\prime }$ is the local concentration of the solute, $w^{\prime }$ is the axial velocity of the fluid in tube, $D_{m}$ is the molecular diffusivity, $t^{\prime }$ is the time and $r^{\prime }$ and $z^{\prime }$ represent the radial and axial directions, respectively.

2.2 Initial and boundary conditions

(I) Initial condition:

Following Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973), the initial solute distribution at $t^{\prime }=0$ is considered by introducing the solute of mass $M$ at the plane $z^{\prime }=0$ in a circular tube over the circular cross-section of radius $R$ . Hence, the initial concentration distribution may be written as

with

and

where ${\it\delta}(z^{\prime })$ is the Dirac delta function.

(II) Boundary conditions:

The following three boundary conditions are relevant for the present study.

1. (i) Condition due to symmetry

   (2.6) $$\begin{eqnarray}\displaystyle \frac{\partial C^{\prime }}{\partial r^{\prime }}(t^{\prime },z^{\prime },0)=0. & & \displaystyle\end{eqnarray}$$

2. (ii) Condition due to the irreversible first-order heterogeneous reaction at the tube wall

   (2.7) $$\begin{eqnarray}\displaystyle -D_{m}\frac{\partial C^{\prime }}{\partial r^{\prime }}(t^{\prime },z^{\prime },R)=kC^{\prime }(t^{\prime },z^{\prime },R), & & \displaystyle\end{eqnarray}$$
   where $k$ is the reaction rate constant.

3. (iii) Since the amount of solute in the system is finite, we shall have

   (2.8) $$\begin{eqnarray}\displaystyle C^{\prime }(t^{\prime },\infty ,r^{\prime })=\frac{\partial C^{\prime }}{\partial z^{\prime }}(t^{\prime },\infty ,r^{\prime })=0. & & \displaystyle\end{eqnarray}$$

2.3 Constitutive equation for Casson fluid and momentum equation

Following Aroesty & Gross (Reference Aroesty and Gross1972), the constitutive equation for Casson fluid in one dimensional shear flow may be written as

where ${\it\tau}^{\prime }$ and $\partial w^{\prime }/\partial r^{\prime }$ are the shear stress and shear rate, respectively, ${\it\tau}_{y}^{\prime }$ is the yield stress, and ${\it\mu}$ is the viscosity of Casson fluid (viscosity at high shear rate).

In the plug flow region in which shear stress ${\it\tau}^{\prime }$ is less than yield stress ${\it\tau}_{y}^{\prime }$ , the velocity gradients become zero. When shear rates in the fluid become very high i.e. for ${\it\tau}^{\prime }>{\it\tau}_{y}^{\prime }$ , Newtonian behaviour induces in the fluid. This behaviour is related to the blood flow in small tubes (Aroesty & Gross Reference Aroesty and Gross1972). The yield stress of the human blood is typically $0.04~\text{dyn cm}^{-2}$ at a hematocrit of 40 % (Merrill Reference Merrill1969).

Due to the pulsatile flow, the pressure gradient at any $z^{\prime }$ is represented by

where $P_{0}$ and $P_{1}$ are the steady and fluctuating components of the pressure gradient acting along the axis of tube, ${\it\omega}$ is the frequency of the pressure pulsation.

The momentum equation for Casson fluid is represented as

with boundary conditions

and

2.4 Non-dimesionalization and velocity distribution

By introducing the following non-dimensional variables

the governing equation (2.1) transforms to

where $w_{0}=(R^{2}/4{\it\mu})P_{0}$ is the characteristic velocity, and $Pe=Rw_{0}/D_{m}$ is the Peclet number.

The initial condition (2.2) and boundary conditions (2.6)–(2.8) can be written in non-dimensional form as:

with

and

where ${\it\beta}=kR/D_{m}$ is the wall absorption parameter.

Using the non-dimensional variables, the constitutive equation (2.9) reduces to

The momentum equation (2.11) in non-dimensional form is given by

with boundary conditions

and

where ${\it\varepsilon}=1/Sc$ , $p(t)=-2(\partial p^{\prime }/\partial z^{\prime })/P_{0}$ , $Sc={\it\nu}/D_{m}$ is the Schmidt number and ${\it\nu}$ is the kinematic viscosity coefficient.

Since the Schmidt number is very large $(O(10^{3}))$ in arterial blood flows (Caro et al. Reference Caro, Pedley, Schroter and Seed1978), the parameter ${\it\varepsilon}$ is very small. In this study, ${\it\varepsilon}$ is treated as a perturbation parameter to obtain the velocity distribution for small as well as large values of Womersley frequency parameter ${\it\alpha}$ . Because of large values of $Sc$ , it is more appropriate to use this non-dimensionalization instead of the one that was given in Nagarani & Sebastian (Reference Nagarani and Sebastian2013).

By solving the nonlinear coupled equations (2.22) and (2.23) with perturbation analysis for small parameter ${\it\varepsilon}$ , the velocity distribution (Aroesty & Gross Reference Aroesty and Gross1972) in the plug flow region $(0\leqslant r\leqslant r_{p})$ is obtained as

and that in the outside of plug region $(r_{p}\leqslant r\leqslant 1)$ is obtained as

In the above equations for velocity, $p(t)=2(1+e\sin ({\it\alpha}^{2}Sc\,t))$ , ${\it\alpha}^{2}={\it\omega}R^{2}/({\it\mu}/{\it\rho})$ , ${\it\alpha}$ is known as Womersley frequency parameter, $e$ is the amplitude of the fluctuating pressure component, $r_{p}={\it\tau}_{y}/p(t)$ is the dimensionless radius of the plug flow region and $p^{\prime }(t)=\text{d}p/\text{d}t$ . The product ${\it\alpha}^{2}Sc$ is the ratio of characteristic time of diffusion to the period of oscillation.

