Soret and Dufour effects on MHD peristaltic transport of Jeffrey fluid in a curved channel with convective boundary conditions

The purpose of present article is to examine the peristaltic flow of Jeffrey fluid in a curved channel. An electrically conducting fluid in the presence of radial applied magnetic field is considered. Analysis of heat and mass transfer is carried out. More generalized realistic constraints namely the convective conditions are utilized. Soret and Dufour effects are retained. Problems formulation is given for long wavelength and low Reynolds number assumptions. The expressions of velocity, temperature, heat transfer coefficient, concentration and stream function are computed. Effects of emerging parameters arising in solutions are analyzed in detail. It is found that velocity is not symmetric about centreline for curvature parameter. Also maximum velocity decreases with an increase in the strength of magnetic field. Further it is noticed that Soret and Dufour numbers have opposite behavior for temperature and concentration.


Introduction
Undoubtedly the peristalsis of fluids in a channel is useful in several applications in engineering and biomechanics. The importance of topic can be recognized by its numerous physiological and industrial applications about swallowing food through esophagus, capillaries and arterioles, in the vasomotion of venules, in sanitary fluid transportation, toxic liquid transport in the nuclear industry, locomotion of worms, roller and finger pumps etc. Peristalsis is a radially symmetrical contraction and relaxation of muscles so as to propagate in a wave down a tube, in an anterograde way. In digestive tract (for instance the human gastrointestinal tract) a peristaltic wave is produced due to smooth muscle tissue contraction which forces a ball of food (entitled a bolus as in the upper gastrointestinal tract, in the esophagus and chyme in the stomach) across the tract. A non-Newtonian fluid model for the study of peristasis in a nonuniform rectangular duct has been investigated by Ellahi et. al [1]. Another useful article, peristaltic flow with thermal conductivity of water with copper nanofluid is done by Akbar et. al [2]. Peristaltic transport of magnetohydrodynamic (MHD) physiological fluids are important a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 in medicine and bioengineering. In fact electric current is induced due to movement of conducting liquid across the magnetic field. Fluid flow is modified because of mechanical forces arising in view of magnetic field on these currents. MHD compressor operation, blood pump machines, design of heat exchangers, flow meters, power generators, radar systems etc are based upon MHD principles. Such principles in bioengineering and medical sciences have been utilized for targeted drugs transport, bleeding reduction during surgeries, magnetic devices development for cell separation, magnetic tracers development, hyperthermia etc. Magneto therapy largely involve MHD non-Newtonian materials. In particular, the MHD peristaltic flows have relevance for problems about urinary tract, cells and tissues behavior modification and cure of gastrointestinal motility related disorders. Having all such in mind, some advancements have been made for peristaltic flows of MHD fluids in a channel (see [3][4][5][6][7][8][9][10][11][12][13][14][15]). It should be noted that much alteration in the past has been focused to such flows using constant applied magnetic field. Distinct from the previous studies are in present attempt will deal with the peristalsis through non-uniform applied magnetic field in the radial direction. We believe that such consideration in the peristaltic flows of physiological fluids is more realistic.
Impact of heat transfer in peristaltic transport of fluid is quite significant in food processing, oxygenation, hemodialysis, tissues conduction, heat convection for blood flow from the pores of tissues and radiation between environment and its surface. Mass transfer is useful in the aforementioned processes. Especially mass transfer cannot be under estimated when nutrients diffuse out from the blood to neighboring tissues. Further mass transfer involvement is quitee prevalent in distillation, chemical impurities diffusion, membrane separation and combustion process. It should be noted that relationships between fluxes and driving potentials occur when both heat and mass transfer act simultaneously. Here temperature gradient generates energy flux. However mass flux and composition gradients are due to temperature gradient (which is called Soret effect). Thermal diffusion (or Dufour effect) is the energy flux induced by concentration gradient. Although sizeable information exists about peristaltic flows in presence of heat and mass transfer but Soret and Dufour effects are less emphasized (see [16][17][18][19][20][21][22][23][24][25]).
Previous literature on the topic witnesses that peristaltic flows of fluids in a curved geometrical configurations have been scarcely examined. To our knowledge there are only few attempts [26][27][28][29][30][31][32][33][34] which address this aspect. Also there is not any attempt available which investigates the effect of radial magnetic field on the peristaltic flow of Jeffrey liquid in a curved complaint wall channel. The objective here is to address this problem. Thus relevant equations are modeled. These equations are then reduced subject to lubrication approach. The resulting problems for heat and mass transfer in a curved channel with convective conditions are also considered. Function formulation are adopted. Stream results for stream function, temperature, concentration and heat transfer coefficient are obtained and discussed. Streamlines are plotted and studied.

