Approximation of fixed point of generalized non-expansive mapping via new faster iterative scheme in metric domain

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1.   Introduction

Demand of getting insight of the peristaltic flow is increasing day by day due to its applications in many fields of science and engineering especially in the fields of biology and medical engineering. Spectrum of research on the topic of peristaltic flow got widen by involving non-Newtonian fluids since most of the fluid involved in different biological systems and other field of engineering are non-Newtonian in nature. Peristalsis is a phenomenon in which the sinusoidal waves arise due to the continuous relaxation and diminution of smooth muscular tissues. Some of the natural and engineering systems involving the application of peristalsis motion are locomotion of earthworm, artificial heart pump, transportation of urine starting from the kidneys to the urinary bladder, vasomotion of tiny blood arteries, motion of the food along the digestive tube and blood transport vessels. The Rotho-peristaltic pump is also one of the applications of peristalsis in chemical industry.

Before Thurston ^[1], blood was treated as a Newtonian fluid. However, viscoelasticity is a basic rheological property of blood which makes the human blood non-Newtonian. Recently interest in problems of non-Newtonian fluid has grown and many mathematical models for describing the rheological behavior of blood have been extensively developed. Many mathematical structures for elaborating the rheological characteristics of blood have been briefly investigated in references ^[2,3,4,5]. Latham ^[6] and Shapiro et al. ^[7] primary reported the fluid dynamical simulation of peristalsis for 2D Newtonian fluid. Tripathi ^[8] addressed the peristaltic transport of high viscosity across a finite length of channel under the influence of porosity and heat transmission. Hayat et al. ^[9] explored the significance of hall current on peristaltic flow of Maxwell's fluid passing via porous medium. The latest research on the peristaltic subject can also be seen through references ^{[10,11,12,13,14]}.The Reiner-Philippoff fluid model is one of the most challenging non-Newtonian fluid models due to implicit relationship of shear stress and strain. Due to this mathematical difficulty, limited literature is available on Reiner-Philippoff fluid model. In 1994, Na ^[15] was the first to examine the boundary layer flow of Reiner-Philippoff fluid due to free stream velocity. The boundary layer flow of Reiner-Philippoff fluid over a stretching sheet was studied by Yam et al. ^[16] Ahmad ^[17] investigated the boundary layer flow of Reiner-Philippoff based nanofluid past a non-linear stretching sheet. Impact of Brownian and thermophoresis diffusions on thin film flow of Reiner-Philippoff fluid was studied by Asad Ullah ^[18]. Sajid et al. ^[19] investigates the impact of triple diffusive convective flowing on Reiner-Philippoff (RPh) fluid past a stretching surface with Avramenko-Blinov-Shevchuk boundary restrictions. Also Reiner-Philippoff (RPh) fluid model is discussed by considering blood as a base fluid by Sajid et al. ^[20]. Tahir and Ahmad ^[21] initiated the study of peristaltic transportation of Reiner-Philippoff fluid.

