3.1. The effect of $\beta, e, a, n, \eta, We, Pe, \alpha, \mathcal {P}^2$ on $K_0(t)$, $K_1(t)$ and $K_2(t)$ coefficients

Much discussion on $K_i(t),\ i=0,1,2$, is seen even for various non-Newtonian fluids such as the yield stress Casson, H–B, K–L and non-yield stress C–Y fluids in the literature (Rana & Murthy Reference Rana and Murthy2016a,Reference Rana and Murthyb,Reference Rana and Murthyc, Reference Rana and Murthy2017; Singh & Murthy Reference Singh and Murthy2022a,Reference Singh and Murthyb). What follows is a brief discussion on these coefficients with the parameters due to the pulsatile fluid nature. Because of the absorption at the boundary, the volume of solute in the tube decreases steadily. The coefficient $-K_0(t)$ is dependent on $\beta$. For small $\beta$ in figure 4, there is no significant change in $-K_0(t)$, but for higher values of $\beta$, the magnitude of $-K_0(t)$ decreases with increasing $t$, and eventually it reaches the steady state as seen in figure 4. This was noticed in Sankarasubramanian & Gill (Reference Sankarasubramanian and Gill1973), Singh & Murthy (Reference Singh and Murthy2022a,Reference Singh and Murthyb) and Rana & Murthy (Reference Rana and Murthy2016a,Reference Rana and Murthyb,Reference Rana and Murthyc). Also, $-K_0$(t) is unaffected by the pulsatility and the type of the fluid, whether it is a Newtonian or non-Newtonian fluid.

Figure 4.

Variation of negative exchange coefficient $-K_0(t)$ with time $t$ for different $\beta =0.01,1,10,100$ and $Pe=1000$.

Newtonian and various non-Newtonian fluid models, C–Y, Carreau, simplified Cross fluids, are presented in figure 3(a,b) for small time variations in $-K_1(t)$ and $K_2(t)$. The data is presented by using both Aris’ and Gill's methods. The amplitude of $-K_1(t)$ and $K_2(t)$ is seen to increase in the following order: Newtonian; Carreau; C–Y 2; simplified Cross; C–Y 1. The explanation for such a behaviour is that the average fluid velocity is seen to increase in this order of the fluid model. The solute is convected with the lowest velocity for Newtonian fluid and with the highest velocity for C–Y fluid 1.

The fluctuations in $-K_1(t)$ are not significant for smaller $\beta$, as evident from figure 5(a), where it is shown for when $\beta = 0.01$ for small times. With the increase in values of $\beta$, the amplitude and oscillations of $-K_1(t)$ increased. The reason for this increase in both amplitude and oscillations is due to the rise in the depletion in the boundary due to wall absorption $\beta$; leading to a lesser amount of solute that is available for convection. 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. It is noticed that these oscillations reach a non-transient state with an increase in time which is shown in figure 6(a). At the small time, $K_2(t)$ is also seen to increase in its oscillations but reaches a non-transient state as time increases which are seen from figure 6(b) and thereafter the dispersion of solute takes place with the period of oscillation at a uniform rate. Time variation of $- K_1(t)$ and $K_2(t)$ are shown for $We$, $n$ and $a$ in figures 7–9. With the increment in the value of $a$ and $n$, the amplitude of $-K_1(t)$ and $K_2(t)$ decreased and these oscillations are suppressed in both magnitude and amplitude. This is because when $a,n$ decreases, the fluid behaviour shifts to a shear thinning nature. So, the rise in fluid velocity leads to an increase in $K_2(t)$. The increasing value of $We$ also has a significant and aiding nature of both the convection and diffusion coefficients, the reason being again that it leads to the shear thinning nature of the C–Y fluid – this is seen from figure 9. The variations in $K_1(t)$ and $K_2(t)$ with time $t$ for different values of $\alpha$ are shown for a large value of $e~(=10)$ and is shown in figure 10(a,b); while the variation for different values of $e$ is shown in figure 10(c,d). At large value of $e$, the convective velocity is seen to be changing its sign in every period of oscillation; which is evident from figure 10(c). The dispersion coefficient seen to change its sign in every period of oscillation for even for small values of $\alpha$ with some higher value of $e>1$. As it is evident that for $e>1$ the oscillatory pressure gradient dominates, for a large value of amplitudes these oscillations of $K_2(t)$ grow in time with increasing value of $e$. Also, it is worth mentioning that the double pulse is seen for an increasing value of $e$ for both $K_1(t)$ and $K_2(t)$. But with increasing value of $\alpha$ that amplitude is seen to be decreasing. The variations in $K_1(t)$ and $K_2(t)$ with time $t$ is seen decreasing with increasing value of viscosity ratio $\eta$, and these variations for $-K_1(t)$ and $K_2(t)$ are shown in figure 11(a,b). The magnitude and amplitude of the convection coefficient $-K_1(t)$ and the dispersion coefficient $K_2(t)$ rises with the fluctuating pressure component $e$ and this is presented in figure 12(a,b). As $e$ increases, the flow velocity increases, resulting in an increase in $-K_1(t)$ and $K_2(t)$.

Figure 5.

