Double diffusion on peristaltic flow of nanofluid under the influences of magnetic field, porous medium, and thermal radiation

The present article investigates the study of double‐diffusive convection on peristaltic flow under the assumption of long wavelength and low Reynolds number. The mathematical modeling for a two‐dimensional flow, along with double diffusion in nanofluids, is considered. The motivation of the present research work is to analyze the effects of the magnetic field and thermal radiation on a peristaltic flow through a porous medium in an asymmetric channel. The heat flux of the linear approximation employs the thermal radiation of the flow problem. The effect of thermal radiation and double diffusion is an important aspect of research due to its application in public health potential. Infrared radiation techniques are indeed used to treat many skin‐related diseases. It can also be used as a measure of thermotherapy in some bones to enhance the blood circulation. It is found that approximately 80% of blood flow increases with radiation. The governing equations are analytically solved by using Homotopy analysis method with the help of the symbolic software Mathematica. The results of the velocity, pressure rise, temperature, solutal (species) concentration, and nanoparticle volume fraction profiles are graphically shown.

absolute temperature variation. Also, the thermal radiation has many applications in biological treatments such as lung cancer, bone cancer, breast cancer, thyroid, blood disorders, radiotherapy, and liver cancer. Research work on peristaltic motion with the influence of thermal radiation is reported in References. [8][9][10][11][12][13] The radiation has been linearized using Roseland approximation. 14 A homogeneous mixture of nanoparticles consists of a base fluid is known as nanofluids. The nanoparticles employed in nanofluids are made up of carbon nanotubes, metals, oxides, and carbides. The base fluid includes oil, water, and ethylene glycol. Nanofluids are produced by dispersing nanoparticles in base fluid; given it is superior thermophysical properties, nanofluids are gaining increasing attention and are showing promising potential in various applications. Choi 15 was first to proposed nanofluids. Nanofluids include the enhanced thermophysical properties compared with ordinary fluids. The different types of nanoparticles are oxide ceramics, carbide ceramics, metals, and carbon nanotubes. 16 In nanofluids, nanoparticles randomly move through liquids and possibly collide with Brownian motion and thus proposed to be one of the possible origins for thermal conductivity enhancement. The connection of peristalsis with the nanofluids has important applications in biomedical science (such as drug delivery and cancer treatment to treat radiotherapy), mechanical engineering, and chemical engineering (transport of chemicals). Recently, the combined study of peristaltic flow of nanofluids 17,18 has been described to examine the role of nanofluids in physiological flows.
The heat and mass transfer occur concurrently with the complicity of the fluid motion is known as double diffusion. Double diffusion has important applications in solid-state physics, chemical engineering, geophysics, oceanography, astrophysics, and biology, 19 as well as many engineering applications such as natural gas storage tanks, solar ponds, metal solidification processes, and crystal manufacturing. Nowadays, researchers have been focused on double diffusive convection of peristaltic transport. The peristaltic flow of double diffusive natural convection in nanofluid studied by Noreen et al. 20 The research work on double diffusion was continued in References. [21][22][23][24][25][26] The porous medium can be defined as solid bodies containing pores structures. The porous medium is the ratio of pore volume to the total volume of given sample materials. The study of porous medium on peristaltic transport has been attracted the many investigators in the field of biomedicine such as gall bladder with stones, human lungs, vascular beds, kidney, small blood vessels, bile duct, and also heat generation in industries. 10,[27][28][29][30][31][32] Limited works are found during the review literature on magnetic field combined with the double diffusive convection on peristaltic flow. So, in the current study, we have considered the magnetic field on peristalsis with double diffusive convection flow. Magnetic field is gained considerable important applications in bioengineering and industry. The important applications of magnetic field are in aerodynamic heating process, fluid droplet sprays, cancer tumor, electrostatic precipitation, removal blockage in the arteries, bleeding reduction in case of surgery, and hyperthermia. The recent research work on MHD. [33][34][35][36] In all above-mentioned investigations reveal that the role of magnetic field and thermal radiation effects on double diffusive convection of nanofluid has not yet been studied in literature. Keeping in this mind, the aim of the current study is focused on the influence of thermal radiation and magnetic field on peristaltic flow with double diffusive convection through porous medium. Double diffusive convection problems have so many practical applications in oceanography, geophysics, biology, and astrophysics. Heat transfer rate is controlling by applying the radiation. The basic governing equations are highly nonlinear, which are solved by using Homotopy analysis method (HAM), 37,38 and analysis of the embedded parameters on velocity, pressure rise, energy, solutal (species) concentration, and nanoparticle volume fraction is shown in the form of graphs.

