Soret and Dufour Effects on MHD Peristaltic Flow of Jeffrey Fluid in a Rotating System with Porous Medium

The objective of present paper is to examine the peristaltic flow of magnetohydrodynamic (MHD) Jeffrey fluid saturating porous space in a channel through rotating frame. Unlike the previous attempts, the flow formulation is based upon modified Darcy's law porous medium effect in Jeffrey fluid situation. In addition the impacts due to Soret and Dufour effects in the radiative peristaltic flow are accounted. Rosseland’s approximation has been utilized for the thermal radiative heat flux. Lubrication approach is implemented for the simplification. Resulting problems are solved for the stream function, temperature and concentration. Graphical results are prepared and analyzed for different parameters of interest entering into the problems.


Introduction
The flow induced by travelling waves along the channel walls has accorded the attention here due to its significance in physiological and industrial applications. Many biological ducts for example the digestive system [1] and the ureter [2], convey their fluid contents by peristalsis. It is also employed for blood transport in capillaries, chyme motion in the gastrointestinal tract, intrauterine fluid motion, movement of ovum in the female fallopian tube etc. In the industry, this mechanism is adopted for controlled transport of fluids inside the tracts. To prevent the blockage and to keep apart the fluid contents from the tract boundaries, the peristalsis also used to provide additional pumping to the flows in heart-lung machines, artificial heart pumps etc. There is ample information on peristalsis now but some recent developments on the topic may be seen through the studies [3][4][5][6][7][8][9][10][11][12].
Heat transfer involved many complicated processes in tissues such as heat conduction in tissues, heat convection due to blood flow through pores of the tissues, metabolic heat generation and external interactions such as electromagnetic radiation emitted from electronic devices. The combined effect of heat and mass transfer is mostly useful in the chemical industry and in reservoir engineering in connection with thermal recovery process and may be found in salty

Modeling
We examine the peristaltic motion of an incompressible Jeffrey liquid in a compliant wall channel. An incompressible liquid saturates the porous space between the flexible walls of channel. Fluid is electrically conducting due to uniform applied magnetic field of strength B 0 Induced magnetic field subject to low magnetic Reynolds number is neglected. Electric field contribution is not taken into account. Effects of Soret and Dufour and thermal radiation are retained. The whole system is in a rotating frame of reference with constant angular velocity O. Flow configuration is presented in Fig 1. The channel walls are taken at z = ±η Shapes of the travelling waves are described by the following expression: where a depicts the wave amplitude, t the time, d the half width of channel, λ the wavelength and c the wave speed. The fundamental equations governing the present flow and heat/mass transfer are represented by in which ρ shows the fluid density, p the pressure, I the identity tensor, S the extra stress tensor, A 1 the first Rivlin Erickson tensor, τ the Cauchy stress tensor, O = Ok the angular velocity, C p the specific heat at constant volume, κ the thermal conductivity, T the temperature of fluid, D the coefficient of mass diffusivity, K T the thermal diffusion ratio, C s the concentration susceptibility, C the concentration of fluid and T m the mean temperature of fluid, η the dynamic viscosity, λ 1 the ratio of relaxation to retardation times and λ 2 the retardation time.

Exact solutions
Solving Eqs (32) and (33) we have the following relations of stream function and secondary velocity Making use of Eqs (34) and (35) into Eqs (28) and (29) and solving the resulting expressions through lubrication approach we have With B 1 ¼ a À 2ib; Heat transfer coefficient at the wall is given below in which A 4 ! A 15 , B 13 ! B 23 , F 1 !F 10 can be calculated algebraically.

