Soret and Dufour features in peristaltic motion of chemically reactive fluid in a tapered asymmetric channel in the presence of Hall current

The present work examines heat and mass transfer characteristics of peristaltic motion of Johnson-Segalman fluid in a tapered asymmetric channel along with chemical reaction, by taking into account the Soret and Dufour effects. Effects of Hall current have also been discussed in mathematical modeling and analysis. Following the peristaltic wave procedure, the tapered asymmetric channel is based on the non uniform boundaries having diverse phases and amplitudes. The channel walls show excellent agreement with more realistic convective conditions. The modeled flow problem is directed into ordinary differential equations set with proper utilization of similarity quantities. The estimation of high wavelength as well as small Reynolds number are acknowledged to deduce the equations of Johnson-Segalman liquid model. The adopted solution procedure is constructed via homotopic algorithm. The results have been analyzed for various parameters of interest and sketched for better understanding. The velocity profile reveals decreasing behavior for increasing values of Weissenberg number and Hartman number while converse behavior is found for mean flow rate and Hall parameter. The temperature profile falloffs for heat transfer Biot number and Hartman number whereas it increases for Prandtl number, Brinkman number, Dufour number and Hall parameter. The concentration profile tends to decrease for mass transfer Biot number and increase for Schmidt constant.


Abstract
The present work examines heat and mass transfer characteristics of peristaltic motion of Johnson-Segalman fluid in a tapered asymmetric channel along with chemical reaction, by taking into account the Soret and Dufour effects. Effects of Hall current have also been discussed in mathematical modeling and analysis. Following the peristaltic wave procedure, the tapered asymmetric channel is based on the non uniform boundaries having diverse phases and amplitudes. The channel walls show excellent agreement with more realistic convective conditions. The modeled flow problem is directed into ordinary differential equations set with proper utilization of similarity quantities. The estimation of high wavelength as well as small Reynolds number are acknowledged to deduce the equations of Johnson-Segalman liquid model. The adopted solution procedure is constructed via homotopic algorithm. The results have been analyzed for various parameters of interest and sketched for better understanding. The velocity profile reveals decreasing behavior for increasing values of Weissenberg number and Hartman number while converse behavior is found for mean flow rate and Hall parameter. The temperature profile falloffs for heat transfer Biot number and Hartman number whereas it increases for Prandtl number, Brinkman number, Dufour number and Hall parameter. The concentration profile tends to decrease for mass transfer Biot number and increase for Schmidt constant.

