Exploration of thermal transport for Sisko fluid model under peristaltic phenomenon

The present analysis focuses on the thermal radiation and slips effects on the peristaltic movement of Sisko fluid with the symmetric compliant channel with rheological properties, enhancing damping tools, protection apparatus individuals and in various distinct mechanical procedures. Keeping in mind the considered problem assumptions (Non-Newtonian Sisko fluid model, power-law model, Prandtl number, wall properties, porous space, etc) it is found that the modeled equations are coupled and non-linear. Using low Reynold’s number and long wavelength assumptions, the governing system of a nonlinear coupled system of equations with appropriate boundary constraints is solved with the perturbation technique. Due to convectively heated surface fluid between the walls having a small temperature. Sherwood and Nusselt numbers both deduce for fixed radiation values and different Sisko fluid model quantity. Skin friction is maximum in the case of Newtonian, while minimum in case of dilatant model and pseudoplastic models. The influence of numerous parameters associated with flow problems such as Hartman number, Prandtl number, and slip parameters are also explored in detail and plotted for concentration profile, thermal distribution, and momentum distribution profile. From this analysis, it is concluded that velocity escalates for the larger slip. Also, skin friction is found similar for Newtonian and pseudoplastic models where in case of dilatant model it is little different but it increases for these three cases when Schmidt number is increased. Moreover, the shear-thinning and shear-thickening behavior of the fluid model is also explained in detail. Industrial applications of the Sisko model include minimum friction, reduction in oil-pipeline friction, uses as a surfactant for comprehensive scales cooling and heating systems, scale-up, flow tracers, and in several others.


Abstract
The present analysis focuses on the thermal radiation and slips effects on the peristaltic movement of Sisko fluid with the symmetric compliant channel with rheological properties, enhancing damping tools, protection apparatus individuals and in various distinct mechanical procedures. Keeping in mind the considered problem assumptions (Non-Newtonian Sisko fluid model, power-law model, Prandtl number, wall properties, porous space, etc) it is found that the modeled equations are coupled and non-linear. Using low Reynold's number and long wavelength assumptions, the governing system of a nonlinear coupled system of equations with appropriate boundary constraints is solved with the perturbation technique. Due to convectively heated surface fluid between the walls having a small temperature. Sherwood and Nusselt numbers both deduce for fixed radiation values and different Sisko fluid model quantity. Skin friction is maximum in the case of Newtonian, while minimum in case of dilatant model and pseudoplastic models. The influence of numerous parameters associated with flow problems such as Hartman number, Prandtl number, and slip parameters are also explored in detail and plotted for concentration profile, thermal distribution, and momentum distribution profile. From this analysis, it is concluded that velocity escalates for the larger slip. Also, skin friction is found similar for Newtonian and pseudoplastic models where in case of dilatant model it is little different but it increases for these three cases when Schmidt number is increased. Moreover, the shearthinning and shear-thickening behavior of the fluid model is also explained in detail. Industrial applications of the Sisko model include minimum friction , reduction in oil-pipeline friction, uses as a surfactant for comprehensive scales cooling and heating systems, scale-up, flow tracers, and in several others. Electrical conductivity (s) Permeability of porous medium (k)