Variation of (a) negative convection coefficient $-K_1(t)$; (b) dispersion coefficient $K_2(t)$; (c) skewness $K_3(t)$; (d) kurtosis $K_4(t)$ with time $t$ for different $\beta$, when $\alpha = 0.5$, $n = 0.2$, $a=0.9$, $We = 0.025$, $\eta =0.1,\ Pe=1000,\ Sc=1000$ and $e=0.5$.

Figure 6.

Variation of (a) negative convection coefficient $-K_1(t)$; (b) dispersion coefficient $K_2(t)$ with large range of time $t$, when $\alpha = 0.5$, $n = 0.2$, $a=0.9$, $We = 0.025$, $\eta =0.1,\ Pe=1000,\ Sc=1000$ and $e=0.5$.

Figure 7.

Variation of negative convection coefficient $-K_1(t)$ at (a) $\alpha = 1$; (b) $\alpha = 4$; (c) $\alpha = 6$; and dispersion coefficient $K_2(t)$ at (d) $\alpha = 1$; (e) $\alpha = 4$; ( f) $\alpha = 6$; with time $t$ for different $a$, when $We =0.025,\ n =0.2$, $\beta =0.01$, $e=0.5, Pe=1000,\ Sc=1000$ and $\eta =0.1$.

The parameter $\alpha$ is seen to have a substantial impact on $-K_1(t)$ and $K_2(t)$ in the non-Newtonian fluids. With a rise in the value of $\alpha$, the amplitude of $-K_1(t)$ decreases for all $(a,n,We)$ which is evident from figures 7–9. As $\alpha$ increases, the phase lag gets larger. It can also be observed that with a rise in the value of $\alpha$, $-K_1(t)$ oscillates with positive values over each oscillation period with increased phase lag. The amplitude of oscillations of $K_2(t)$ reduces as the value of $\alpha$ rises, and $K_2(t)$ turns positive with phase lag all along each of the oscillation period. It is worth mentioning that the dispersion coefficient $K_2(t)$ attains higher phase lag in three cycles of oscillations when $\alpha$ = 6 for various values of ($a$, $n$, $We$). The amplitude of $K_2(t)$ decreases consistently with the phase lag with time at each cycle, this is clear from the figures 7( f), 8( f) and 9( f).

Figure 8.

Variation of negative convection coefficient $-K_1(t)$ at (a) $\alpha = 1$; (b) $\alpha = 4$; (c) $\alpha = 6$; and dispersion coefficient $K_2(t)$ at (d) $\alpha = 1$; (e) $\alpha = 4$; ( f) $\alpha = 6$; with time $t$ for different $n$, when $We =0.025,\ a = 0.90$, $\beta =0.01$, $e=0.5, Pe=1000,\ Sc=1000$ and $\eta =0.1$.

Figure 9.

Variation of negative convection coefficient $-K_1(t)$ at (a) $\alpha = 1$; (b) $\alpha = 4$; (c) $\alpha = 6$; and dispersion coefficient $K_2(t)$ at (d) $\alpha = 1$; (e) $\alpha = 4$; ( f) $\alpha = 6$; with time $t$ for different $We$, when $n =0.2, a = 0.90$, $\beta =0.01$, $e=0.5, Pe=1000,\ Sc=1000$ and $\eta =0.1$.

Figure 10.

Variation of (a) negative convection coefficient $-K_1(t)$ for varying $\alpha$ at large $e=10$; (b) dispersion coefficient $K_2(t)$ for varying $\alpha$ at large $e=10$; (c) negative convection coefficient $-K_1(t)$ for varying large values of $e$ at $\alpha =0.5$ and (d) dispersion coefficient $K_2(t)$ for varying large values of $e$ at $\alpha =0.5$ with time $t$, when $We=0.2,\ n =0.9,\ a =0.9,\ \eta =0.1,$ $Sc=1000$, $\beta =0.01$ and $Pe=1000$.

Figure 11.

Variation of (a) negative convection coefficient $-K_1(t)$; (b) dispersion coefficient $K_2(t)$; (c) skewness $K_3(t)$; and (d) kurtosis $K_4(t)$ with time $t$ for different $\eta$, when $We =0.025,\ n =0.2,\ a = 0.90$, $\beta =0.01$, $e=0.5,\ Pe=1000,\ Sc=1000$ and $\alpha = 0.5$.

Figure 12.

Variation of (a) negative convection coefficient $-K_1(t)$; (b) dispersion coefficient $K_2(t)$; (c) skewness $K_3(t)$; and (d) kurtosis $K_4(t)$ with time $t$ for different $e$, when $We =0.025,\ n =0.2,\ a = 0.90$, $\beta =0.01$, $\alpha = 0.5,\ Pe=1000,\ Sc=1000$ and $\eta =0.1$.

Figure 13.

Variation of (a) skewness coefficient $K_3(t)$; (b) kurtosis coefficient $K_4(t)$ with all time $t$ for different $\beta$, when $n = 0.2$, $a=0.9$, $We = 0.025$, $\eta =0.1,\ Pe=1000, Sc=1000$, $\alpha = 0.5$ and $e=0$.