3.1 Estimation of $f_{0}(t,r)$ and $K_{0}(t)$

The equation involving $f_{0}(t,r)$ and exchange coefficient $K_{0}(t)$ can be written from (3.10) as

with the initial and boundary conditions (3.12), (3.14), (3.15) for $n=0$ , and an additional condition obtained from (3.17) as

Using the method of eigenfunction expansion, the solution of non-homogeneous boundary value problem (3.18) for $f_{0}(t\text{,}\,\,r)$ satisfying the initial and boundary conditions (3.12), (3.14), (3.15) and the condition (3.19) is obtained as (Boyce, DiPrima & Haines Reference Boyce, DiPrima and Haines1992)

where $\text{J}_{0}$ and $\text{J}_{1}$ are the Bessel functions of the first kind of order zero and one, respectively and ${\it\mu}_{n}$ are the roots of transcendental equation

and

From (3.8) and (3.20)–(3.22), we obtain the exchange coefficient $K_{0}(t)$ as

which is exactly the same as derived by Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973). It is seen that $K_{0}(t)$ is independent of velocity field.

3.2 Estimation of $f_{1}(t,r)$ and $K_{1}(t)$

The governing equation for $f_{1}(t,r)$ and the convection coefficient $K_{1}(t)$ is derived from (3.10) by taking $n=1$ and is written as

The appropriate initial and boundary conditions are given in (3.13)–(3.15) and we have also an additional condition obtained from (3.17) as

The eigenfunction expansion solution for $f_{1}(t,r)$ satisfying the initial and boundary conditions (3.13)–(3.15) and the condition (3.25) is obtained as

where

$m=0,1,2,\ldots$  , $n=0,1,2,\ldots$  , and ${\it\mu}_{n}$ are the roots of transcendental equation

From (3.8), $K_{1}(t)$ is obtained and it is represented as

where

Now from (3.31) using (3.20), (3.27)–(3.30) and (3.32), the convective coefficient $K_{1}(t)$ is obtained numerically using the Simpson’s $1/3$ rule for integration.

3.3 Estimation of $f_{2}(t,r)$ and $K_{2}(t)$

By taking $n=2$ in (3.10), we obtain the equation for $f_{2}(t,r)$ involving the dispersion coefficient $K_{2}(t)$ as

The initial and boundary conditions are obtained from (3.13)–(3.15) for $n=2$ , and the addition condition is given by

Using the method of eigenfunction expansion, the solution of (3.33) for $f_{2}(t,r)$ satisfying the initial and boundary conditions (3.13)–(3.15) and the additional condition (3.34) is obtained as

where

$m=0,1,2,\ldots$  , $n=0,1,2,\ldots$  , and ${\it\mu}_{n}$ are the roots of transcendental equation

From (3.8), $K_{2}(t)$ is obtained as

where

and

Now from (3.44) using (3.26), (3.36)–(3.43), (3.45) and (3.46), the dispersion coefficient $K_{2}(t)$ is obtained numerically using the Simpson’s $1/3$ rule of integration.

3.4 Solution for mean concentration $C_{m}(t,z)$

The solution of (3.7) satisfying the initial and boundary conditions (3.11), (3.16) is given by

where

and

4.1 Exchange coefficient $K_{0}(t)$

The amount of solute is continuously reduced in the tube due to the irreversible first-order reaction occurring at the tube wall. From (3.15) for $n=0$ and (3.23), it is seen that the exchange coefficient $K_{0}(t)$ directly depends on the wall absorption parameter ${\it\beta}$ . The coefficient $K_{0}(t)$ accounting for the non-zero solute flux at the tube wall will be negative for the depletion of solute in the system. At the time $t=0$ , $K_{0}$ becomes zero due to the absence of solute at the tube wall. As the time increases, solute moves along the tube and diffuses to the wall region. Due to which, the concentration of solute increases at the wall region. An equilibrium between the diffusive transport from the centre of the tube to the wall region and the reaction at the wall occurs due to which there is a steady state of the process and from that instant, $K_{0}$ becomes constant. Figure 2 shows the variation of negative exchange coefficient $-K_{0}$ with time for ${\it\beta}=0.01$ , 1 and 100, respectively. From this figure, it is seen that there is no significant change in $-K_{0}$ with time $t$ for ${\it\beta}=0.01$ and ${\it\beta}=1$ . But the magnitude of $-K_{0}$ decreases rapidly as $t$ increases for large values of ${\it\beta}$ ( ${\it\beta}=100$ is shown in the figure 2) and after a certain time it reaches a constant value. For small values of ${\it\beta}$ , an equilibrium between the diffusion of solute and the wall reaction occurs within small time and $K_{0}$ reaches the constant value. But, for large ${\it\beta}$ , as the time increases, solute in the system is reduced more rapidly due to the reaction at the wall and large time is taken to reach a steady state. From figure 2, it is also seen that as ${\it\beta}$ increases, $-K_{0}$ increases at initial time. The same phenomenon for $-K_{0}$ was reported for ${\it\beta}=0.01$ , 1 and 100 by Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973).