Heat transfer plays a significant role in equipment efficiency and material selection. The importance of peristaltic fluid motion with heat transfer is noticeable in biomedical sciences, metabolic heat generation, blood transport, perfusion hypothermia, thermoregulation and in bio-heat exchange processes. The working capability of every biological and mechanical process decays time to time due to the production of heat by such processes. To improve the efficiency of such mechanism it is compulsory to eliminate the heat regularly from the system. Nanofluids are innovative compounds that can control the heat transfer. Nanofluids are made by nanoprecipitation method in ordinary fluids. The ordinary fluids like water, polymer solutions, engine oil and some lubricants have less thermal conductivity values as compared to solids. Including a specific quantity of solid particles to these fluids, can easily enhance the heat conductance of these fluids. These particles are usually of sizes ranging 1-100 nanometer and known as nanoparticles. The consequential mixture of fluid and nanoparticles is called nanofluid. This idea of constructing nanofluids was primarily introduced by Choi ^[22,23] in 1995. Xuan ^[24] talked about the improvements of heat transfers for nanofluids. Buongiorno ^[25] proposed a model that incorporates thermophoresis and Brownian motion phenomena to evaluate the transport of nanofluids. Nanofluids are also helpful and applicable in the understanding of a variety of phenomenon such as the minimization/enhancement of the thermal magnitude of the system, microchannel clogging, miniaturization of the system and minimal clogging. Recently Rasool et al. ^[26] investigates and provides a numerical analysis of electro-magneto-hydrodynamic (EMHD) nanofluid flows past a Riga pattern embedded horizontally in a Darcy-Forchheimer porous medium. The more literature regarding heat transfer through nanofluids can be seen through references ^{[27,28,29,30,31,32,33]}. Many academics are interested in the combined impacts of heat and mass transfer because of its vast spectrum of uses in industry, science, and engineering. These procedures play a critical role in enhancing the product quality. They're used in a variety of pharmaceutical and chemical operations, as well as magnetic drug targeting and electronic devices cooling. Thermal insulation, thermal energy storage, fossil fuel extraction, the paper industry, food processing, oxygenation, porous solid drying, hemodialysis, subterranean energy transmission and geothermal energy recovery are some additional physical examples. Hina ^[34] in 2012 explored how the compliant walls affected peristaltic flow, heat transfer, mass transport and chemical reactions. Peristaltic movement study of Eyring-Prandtl fluid in a tube together with heat and mass transfer analysis was investigated by Iftikhar et al. ^[35]. Researcher's contributions to peristaltic flows are fast expanding due to the relevance of heat and mass transport in several fields of engineering and science (see citations ^{[36,37,38,39]}).

As discussed earlier that negligible research was available on non-Newtonian Reiner-Philippoff fluid model. And particularly no literature was available on peristaltic motion of Reiner-Philippoff fluid. In previous work ^[40] heat transfer analysis of Renner-Philippoff nanofluid was discussed by using conventional nanofluid model. Considering the importance of non-Newtonian fluid in peristaltic flow and role of nanoparticles as an efficient agent of heat transfer control, the aim of this article is to explore heat and mass transfer analysis of non-Newtonian Reiner-Philippoff fluid in peristaltic motion in the presence of nanoparticles by using Buongiorno's nanofluid model. The impact of thermophoretic diffusion parameter, Brownian diffusion parameter and Reiner-Philippoff fluid parameter on different quantities of interest is discussed in detail. I hope this study will be step forward in the field of peristaltic flow, heat and mass transfer of non-Newtonian fluids and also provides new avenues of research for scholars to the way of RPh fluid academics.

2.   Mathematical analysis

Consider Reiner-Philippoff (RPh) fluid flowing through a non-uniform (tapered) two-dimensional asymmetric channel to be incompressible. The fluid movement is caused by the sinusoidal waves having wavelength $\lambda$ and constant speed c along the channel walls. The walls of the non-uniform channel are separated by width ${d}_{1}\left(\overset\smallfrown{X}\right)+{d}_{2}\left(\overset\smallfrown{X}\right)$. The coordinate system for the given problem is Cartesian system $(\overset\smallfrown{X}, \overset\smallfrown{Y})$ introduced in such a way that $\overset\smallfrown{Y}-axis$ is chosen alongside the width and the $\overset\smallfrown{X}-axis$ is along the length of the channel. Geometrical deformation of the walls for discussed problem can be put mathematically as ^[41]:

where, ${d}_{i}\left(\overset\smallfrown{X}, \overset\smallfrown{t}\right) = {d}_{i}+\left(\overset\smallfrown{X}-c\overset\smallfrown{t}\right)\text{tan}\left(\alpha \right), \left(\text{i = 1, 2}\right),$ is the half width of channel, $\alpha$ is the non-uniformity angle of the channel, ${\overset\smallfrown{H}}_{1}(\overset\smallfrown{X}, \overset\smallfrown{t})$ and ${\overset\smallfrown{H}}_{2}(\overset\smallfrown{X}, \overset\smallfrown{t})$ symbolized to show the geometry of the said problem for topper and lower walls respectively corresponding to the coordinate $\overset\smallfrown{Y} > 0$ and $\overset\smallfrown{Y} < 0.$ ${a}_{1}$ and ${b}_{1}$ are the amplitudes of the topper and lower waves respectively, $\overset\smallfrown{t}$ is for the time and $\zeta (0\le \zeta \le \pi)$ represents phase difference between the waves, for $\zeta = 0$ the channel is reduced to symmetric in which waves are totally out of phase and for $\zeta = \pi$ waves are totally in phase (see Figure 1). Superscript $\overset\smallfrown{.}$ is used to indicate the dimensional quantities. Further, ${d}_{1}, {a}_{1}, {d}_{2}$ and ${b}_{1}$ satisfies the following ^[42]:

Two-dimensional velocity fields are taken with the velocity components as $\overset\smallfrown{U}(\overset\smallfrown{X}, \overset\smallfrown{Y}, \overset\smallfrown{t})$ and $\overset\smallfrown{V}(\overset\smallfrown{X}, \overset\smallfrown{Y}, \overset\smallfrown{t})$ along the $\overset\smallfrown{X}-axis$ and $\overset\smallfrown{Y}-axis$ respectively.

The governing equations for an incompressible RPh fluid flow can be expressed as the following set of equations ^[43]:

where, ${\rho }_{f}$ signifies the base fluid density and ${\rho }_{p}$ is the density of particles, $u = u\left(\overset\smallfrown{U}(\overset\smallfrown{X}, \overset\smallfrown{Y}, \overset\smallfrown{t}), \overset\smallfrown{V}(\overset\smallfrown{X}, \overset\smallfrown{Y}, \overset\smallfrown{t}), 0\right)$ indicate velocity vectors, $\overset\smallfrown{t}$ is for time, ${\stackrel{̄}{C}}_{f}$ is specific heat of base fluid and ${\stackrel{̄}{C}}_{p}$ is particles specific heat, ${K}_{f}$ indicates the thermal conductivity, $T$ is temperature, $C$ is concentration, ${D}_{B}$ and ${D}_{T}$ are stands for Brownian and thermophoretic diffusion coefficients respectively. ${S}_{ij}$ and $\Phi$ are the Cauchy stress tensor and viscous dissipation terms and are given as ^[44]:

Where for the Reiner-Philippoff (RPh) fluid model, the stress strain relation is specified as ^[18]:

with,

Using ${\overset\smallfrown{\tau }}_{12} = {\overset\smallfrown{\tau }}_{21}$.

The RPh fluid model (Eq (10)) contain three positive parameters ${\mu }_{\infty }, {\mu }_{0}$ and ${\overset\smallfrown{\tau }}_{0}$. Here ${\mu }_{\infty }$ represents the higher Newtonian limiting viscosity, ${\mu }_{0}$ is the viscosity at zero shear stress rate and ${\overset\smallfrown{\tau }}_{0}$ indicates the reference shear stress.

In two dimensional Cartesian coordinates system, the continuity Eq (4), momentum Eq (5), the thermal energy Eq (6) and the concentration Eq (7) in laboratory frame are expressed as:

where, $\Gamma = \frac{{\rho }_{p}{\stackrel{̄}{C}}_{p}}{{\rho }_{f}{\stackrel{̄}{C}}_{f}}$ and Eq (10) in component form can be written as:

where,

The subsequent transformations are used to change the investigated problem from stationary reference frame to moving frame ^[45]:

And the following dimensionless quantities are used to convert the above dimensional system in dimensionless form:

Here ${h}_{1}$ and ${h}_{2}$ represents the dimension free forms of both walls, $\nu$ is kinematics viscosity, $d$ is the channel width ratio, $\phi$ indicates the concentration of nanoparticles, ${N}_{b}$ is the parameter of Brownian motion, ${N}_{t}$ is indicate the thermophoresis diffusion parameter, $\theta$ correspond to dimension free temperature, $\text{Pr, Re, Ec}$ and $Br$ are the Prandtl number, Reynolds number, Eckert number and Brinkman number respectively.