Introduction
Any exchange on drug intricacies must pay courtesy to serval type of motion during biological functions. Amongst these biotic flow mechanisms, the peristaltic mechanism is one of them. Peristalsis is used in several physiological, industrial, and biomedical processes and is experimented by many researchers. This flow mechanism occurs because of continuous symmetrical and asymmetrical propulsion of smooth channel walls. Peristalsis is a very significant mechanism for carrying the drug and other materials during sensitive diseases treatments (e.g. cancer, tumor, heart surgery, etc) and also in several engineering and industrial processes because in peristalsis movement of walls fluid material does not stick with channel walls. In the Peristaltic flow, the fluid move by way of relaxation/ shrinking of liberal waves. Physiologically it studies the evacuation from the kidney to sac, spermatozoa motion, food consuming, gastrointestinal chyme activity and moving of ovum, etc are due to peristalsis mechanism. Peristaltic pumps are used in many industrial and engineering processes especially in chemical-pharmaceutical industries. In the biomedical field, it helps in pumping blood through the arteries. Peristaltic flow has many applications in industries as well, such as the motion of corrosive fluids, movement of sanitary liquid, transport of toxic liquid. Many biotic systems also invoke peristalsis mechanisms like the esophagus, small intestine, large intestine. Peristalsis is used in several physiological, industrial, and biomedical processes and is experimented by many researchers. Bhatti et al [1] elucidated the fundamental procured and principal evidently impacts of innumerable physical factors that govern the fluid flow. Bhatti et al [2] investigated the thermal radiation effect for electro-hydrodynamic through the Riga plate. They solved the fluid flow expressions with numerical shooting techniques. They also incorporated the in this study, the effect of Nusselt and Sherwood numbers. The main outcomes of this analysis are that, temperature profile boots up for higher values of thermal radiation effect and observed opposite nature for Prandtl number. Riaz et al [3] investigated the entropy generation impact on the peristaltic transport of the Non-Newtonian Williamson fluid model in an asymmetric complaint channel. They employed perturbation techniques for the solution of governing expressions of fluid transport. Form this study they obtained that the thermal profile id enhancing for Biot and Brinkman numbers. Latham [4] and Shaprio et al [5] gives the idea of low Reynolds number and long wavelength analysis. They analyzed the attributes of fluid movement in the peristaltic pump. Fung [6] explained the concept of ureteral muscles through peristalsis. The mathematical model of peristaltic transport of blood with the magnetic field is presented by Mishra et al [7]. Transportation of peristalsis in a porous annulus with heat transfer has been analyzed by Vajravelu et al [8]. Srinivas and Kothandapani [9] explored the heat transfer study of peristaltic flow in an asymmetric path. Hayat et al [10] explored the MHD peristaltic flow of Newtonian fluid in porous space under the consequence of heat transfer. Aly and Ebaid [11] derived the precise methodical solution of nanofluid under the effects of a slip in an asymmetric channel in presence of peristalsis. Carreau fluid under convective conditions in presence of peristalsis is examined by Hayat et al [12]. The viscous Newtonian fluid that depends on radial and axial components in the presence of peristalsis was analyzed by Lachiheb [13]. A precise solution of Jeffery nano-fluid in peristalsis was derived by Ebaid et al [14]. Recent exploration in this regard and some references therein can be mentioned in [15][16][17][18][19].
The study of MHD channel transport is of countless contemplation due to its real-world applications in industrial and biomedical processes, such as magneto-hydrodynamic pumps, tumor treatment, petroleum industry, bleeding reduction during surgeries, aerodynamics heating and cancer therapy. MHD is used in several physiological, industrial, and biomedical processes and is experimented by many researchers. Hina et al [20] investigated the Laminar flow of the MHD Oldroyd-B fluid model with a non-Fourier flux effect. Abbasbandy et al [21] discussed the transport of non-Newtonian Oldroyd-B fluid-induced due to non-linear deforming sheets in the existence of a strong magnetic field.