Discussion
The main objective here is to predict the impact of sundry parameters on the velocity, temperature and concentration profiles. The theme of present study is to analyze the influence of rotation in the presence of Soret and Dufour effects. Here  Fig 2a shows that velocity enhances when elasticity parameters E 1 and E 2 are increased. There is decrease in velocity for larger E 3 , E 4 and E 5 . Since E 1 and E 2 represent the elastic parameters therefore increasing elasticity offers less resistance to the flow and hence velocity increases. On the contrary the wall damping creates a resistive type force and so the velocity decreases when E 3 increases. Similar behavior is noticed for E 4 and E 5 in the presence of damping. Velocity enhances when the value of K 1 is increased (see Fig 3b). Porosity parameter depends on the permeability parameter K. Increase K 1 in leads to the higher permeability parameter. Ultimately the velocity thus increases through larger K 1 .  shows that the secondary velocity ν increases for larger λ 1 . Similar effect is shown by K 1 which is observed in Fig 5b. Impact of different parameters on temperature profile can be seen from Figs 6-8. It is a known fact that temperature is the average kinetic energy of particles which in turn depends on the velocity. Therefore an increase in temperature θ is noticed for increasing values of E 1 and E 2 . On the contrary decrease in temperature is noticed for increasing values of E 3 , E 4 and E 5 (see Fig 6a). Fig 6b reveals that an increase in T ' causes decrease in θ. It is noticed that temperature enhances when we increase Sr and Du(see Fig 7a and 7b). In fact for increasing Sr and Du the thermal diffusion is increased and consequently the temperature enhances. Physically the diffusion-thermo or Dufour effect defines a heat flux produced when a chemical system    undergoes a concentration gradient. These effects depend upon thermal diffusion which is though very small, but sometimes become substantial when the partaking species differ by molecular weights. Mass diffusion follows by the uneven distribution of species creating a concentration gradient. A temperature gradient can also work as a driving force for mass diffusion called thermo-diffusion or Soret effect. Therefore the higher the temperature gradient, the larger the Soret effect. Fig 7c reveals that temperature increases when the value of λ 1 is increased. In fact higher λ 1 corresponds to larger relaxation time which provides more resistance to the fluid motion and thus the temperature profile enhances. As by increasing the value of porosity parameter K 1 the permeability of the medium increases which accelerates the fluid and thus temperature enhances (Fig 7d). Fig 8a and 8b display that temperature is decreasing function of Sc and R. Influence of Eckert number Ec on θ is displayed in Fig 8c. It is depicted from this Fig. that temperature enhances by increasing Ec. The heat generation due to internal friction caused by the shear in the flow is the reason behind such increase. Similar behavior is observed for Prandtl number Pr (Fig 8d).    values of λ 1 and K 1 . These Figures witness that ϕ decreases by increasing λ 1 and K 1 . Fig 11a  depicts that the concentration profile decreases when Sc increases. As Schmidt number is defined as the ratio of momentum diffusivity (viscosity) to mass diffusivity. Therefore increasing Sc decreases the mass diffusion which in turn reduces the concentration. For larger R concentration increases (see Fig 11b). It can be observed through the graphical results that concentration field have opposite effect when compared with temperature.
Behavior of heat transfer coefficient Z for various parameters is shown in the Figs 12-15. The heat transfer coefficient is represented by Z(x) = η x θ z (η) which defines the rate of heat transfer or heat flux at the walls. As expected Z shows an oscillatory behavior which is because of the propagation of sinusoidal waves along the channel walls. Fig 12a and 12b explore the effect of wall properties and T ' on ϕ. It can be noticed that there is an increase in rate of heat transfer for E 1, E 2 and T ' whereas decrease in the heat transfer rate is observed for E 3, E 4 and E 5.  Effects of Sr, Du, λ 1 and K 1 on Z can be observed through Figs 13-14 respectively. By increasing these parameters the absolute value of Z increases. Effect of radiation parameter R can be seen in Fig 15a. Here increase in R enhances Z. It is clear from Fig 15b that

Conclusions
Soret and Dufour effects in peristaltic motion of Jeffrey liquid in a channel with thermal radiation and porous medium are discussed in a rotating frame. The key findings of present study can be listed below.
1. Behaviors of wall parameters and Taylor number on axial and secondary velocities are opposite.