The dimensionless form of the governing equations in the wave frame in accounts of Eq (20) and the quantities (21) is as:

and corresponding dimensionless form of Eqs (16)-(18) are as under:

In above equations ${\lambda }_{1} = \frac{{\mu }_{0}}{{\mu }_{\infty }}.$ Now by using the estimations of lengthy wavelength $(\delta < < 1),$ small Reynolds number and by using the description of following stream functions:

Equation (22) satisfies identically and Eqs (23)-(29) simplifies to the following form:

and

From Eqs (31) and (32) we have:

Which depicts that ${\tau }_{xy}$ is a linear function of $y$ and can be written as ${\tau }_{xy} = {A}_{1}y+{B}_{1},$ where ${A}_{1}$ and ${B}_{1}$ are arbitrary parameters. And Eq (36) can be written as:

This is explicit relationship between deformation rate and shear stress for RPh nanofluid. Where, ${\lambda }_{1} = \frac{{\mu }_{0}}{{\mu }_{\infty }}$ and $\gamma = {\tau }_{0}^{2}$ are RPh fluid parameter and shear stress parameter respectively. For ${\lambda }_{1} = 1$ this relation converges to Newtonian fluid.

The dimension freeform of the walls ${h}_{1}\left(x\right)$ and ${h}_{2}\left(x\right)$ are is given as:

where, ${k}_{1} = \frac{\lambda \mathit{tan}\alpha }{{d}_{1}}$ is the non-uniformity parameter of the channel.

2.1.   Pumping characteristics

The instantaneous volumetric flow rate in the laboratory frame of reference is given by:

where, ${\overset\smallfrown{H}}_{1}$ and ${\overset\smallfrown{H}}_{2}$ are the functions of $\overset\smallfrown{X}$ and $\overset\smallfrown{t}.$ Now using Eq (19), Eq (40) becomes:

where, ${\overset\smallfrown{h}}_{1}\left(\overset\smallfrown{x}\right)$ and ${\overset\smallfrown{h}}_{2}\left(\overset\smallfrown{x}\right)$ represents the upper, lower channel walls for wave frame and $\overset\smallfrown{q}(\overset\smallfrown{x}, \overset\smallfrown{y})$ is the instantaneous volumetric flow rate in the moving frame and is given by:

The time average flow rate (over one period T) is given by:

Using Eq (41) in Eq (43) we have:

We note that ${h}_{1}\left(x\right)$ and ${h}_{2}\left(x\right)$ given in Eq (39) are the corresponding dimension free forms of the non-uniform peristaltic channel. So the corresponding dimension free form of Eq (44) is given as:

where, $\eta = \frac{\stackrel{̄}{Q}}{c{d}_{1}}$ and $F = \frac{\overset\smallfrown{q}}{c{d}_{1}}$ are the dimension free mean flow rates in the fixed and moving frame respectively and the expression of $F$ is given as:

The associated dimensionless boundary conditions are:

3.   Result and discussions

Equation (31) depicts that the expression of the pressure function is free from 'y' variable and Eq (36) demonstrates that ${\tau }_{xy}$ is a linear function of 'y'. Thus the solution of Eqs (32), (33) and (37) corresponding to the boundary conditions given in Eq (46) is retrieved numerically by the help of built in NDSolve function of MATHEMATICA software. Outcomes of the study are discussed thoroughly by plotting the graphs in rest of this section. We will talk about the effects of RPh nanofluid properties on physical quantities of interest like the axial velocity $u,$ streamline functions $\psi,$ concentration of nanoparticles $\phi,$ temperature profiles $\theta,$ heat transfer rate $-{\theta }^{\text{'}}\left(h\right)$ and mass transportation rate $-{\phi }^{\text{'}}\left(h\right)$ at the upper boundary and pressure gradient $\frac{dp}{dx}$.