Figure 2.

Variation of negative exchange coefficient $-K_{0}(t)$ with time for wall absorption parameter ${\it\beta}=0.01$ , 1 and 100.

Figure 3 shows the variation of negative exchange coefficient $-K_{0}$ with ${\it\beta}$ at large time $t$ (Here the time $t=1$ is taken as very large). As ${\it\beta}$ increases, the magnitude of $-K_{0}$ increases continuously and becomes $5.669$ at $t=1$ for ${\it\beta}=100$ . Further, for very large values of ${\it\beta}$ , it approaches the value $5.78$ , as discussed in Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973). The reason for this is because at sufficiently large value of ${\it\beta}$ , solute is consumed rapidly by the first-order reaction at the wall than it is supplied by the molecular diffusion and as a result the diffusion process is stabilized. It is worth mentioning that the exchange coefficient $K_{0}(t)$ is independent of the nature of the fluid (Newtonian/non-Newtonian) and the pulsatility of the vessels. Hence, the coefficient $K_{0}(t)$ is independent of ${\it\tau}_{y}$ , $e$ and  ${\it\alpha}$ .

Figure 3.

Variation of negative exchange coefficient $-K_{0}(t)$ at large time $t=1$ with wall absorption parameter ${\it\beta}$ .

4.2 Convection coefficient $K_{1}(t)$

The convective coefficient $K_{1}$ is arising because of the fluid movement, unlike $K_{0}$ , the convection coefficient depends on yield stress of the fluid ${\it\tau}_{y}$ , amplitude of the fluctuating pressure component $e$ and frequency parameter ${\it\alpha}$ . In this analysis, the effects of ${\it\tau}_{y}$ , $e$ , ${\it\beta}$ and ${\it\alpha}$ on negative convection coefficient $-K_{1}$ have been discussed for small and large time.

The variation of negative convection coefficient $-K_{1}$ with small and large time for different values of yield stress ${\it\tau}_{y}$ has been shown in figure 4(a,b). From these figures, it is seen that as ${\it\tau}_{y}$ increases, both the amplitude of fluctuations and the magnitude of $-K_{1}$ decreases. The coefficient $-K_{1}$ oscillates and reaches a non-transient state after certain period of time. For yield stress ${\it\tau}_{y}=0$ , 0.02 and 0.05, $-K_{1}$ starts to oscillate at initial time $t=0$ from the values 0.49, 0.384 and 0.329, respectively. So, solute is started initially to convect with the velocity of the fluid 0.49 along the tube for ${\it\tau}_{y}=0$ . At non-transient state in each period of oscillation, $-K_{1}$ reaches a constant value 0.74, 0.62 and 0.549 for ${\it\tau}_{y}=0$ , 0.02 and 0.05, respectively. Due to increase in ${\it\tau}_{y}$ or in other words, increase in the non-Newtonian parameter decreases the velocity of the fluid, which in turn caused a reduction in the value of $-K_{1}$ .

Figure 4.

Variation of negative convection coefficient $-K_{1}(t)$ with $t$ for different values of ${\it\tau}_{y}$ when ${\it\beta}=0.01$ , ${\it\alpha}=0.5$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 5(a,b) show the variation of $-K_{1}$ with small and large time for different values of fluctuating pressure component $e$ . It is observed from these figures that the amplitude of fluctuations of $-K_{1}$ increases with $e$ . The flow is steady for the case $e=0$ . In steady flow at non-transient state, $-K_{1}$ reaches a constant value 0.336, that means the solute is convected along the tube with the constant velocity of the fluid 0.336 for ${\it\tau}_{y}=0.05$ . For unsteady flow at non-transient state, the amplitude of fluctuations of $-K_{1}$ reaches a constant value 0.549 and 0.421 in each cycle of oscillation when $e=0.5$ and $e=0.2$ , respectively. The critical time for which $K_{1}$ reaches a non-transient state is independent of $e$ . As $e$ increases, fluctuating component of the pressure gradient increases and hence flow velocity increases which leads to increase in $-K_{1}$ .

Figure 5.

Variation of negative convection coefficient $-K_{1}(t)$ with $t$ for different values of $e$ when ${\it\tau}_{y}=0.05$ , ${\it\beta}=0.01$ , ${\it\alpha}=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

The variation of $-K_{1}$ has been shown at small and large time for different values of wall absorption parameter ${\it\beta}$ in figure 6. For ${\it\beta}=0.01$ , the fluctuation of $-K_{1}$ does not increase significantly and reaches a stationary state at small time after certain period of oscillation. But for ${\it\beta}=1$ and ${\it\beta}=100$ , the amplitude of oscillations of $-K_{1}$ increases at small time and becomes stable at large time. At each period of oscillation in stationary state, $-K_{1}$ reaches a constant value $0.809$ , $0.61$ and $0.549$ for ${\it\beta}=100$ , 1 and 0.01, respectively. It is evident from these figures that as ${\it\beta}$ increases, the amplitude of oscillations of $-K_{1}$ increases at small and large time. The growth of oscillations of $-K_{1}$ at small time becomes larger for larger values of ${\it\beta}$ . The reason for this enhancement of $-K_{1}$ at large value of ${\it\beta}$ is due to the consumption of solute in the presence of boundary absorption at the wall. Higher values of ${\it\beta}$ satisfies more absorption at the wall region and lesser solute is available for convection compared to 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. From the figures 7–9, it is also seen that as ${\it\beta}$ increases, the amplitude of oscillations of $-K_{1}$ increases for different values of ${\it\alpha}$ . But the difference between the amplitude of fluctuations of $-K_{1}$ for different values of ${\it\beta}$ decreases as ${\it\alpha}$ increases and becomes very small at large value of  ${\it\alpha}$ .