Introduction
The idea of peristalsis is deduced from a Greek word 'per stalt l lkos' which represents the phenomenon of condensing as well as embracing. The novel implementation of this phenomenon is to evaluate the revolutionary wave contraction associated with a non-uniform cross sectional area of tube or channel. In physiology, it is used with the aid of the physique to drive or blend the tube contents as in gastro-intenstinal tract, ureter, bile and different glandular ducts. The mechanism of peristaltic movement is being abused for industrial purposes including transport of sanitary and corrosive fluids where the contact of the fluid with the apparatus components is denied. The mechanism of peristalsis contains wave like series of muscle contractions and relaxations that causes the motion of bio-fluid in diverse processes. Peristalsis is involved in various phenomenon's such as movement of urine from kidney to gallbladder, transport of chime in small intestine, food swallowing through esophagus, bile duct applications, vasomotion of small blood vessels, spermatozoa transport, locomotion of warms, pumping blood in dialysis etc. In medical and industrial systems, peristalsis has its numerous bio-medical applications like kidney to the intestine, tube pumps, dialysis heart lung equipments and open-heart surgery etc. Latham's [1] first attempt was conducted to figure out the peristaltic action of different fluids under various conditions along with various assumptions of long wave length, Reynolds number, material parameters of the fluid and wave amplitude etc. with reference to mechanical and physical situations. Later on Shapiro et al [2] used the concept of low Reynolds and long wavelength assumptions to explore the peristaltic pumping configured by an asymmetric channel by using linear fluid model. Now abundant information is present on peristalsis in the existing literature. The magnetohydrodynamic flow of blood under long wave-length estimation is made by Agrawal et al [3]. Mekheimer [4] worked on the utilization of magnetic field impact for mechanism of nonlinear peristaltic in inclined channel. In literature, various of theoretical Williamson nanofluid between rotating disks. The swirling flow in presence of entropy generation, Soret and Dufour consequences was analyzed by Qayyum et al [20].
The peristaltic flows with heat transfer are of great importance in the processes like hemodialysis and oxygenation. Such motivation dragged the attention of many researchers to examine the interaction of heat transfer with peristalsis. Radhakrishnamacharya and Srinivasulu [21], made their efforts to see the influence of wall properties on peristaltic flow with heat transfer. Vajravelu et al [22], studied peristaltic flow with heat transfer in a vertical annulus with long wave approximation. The assessment of wall features and heat transport in peristaltic investigation having different wall features was worked out by Kothandapani and Srinivas [23]. Mekheimer and AbdElmaboud [24], studied the influence of heat transfer and magnetic field on peristaltic transport of Newtonian fluid in a vertical annulus. Srinivas and Kothandapani [25], involved heat features on peristaltic prospective in asymmetric tube. Nadeem and Akbar [26], made findings on the influence of heat transfer on peristaltic flow of Johnson-Segalman fluid in a non-uniform tube. Hayat et al [27], examined the simultaneous effects of slip and heat transfer on the peristaltic flow. A relatively intricate relationship occurs between fluxes and driving potentials during heat/mass process. The energy flux is appeared because of join utilization of heat and mass gradients. The existing literature comprises few other attempts on peristaltic flow with heat and mass transfer.
The utilization of chemical reaction in combined heat/ mass transportation systems is another diligent feature which specified valuable applications in the manufacturing and chemical technologies. The chemical reaction reports many physical applications in the petroleum recovery, nuclear reaction cooling, fission and fusion processes, thermal insulation, drying and desert cooler applications. In various practical diffusive operations, the molecular diffusive species with associated chemical reactions play vital role [28][29][30][31][32][33].
It is often observed that during the process of joint heat and mass transportation, the temperature gradient is altered not only from temperature flux but also due to concentration flux. Similarly, the concentration gradient is affected due to impact of concentration and temperature fluxes. The energy flux has been induced due to composition gradient termed as Dufour effects. Beside this, the temperature gradient also deduced mass flux namely Soret features. It is emphasized that these thermo-diffusion features are of lesser order magnitude as compared to the features prearranged is case of Fourier's or Fick's theories and are consistently disregard to perform combined heat/mass analysis. The Soret features can be efficient to distinguish the isotopes separation and in gasses mixture when light molecular weight (H 2 , He) becomes extremely high. The Dufour effects play a significant role in medium molecular weight (N 2 , air) havinga desirable magnitude. It is noticed that these thermo-diffusion features are not carried out properly in most of recent investigations. In area of thermal field flow fractioning, the different size of molecules is split up from solvent by using these thermo features. This useful application can be seen in the biological samples like DNA, cells and proteins for which higher temperature difference can disintegrate these samples. Many physical applications of these thermo-diffusion characteristics include the Haber process which involves the nitrogen binding from air to prepare the ammonia. Similarly, the process of disinfection in involves the killing of bacteria and other viruses are also leads to these applications. Hayat et al [34] intended the Soret and Dufour attribute for convectively heated curved channel in peristaltic flow of Jeffrey liquid.
There are different analytical methods like differential transformation method (DTM), homotopy perturbation method (HPM), adomian decomposition method (ADM), homotopy analysis method (HAM) for solving the physical and engineering problems. The utmost competent method in solving different type of nonlinear equations such as homogeneous, non-homogeneous and coupled systems is homotopic procedure. The disadvantage of many other analytic methods is that they have some limitations for solving nonlinear equations. Unlike other analytical methods, HAM is independent of any small or large parameter. The proper utilization of initial guesses and auxiliary parameters improve the convergence procedure in contrast to other analytical techniques. Primarily, it is developed by Liao [35]. He further modified with a non-zero auxiliary parameter which is also acknowledge as convergence control parameter h. It is a non-physical variable which shows a convenient way to ensure the convergence of approximation series. This method is being used by many researchers and observed to be very operative in deriving an analytic solution particularly for non-linear differential equations [36][37][38][39][40][41][42][43].
Above all, the major attention of this study is to explore the impacts of heat and mass transfer with chemical reaction on peristaltic flow of Johnson-Segalman fluid in a tapered asymmetric channel. Additionally, Soret and Dufour effects are considered here along with Hall current. Although some studies regarding flow of non-Newtonian fluids in tapered asymmetric channel are already reported in the literature. However, investigation regarding heat and mass transfer characteristics in presence of Soret and Dufour effects has not been investigated yet. The Soret and Dufour features involve diverse applications in chemical engineering and geo-sciences. Moreover, the analysis is performed by using convective boundary conditions. The results are obtained and analyzed by using homotopy analysis method under the assumptions of long wavelength and low Reynolds number.