3.1.   Axial velocity

Graphical visualization of velocity profiles against dissimilar values of RPh fluid parameter ${\lambda }_{1},$ shear stress parameter $\gamma,$ for both pseudoplastic and dilatants fluids and arbitrary parameter ${A}_{1}$ is shown in Figures 2-5. Reiner-Philippoff (RPh) fluid behaves like dilatants fluid for ${\lambda }_{1} < 1$, as pseudoplastic for ${\lambda }_{1}$ > 1 and as Newtonian fluid for ${\lambda }_{1} = 1$. The influence of RPh fluid parameter ${\lambda }_{1}$ on the fluid's velocity profiles is drawing in Figure 2 the blue line in the figure against the value ${\lambda }_{1} = 1$ is showing the velocity of Newtonian fluid. Further it is shown that the maximum velocity for pseudoplastic (shear thinning) fluid, corresponding to the value ${\lambda }_{1} > 1,$ is larger than the velocity of dilatants fluid corresponding to the value of ${\lambda }_{1} < 1$. As RPh fluid proceeds as shear thinning (pseudoplastic) fluid for the value ${\lambda }_{1} > 1,$ therefore its viscosity decreases as compared to shear thickening (dilatants) fluid and that's why the fluid's deformation rate increases and consequently velocity profile improves. RPh fluid model (see Eq (35)) behaves like Newtonian fluid for extreme shear rates. The same case is analyzed in Figures 3 and 4. From these two figures we can see that for $\gamma \to 0$ and for $\gamma \to \infty.$ in both cases the fluid's velocity is same for both pseudoplastic $({\lambda }_{1} = 2.0)$ and dilatants $({\lambda }_{1} = 0.25)$ fluid. The other survey from these two graphs is that velocity is increasing function of shear stress parameter $\gamma$ for pseudoplastic fluid and decreasing function for dilatants fluid. As in shear thinning case, fluid decreases its viscosity against applied shear stress and in case of shear thickening fluid, fluid gains viscosity that's why in case of pseudoplastic fluid velocity increases and in case of dilatants fluid velocity decreases against shear stress parameter. Figure 5 depicts that by escalating the value of arbitrary parameter ${A}_{1}$ the fluid's velocity increase. Initially this increment is notable but after that there is no significant effect of arbitrary parameter ${A}_{1}$ on velocity graph.

3.2.   Temperature

This section covers physical assessment of parameters like RPh fluid parameter ${\lambda }_{1}$ shear stress parameter $\gamma,$ Brownian motion parameter ${N}_{b},$ thermophoresis diffusion parameter ${N}_{t},$ Brinkman number $Br$ and Prandtl number $\text{Pr}$ on temperature profiles. As a result, Figures 6-12 are required to look for such an analysis briefly. Figure 6 demonstrates that by ascending the value of RPh fluid parameter ${\lambda }_{1}$ the fluid's temperature increases. As at ${\lambda }_{1} = 1$ the fluid behaves likes the Newtonian fluid. And so the study shows that in same circumstances the temperature for dilatants fluid is less than the Newtonian's fluid temperature and Newtonian's fluid temperature is smaller than the pseudoplastic fluid's temperature. In Figures 7 and 8 the influences of shear stress parameter $\gamma$ are deliberated on temperature profiles for pseudoplastic and dilatants fluids respectively. These graphs reveal that the temperature is an increasing function of shear stress parameter $\gamma$ for pseudoplastic fluids and it behaves totally opposite for dilatants fluids for same values of shear stress parameter $\gamma$ as decreasing function. The relevance of Brinkman's number on fluid hotness can be seen through Figure 9. This figure proves that fluid becomes hotter by enlarging the value of Brinkman number. Because the Brinkman number occurs as a result of viscous dissipation therefore, extra heat is created with greater Brinkman number, and therefore the temperature rises and hence temperature profile get enlarged. Also see reference ^[45]. Figure 10 represents that for greater Brownian motion parameter the temperature of the liquid increases. The consequence of Brownian scale tends to increase the relative mobility of nanoparticles compounds from the walls to the substance in response to which an increment in the temperature profile occurs. Figure 11 confirms a similar pattern of temperature distributions in relation to the thermophoresis parameter ${N}_{t}$. Physically greater ${N}_{t}$ implies the occurrence of temperature differential phenomenon in which high-temperature nanoparticles transported to cooler region of the channel. As a result temperature is intensified. Figure 12 signifies that by increasing the amount of Prandtl number $\text{Pr}$ the temperature of the nanofluid also increases. Because Prandtl number $\text{Pr}$ has direct connection with specific heat of the liquid so increased specific heat raise the temperature of fluid.