Figure 6.

Variation of negative convection coefficient $-K_{1}(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=0.5$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 7.

Variation of negative convection coefficient $-K_{1}(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=1$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 8.

Variation of negative convection coefficient $-K_{1}(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=4$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 9.

Variation of negative convection coefficient $-K_{1}(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=10$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

The Womersley frequency parameter ${\it\alpha}$ is seen to have significant influence on $-K_{1}$ in the case of non-Newtonian fluid. Figures 6–9 show the variation of $-K_{1}$ with small and large time for different values of ${\it\alpha}=0.5$ , $1$ , $4$ and $10$ . It can be observed that the amplitude of fluctuation of $-K_{1}$ increases with ${\it\alpha}$ . The period of oscillation becomes short at large value of ${\it\alpha}$ . At small time, the fluctuation of $-K_{1}$ does not increase significantly for large values of ${\it\alpha}$ and $-K_{1}$ reaches a stationary state at small time. In each period of oscillation for ${\it\beta}=0.01$ , $-K_{1}$ reaches a constant value $0.549$ , $0.551$ , $0.845$ and $3.195$ at non-transient state for ${\it\alpha}=0.5$ , $1$ , $4$ and $10$ , respectively. It also seen that as ${\it\alpha}$ increases, the magnitude of $-K_{1}$ decreases initially at time $t=0$ but after that the amplitude of fluctuations of $-K_{1}$ increases more and becomes positive and negative during the period of oscillation. The negative value of $-K_{1}$ occurs due to the flow reversal in the period.

When $e=0$ and time $t$ is very large, the variation of negative convection coefficient $-K_{1}$ with ${\it\beta}$ for various values of the non-Newtonian parameter ${\it\tau}_{y}$ has been shown in figure 10. It is observed that as ${\it\beta}$ increases, the magnitude of $-K_{1}$ increases and it decreases as ${\it\tau}_{y}$ increases (Nagarani et al. Reference Nagarani, Sarojamma and Jayaraman2004). The fluid velocity in Casson fluid is less compared to that of the Newtonian fluid and this leads to lesser convective coefficient in the Casson fluid. As ${\it\beta}\rightarrow \infty$ , $K_{1}\rightarrow -0.78$ for ${\it\tau}_{y}=0$ , i.e. as ${\it\beta}$ increases, the coefficient $-K_{1}$ increases and possesses to the value $0.78$ for ${\it\tau}_{y}=0$ . In the absence of wall absorption, i.e. for ${\it\beta}=0$ , the value of $-K_{1}$ is $0.5$ which means that solute is moved along the tube with the average velocity of the flow. As ${\it\beta}$ increases, solute is transported with the velocity higher than the average velocity of the flow. Similar profile for $-K_{1}$ with ${\it\tau}_{y}=0$ is seen in Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973) and Nagarani et al. (Reference Nagarani, Sarojamma and Jayaraman2004).

Figure 10.

Variation of negative convection coefficient $-K_{1}(t)$ for different values of ${\it\tau}_{y}$ with wall absorption parameter ${\it\beta}$ at large time $t=1$ when $e=0$ .

4.3 Dispersion coefficient $K_{2}(t)$

The dispersion coefficient $K_{2}$ depends on the yield stress of the fluid ${\it\tau}_{y}$ , amplitude of the fluctuating pressure component $e$ , wall absorption parameter ${\it\beta}$ and frequency parameter ${\it\alpha}$ . In the present study, the effects of ${\it\tau}_{y}$ , $e$ , ${\it\beta}$ and ${\it\alpha}$ on solute dispersion have been discussed for small as well as large time.

Figure 11(a,b) show the variation of dispersion coefficient $K_{2}$ (from which the additive contribution of the axial diffusion $1/Pe^{2}$ is deducted) with small and large time for different values of ${\it\tau}_{y}$ . From these figures, it is noticeable that as ${\it\tau}_{y}$ increases, both the amplitude of fluctuations and the magnitude of $K_{2}$ decreases. With the injection of solute in the blood flow, the solute dispersion starts within small time. At small time, $K_{2}$ increases periodically with wavy nature and thereafter decreases as time increases and reaches a non-transient state at large time $t$ , which is related to the characteristic time of transverse diffusion $R^{2}/D_{m}$ . After the initial increment of the amplitude of oscillation for small time as seen in figure 11(a), these oscillations become stable at large time as large as $t\geqslant 0.4$ (as shown in figure 11 b) and thereafter the dispersion of solute takes place with the period of oscillation at a uniform rate. Due to increase in ${\it\tau}_{y}$ , the velocity decreases and hence a decrease in the dispersion coefficient is noticed.