Mathematical formulation
The motion of an incompressible Johnson-Segalman fluid in two dimensional infinite tapered asymmetric channel is intended in this researcher. The constituted coordinate system for current analysis has been imposed such that X is taken in parallel direction while Y is imposed normally. The walls of channel satisfy the convective conditions of heat and mass transfer. Moreover, along the channel walls an infinite peristaltic wave train travelling with velocity c has been considered which generates the flow. It is assumed a peristaltic wave in a channel walls in order to induce asymmetry in the channel which results in different amplitudes and phase as shown in figure 1. Soret and Dufour effectshave also been analyzed in mathematical modeling and analysis. The Hall current impact are also incorporated in current investigation [26,34]: Equation (1) reports the upper wave in channel flow while mathematical expression given in equation (2) are associated with lower waves of the tapered configuration.
In above expressions, l, ( ) ¢ k a a , , 1 2 and stands for wavelength, non-uniform parameter, wave amplitudes and phase difference, respectively. It is remarked that channel waves in phase and out of phase are generated for j = 0 and j p = , respectively. Furthermore the following inequality is satisfied by a a d d , , , and j at the outlet of convergent or inlet of divergent channel, otherwise collision is set between both the walls.
The equations representing the present flow analysis are For Johnson-Segalman fluid, the Cauchy stress tensor is given as [26] ( ) , ,symmetric and skew symmetric parts of the velocity gradient ( ) D W , .
where ¢ d dt represents the material time derivative, k* the thermal conductivity, T is temperature, S the extra stress tensor, C is concentration, c s reflects the concentration susceptibility, D 1 mass diffusivity coefficient, T m is for mean temperature, K T thermal diffusion rate and k 1 the chemical reaction parameter.
We consider the uniform magnetic field of the form From the application of generalized Ohm's law, we have where s , 0 V, n, e , 0 and J are respectively symbolized the electrical conductivity, velocity vector, number density of electrons , electric charge and current density. Since the electric field features are neglected so = E 0. For the present flow analysis, the velocity is same as defined in previous chapter. i.e.
Under the above velocity field, the equations (11) and (12) give The corresponding equations for the present flow analysis are as follows [26,34]: The corresponding boundary conditions for the present flow analysis of tapered asymmetric channel are given below: The boundary assumptions in both walls channel in convective forms are peculated as:    Followings are the reduced boundary assumptions.

HAM solution
Let us find out the solution of constructed flow equations via homotopy analysis procedure by assuming following initial guesses for ( ) y y , ( ) q y , and ( ) s y Where y  , q  and s  are auxiliary parameters and [ ] Î q 0, 1 is embedding parameter.   For = q 0, we develop the initial guess and for = q 1, we meet with the solution of the problem under consideration.
At mth order the current problem takes the form 1 3 and the boundary at this order are given as We reached at the solution of the considered problem iteratively for = m 1, 2, 3.... using Mathematica.

Convergence exploration
Homotopy analysis method plays a vivacious role in order to get convergent series solution. Auxiliary parameters y h , q h and s h have been used here to tackle and settle the required convergent region. The ranges of these auxiliary parameters detect primary contribution to achieve convergent series solution. The homotopic convergent is achieved from ranges of ξ( The tolerable ranges of values for concerned auxiliary parameters are - Table 1 depicts the convergence of stream function, temperature and concentration. It is noticed that 14th order of approximation is sufficient for ( ) y h , 1 ( ) q¢ h 1 and ( ) s¢ h 1 which have been sketched graphically in the figure 2 for stream function, temperature and concentration profiles respectively.

