Heat generation and chemical reaction in flow propagated by peristalsis incidence of radiation in MHD Jeffrey nanofluid

Objectives: In the present study, the heat Source and chemical reaction inMHD Jeffrey nanofluid driven by means of peristalsis through asymmetric channel has been addressed. Methods: Considering low Reynolds number and long wave length assumptions the fundamental equations of mathematical model are obtained. By applying long-wavelength and lowReynolds’s number thenonlinear partial differential equations are transformed into ordinary differential equations. The simplified equations are solved by using Mathematica software and various effects are discussed in the form graphs. Findings: It is observed that influence of chemical reaction parameter on temperature and concentration profiles are increasing. The converse behavior is found in the case of Radiation and heat source parameters. Novelty: The present analysis is applicable for magnetic drug targeting for various treatments in a human body. The present study addresses the influence of heat source and chemical reaction in peristaltic pumping of Jeffrey nanofluid incidence of thermal radiation.


Introduction
Improvements to create heat transfer utensils more energy proficient would need to concentrate on miniaturization on the one hand and an astronomical raise in heat flux on the other. Heat transmit fluid such as mineral oil, ethylene glycol , and water play a vital role in various engineering processes, include microelectronics, chemical, heating processes, and power production. This encounters in approximately every branches of Engineering and Biosciences. Jeffrey model flow past a fixed sheet focused on power law temperature in the incidence of heat source/sinks explored (1) . The collective consequence of heat and mass transmit in Jeffrey fluid is described in this article. The heat transfer effects on the peristaltic flow of Jeffrey six-constant fluid representation have been studied (2) . In the paper boundary layer flow of nanofluid and heat transfer characteristics past a stretching sheet Studied (3) . Thermal radiation, Brownian motion, and slip boundary condition effects are examined. (4) Look into The property of soret and chemical effect on steady MHD mixed convective heat and mass transfer flow embedded in a porous medium in the incidence of heat source, viscous and Joules dissipation.
Significant consideration in modern times pooled the problems with chemical reaction, heat and mass transfer, which gained much significance in several processes. At the same time heat and mass transfer take place in process such as transport of energy in a wet cooling tower, geothermal reservoirs, and the flow in a desert cooler. J. Prakash et al. (5) presented a mathematical model to study the nanofluids flow driven by peristalsis mechanisms through asymmetric channel. Furthermore thermal radiative flux replica is deployed, to examine effects of the thermal radiation. In the article (6) Cat taneo-Christov diffusion model is introduced in describing the temperature and concentration diffusions with thermal and solutal relaxation times respectively.. Anjali Devi and Kandasamy (7) focused to find approximate solution for MHD boundary layer fluid with the effects of heat and mass transfer, chemical reaction over a block. Muthucumaraswamy (8) dealt with effects on a moving vertical plane with chemical reaction. The study (9) investigates the impact of Joule heating and velocity slip on MHD peristaltic flow in a porous space with chemical reaction. It is noticed that a generative chemical reaction is superior to the destructive one on the concentration. M. Ganapathirao (10) studied the effects of chemical reaction, heat and mass transfer on an unsteady mixed convection boundary layer flow over a vertical wedge with heat generation/absorption in the presence of uniform suction or injection. Srinivasa Raju (11) studied the effect of chemical reaction on unsteady, incompressible, viscous fluid flow past an exponentially accelerated vertical plate with heat absorption and variable temperature in a magnetic field.
Mixed convective peristaltic flow of Jeffrey nanofluid variable viscosity is studied with the consideration of viscous dissipation and Joule heating effects. In this study the viscosity is taken as temperature dependent (12) . Influences of thermal radiation and thermophoresis on peristaltic flow in a rotating frame are discussed (13) . The steady laminar 2D flow and heat transfer characteristics by boundary-layer of one phase Sisko bio-nanofluid model are discussed (14) . In the article, non-uniform hemodynamic nanofluid flow in the presence of an external magnetic field studied. Srinivasa Raju (15) studied thermal diffusion and diffusion thermo effects on an unsteady two-dimensional heat and mass transfer radiative MHD natural convective Couette flow of a viscous, incompressible, electrically conducting fluid between the two vertical parallel plates. Jithender Reddy et al. (16) studied the hall current and rotation effects on MHD free convection flow past a moving vertical plate with the presence of thermal diffusion and diffusion thermo for isothermal and ramped temperature. In the paper, influence of thermal radiation on peristaltic motion with double diffusive convection of gold nano particles through an asymmetric channel (17) .
The peristaltic transport of liquid has gained exceptional interest in the recent time owing to its extensive application in physiology as it is considered a vital method for flow in bio fluids. This course is exceedingly essential in numerous physiological systems and in engineering includes swallowing food through esophagus, arterioles and capillaries, in sanitary fluid transportation, toxic fluid move in the nuclear engineering etc. Several investigators have analyzed the peristaltic motion of both Newtonian and non-Newtonian fluids in mechanical as well as physiological systems (18)(19)(20) .
To acquire improved quantitative and qualitative characteristics concerning the rheological activities of blood, several authors presented numerous non Newtonian fluid models. Jeffrey fluid amongst them is a well-known model in studying vascular dynamics. The magnetic field effect in two-dimensional mixed convection boundary layer flow and heat transfer of a Jeffreyfluid over a stretched sheet immersed in a porous medium is studied by Kartini Ahmad, Anuar Ishak (21) . M.M. Bhatti, M. Ali Abbas (22) reported concurrent slip and hydro magnetic effects on peristaltic blood transport of Jeffrey model fluid through a non-uniform channel in porous medium. M. Gnaneswara Reddy (23) investigated the impact of velocity slip on MHD peristaltic flow through a porous medium with heat and mass transfer.
The published studies about the heat generation in peristaltic MHD flows of non-Newtonian fluids are still scarce. In fact heat generation and absorption concepts in fluids have relevance in problems dealing with chemical reactions, geo-nuclear repositions and these concerned with dissociating fluids. The nanofluid flow with thermal and chemical reaction has broad applications in cooling, expulsion procedures and polymer industry. The magnetic field plays an essential role in targeting drugs by magnetic nano particles for different kinds of diseases in a human body. The Jeffrey fluid model is employed to simulate non-Newtonian characteristics. Hence the current model is to describe heat generation and chemical reaction peristaltic pumping of MHD Jeffrey nanofluid. Thermal radiation in addition to chemical reaction effects has been addressed.