Figure 11.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\tau}_{y}$ when ${\it\beta}=0.01$ , ${\it\alpha}=0.5$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

In figure 12(a,b), the variation of dispersion coefficient $K_{2}$ at all times is shown for steady flow $(e=0)$ of Newtonian fluid $({\it\tau}_{y}=0)$ and Casson fluid $({\it\tau}_{y}=0.05)$ with and without the effect of wall absorption ( ${\it\beta}=0.01$ and ${\it\beta}=0$ ). Present result (shown in the figure 12 a) for steady flow of Newtonian fluid in the absence of wall absorption matches exactly with the result of Gill & Sankarasubramanian (Reference Gill and Sankarasubramanian1970) (figure 1) at small and large time. Also, our result for steady flow of Newtonian fluid with the effect of wall absorption is found to be in best agreement with Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973) at very large time. From figure 12(a,b), it is seen that as time increases, $K_{2}$ increases rapidly at small time and reaches a constant value at large time $(t\geqslant 0.5)$ . For ${\it\tau}_{y}=0$ and ${\it\tau}_{y}=0.05$ with ${\it\beta}=0$ , $K_{2}$ reaches a constant value $5.21\times 10^{-3}$ and $2.13\times 10^{-3}$ , respectively at large time $(t\geqslant 0.5)$ . It is also observed that $K_{2}$ decreases as ${\it\tau}_{y}$ increases at small as well as large time. Also, $K_{2}$ decreases with the increment of wall absorption parameter ${\it\beta}$ . But the difference of $K_{2}$ between the case of ${\it\beta}=0$ and ${\it\beta}=0.01$ is small and for each case $K_{2}$ attains a same constant value at very large time $(t=10)$ .

Figure 12.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\tau}_{y}$ and ${\it\beta}$ at small and large time when $e=0$ . The data of Gill & Sankarasubramanian (Reference Gill and Sankarasubramanian1970) for $K_{2}$ is obtained from the equations (19) and (20) of their paper. (a)  ${\it\tau}_{y}=0$ with ${\it\beta}=0$ ; (b)  ${\it\tau}_{y}=0$ and 0.05 with ${\it\beta}=0$ and 0.01.

At small as well as large time, the variation of $K_{2}$ for different values of $e$ and ${\it\tau}_{y}$ has been presented for ${\it\beta}=0$ and ${\it\beta}=0.01$ in figures 13 and 14, respectively. For the clarity in comparison, these results are presented here for four cycles of oscillation for both Newtonian fluid $({\it\tau}_{y}=0)$ and Casson fluid $({\it\tau}_{y}=0.05)$ . From these figures, it is seen that as $e$ increases, both the amplitude of fluctuations and the magnitude of $K_{2}$ increases. For the steady flow, $K_{2}$ increases at small time and attains a value $4.04\times 10^{-3}$ at $t=0.1$ and $5.19\times 10^{-3}$ at $t=0.4$ for ${\it\beta}=0$ and ${\it\tau}_{y}=0$ . For ${\it\tau}_{y}=0$ with ${\it\beta}=0.01$ , $K_{2}$ reaches a value $3.89\times 10^{-3}$ at $t=0.1$ and $5.04\times 10^{-3}$ at $t=0.4$ . For the unsteady flow $(e>0)$ , $K_{2}$ increases cyclically at small time and reaches a stationary state at large time. At each cycle of oscillation in large time with ${\it\tau}_{y}=0$ and ${\it\beta}=0.01$ , $K_{2}$ reaches a constant value $6.1\times 10^{-3}$ and $7.64\times 10^{-3}$ for $e=0.2$ and $e=0.5$ , respectively. A similar nature is seen for other values of ${\it\tau}_{y}$ . The critical time at which $K_{2}$ reaches a non-transient state becomes independent of $e$ . It is worth noting that the value of $K_{2}$ at small time $t=0.1$ for steady flow of Newtonian fluid without wall absorption as observed by Nagarani & Sebastian (Reference Nagarani and Sebastian2013) (referred to their figure 5) is near about $1\times 10^{-3}$ which deviates much from the value $(4.04\times 10^{-3})$ of Gill & Sankarasubramanian (Reference Gill and Sankarasubramanian1970) (equations (19) and (20) therein, for $K_{2}$ ).

Figure 13.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of $e$ and ${\it\tau}_{y}$ when ${\it\beta}=0$ . (a) For small times; (b) for large times.

Figure 14.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of $e$ and ${\it\tau}_{y}$ when ${\it\beta}=0.01$ . (a) For small times; (b) for large times.

The effect on $K_{2}$ has been studied for different values of absorption parameter ${\it\beta}=0$ , 0.01, 1 and 100 with small and large time and it is shown in figures 15(a) and 15(b), respectively. It is observed from these figures that as ${\it\beta}$ increases, both the amplitude of fluctuations and the magnitude of $K_{2}$ decreases. Initially, the amplitude of oscillation of $K_{2}$ increases and reaches a constant value $3.56\times 10^{-3}$ , $3.42\times 10^{-3}$ , $3.2\times 10^{-3}$ and $0.81\times 10^{-3}$ at each period of oscillation in large time for ${\it\beta}=0$ , 0.01, 1 and 100, respectively. With increase in the value of ${\it\beta}$ , the dispersion coefficient $K_{2}$ decreases because number of moles of reactive material at the wall increases to undergo the chemical reaction or absorption and hence, more amount of solute near the wall is continuously depleted 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 the 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. This could be a reason for decrease in axial dispersion when ${\it\beta}$ increases. Also, 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. From figures 16, 17 and 20–25, it is also noticed that the increase in ${\it\beta}$ tends to decrease the axial dispersion for various values of  ${\it\alpha}$ .

Figure 15.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=0.5$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 16.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=1$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 17.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=2$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 18.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with small time $t$ for different values of ${\it\tau}_{y}$ when ${\it\beta}=0.01$ , ${\it\alpha}=3$ , $e=0.5$ and $Sc=1000$ .