Problem development
Consider an incompressible Jeffrey liquid in curved channel of thickness 2d and mean radius R Ã . The wave is propagating along the walls of channel with velocity c. Let u(r, x, t) and v(r, x, t) represent the components of velocity in the axial x and radial r directions respectively (see refs. [28] and [29]). The peristaltic wave shape is represented by where a is the amplitude of wave, λ the wavelength and t the time. The displacements of the upper and lower walls are represented by ±η respectively. Further in radial direction we applied a magnetic field B by which the fluid is electrically conducting. The magnetic field in radial direction is defined by where B 0 shows the magnetic field strength and e r is unit vector in the radial direction. Utilization of Ohm's law gives the following expression in which J represents the current density, σ the electrical conductivity, u the velocity component in axial direction and e x corresponds to unit vector in axial direction. The constitutive equations for Jeffrey model are given by [9]: and in component form, S can be written as follows: where τ is the Cauchy stress tensor, p the pressure, I the identity tensor, S the extra stress tensor, A 1 the first Rivlin Ericksen tensor, μ the fluid dynamic viscosity, β ratio of the relaxation to retardation times and λ 3 the retardation time.
The governing equations in absence of body forces are given by Equation of Continuity: Component of momentum equation in radial direction: Component of momentum equation in axial direction: Energy equation comprising viscous dissipation and Dufour effects: Concentration equation with Soret effect is In above equations ρ the density of fluid, ν the kinematic viscosity, T the fluid temperature, C the concentration of fluid, T 0 and C 0 the temperature and concentration at the lower and upper walls respectively, c p the specific heat at constant pressure, C S the concentration susceptibility, D the mass diffusivity coefficient, k T the thermal diffusion ratio and κ t the thermal conductivity. Here L = rV and T m the mean fluid temperature.
The convective boundary conditions for the exchange of heat and concentration, no slip condition and compliant nature of the walls are described through the expressions In above expressions h 1 and h 2 are the heat and mass transfer coefficients at the upper and lower walls of the channel respectively, τ the elastic tension in the membrane, m Ã 1 the mass per unit area and d 0 the coefficient of viscous damping.
On setting velocity in terms of stream function in polar cordinates and using the non-dimensional variables The long wavelength and low Reynolds number assumptions are commonly used in the analysis of peristalsis flow [35][36][37][38]. Using this approach Eqs (5)-(10) becomes with the dimensionless conditions In above equations, δ corresponds to the dimensionless wave number, Re the Reynolds number, Pr the Prandtl number, H the Hartmann number, E i (i = 1 − 3) the non-dimensional elasticity parameters, E the Eckert number, Br the Brinkman number, Sr the Soret number, Du the Dufour number, the amplitude ratio parameter and Bi{ð^¼ 1; 2Þ the heat and mass transfer Biot numbers respectively. The values of these parameters can be defined as follows: Through Eqs (17) and (18) we have @ @r where non-dimensional tensor is

Method of solution
The closed form solutions of Eqs (20), (21) and (28) are and the heat transfer coefficient is defined by and Z are mentioned in appendix.

Results and discussion
In this section the results of velocity, temperature, concentration, heat transfer coefficient and streamlines are discussed physically.

Velocity profile
The impacts of elasticity parameters E 1 , E 2 and E 3 on the velocity distribution is presented in  11). The behavior of Hartmann number H is decreasing (see Fig 12).
Since temperature and velocity are in direct relation so similar results are obtained. It is seen from Fig 13 that Fig 23). However β increases the heat transfer coefficient near centerline and it decreases along the boundary walls (see Fig 24). Fig 25 elucidates    Brinkman number Br. It is because of the fact that the concentration distribution is higher when compared to straight channel in the upper half portion.

Concluding remarks
Peristaltic flow of Jeffrey fluid in curved channel with compliant walls is studied in the presence of convective boundary conditions and radial magnetic field effect. The major findings of presented analysis are listed below: • Similar response of wall parameters on the velocity, temperature and heat transfer coefficient is noticed inside the curved channel.
• Temperature is increasing function of Brinkman, Soret, Dufour, Prandtl, and Schmidt numbers.
• Temperature decreases for larger heat and mass transfer Biot numbers.
• Concentration drops with the increase in Soret, Dufour, Prandtl, Schmidt and Brinkman numbers and it increases for heat and mass transfer Biot numbers.
• Opposite effects of Hartmann number and ratio of relaxation and retardation times parameter β are observed for the velocity, temperature, concentration and heat transfer coefficient.
• Similar effects of curvature parameter are seen on the velocity, temperature, concentration and heat transfer coefficient.
• Behavior of streamline pattern is increasing with ratio of relaxation and retardation times parameter. • The Hartmann number on streamline pattern has opposite effects when compared with the ratio of relaxation and retardation times parameter.