MATHEMATICAL ANALYSIS
We have considered the two-dimensional peristaltic flow of uniform thickness d 1 + d 2 on asymmetric channel. The fluid flow in the channel walls is produced, when the sinusoidal waves of the small amplitudes a 1 and b 1 propagates the speed of the channel walls. c is the constant speed of the channel (Figure 1). The physical model of the channel is defined as F I G U R E 1 Physical model of the given problem Amplitudes of the waves are a 1 and b 1 , is the wavelength of channel walls, is the phase difference, considering the cartesian coordinate system ( ′ , ′ ), where ′ and ′ are perpendicular to each other. Here, d 1 and d 2 are satisfies the below condition.
Vector form of the velocity field Vis given as, where U ′ ( ′ , ′ , t ′ ) and V ′ ( ′ , ′ , t ′ ) are the velocity components. The radiative heat flux q r can be written as In the above equation, k* is the Rosseland mean absorption coefficient and * denotes the Stefan-Boltzmann constant. Considering the nanofluid flow temperature is very small, therefore the term T ′ 1 3 is a linear function of temperature.
The basic governing equations describing the peristaltic flow patterns for the nanofluid are as follows: where ( c) f is the heat capacity of the fluid, D TC is the Dufour diffusivity, k T is the thermal conductivity of the fluid, ( c) p is the effective heat capacity of the nanoparticle material, gis the gravity, ' is the solutal (species) concentration, K 0 is permeability constant of the porous medium, and is electrically conductivity of the fluid. Furthermore, f is the effective density, D T is thermophoresis diffusion coefficient, D S is the solutal diffusivity, C p is the specific heat at constant pressure, is the thermal diffusion ratio, C s is the concentration susceptibility, D CT is the Soret diffusivity, and T m is the mean fluid temperature.
The corresponding boundary conditions The connection between the wave frame and laboratory frame are introduced through Introducing the following nondimensional variables where M is the magnetic field, is the dimensionless wave number, is the dimensionless temperature, is the nanoparticle volume fraction, and the stream function taken as v = − and u = . By using Equation (13), Equations (6) to (12) can be written as Boundary can be reduces to Here, we have considering f * is the mean flow over a period is where Θ = Q 1 and f * = q 1 . The coupled partial differential Equations (14) to (19) with the boundary conditions (20) are solved by using HAM with symbolic software Mathematica and the analysis of the present method of solution is mentioned the below section.

METHOD OF SOLUTION
The governing Equations (14) to (19) are evaluated by Homotopy Analysis Method (HAM). This method is an analytical technique, which can be used to compute the nonlinear problems that contains small and large physical parameters (solution is obtained in terms of convergent series solutions). The method provides great freedom to select the initial approximations and auxiliary linear operators. By using this, any complicated nonlinear problems can be transformed into linear subproblems. Initial approximations are chosen as follows: Furthermore, the auxiliary linear operator of the problem is taken as which satisfies the properties According to the methodology, the zeroth-order deformations of the problems are In the above equations, q ∈ [0, 1] is the embedded parameter, h , h , h , and h are the nonzero auxiliary linear parameters, L , L , L , and L are the auxiliary linear operators, and H , H , H , and H are the auxiliary functions. The  nonlinear operators N , N , N , and N are written as.
Differentiating zeroth-order deformation n-times and setting q = 0, we obtain nth-order deformation equations where Using Taylor series expansion, the equation of ( , q), ( , q) ( , q) , and ( , q) with embedding parameter q can be written as The above solutions are easily found coupled equations together with boundary conditions. The methodology of the given method, the solutions of equations (14) to (19) are written as follows.

DISCUSSION
The nonlinear partial differential equations are solved by utilizing the HAM. In the present article, we have used Mathematica as a computational software. The solutions of the HAM are obtained through code of the software Mathematica for velocity, pressure rise, temperature, nanoparticles volume fraction, and solutal (species) concentration profiles, and graphical results are plotted in Origin. This section represents the analysis of different values of physical parameters for velocity, pressure rise, temperature, solutal concentration, and nanoparticle volume fraction.