Figure 19.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with small time $t$ for different values of ${\it\tau}_{y}$ when ${\it\beta}=0.01$ , ${\it\alpha}=3.05$ , $e=0.5$ and $Sc=1000$ .

Figure 20.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with small time $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=3$ , $e=0.5$ and $Sc=1000$ .

Variation of $K_{2}$ for small and large values of Womersley frequency parameter ${\it\alpha}$ is shown through the figures 15–17 and 22–25 for small and large time. From these figures, it is seen that as ${\it\alpha}$ increases, the axial dispersion decreases. The amplitude of fluctuations of $K_{2}$ decreases at both small and large time as ${\it\alpha}$ increases. For all values of ${\it\alpha}$ , $K_{2}$ oscillates in increasing manner at small time and reaches a static state in oscillation at large time. In each period of oscillation for ${\it\beta}=0.01$ , the dispersion coefficient becomes stable at large time with a constant value $5\times 10^{-3}$ , $3.42\times 10^{-3}$ , $3.28\times 10^{-3}$ , $2.8\times 10^{-3}$ , $2.1\times 10^{-3}$ , $1.75\times 10^{-3}$ and $1.5\times 10^{-3}$ for ${\it\alpha}=0.1$ , 0.5, 1, 2, 4, 6 and 10, respectively. As ${\it\alpha}$ increases, amplitude of oscillations of $K_{2}$ decreases and $K_{2}$ becomes negative during the period of oscillation for large values of ${\it\alpha}$ . This nature also depends on ${\it\tau}_{y}$ , $e$ and ${\it\beta}$ . From the figures 18 and 19, it is seen that $K_{2}$ attains the negative value within three cycles of oscillation in small time at ${\it\alpha}=3$ for different values of ${\it\tau}_{y}$ and as ${\it\tau}_{y}$ increases, the peak of negative dispersion increases for a value of ${\it\alpha}$ around ${\it\alpha}=3$ . With ${\it\alpha}=3.05$ , this phenomenon is shown in figure 19. Similar observations w.r.t.  ${\it\beta}$ are shown in figures 20 and 21. From these figures, it is observed that for ${\it\beta}=0.01$ , 1 and 100, $K_{2}$ becomes negative at ${\it\alpha}=3$ but for ${\it\beta}=0$ , $K_{2}$ becomes negative at ${\it\alpha}=3.05$ within three cycles of oscillation in small time. So, negative dispersion is seen at ${\it\alpha}=3$ and the peak of negative dispersion increases with ${\it\alpha}$ . Womersley parameter directly depends on the radius of the vessel and its value is high in large vessels. The reversal of blood flow occurs in large vessels undergoing peristalsis (Craciunescu & Clegg Reference Craciunescu and Clegg2001). The backward flow is experienced during late systole and early diastole in the arteries such as carotid, brachial, femoral arteries, renal arteries and abdominal aorta in which the values of ${\it\alpha}$ are large (Ku Reference Ku1997). Due to the flow reversal in period, negative dispersion occurs and the material is moved back with the flow during the cycle of oscillation for large values of ${\it\alpha}$ (Mazumder & Das Reference Mazumder and Das1992). During the period of oscillation at each reversal of flow, the material dispersion is contracted (Smith Reference Smith1982).

Figure 21.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with small time $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=3.05$ , $e=0.5$ and $Sc=1000$ .

Figure 22.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=4$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

From figures 15–17, it can be seen that fluctuation of $K_{2}$ increases cyclically in small time with single frequency at each cycle of oscillation for ${\it\alpha}=0.5$ , 1 and 2. But, from the figures 23 and 24, it is observed that in small time, the amplitude of fluctuations of $K_{2}$ increases periodically with double frequency at first cycle for ${\it\alpha}=6$ and at first two cycles of oscillation for ${\it\alpha}=10$ . The double frequency period was also observed by Mazumder & Das (Reference Mazumder and Das1992) for Newtonian fluid. For ${\it\alpha}=4$ , very small double frequency period is seen in small time at first cycle of oscillation (see figure 22). So, double frequency period can be seen in the oscillation of $K_{2}$ at small time for large values of ${\it\alpha}$ ( ${\geqslant}\,4$ ) and becomes more frequent with the increase in ${\it\alpha}$ . From the figures 15–25, it is noticed that as ${\it\alpha}$ increases, the period of oscillations become shorter and the critical time for $K_{2}$ to reach a stationary state decreases.

Figure 23.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=6$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 24.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=10$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Figure 25.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\beta}$ in small and large time when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=0.1$ , $e=0.5$ and $Sc=1000$ .

From the figures 22 and 26, it is evident that for ${\it\alpha}=4$ , $K_{2}$ becomes both positive and negative during the period of oscillation. It is worth mentioning that Nagarani & Sebastian (Reference Nagarani and Sebastian2013) have not been able to notice this in their study, (see the figure 4 of Nagarani & Sebastian (Reference Nagarani and Sebastian2013)). This could be due to the error associated with the numerical integration of their equation (32) for finding $K_{2}$ . This figure is reproduced in this work for the same set of parameter values and is shown in figure 26. Also, the expression given for $K_{1}$ is erroneous. In fact, the perturbation solution that was presented in Nagarani & Sebastian (Reference Nagarani and Sebastian2013) is valid only for small values of the Womersley frequency parameter ${\it\alpha}<1$ , but they have considered the values as large as 4, which is not acceptable.