3.2. The effect of $\beta, e, a, n, \eta, We, Pe, \mathcal {P}^2, \alpha$ on skewness $K_3(t)$ and kurtosis $K_4(t)$

The time dependent data for the exchange $K_0(t)$, convection $K_1(t)$ and dispersion coefficients $K_2(t)$ all together does not provide the complete information about the unsteady solute distribution in any fluid flow. The degree of symmetry of the axial mean concentration distribution is measured by skewness. Negative kurtosis represents a platykurtic distribution. When kurtosis is negative, the tail of the platykurtic distribution will be thinner than that of a Gaussian distribution (which is known as a sub-Gaussian distribution). The curve peak upstream of the mean concentration distribution profile is lowered by solute consumption in the rear section. The large-time behaviour of skewness and kurtosis coefficients is that both tend to zero, and both of these coefficients will be $0$ if the concentration distributions are perfectly Gaussian. As a result, measuring these parameters is more important in an attempt to achieve Gaussianity of the solute dispersion, making this investigation more significant for both the Newtonian and non-Newtonian flows. A brief discussion on the effect of $\beta$, $e$, $We$, $n$, $a$, $\eta$, $t$, $Pe$ and $\alpha$ on the skewness and kurtosis is presented below.

3.2.1. Effect of wall absorption parameter $\beta$ and fluctuating pressure component $e$

Wall absorption coefficient $\beta$ has a significant effect on the skewness $K_3(t)$ and kurtosis $K_4(t)$ parameters and the results are presented in figure 13(a,b) for the case of static pressure gradient ($e=0$). The large rate of absorption ($\beta =10,100$) is seen to change the sign of $K_3(t)$, as evident from figure 13(a), indicating that the mean concentration distribution upstream-tailed ($K_3(t)>0$) will revert to a downstream-tailed ($K_3(t)<0$) one. The fluid velocity and the concentration distribution are the physical sources of the skewness. The majority of the solute remains in the faster-moving flow region, far from the boundary. As a result, a limited number of moles at the boundary will tend to leave a long tail (i.e. negative skewness). Wall absorption tends to reduce the dispersion that leads to a decrease in the values of $K_3(t)$, which is observed in all these curves. With increasing time $t$, It is observed that $K_3(t)$ is approaching $0$, as expected.

With increase in the wall absorption parameter, kurtosis $K_4(t)$ decreased initially and after a certain time (say $t>125$), it increased and tends to zero limit. At small time of the injection of the solute in the free stream the kurtosis parameter is also seen to change its behaviour. This behaviour is shown in figure 13(b) for large $\beta$ (for example $\beta = 10, 100$). An increase in $\beta$ leads to a reduction in the upstream mean concentration, resulting in a new curve peak in the downstream. As a result, even if the global concentration has dropped, the kurtosis has increased. It is noticed that $K_4(t)$ approached $0$ when $\beta =0.01, 1$ at large time. So, this discussion on non-Gaussian effects cannot be ignored in the early stages of the solute transport (Andersson & Berglin Reference Andersson and Berglin1981; Mazumder & Das Reference Mazumder and Das1992; Wang & Chen Reference Wang and Chen2017; Jiang & Chen Reference Jiang and Chen2018; Guo et al. Reference Guo, Jiang, Zhang, Li and Chen2019; Jiang & Chen Reference Jiang and Chen2021).

For the case of pulsating pressure component $e=0.5$, the effect of wall absorption $\beta$ on both $K_3(t)$ and $K_4(t)$ is presented in figure 5(c,d) for fixed values of other parameters. The pulsation is seen for $K_3(t)$ and $K_4(t)$ as time progresses, and these results are presented for a small time interval $(1, 20)$ with three time periods for the unsteady fluid flow. With increasing values of $\beta$, the skewness coefficient $K_3(t)$ decreases in a pulsating manner. Except the pulsation, qualitative behaviour in the $K_3(t)$ and $K_4(t)$ is similar to the steady flow behaviour as seen from the figure 13(a,b). With increase in the value of $e$, the pulsatile effect is visible for both $K_3(t)$ and $K_4(t)$, the amplitude of fluctuation and magnitude rises with increasing $e$. This is evident from figure 12(c,d).

Figure 14.

Variation of skewness coefficient $K_3(t)$ at (a) $\alpha = 0.5$; (b) $\alpha = 1$; and kurtosis coefficient $K_4(t)$ at (c) $\alpha = 0.5$; (d) $\alpha = 1$; with time $t$ for different $(a,n)$, when $We = 0.025$, $\beta =0.01$, $e=0.5,\ Pe=1000, Sc=1000$ and $\eta =0.1$.

Figure 15.

Variation of skewness coefficient $K_3(t)$ at (a) $\alpha = 0.5$; (b) $\alpha = 1$; and kurtosis coefficient $K_4(t)$ at (c) $\alpha = 0.5$; (d) $\alpha = 1$; with time $t$ for different $We$, when $n = 0.2$, $a=0.9$, $\beta =0.01$, $e=0.5,\ Pe=1000,\ Sc=1000$ and $\eta =0.1$.

Figure 16.

Variation of (a) skewness $K_3(t)$ at $\alpha = 0.5$; and (b) kurtosis $K_4(t)$ with time $t$ for large $(a,n)$, when $\alpha = 0.5$, $We = 0.025$, $\beta =0.01$, $e=0.5,\ Pe=1000, Sc=1000$ and $\eta =5$.

Figure 17.

Variation of skewness coefficient $K_3(t)$ at (a) $\alpha = 4$; (b) $\alpha = 6$; and kurtosis coefficient $K_4(t)$ at (c) $\alpha = 4$; (d) $\alpha = 6$; with time $t$ for different $(a,n)$, when $We = 0.025$, $\beta =0.01$, $e=0.5, Pe=1000,\ Sc=1000$ and $\eta =0.1$.