Formulation of the problem
An incompressible Jeffrey nanofluid in an unsymmetrical channel of non-uniform width is considered in two-dimensional flow.
Flow is considered to be free from the electric field. The movement in the fluid is shaped in reaction to the promulgation of couple of sinusoidal wave trains of stable speed c 0 with wavelength λ along the flexible channel walls. η = η 1 , η = η 2 are considered to be upper and lower walls. A magnetic field of intensity B 0 is applied and the effect of induced magnetic field is negligible for small magnetic Reynolds number. The wave shape is represented by the following equation Where the constitutive relation for Cauchy stress tensor (τ) in an incompressible Jeffrey fluid is https://www.indjst.org/ The equations governing the flow for incompressible nanofluid are given as (5,24,25) ∂U ∂ ξ Radiative heat flux q r is given by The wave and laboratory frame transformation is given by https://www.indjst.org/ Introducing the following non-dimensional quantities: In equations (4)-(7) after omitting primes and writing stream function ψ(ξ , η, t) The above governing equation (5)-(8) after eliminating pressure term using long wavelength and low Reynolds number approximations can be obtained in non dimensional form as follows The relevant boundary conditions are Where q is the flux in the wave frame and the constants a, b, d, φ must satisfy the relation The flow rate in wave frame and fixed frame are related by 3 Discussion

Dimensionless velocity
Variations in magnetic parameter (M) are presented in Figure 2 (a). One can find that the velocity diminishes at hub part of the walls with rising in M. The reason behind this aspect is that application of magnetic field to an electrically conducting fluid gives to a resistive type force called the Lorentz force which opposes the fluid motion. Features of Grash of number (Gr), thermophoresis (Nt) and Jeffrey fluid parameter ( λ ) are plotted through the Figure 2 Figure 3 (a)-(d) is made to see the effects of Radiation parameter (Rn), Jeffrey fluid parameter (λ ), Chemical reaction parameter (γ) and heat source parameter ( β ) on temperature. Influence of radiation parameter Rn, λ is illustrated in Figure 3(a), Figure 3(b). Figure 3(a) shows that the enhancing of radiation parameter Rn leads to decrease in the fluid temperature near the surface region. This is because rises in Rn have the tendency to increase the conduction effects. Therefore, higher value of radiation parameter implies higher surface heat flux and so, decreases the temperature at each point away from the surface. It is observed from Figure 3(b) that temperature (θ ) decreases with λ . This is due to the reason that, the transverse magnetic field gives rise to a resistive force known as the Lorentz force of an electrically conducting fluid. This force makes the fluid experience a resistance by increasing the friction between its layers and thus decreases its temperature. Figure 3(c), Figure 3(d) is made to see the variation of γ, β on θ . It is found that θ increases with increase in γ, β . Obviously in heat source parameter β process more heat is produced which result in the enhancement of temperature field.

Dimensionless concentration
It is clear from Figure 4 (a) that increasing chemical reaction parameter (γ) the concentration would be exhibited. Increase in γ increase in concentration profile (ϕ ). Physically, fluid becomes thicker due to increasing value in γ which is reduced concentration profiles. Similar behavior is found with the case radiation parameter (Rn) Figure 4(d). A converse behavior can be found with the case of thermophoresis (Nt), heat source parameter (β ) Figure 4(b) and Figure 4(c)

Conclusion
The impact of nanofluid particles on the peristaltic bloodstream has been explored scientifically under the effects of thermal radiation and chemical response with the guide of the computational program Mathematica. The base liquid is considered as a Jeffrey fluid model within sight of an applied magnetic field. The administering flow is displayed for low Reynolds number and long wavelength.The present paper focused on heat generation and chemical reaction in flow propagated by peristalsis in hydro magnetic Jeffrey fluid with thermal radiation taking into consideration. The main findings are listed below.
https://www.indjst.org/ • Velocity increases with the larger values of M at the channel walls. Velocity is enhanced in the lower part of the channel with increasing Gr, Nt and λ , whereas it behaves in converse with the chemical reaction parameter. • It is observed that temperature (θ ) decreases with Rn, λ . It is found that θ increases with increase in γ, β . • Increase in γ increase in concentration profile (ϕ ). Similar behavior is found with the case radiation parameter (Rn). A converse behavior can be found with the case of thermophoresis (Nt), heat source parameter (β ).