Graphical discussion of results
The main aim of presenting this section to explore the physical output of parameters when varied against axial velocity, temperature, pressure and concentration profiles. We used homotopy analysis method to acquire the required solutions of axial velocity, pressure gradient, temperature and concentration. HAM is found to be the best convergent method. Intended for such determination, figures 3-17 have been plotted and their preliminary performances are evaluated. Figure 3(a) reflects the altered profile of axial velocity due to amplitude of lower wall α. A axial turn down velocity component is associated with maximum variation of α. Figure 3(b) reveals the influence of amplitude of the upper wall b on the axial velocity. It can be obviously noticed that the axial velocity profile increases on increasing the value of amplitude of the upper wall. The impact of non-uniform parameter k on velocity distribution of the fluid has been plotted in figure 4(a). The observations perceived that an increasing trend is noticed in the lower parts and upper region of tapered asymmetric channel. Conversely, opposite behavior is find out in the core trapped channel region. Figure 4(b) illustrates the dependence of axial velocity on the phase difference j. A decrease in velocity profile is observed when phase difference is enhanced. The effect of Weissenberg number We on the velocity distribution of the fluid has been displayed in figure 5(a) It reveals that an increase in the value of We results as a decrease in the axial velocity. The physical justification behind such trend is associated with the dominant of viscous forces. Figure 5(b) deliberates the physical consequences of mean flow rate Q on component of axial velocity. An arise axial velocity profile has been reported with leading values of Q. Figure 6(a) has been displayed to study the outcome of velocity distribution in the presence of Hartman number m 1 . The results claimed that maximum variation in m 1 is associated with lower axial velocity   component in the center region while it get upshot level near the channel walls. Figure 6(b) aim to highlight the axial velocity change when Hall parameter m assigns maximum variation. The careful observation reported that middle channel region shows increasing behavior due to m.      figure 9(a). In contrast to this figure, a reverse trend (declining behavior) is found out for D u (figure 9(b)). Now we reported the change in temperature due to Hartman number m 1 by sketching figure 10(a). It can be realized that huge estimation of Hartman number reveals a decline in temperature.       Figure 11(a) discloses the effect of non-uniform parameter k on concentration. It is noticed that an increase in the value of k leads to the increase in concentration. The effect of phase difference j on concentration is drafted in figure 11(b). Figure 12(a)is exhibited to study the influence of mean flow rate Q on concentration. It shows that as the value of Q enhances the concentration also enhances. Figure 12 figure 13(b). The progress in magnitude of concentration is regarded by increasing the value of Sc. The effect Soret number on concentration is articulated in figure 14(a). By increasing the value of Sr the concentration gets increased. The appliance of g on concentration is thrashed out in figure 14(b) The observations reveal that by expanding the value of g the concentration distribution also expands.

Pumping characteristics
The effects of amplitude of lower wall a, upper wall b, non-uniform parameter k, Weissenberg number We, Hartman number m 1 and Hall parameter m on distribution of pressure are reported in figures 15-17. Figure 15(a) briefs the impact of amplitude of lower wall a on pressure gradient. It is seen that the pressure gradient tends to enhance as we enhance the values of a. The observations explored in figure 15(b) justify the input of amplitude of upper wall b with pressure gradient graphically. The pressure gradient increases as value of b goes up. Figure 16(a) depicts the effect of non-uniform parameter k on pressure gradient. It is captured that pressure gradient attains minimum value as we increase the values of k. Figure 16(a) represents the effect of Weissenberg number We on pressure gradient. It is pointed out that pressure shows declining trend in the core channel region when we increase the value of We. The impact of Hartman number m 1 on pressure gradient is evoked in figure 17(a) It is interesting to perceive that pressure gradient goes up by increasing the value of m . 1 Figure 17(b) exposes the effect of Hall parameter m on pressure gradient. It shows that pressure gradient declines on increasing the value of m. Table 2 presenst the numerical values of Nusselt number at upper and walls of disks. It is noted that Nusselt number at upper wall increases with Pr and Br.

Concluding remarks
In contemporary work, a mathematical model is developed to observe heat and mass transfer characteristics in the peristaltic movement of Johnson-Segalman fluid. The motion takes place in tapered asymmetric channel in the presence of Hall current. Furthermore, Sore and Dufour effects have also been discussed and analyzed in mathematical modeling and analysis. Instead of diverse phase and amplitudes on non-uniform channel boundaries, we have considered here the peristaltic wave train which gives rise to tapered asymmetric channel. We have treated the model with the assumptions of peak wavelength and smaller Reynolds theories. The resulting differential equations have been solved for stream function, axial velocity, temperature, concentration and pressure gradient with the aid of convergence procedure. Various parameters effects are graphically underlined for respective profiles. The outcomes have been summarized as follows: • The velocity profile exhibits decreasing behavior for increasing values of amplitude of lower wall, nonuniform parameter, Weissenberg parameter and Hartman number.

•
The fluid velocity increases for amplitude of upper wall, mean flow rate and Hall parameter.

•
The temperature profile decreases for heat transfer Biot number and Hartman number.
• Phase difference, mean flow rate, Prandtl number, Brinkman number, Dufour number, and Hall parameter enhance the temperature profile.
• The concentration profile tends to decrease for phase difference and mass transfer Biot number.
• The concentration profile enhances for non-uniform parameter, mean flow rate, Brinkman number, Schmidt number, Soret number and chemical reaction parameter.

•
The pressure gradient shows decreasing behavior for non-uniform parameter, Hall parameter and Weissenberg number.

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The pressure gradient enlarges Hartman number.
• The transfer Biot number variation results decreasing heat transpiration are lower wall as compared to upper wall channel.