Figure 18.

Variation of skewness coefficient $K_3(t)$ at (a) $\alpha = 4$; (b) $\alpha = 6$; and kurtosis coefficient $K_4(t)$ at (c) $\alpha = 4$; (d) $\alpha = 6$; with time $t$ for different $We$, when $n = 0.2$, $a=0.9$, $\beta =0.01$, $e=0.5, Pe=1000,\ Sc=1000$ and $\eta =0.1$.

3.2.2. Effect of Weissenberg number $We$, power law exponent $n$, Yasuda parameter $a$, Peclet number $Pe$ and Womersley frequency parameter $\alpha$

The temporal evolution of $K_3(t)$ and $K_4(t)$ for Newtonian fluid, C–Y fluid, simplified Cross fluid and Carreau fluid is given in figure 3(c,d) for absorption parameter $\beta =0.01$. The minimum values for $K_3(t)$ and $K_4(t)$ are seen for C–Y fluid 1 and maximum values for $K_3(t)$ and $K_4(t)$ are noticed for the Newtonian fluid because of the velocity difference between these two fluids is the maximum. The effect of the flow pulsation on the skewness and kurtosis coefficients for varying value of $n$ and $a$ is presented in figure 14(a–d) for fixed values of other parameters. As the values of $a$ and $n$ increases in their physically realistic range, both $K_3(t)$ and $K_4(t)$ increased. Similar behaviour is seen for varying values of $\eta$ from figure 11. Reverse behaviour is seen for varying values of $We$. Velocity increases as $a$ and $n$ decreases, so the skewness for smallest values of both $a$ and $n$ will be maximum. Variation of both $K_3(t)$ and $K_4(t)$ with time at different $We$ is presented in figure 15(a–d) and indicates that for larger values of $We$, the values for both $K_3(t)$ and $K_4(t)$ are smaller.

As reported in the earlier section, $\alpha$ has a significant effect on the dispersion coefficient $K_2(t)$. It is natural to expect that this effect should be reflected on the skewness $K_3(t)$ and kurtosis $K_4(t)$ with varying $\alpha$. Figures 14–18 bring out the time variation of $K_3(t)$ and $K_4(t)$ for different values of $\alpha =0.5,1,4,6$ with varying values of $a,n,We$. The magnitude of $K_3(t)$ and $K_4(t)$ for large $\alpha$ is less than that for small $\alpha =0.5$ (as can be seen from figures 14–18). It is observed from these results that $K_3(t)$ and $K_4(t)$ increased with a single frequency for every oscillation cycle. Similar variation of $K_3(t)$ and $K_4(t)$ is seen for varying $a,n,We$ from these figures 14–18. This is because of phase lag and low amplitude in the flow at the high $\alpha$.

The effect of the Péclet number $Pe$ in the range of $10\unicode{x2013}10^3$ on $-K_1(t)$ and $K_2(t)$ is shown in figure 19(a,b). It is noticed from these two figures that there are no significant changes in $-K_1(t)$ for varying Péclet number $Pe$; this can be understood from the mathematical expression that is given for $-K_1(t)$. The dispersion coefficient $K_2(t)$ continuously decreases with increasing value of $Pe$ as seen in figure 19(b). For all $Pe \geq 500$, no significant changes have been seen.

The Péclet number $Pe$ has a significant impact on $K_3(t)$ and $K_4(t)$. This is seen from figure 19(c,d). The skewness $K_3(t)$ increases continuously with increasing value of $Pe$. The kurtosis $K_4(t)$ is decreasing with increasing values of $Pe$. But for the relatively larger value of $Pe$, the change in $K_3(t)$ and $K_4(t)$ is insignificant.

Figure 19.

Variation of (a) negative convection coefficient $-K_1(t)$; (b) dispersion coefficient $K_2(t)$; (c) skewness coefficient $K_3(t)$; (d) kurtosis coefficient $K_4(t)$ with time $t$ for different Péclet number $Pe$, when $We=0.025,\ n =0.392,\ a =0.644,\ \eta =0.1,$ $\alpha = 0.5,\ Sc=1000$, $\beta =0.01$ and $e=0.5$.

Higher moments provide more precise information regarding the unsteady solute dispersion in the flow field, whether it is Newtonian fluid or non-Newtonian fluid.

3.3. Flow and dispersion regimes

To have a better understanding of the solute dispersion in the tube, the flow and dispersion regimes are derived here for moments considering the limiting behaviour of the Womersley frequency parameter $\alpha$ and the other parameter $\mathcal {B} ^2\ (=\mathcal {P} ^2 {Pe}^2)$. Following Sharp et al. (Reference Sharp, Carare and Martin2019), we present below the flow and concentration regimes characterized by the Womersley frequency parameter $\alpha$. When $\alpha ^2 <\;<1$ the flow is viscous dominant while when $\alpha ^2 >\; >1$ the flow is more unsteady. The pressure gradient which is the combination of static and oscillatory terms drives the flow. The character of the flow depends mostly on unsteady and viscous terms. The velocity profiles that are obtained using perturbation solution ($\alpha <1$) and those are obtained numerically using new class of CERK method (for other values of $\alpha$) are compared with the results presented in Uchida (Reference Uchida1956) for Newtonian fluid and this comparison is shown in figure 20. This comparison shows a very good agreement. The pulsatile nature with phase lag is noticed for Newtonian fluid in an unsteady regime at large values of $\alpha$, as seen in Uchida (Reference Uchida1956).