The study of flows through porous medium also has real-world applications especially, in the geophysical fluid dynamics. Natural porous medium includes bile duct, beach sand, limestone, sandstone, rye bread, wood, the human lung, small blood vessels, and gall bladder with stones. Because of this, many researchers examined the porous impact of the peristaltic flow of fluid models in different geometry/channels. Sanked et al [22] studied the impact of the flexible wall in the MHD peristaltic movement of different couple stress fluid models in the porous medium using thermal and momentum slip conditions. Veera et al [23] discussed the peristaltic movement of electrically conducting and incompressible fluid with viscous dissipation and Joule heating impact for the Williamson fluid model in the uniform symmetric flexible channel with the inclined magnetic field. Elangovan et al [24] investigated the Non-Newtonian blood movement for velocity slip with periodic body acceleration. Sankad and Nagathan [25] explained the peristaltic transport phenomena for MHD, heat transfer, thermal slip, wall properties, and porous medium for couple stress fluid mode. Ramesh and Devakar [26] studied the mass and heat transfer for peristaltic transport phenomena due to myometrial contractions occur in both symmetric and asymmetric channels.
The concept of wall properties, wall rigidity, and wall tension has great importance in peristalsis. Radhakris et al [27] explored peristaltic flow with plaint walls under the influence of heat transfer. The influence of micropolar fluid in cylindrical tubes having complaint walls in peristalsis was discussed by Muthu et al [28]. Hayat et al [29] described the influence of Johnson-Segalman fluid on peristaltic momentum having complaint walls. Hina et al [30] described the peristaltic flow with flexible walls channel under the chemical effect reaction. Under the impact of slip for peristaltic flow of wall properties and Nanofluid were analyzed by Mustafa et al [31]. Bhatti et al [2] investigated the teradiation effect for electro-hydrodynamic through the Riga plate. They solved the fluid flow expressions with numerical shooting techniques. They also incorporated the in this study, the effect of Nusselt and Sherwood numbers. The main outcomes of this analysis are that, temperature profile boots up for larger values of thermal radiation effect and observed opposite nature for Prandtl number.
The concept of fluid flow subject to heat and mass transfer has significant importance in vaporization, biomedical production, desert chiller, and microfabrication technologies. Especially the heat transfer in peristaltic transport has an important character in homo dialysis processes and oxygenation. Hayat et al [32] addressed the transport of Jeffery nanofluid in complaint vertical channels. In the study, they also incorporated the impact of thermal radiation, chemical reaction, and Brownian motion. They observed that momentum profile raises for Darcy and Hall effect. Hayat et al [33] investigated the mixed convective transport of MHD Jeffery fluid in an asymmetric porous channel. They handle the expression of fluid flow using the ND solver program. The velocity profile of the fluid motion is directly proportional to the Soret and Dufour effects. Some references from recent exploration are explained in [34][35][36][37].
It is evident from the literature survey, the influence of the Sisko fluid model with a compliant channel under the effects of thermal slips with chemical reaction and thermal radiation for power index, upper Newtonian regions model and power-law model for heat and mass transfer in peristalsis is still unexplored. Therefore, our current study targeted to explore heat and mass transfer effect for chemical reaction and thermal radiation effects on MHD peristaltic transport in a compliant channel. The solution of non-linear fluid flow equations has been computed using the perturbation technique and effects of controlling factors are analyzed through graphs for the thermal profile, concentration distribution profile, momentum distribution, and coefficient of the rate of heat transfer.