3.3.   Concentration

The influence of sundry factors on the concentration of nanoparticles graphs are scrutinized through Figures 13 to 19. These graphs pointed out that the concentration of nanoparticles at the center of the channel is highly effected by the changing the values of implanted parameters. Figure 13 is plotted to show the effect of RPh fluid parameter ${\lambda }_{1}$ on the concentration of nanoparticles. A decreasing tendency in nanoparticles mass transport rate is observed for enhancing values of ${\lambda }_{1}$. From this we can conclude that concentration of nanoparticles for dilatants fluids is larger as compared to pseudoplastic fluids. Shear stress parameter $\gamma$ have opposite concern on concentration for pseudoplastic and dilatants fluid which can be seen through Figures 14 and 15. For pseudoplastic fluids the concentration of nanoparticles is decreasing graph of shear stress parameter while for dilatants fluids it depicts increasing behavior. Figure 16 illustrates that with an increment in Brinkman number $Br$ results with a decrease in nanoparticles mass transfer. Figures 17 and 18 reveal the antagonistic effects of Brownian motion parameter ${N}_{b}$ and thermophoretic diffusions parameter ${N}_{t}$ on concentration graphs. Declining impact is marked by $\phi$ for increasing thermophoretic diffusion parameter because with the strength of thermophoretic forces the action of diffusion is turned on which depressed the concentration profile of nanofluid. Contrary to this a rising effect in concentration is achieved by enhancing Brownian motion as nanoparticles attained higher densities for stronger Brownian motion which in turn develops concentration. Figure 19 depicts the Prandtl number $\text{Pr}$ effects on concentration profile. From figure we can see that by growing values of Prandtl number $\text{Pr}$ yields a depressed behavior of concentration profiles near the upper wall but this observation is not hold near lower wall where the significance of $\text{Pr}$ is directly proportional to concentration profile.

3.4.   Pressure gradient

In Figures 20-23 we will discussed about the effects of and on the pressure gradient profiles within channel. The common observation from these graphs is that the pressure gradient gains its maximum value at the narrowest section of the channel. Figure 20 demonstrates that by intensifying the value of the pressure gradient of RPh nanofluid increases. The rising behavior of pressure gradient field is larger in the contracted as compared to the broader section of the channel. In this figure the blue thick line is the representation of Newtonian fluid's pressure gradient. The lines upper the blue lines against the values correspond to the pseudoplastic fluid and the lines below the blue line against the values characterized the dilatants fluid. In Figure 21 the significance of on the pressure gradient is evaluated for shear thinning fluid. It is investigated from this sketch that pressure gradient of RPh fluid raises by increasing the value of. In Figure 22 the same significance of on pressure gradient is studied for the dilatants fluid. The influence of on pressure gradient for dilatants fluid is quite dissimilar as compared to the pseudoplastic fluid. In the case of dilatants, by increasing the value of the amount of pressure gradient reduces. Figure 23 examined that for, the height of every crest of the pressure gradient is same at narrow section of the channel. However, as we increase the value of parameter, that is, when we increase the width of channel, the height of each succeeding crest of the pressure gradient decreases than the preceding crest. This is because of by adding the value of the width of the channel increases and so the pressure gradient decreases in the wider part. This whole phenomenon can be seen in Figure 23.

3.5.   Streamlines

This part consists of the physical study of parameters like RPh fluid parameter ${\lambda }_{1},$ phase difference of the waves $\zeta,$ shear stress parameter $\gamma$ and non-uniformity parameter ${k}_{1}$ on stream line contours. From Figure 24, it is experimented that by improving the value of ${\lambda }_{1}$ the size of the bolus increases. Bolus is the result of fluid's trapped volume by streamlines with the motion of the fluid, this generates an intriguing phenomenon known as trapping, the construction of an inner flowing bolus of nanofluid which moved together with the waves with wave speed at a fixed mean flow rate. Figure 25 indicates that phase difference of the channel has opposite effect on fluid trapped volume than the ${\lambda }_{1}$ as by increasing the angle of phase difference, the volume of the bolus decreases. Figures 26 and 27 are drawn to understand the impact shear stress parameter $\gamma$ on streamline functions for shear thinning fluids and shear thickening fluids respectively. From Figure 26 we can see that bolus for pseudoplastic fluids have increasing behavior with greater shear stress parameter. As this fluid is shear thinning fluid so by increasing shear stress parameter the fluid's viscosity decreases and fluid moves more freely, and more fluid's volume get trapped. Contrary to this fluid the size of bolus decreases in case of dilatants fluids for same values of shear stress parameter as we can observe from Figure 27. Figure 28 signifies that the amount of trapping volume of fluid get increases by increasing the non-uniformity of the channel and consequently the dimension of the bolus get increased.