Figure 20.

Velocity profiles $w(r,t)$ versus time $t$ for the different regimes (viscous $\alpha ^2<<1$ and unsteady $\alpha ^2 >\;>1$ regimes) of flow. Comparison of this study with Uchida (Reference Uchida1956) (triangular marker) results for Newtonian fluid ($n=1$) at $e=0.5$ and varying $\alpha$ at (a) $\alpha \leq 1$; (b) $\alpha \geq 1$.

The radial variation of the concentration distribution at different $\alpha$ and $\mathcal {B}^2\ (=\mathcal {P}^2 {Pe}^2)$ is shown in figure 21 for the three possible regimes: (a) viscous flow ($\alpha ^2 <<1$) and diffusive dispersion ($\mathcal {B} ^2<<1$); (b) viscous flow ($\alpha ^2 <<1$) and unsteady dispersion ($\mathcal {B} ^2 >\;> 1$); and (c) unsteady flow ($\alpha ^2 >\;> 1$) and unsteady dispersion ($\mathcal {B} ^2 >\;> 1$). These results are qualitatively in agreement with similar results presented by Sharp et al. (Reference Sharp, Carare and Martin2019). The purpose of presenting these results for velocity and concentration in different flow regimes is to emphasize that the numerical calculation performed in this investigation for solute dispersion are agreeing qualitatively with similar results reported in Watson (Reference Watson1983) and Sharp et al. (Reference Sharp, Carare and Martin2019); it is not the claim that these are the sharp estimates for convective, dispersive regimes.

Figure 21.

Concentration profiles $C(r,z,t)$ versus radius $r$ for the regimes of dispersion. (a) Diffusive dispersion ($\mathcal {B}^2<<1$) at $\alpha =0.5, 10$ and $Pe=0.01$; (b) unsteady dispersion ($\mathcal {B}^2 >\;>1$) at $\alpha =0.5,4$ and $Pe=1000$. Here $z=0.194,\ \beta = 0.01,\ We=0.025,\ n=0.392$, $a=0.9,\ \eta =0.1, e=0.5,\ Sc=1000$.

Following Chatwin (Reference Chatwin1975), Pedley & Kamm (Reference Pedley and Kamm1988) and Sharp et al. (Reference Sharp, Kamm, Shapiro, Kimmel and Karniadakis1991, Reference Sharp, Carare and Martin2019), the estimates for moments in each regime are presented below. The mean concentration $C_m(z, t)$ can be interpreted as the probability density function of the longitudinal displacement of a molecule of solute. Let $Z(t)$ and $W(t)$ denote the axial displacement and velocity of a molecule of solute at time $t$. Then we have $Z(t) = \int _{0}^{t} W(s)\,\textrm {d} s$ and $\mu _g=\langle Z (t) \rangle =\int _{0}^{t} \langle W(s)\rangle \,\textrm {d} s=\int _{- \infty }^{+\infty } z C_m\,\textrm {d} z$, where $\langle {\cdot } \rangle$ has standard meaning in statistics. From the above we can write

(3.1)\begin{align} \langle\{Z (t)-\langle Z (t) \rangle \}^n \rangle &= \int_{- \infty}^{+\infty} (z-\mu_g)^n C_m \,{\rm d} z = \int_{0}^{1} \int_{-\infty}^{+\infty} 2 r (z-\mu_g)^n C\,{\rm d} z\,{\rm d} r \nonumber\\ &= \left \langle \left \{ \int_{0}^{t} W(s)-\langle W(s) \rangle\,{\rm d} s\right\}^n \right \rangle. \end{align}

Now, we can write the various order moments as

(3.2)\begin{align} M_n(t) &= \frac{\displaystyle\int_{0}^{1} \int_{-\infty}^{+\infty}2 r(z-\mu_g)^n C\,{\rm d} z\,{\rm d} r}{\displaystyle\int_{0}^{1} \int_{-\infty}^{+\infty}2 r C\,{\rm d} z\,{\rm d} r} \nonumber\\ &=\frac{\displaystyle\left \langle \left\{\int_{0}^{t} W(s)-\langle W(s)\rangle \,{\rm d} s\right\}^n \right \rangle}{\displaystyle\left \langle \left \{ \int_{0}^{t} W(s)-\langle W(s)\rangle\,{\rm d} s\right\}^0 \right \rangle} \frac{\displaystyle\int_A {\rm d} A}{A}= (w_{rel} t_c)^n \mathcal{A}. \end{align}

Now, the $n$th central moment about the mean concentration distribution $M_n$ is scaled as

(3.3)\begin{equation} \mathcal{M}_n = \frac{M_n}{(W_{0}T)^n}=\frac{(w_{rel} t_c)^n}{(W_{0} T)^n} \mathcal{A}, \end{equation}

where $w_{rel}$ is the characteristic axial velocity of diffusing molecules relative to the average velocity, $t_c$ is the time during which the velocity of the molecules remains correlated and $\mathcal {A}$ is the fraction of the cross-section over which molecules experience relative motion. The cycle period scales as $T \sim 1 /\omega$. What follows is the estimates of moments in all the three flow and dispersion regimes.