Figure 26.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\tau}_{y}$ when ${\it\beta}=0$ , ${\it\alpha}=4$ , $e=0.5$ and $Sc=1000$ . (a) For small times; (b) for large times.

Limiting to smaller values of Womersley frequency parameter ${\it\alpha}<1$ , Sebastian & Nagarani (Reference Sebastian and Nagarani2015) studied the effect of wall absorption on the solute dispersion in pulsatile Casson fluid flow numerically using the Crank–Nicolson method adopted at each time step. The present investigation is a thorough analytical study considering the small and large values of Womersley frequency parameter ${\it\alpha}$ along with the effects of ${\it\tau}_{y}$ , $e$ and ${\it\beta}$ on both the convective coefficient $K_{1}$ and the dispersion coefficient $K_{2}$ and the present results are discussed already earlier in this paper. By considering the same set of parameters, we have re-examined the results that were presented in Sebastian & Nagarani (Reference Sebastian and Nagarani2015). Our results for small values of ${\it\alpha}$ are shown in figures 27–30. Comparison of the two calculations for the dispersion coefficient against the Womersley frequency parameter ${\it\alpha}$ between Sebastian & Nagarani (Reference Sebastian and Nagarani2015) and the present work revealed the following discrepancies.

Figure 27.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\tau}_{y}$ when ${\it\beta}=1$ , ${\it\alpha}=0.1$ , $e=0.1$ and $Sc=1000$ .

Figure 28.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of $e$ when ${\it\tau}_{y}=0.02$ , ${\it\beta}=1$ , ${\it\alpha}=0.1$ and $Sc=1000$ .

Figure 29.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\beta}$ when ${\it\tau}_{y}=0.05$ , ${\it\alpha}=0.1$ , $e=0.2$ and $Sc=1000$ .

Figure 30.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ with $t$ for different values of ${\it\alpha}$ when ${\it\tau}_{y}=0.05$ , ${\it\beta}=1$ , $e=0.2$ and $Sc=1000$ .

It is seen from the figure (a) of Sebastian & Nagarani (Reference Sebastian and Nagarani2015) that $K_{2}$ becomes negative at large time $t=1$ for ${\it\tau}_{y}=0.1$ , but the present investigation shows that dispersion coefficient is non-negative for those chosen values of the parameters (see figure 27). Similarly, as ${\it\beta}$ increases in its range, $K_{2}$ is shown to decrease and become negative (see figure (c) of Sebastian & Nagarani (Reference Sebastian and Nagarani2015)). But it is noticed from the present investigation that as ${\it\beta}$ increases, $K_{2}$ decreases but does not become negative for ${\it\beta}=1$ at large time $t=1$ and this decrement of $K_{2}$ is small for small values of ${\it\beta}$ as shown in figure 29. The reason for decrement of dispersion coefficient has been discussed earlier in this study. Similar discrepancy is noticed for the dispersion coefficient $K_{2}$ when plotted against ${\it\alpha}$ , which are shown in figure (d) of Sebastian & Nagarani (Reference Sebastian and Nagarani2015) and figure 30 of the present investigation. The period of oscillation becomes shorter and $K_{2}$ reaches early a non-transient state as ${\it\alpha}$ increases. It is seen that for ${\it\alpha}=0.2$ , the amplitude of oscillations of $K_{2}$ becomes stable after four cycles of oscillation. But figure (d) of Sebastian & Nagarani (Reference Sebastian and Nagarani2015) indicated that $K_{2}$ decreases sharply at large time for ${\it\alpha}=0.2$ and does not reach a stationary state at large time $(0.5<t<1)$ . Figure 25 also shows that $K_{2}$ is positive for small value of ${\it\alpha}$ (say 0.1). This result is shown in three cycles of oscillation. As time increases, the amplitude of oscillations of $K_{2}$ increases and becomes stable at large time after first cycle of oscillation. All these above mentioned discrepancies are probably associated with the numerical treatment made in Sebastian & Nagarani (Reference Sebastian and Nagarani2015). Also, by considering ${\it\tau}_{y}=0$ (Newtonian fluid case), the variation of $K_{2}$ with small time $t$ for different values of ${\it\beta}$ and $e=1.5$ has been shown in figure 31. These results coincide exactly with those presented in figure 10 of Mazumder & Das (Reference Mazumder and Das1992) for all values of ${\it\beta}$ . To make this comparison, we set ${\it\alpha}$ as ${\it\omega}R^{2}/({\it\mu}/{\it\rho})$ instead of ${\it\alpha}^{2}$ which is used throughout this investigation. This comparison also shows that the present results are consistent for small and large times.

Figure 31.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ for different values of ${\it\beta}$ with small time $t$ when ${\it\tau}_{y}=0$ , $e=1.5$ , ${\it\alpha}=0.5$ and $Sc=1000$ .

Figure 32.

Variation of dispersion coefficient $(K_{2}-1/Pe^{2})(t)$ for different values of ${\it\tau}_{y}$ with wall absorption parameter ${\it\beta}$ at large time $t=1$ when $e=0$ .

Figure 33.

Variation of axial mean concentration $C_{m}(t,z)$ with axial distances $z$ for different values of ${\it\tau}_{y}$ when ${\it\beta}=0.01$ , $t=0.2$ , ${\it\alpha}=0.5$ , $e=0.5$ and $Sc=1000$ .