Problem formulation
Consider a viscous incompressible Sisko fluid model moving in a two-dimensional flexible channel of width. Homogeneous liquefied saturates with a strong magnetic field in a porous medium. The compliant walls temperature and concentration are T , 0 C 0 and T , 1 C 1 respectively. The geometry and coordinate system are illustrated in figure 1. The sinusoidal wave propagating for the small amplitudes along flexible walls of the channel is where a, l and c are the wave amplitude, wavelength and wave speed. The basic flow governing equations for viscous fluid model are: the corresponding boundary conditions for equations (2)-(6) are: here ( ) s expressed the Stefan-Boltzmann constant. By expanding T 4 about T 0 and ignoring higher-order terms one obtained where ⁎ v and ⁎ u are the velocity components. The displacement compliant wall is given as where L represents the movement of a flexible walls of the channel, where t, C and m 1 indicated the elastic membrane tension, viscous damping effect and plate mass/unit area. ⎛ where S , x x * * S x y * * and S y y * * are the component of extra stress tensor of sisko model, defined as  Denoting acquainting stream function by Y as using approximations of low Reynolds number Re and long wavelength ( )  d 1 and equations (14) and (15), equation (2) is satisfied identically and equations (3)-(12), one has The suitable boundary conditions as follows:

Solution methodology
Using perturbation technique on equations (16)- (19) for small Sisko fluid parameters ( ) A , for along with corresponding boundary condition equations (20)- (23). We obtained the following expressions After applying equation (25), to equations (15)- (18), one has, 3.1. Zero order syste with subject boundary conditions as  with respective Boundary conditions    The rate of heat transfer coefficient at flexible channel is expressed as

Graphical Investigation and discussion
In this section, we explored the behaviors of numerous effective parameters on thermal profile, momentum profile, concentration profile, rate of heat transfer coefficient, skin friction coefficient, and Sherwood number. In fact, that for larger values of Sisko model parameter A, it decreases the effective conductivity of fluid, and consequently it reduced the magnetic damping effect. Therefore, velocity is increases. Figures 2(e) and (f) indicated reduction in momentum profile for both Newtonian and Non-Newtonian Sisko fluid model. The impact of elastic parameter E1, E2 and E3 are plotted in figures 2(g) and (h) it is exciting to note that the momentum distribution profile is enhancing for larger values of elastic parameters. It is also observed that for non-Newtonian case fluid flow is rapidly increases. Furthermore momentum profile contour a parabolic shape for the different amount of parameter and having extreme magnitude in bordering to center of the flexible channel. Figures 3(a) and (l), shows impact of the different parameters on thermal distribution for Newtonian and non-Newtonian fluid model. Figures 3(a) and (b), indicated that, temperature profile ( )    Table 1 include the comparison of the present analysis of local Nusselt number for various values of Prandtl number Pr with the results of Wang and Asghar. Figure 8 indicated that for larger amount of rigidity effect (E1), the stream lines get closer and the shaped of the trapped bolus enhanced in some region. Figure 9 show that for higher values of Hartman number (M), the trapped bolus and size of stream lines enhanced.

Key findings of accomplished investigation
In the present analysis, we have explores the role of slip effects, compliant wall, and thermal radiation effects on the peristaltic mechanism for the non-Newtonian Sisko fluid model in a porous symmetric channel. More explicitly, the authors examined the impact on concentration distribution, momentum profile, thermal analysis, Nusselt number, chemical reaction, Sherwood number, and skin friction coefficient profiles. The variations of streamline are also presented for the Hartman number and elastic parameters. The scope of the present article is valuable in explaining the chyme motion in the gastrointestinal tract and blood transport dynamics in small vessels while considering the important wall features and chemical reaction. Graphical sketches are plotted for physical parameters. Noticeable features discoveries can be summarized below • In the center of the channel and near the walls, the longitudinal velocity is reduced for larger values of magnetic effect, while a conflicting performance noted for Sisko fluid parameter.
• Fluid temperature is increased by mounting the value of fluid parameter and radiation parameter.
• Fluid velocity raises for Darcy parameter in case of the Newtonian model, while opposite tends for non-Newtonian model.
• Growing values of elastic parameters upsurge the thermal distribution and velocity field.
• Thermal profile reduces for a larger amount of magnetic parameter, while an inverse tends for Darcy number. • Similar important of Darcy number on velocity and temperature fields have been reported.
• Concentration profile reduced for the larger amount of Brinkman number for both cases.
• Concentration profile show opposite behavior for Newtonian and non-Newtonian models for the Darcy number.   • Influence of larger value elastic parameters show a decrease in concentration profile.
• Concentration decreased with an increase in chemical reaction and concentration slip parameters.
• Skin friction coefficient is increasing function of magnetic parameter and elastic parameters.
• Magnitude of coefficient of heat transfer rises at the upper part of flexible channel for Sisko fluid parameter, while it reduced for larger amount of momentum slip parameter.