Velocity distribution
The effects of Hartmann number M, Darcy number Da, thermal Grashof number Gr T , solutal Grashof number Gr C, and nanoparticle Grashof number Gr F on velocity profile are representing through the Figures 3-7. Figure 3 shows a decreasing of the velocity profile when Hartmann number M is enhanced. Physically, Lorentz force acts like retarding force for the fluid flow; hence, the magnetic force is slowing down the fluid flow. Such results found useful applications in medicine, that is, it regulates the blood flow and avoid the blood clotting. The influence of Darcy number Da on velocity profile is presented in Figure 4. By enhancing Darcy number Da, then the velocity profile is increased. It is observed in Figure 4 that Physically, Darcy number Da will be provide less resistance to the fluid motion. Figures 5 and 6 are plotted to show the effects of thermal Grashof number Gr T , solutal Grashof number Gr C on the velocity. Opposite behavior can be observed in both Gr T and Gr C . The magnitude of the velocity decreases with an increasing the thermal Grashof number Gr T , and also the velocity profile increases, when higher values of solutal Grashof number Gr C . Due to facts that the thermal Grashof number Gr T satisfies the relative influence of thermal buoyancy force and viscous hydrodynamic force. Thus, increasing of Gr T cause the viscosity reduces. The opposite behavior can be seen in the case of solutal Grashof number Gr C . Figure 7 shows the effect of nanoparticle Grashof number Gr T on velocity profile. It is obvious that the velocity profile decreases with increasing the Gr F . It is due to the fact that Gr F increases the viscosity of nanoparticles reduces, which leads to reduction in velocity for M = 0.5 and Da = 0.5.  Figures 8 and 9, we observed that for higher values of Hartmann number M, the pressure rise is enhanced, whereas decay in the pressure rise with higher values of Darcy number is observed in Figure 9. Figure Figures 13 and 14, it is observed that there is enhancement in temperature profile for different values of Brownian motion parameter Nb and thermophoresis parameter Nt. The increase of Brownian motion parameter Nb cause the random motion of fluid particles that produce more heat, so there is temperature rises in the system, which can be seen in Figure 13, whereas, in Figure 14, the temperature profile increases as the fluid particles are moved away from the cold surface to hot surface by increasing the thermophoresis parameter Nt. The same behavior can be observed in Dufour effect N TC and Soret number N CT . In Figure 17, the opposite behavior can be seen, that is, the decay in the temperature profile with an increase of thermal radiation Rd. It is due to fact that the thermal radiation is inversely proportional with thermal conduction parameter T , therefore maximum heat is radiated away from the system, which leads to reduction in the heat conduction of the fluid.    Figure 22 shows that the nanoparticle volume fraction of fluid increases with increase in Brownian motion parameter because the temperature distribution is large in the case of nanofluids, which can lead to the distribution of the system. However, the opposite results are seen in the case of thermophoresis parameter  Figure 23). Figure 23 represents the nanoparticle volume fraction, and the nanoparticle volume fraction reducing when Nt is enhanced. Figure 24 described the influence of N CT on nanoparticle volume fraction; here, nanoparticle volume fraction increases when Soret parameter N CT is increased. Figure 25 shows that the influence of N TC on nanoparticle volume fraction of fluid enhanced with the enhancing of the Dufour effect N TC .

CONCLUSION
The current research work analyzes the double diffusion on peristaltic flow of nanofluid in presence of porous medium, magnetic field, and thermal radiation through asymmetric channel. The important points are listed below.
• The current research work has potential in biomedical, engineering, and industrial applications.
• The behavior of relaxation to retardation time on velocity and pressure rise are opposite.
• Opposite behaviors on temperature is noted for Nb and Rd.
• Behavior of Gr F and Gr C on pressure rise is too similar, that is, increases for N TC and N CT slightly enhanced the temperature of the wall surface.
• Opposite behavior of Gr T and Gr C on velocity profile and pressure rise profile.
• The behavior of Gr T and Gr C on pressure rise are similar.
• The similar behavior of Nb, Nt, N TC , and N TC on solutal (species) concentration profile. Here, solutal concentration profile enhanced with higher values of Nb, Nt, N TC , and N CT .
• Opposite behavior of Nt and Nb on nanoparticle volume fraction profile.