Yet other comparison of the present results is made with Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973) (see figure 4) and Nagarani et al. (Reference Nagarani, Sarojamma and Jayaraman2004) (see figure 4a), which are in the best agreement, for the variation of axial dispersion coefficient $K_{2}$ with ${\it\beta}$ for different values of ${\it\tau}_{y}$ when $e=0$ and time $t$ is very large and is shown in figure 32. As ${\it\beta}$ increases, $K_{2}$ decreases and it also decreases as ${\it\tau}_{y}$ increases. The physical reason for this behaviour is discussed already earlier in this paper. This comparison also reveals that the present results are reliable.

4.4 Mean concentration $C_{m}(t,z)$

The axial distribution of mean concentration $C_{m}$ depends on the time $t$ , yield stress ${\it\tau}_{y}$ , absorption parameter ${\it\beta}$ , amplitude of the fluctuating pressure component $e$ and frequency parameter ${\it\alpha}$ . In this study, the effects of $t$ , ${\it\tau}_{y}$ , ${\it\beta}$ , $e$ and ${\it\alpha}$ on the axial distribution of mean concentration have been analysed. The equation (3.7) involving exchange coefficient $K_{0}$ , convection coefficient $K_{1}$ and dispersion coefficient $K_{2}$ is solved to obtain the mean concentration $C_{m}$ . From (3.47) and (3.48a–c ), $C_{m}$ is obtained and it directly depends on $K_{0}$ , $K_{1}$ and $K_{2}$ .

The distribution profiles of mean concentration $C_{m}$ along the axial distance $z$ with different ${\it\tau}_{y}$ have been shown in figure 33. The increase in the non-Newtonian nature of the fluid is seen to have increasing value of the mean concentration. It is seen that as ${\it\tau}_{y}$ increases, the peak of mean concentration increases. Due to increase in yield stress of the fluid, flow velocity decreases and hence decreases the solute dispersion. Therefore, the mean concentration of solute increases in the axial direction. Figure 34 shows the distribution of mean concentration along the axial distance  $z$ at different times $t$ . It is noticed that as $t$ increases, the peak of mean concentration decreases. Due to the presence of the boundary absorption at the tube wall, the solute in the system is continuously depleted and more amount of solute from the system is absorbed with the increasing of time $t$ . So, the solute concentration decreases as time increases. At small time after the injection of solute in the system, solute dispersion is controlled by convection and the molecular diffusion is not quite effective. Hence, the distribution profile of mean concentration at small time $(t=0.1)$ is more peaky. As the time increases, the dispersion becomes diffusion controlled at large time $(t=0.3)$ and the concentration profile becomes more blunt and dispersed. From figure 35, it is noticed that as ${\it\beta}$ increases, the peak of mean concentration decreases along the axial distance  $z$ . Due to increase in ${\it\beta}$ , more amount of solute is supplied to conduct the boundary absorption or reaction at the tube wall and it reduces the concentration in the tube. From figure 36, it is observed that the peak of $C_{m}$ decreases as $e$ increases. As $e$ increases, both the flow velocity and the dispersion coefficient increases. Hence it decreases the solute concentration. But the decrement of $C_{m}$ is very small and it can be neglected. Figure 37 shows the axial distribution of mean concentration along the axial distance $z$ with different values of ${\it\alpha}$ . It is seen that the peak of $C_{m}$ increases with ${\it\alpha}$ but this increment is also very very small and it can be neglected.

Figure 34.

Variation of axial mean concentration $C_{m}(t,z)$ with axial distances $z$ for different values of time $t$ when ${\it\tau}_{y}=0.05$ , ${\it\beta}=0.01$ , ${\it\alpha}=0.5$ , $e=0.5$ and $Sc=1000$ .

Figure 35.

Variation of axial mean concentration $C_{m}(t,z)$ with axial distances $z$ for different values of ${\it\beta}$ when $t=0.2$ , ${\it\alpha}=0.5$ , ${\it\tau}_{y}=0.05$ , $e=0.5$ and $Sc=1000$ .

Figure 36.

Variation of axial mean concentration $C_{m}(t,z)$ with axial distances $z$ for different values of $e$ when $t=0.2$ , ${\it\beta}=0.01$ , ${\it\alpha}=0.5$ , ${\it\tau}_{y}=0.05$ and $Sc=1000$ .

Figure 37.

Variation of axial mean concentration $C_{m}(t,z)$ with axial distances $z$ for different values of ${\it\alpha}$ when $t=0.2$ , ${\it\beta}=0.01$ , $e=0.5$ , ${\it\tau}_{y}=0.05$ and $Sc=1000$ .

When $e=0$ and $t$ is very large, the distribution profiles of mean concentration $C_{m}$ at $z=0.5$ with $t$ for various values of ${\it\beta}$ and ${\it\tau}_{y}$ are presented in figure 38. It is observed that as ${\it\beta}$ increases, the peak of $C_{m}$ decreases as also seen in Nagarani et al. (Reference Nagarani, Sarojamma and Jayaraman2004). The mean concentration $C_{m}$ decreases with the time due to constant depletion of solute occurring at the tube wall. These results are found to be in agreement with Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973) for ${\it\tau}_{y}=0$ and Nagarani et al. (Reference Nagarani, Sarojamma and Jayaraman2004).

Figure 38.

Variation of axial mean concentration $C_{m}(t,z)$ with time $t$ for different values of ${\it\tau}_{y}$ and ${\it\beta}$ at large time $t=1$ when $e=0$ and $z=0.5$ .