Figure 22.

Variation of (a) dispersion coefficient $K_2(t)$; (b) skewness $K_3(t)$; (c) kurtosis $K_3(t)$; (d) mean concentration $C_m( z,t)$ with $\mathcal {B}^2 = [2.5\times 10^{-6} \mbox { to } 2.5\times 10^{14}]$; when $z=0.1,\ t=12.5, \beta = 0.01,\ e=0, \alpha =0.5,\ We=0.025$, $n =0.392,\ a =0.644, Sc=1000$ and $\eta =0.1$.

1. (a) Viscous flow ($\alpha ^2 <<1$) and diffusive dispersion ($\mathcal {B}^2<<1$): following Pedley & Kamm (Reference Pedley and Kamm1988), Sharp et al. (Reference Sharp, Kamm, Shapiro, Kimmel and Karniadakis1991) and Sharp et al. (Reference Sharp, Carare and Martin2019) the relative velocity $w_{rel}$, and the correlation time $t_c$ scales as $w_{rel} \sim W_0$ and $t_c \sim R^2/D_m$, respectively. The entire cross-section is involved so $\mathcal {A} \sim 1$. The $n$th scaled central moment from (3.3) can be written as

   (3.4)\begin{equation} \mathcal{M}_n =\left(\frac{w_{rel} t_c}{W_{0} T}\right)^n \mathcal{A} \sim \left(\frac{W_0 (R^2/D_m)}{W_{0} (1/\omega)}\right)^n =\left(\frac{R^2 \omega}{\nu} \frac{\nu}{D_m}\right)^n=(\alpha^2 Sc)^n. \end{equation}

2. (b) Viscous flow ($\alpha ^2 <<1$) and unsteady dispersion ($\mathcal {B}^2>\;>1$): the relative velocity is limited to the velocity difference across a characteristic diffusion distance $w_{rel} \sim W_0 \sqrt {D_m T}/ R$, $t_c \sim T$, $\mathcal {A} \sim 1$. The $n$th scaled central moment can be written as

   (3.5)\begin{align} \mathcal{M}_n &= \left(\frac{w_{rel} t_c}{W_{0} T}\right)^n \mathcal{A} \sim \left(\frac{(W_0 \sqrt{D_m T}/ R)T}{W_{0} T}\right)^n = \left(\frac{\sqrt{D_m /\omega}}{R}\right)^n \nonumber\\ &= \left(\frac{\nu}{R^2 \omega}\frac{D_m}{\nu}\right)^{n/2}=(\alpha^2 Sc)^{{-}n /2}. \end{align}

3. (c) Unsteady flow ($\alpha ^2 >\;>1$) and unsteady dispersion ($\mathcal {B}^2>\;>1$) –

   1. (i) At large Schmidt number $Sc$, the molecular diffusion distance is smaller than the viscous diffusion distance. So $w_{rel} \sim W_0 \sqrt {D_m /\nu }$, $t_c \sim T$, and $\mathcal {A} \sim \sqrt {\nu T}/R$. The $n^{th}$ scaled central moment can be written as

      (3.6)\begin{align} \mathcal{M}_n &= \left(\frac{w_{rel}t_c}{W_{0}T}\right)^n \mathcal{A} \sim \left(\frac{(W_0 \sqrt{D_m /\nu})T}{W_{0}T}\right)^n (\sqrt{\nu T}/R) = \left(\sqrt{\frac{D_m}{\nu}}\right)^n \sqrt{\frac{\nu }{\omega}}\frac{1}{R} \nonumber\\ &= Sc^{{-}n/2} \alpha^{{-}1}. \end{align}

   2. (ii) At small Schmidt number $Sc$, the molecular diffusion distance is greater than viscous diffusion distance. Hence $w_{rel} \sim W_0$, $t_c \sim \nu T/D_m$, and $\mathcal {A} \sim \sqrt {\nu T}/R$. The $n$th scaled central moment from (3.3) $\mathcal {M}_n$ can be written as

      (3.7)\begin{align} \mathcal{M}_n &= \left(\frac{w_{rel} t_c}{W_{0}T}\right)^n \mathcal{A} \sim \left(\frac{W_0 \nu T/D_m}{W_{0}T}\right)^n \frac{\sqrt{\nu T}}{R} = \left({\frac{\nu}{D_m}}\right)^n \sqrt{\frac{\nu }{\omega}}\frac{1}{R} \nonumber\\ &= Sc^n\alpha^{{-}1}. \end{align}

These estimates for the Newtonian flow agree with those presented in sharp 2019. The estimates for these moments in each regime help us in having a rigorous understanding of the order estimates on the convection coefficient $K_1(t)$ and dispersion coefficient $K_2(t)$ of the solute. These estimates will be useful in deciding in variations in skewness $K_3(t)$ and kurtosis $K_4(t)$.