3.6.   Heat and mass transfer rates

Table 1 presents the numerical assessment of heat and mass transfer rates at the upper border of the channel for various physical parameters. First four lines of the table exhibits that heat transfer $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ and mass transfer $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ rates are increasing function of RPh fluid parameter ${\lambda }_{1},$ which concludes that RPh fluid parameter improves heat and mass transfer process between the solid boundaries and the base fluid. Secondly, the table 1 predicts that in case of shear thinning fluids, heat $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ and mass transfer $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ rates raise for greater shear stress parameter and on the other hand for shear thickening fluid these transfer rates decline. Next four lines represent that heat transfer $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ and mass transfer $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ rates are increases by enlarging ${N}_{t}.$ Moreover the ratio of mass transfer $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ is much significant as compared to heat transfer $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ rate. The table's following four lines anticipated that the Brownian motion parameter has opposing impacts on heat $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ and mass transfer $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ rates. As with increased Brownian motion parameter, the mass transfer rate $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ drops drastically while the heat transfer rate $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ increases steadily. The very next four lines of the table demonstrate that the Brinkman's number has a direct influence on heat $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ and mass transfer $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ rates. As by enlarging the Brinkman number, both the heat $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ and mass transport $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ rates improved. The last four lines of the table reveals that raising the Prandtl number improved the heat $(-{\theta }^{\text{'}}({h}_{1}\left)\right)$ and mass transfer $(-{\phi }^{\text{'}}({h}_{1}\left)\right)$ rates.

4.   Conclusions

Peristaltic flow of non-Newtonian based nanofluid in a tapered asymmetric channel is examined in this research. The governing equations incorporates dispersion model for the nanofluid and RPh fluid model for non-Newtonian fluid. Simplified form of governing equation based on long wavelength and small Reynolds number approximations are solved numerically. The consequential effects of different physical quantities on the velocity of the nanofluid, pressure gradient, temperature and concentration profiles are portrayed graphically. The most important outcomes of the study are given below:

• Velocity is increasing function of shear stress parameter for pseudoplastic fluid and decreasing function of shear stress parameter for dilatants fluid.

• It is noted that by increasing the value of RPh fluid parameter ${\lambda }_{1}$ the temperature of the nanofluid increases.

• It is worth noting that both thermophoretic and Brownian diffusions causes temperature graphs to climb.

• A decreasing tendency in nanoparticles mass transfer is observed for enhancing values of RPh fluid parameter. From this we can conclude that concentration of nanoparticles for dilatants fluid is greater as compared to pseudoplastic fluid.

• Thermophoretic diffusion and Brownian motion factors have opposing influence on concentration of nanoparticles profiles. Concentration graph decreases by increasing thermophoretic diffusion parameter and increases by enhancing Brownian motion parameter.

• Further it is noticed that the addition of Brinkman number results in an enhancement in the nanofluid temperature.

• The pressure gradient of the nanofluid is higher in the occluded section of the channel as compared to the broader section of the channel.

• By increasing the non-uniformity of the channel, the pressure gradient of the nanofluid decreases as the width of the channel increases.

• The addition in the value of RPh fluid parameter ${\lambda }_{1}$ the trapped volume of the nanofluid increases.

• It is also important to observe that the RPh fluid parameter enables heat and mass transfer rates to increase.

• Brownian motion parameter has opposing impacts on heat and mass transfer rates. As with increased Brownian motion parameter, the mass transfer rate drops drastically while the heat transfer rate increases steadily.

• It is analyzed from the study represented that heat transfer and mass transfer rates are increased by enlarging the thermophoretic diffusion parameter.

I hope this study will be step forward in the field of peristaltic flow, heat and mass transfer of non-Newtonian fluids and also provides new avenues of research for scholars to the way of RPh fluid academics.

Acknowledgments

This research work was funded by institutional fund projects under no. (IFP-A-2022-2-5-24). Therefore, authors gratefully acknowledge technical and financial support from the ministry of education and University of Hafr Al Batin, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.