Variations of $K_2(t)$, $K_3(t)$, $K_4(t)$ and $C_m(z, t)$ with $\mathcal {B}^2$ are shown in figure 22. Variation of $K_2(t)$ continuously decreases with $\mathcal {B}^2$ and reaches a constant value. For small $\mathcal {B}^2$ variation of $K_3(t)$ are insignificant but when $\mathcal {B}^2$ moves from diffusion to an unsteady regime a steep rise in $K_3(t)$ is noticed. As $\mathcal {B}^2$ approaches infinity, $K_3(t)$ becomes constant with its maximum value. Kurtosis $K_4(t)$ is approximately zero and does not change for $\mathcal {B}^2 <1$ but as it reaches the unsteady dispersion regime kurtosis decreased. At large $\mathcal {B}^2$, $K_4(t)$ become constant. As expected mean concentration $C_m$ is the maximum for the diffusion dispersion regime and it is minimum for the unsteady dispersion regime.

3.4. Axial mean concentration $C_m(z,t)$ and two-dimensional concentration $C(r,z,t)$

In the preceding subsections, evolution of the coefficients $K_0(t)$, $K_1(t)$, $K_2(t)$, $K_3(t)$, $K_4(t)$ and also the effect of various flow governing parameters on these coefficients is presented. The solute dispersion in the non-Newtonian fluid flow is the culmination of all these effects as a whole, and it is being presented in this subsection. To show the significance of the present investigation, the peak of $C_m$ calculated up to the second moment and the same calculated up to the fourth moment have been presented in table 2 with varying $We$, $n$, $a$, $\beta$, $e$ and $\alpha$. As is evident from this, the difference in the peak of $C_m$ with the second moment and $C_m$ with the fourth moment is large, indicating that higher moments play a role in the accuracy of solute distribution. Therefore, the study of the third and fourth moments is important for a better understanding of solute concentration distribution.

Table 2.

Maxima of mean concentration $C_m(z,t) \times Pe$ for different fluids at $t =75,\ \eta =0.1$ and $Pe=1000$.

A typical presentation of concentration profile in the tube for three progressive time levels is made in figure 23(a–c), considering up to the second moments, up to third moments, and considering all the four moments (or in other words, with the dispersion, skewness and kurtosis coefficients). As time progresses, the axial mean concentration decreases drastically, with the inclusion of the $K_3(t)$ shifting the $C_m$ curve to the left, while the inclusion of the kurtosis coefficient reduces the peak of this mean concentration curve in each case. Pure diffusion of solute results in a Gaussian distribution and the curves are more compatible with the Gaussian distribution at large time. In the early stages of the injection of the solute, these distribution characteristics provide a useful estimate of blood flow. For $t=5$, $C_m$ distribution is nearly Gaussian, as seen from the figure 23(c). It is advised to evaluate the approximations up to the fourth moment. The cumulative effect of all five coefficients on concentration for varying values of $\beta =0.01,1,10,100$ is investigated, and the same is presented in figures 24 and 25. The more solute is given to deplete at the boundary, the less there is concentration along the regions of the central line of the tube, and the less solute is accessible for convection/diffusion in the non-Newtonian fluid flow. More significantly, the analysis of skewness and the kurtosis coefficients clearly indicated that the $C_m$ is skewed towards the centre zone. The concentration contours presented in figure 25(a,b) clearly show that the solute convection is more in the central region than at the boundary of the tube and figure 25(b) gives an accurate solution due to the consideration up to fourth moments.

Figure 23.

Axial mean concentration $C_m(z,t)\times Pe$ variation with $z$ including (i) second moment, (ii) third moment and (iii) fourth moment at time (a) $t = 0.01 \times \alpha ^2 Sc$, (b) $t =0.1 \times \alpha ^2 Sc$ and (c) $t =5 \times \alpha ^2 Sc$ when $\beta =0.01$, $n = 0.2$, $a=0.9$, $We = 0.025$, $\alpha = 0.5, Sc=1000,\ Pe=1000$ and $e= 0.5$.

Figure 24.

Axial mean concentration $C_m(z,t)\times Pe$ variation with $z$ including (i) second moment, (ii) third moment and (iii) fourth moment at time (a) $\beta =0.01$, (b) $\beta =1$ and (c) $\beta =100$ when $t= 0.1 \times \alpha ^2 Sc$, $n = 0.2$, $a=0.9$, $We = 0.025$, $\alpha = 0.5,\ Sc=1000,\ Pe=1000$ and $e= 0.5$.

Figure 25.

(a) Concentration contour $C(r,z,t)\times Pe$ up to second moment; (b) concentration contour $C(r,z,t)\times Pe$ up to fourth moment for different values of $\beta =0.01,1,10,100$, when $t= 0.3 \times \alpha ^2 Sc$, $n = 0.2$, $a=0.9$, $We = 0.025$, $\alpha = 0.5, Sc=1000,\ Pe=1000$ and $e= 0.5$.

Figure 26.

Axial mean concentration $C_m(z,t)\times Pe$ variation with $z$ including (i) second moment, (ii) third moment and (iii) fourth moment at time (a) $We=0.025$, (b) $We=0.09$ and (c) $We=0.2$ when $t= 0.1 \times \alpha ^2 Sc$, $n = 0.2$, $a=0.9$, $\alpha = 0.5$, $\beta =0.01,\ Sc=1000,\ Pe=1000$ and $e= 0.5$.

Figure 27.

(a) Concentration contour $C(r,z,t)\times Pe$ up to second moment; (b) concentration contour $C(r,z,t)\times Pe$ up to fourth moment for different values of $We$, when $t= 0.3 \times \alpha ^2 Sc$, $a=0.9$, $n = 0.2$, $\beta =0.01$, $\alpha = 0.5,\ Sc=1000, Pe=1000$, $\eta =0.1$ and $e= 0.5$.

Figure 28.

Axial mean concentration $C_m(z,t)\times Pe$ variation with $z$ including second moment and fourth moment (with circular marker) at varying (a) $a$, and (b) $n$ when $t=0.1 \times \alpha ^2 Sc$, $n = 0.2$, $a=0.9$, $We=0.025$, $\alpha = 0.5$, $\beta =0.01,\ Sc=1000,\ Pe=1000$ and $e= 0.5$.

Figure 29.

(a) Concentration contour $C(r,z,t)\times Pe$ up to second moment; (b) concentration contour $C(r,z,t)\times Pe$ up to fourth moment for different values of $(n,a)$, when $t=0.3 \times \alpha ^2 Sc$, $We = 0.025$, $\beta =0.01$, $\alpha = 0.5,\ Sc=1000, Pe=1000$, $\eta =0.1$ and $e= 0.5$.

Figure 30.

Concentration contour $C(r,z,t)$ for different Péclet number $Pe$, when $t=0.3 \times \alpha ^2 Sc$, $a=0.9$, $n = 0.9$, $\beta =0.01$, $\alpha = 0.5,\ Sc=1000,\ We=0.2$, $\eta =0.1$ and $e= 0.5$.

Figure 31.

Axial mean concentration $C_m(z,t)\times Pe$ variation with $z$ including second moment and fourth moment (with circular marker) at varying (a) $e$ at $\eta =0.1$, and (b) $\eta$ at $e=0.5$ when $t=0.1 \times \alpha ^2 Sc$, $n = 0.2$, $a=0.9$, $We=0.025$, $\alpha = 0.5$, $\beta =0.01,\ Sc=1000,\ Pe=1000$ and $e= 0.5$.

The concentration $C$ and maxima of $C_m$ are seen to decrease with increasing values of Weissenberg number $We$ for solute dispersion in figures 26–27 and maxima of mean concentration in table 2. The maxima of $C_m$ are seen to grow with increasing values of $a$ and $n$ as seen in table 2, and solute dispersion decreases with increasing values of $a$ and $n$ in figures 28 and 29. The fluid velocity reduces with an increase in the $a$ and $n$ at fixed Péclet number $Pe=1000$, and it lowered the solute dispersion. The mean concentration peak falls with increasing time $t$. The concentration contours $C$ presented in figure 30 for increasing values of Péclet number $Pe$ in the range of [0.01–1000] clearly depict the diffusion dominance at low Péclet number $Pe\ (=0.01,0.1)$, the significance of convection and the root to the solute dispersion with increasing value of $Pe$. For $Pe=[0.01\unicode{x2013}1000],\ \alpha =0.5$ and $Sc=1000$, the range of $\mathcal {B}^2\ (=\mathcal {P}^2 Pe^2= \alpha ^2 Sc Pe^2)$ is $[0.025\unicode{x2013}2.5 \times 10^{8}]$. These calculations also ascertain that the diffusive dispersion regime is seen for $\mathcal {B}^2<1$ and unsteady dispersion regime is seen for $\mathcal {B}^2>1$. The velocity and the dispersion coefficients grow as $e$ rises. As a result, the peak of $C_m$ lowers as $e$ increases.

The convective and the dispersion coefficients are changing with $e$, especially when the value of $e>1$ a double pulsation is seen for these coefficients. This will alter the mean concentration significantly. The axial mean concentration shown in figure 31(a) clearly indicates the effect of the skewness and kurtosis. Consideration of the skewness and kurtosis lowered the axial mean concentration for all values of $e$. Axial mean concentration is seen to be increasing with an increasing value of the viscosity ratio parameter $\eta$. The variation of the axial mean concentration with increasing value of $\alpha$ is shown in figure 31(a) for a relatively large value of $e (=10)$. With increasing value of $\alpha$, the axial mean concentration is seen to increase with increasing value of $\alpha$ with a sharp peak. The variation in axial mean concentration for large values of $e$ is shown in figure 32(b). These curves are plotted considering the value of $\alpha =0.5$. The comparative study of $C_m$ for non-Newtonian models (C–Y fluid, Carreau fluid, simplified Cross fluid and Newtonian fluid) are presented in table 2. Among all these fluid models considered, it is noted that $C_m$ has the highest peak for Newtonian fluid and the lowest peak for $C_m$ is observed for the C–Y fluid model. In addition, the $C_m$ profile is scattered axially more for the C–Y fluid and Carreau fluid model. The fluid velocity and dispersion are relatively more in the C–Y fluid and Carreau fluid models, resulting in a reduction in the $C_m$ distribution. The difference in peak of $C_m$ with and without the third and fourth moments is large, indicating that higher moments, such as the skewness and kurtosis, are important for accurately estimating the solute distribution in the flow.

Figure 32.

Variation of axial mean concentration $C_m(z,t)\times Pe$ variation with $z$ (a) for varying $\alpha$ at large $e=10$; (b) for varying large values of $e$ at $\alpha =0.5$, when $We=0.2,\ n =0.9,\ a =0.9,\ \eta =0.1,$ $Sc=1000$, $\beta =0.01$ and $Pe=1000$.