DIFFUSION-THERMO AND THERMAL-DIFFUSION EFFECTS ON RIVLIN-ERICKSEN ROTATORY CONVECTIVE FLOW PAST A POROUS VERTICAL PLATE

Diffusion-thermo and thermal-diffusion effects on unsteady, incompressible Rivlin-Ericksen rotatory convective flow of a magnetic conducting electrical fluid with time dependent suction between two vertical plates of which one is permeable are investigated. The uniform angular velocity rotates about an axis normal to the plate. The equations governing the flow model are non-dimensionalised, perturbed for simplification and solved by Adomian decomposition method. Graphical illustrations of the fluid parameters on velocity, temperature, concentration are presented and discussed. The effect of skin-friction, Nusselt and Sherwood numbers are presented in tabular forms and it is discovered from the results that a rise in thermal-diffusion parameter speedup the skin-friction, while increasing diffusion-thermo parameter slowdown the skin-friction.


INTRODUCTION
Forced and free convection mechanisms contribute significantly to heat transfer. The phenomenon occurs in both industrial and technical problems such as solar collectors, in cooling of electronic devices and nuclear reactors resulting in an emergency shutdown etc. The significance of these applications led some researchers to study natural, forced and mixed convective flows in the presence of heat and mass transfer. Deepthi and Prasada (2017) considered heat and mass transfer with mixed convective flow in the presence of radiation and Soret. In the investigation, rotatory and Dufour effects were considered insignificant. The result shown that a rise in Soret parameter decreased the heat and mass transfer rate on the walls. Soret effect on mixed convection viscoelastic fluid flow in the presence of heat and mass transfer was studied by Devasena and Ratmat (2014). The effects of Dufour and thermal radiation were not considered. Dada and Agunbiade (2016) examined the effects of chemical reaction and radiation on convective non-rotatory Rivlin-Ericksen fluid flow in a vertical porous plate. It was discovered that temperature and velocity decreased as radiation parameter increased. Aruna et al. (2015) investigated the influence of both thermal-diffusion and diffusion-thermo of non-rotatory mixed convective hydromagnetic fluid flow through a vertical wavy porous plate. The finite difference method was used to obtain the solution.
However, the study of rotating medium is of great importance in fluid dynamics as a result of its relevance in many natural phenomena and its applications in technology relating to Coriolis force. Some of the applications of rotating flow, particularly in porous media in the field of engineering, to mention but a few are rotating machinery, food and * Corresponding author. Email: dadamsa@gmail.com chemical processing industries. The study of rotating flow has gained the interest of many researchers due to its importance. Sibanda and Makinde (2010) examined steady MHD flow with heat transfer as a result of rotating disk in a porous fluid in the presence of viscous dissipation. Mutua et al. (2013) considered MHD free convection flow of a Newtonian fluid with variable suction through porous plate and the result revealed that skin friction increased both along x and y axes due to a decrease in rotation parameter.
In addition, Singh (2013) studied thermal radiation effects on rotatory viscoelastic MHD flow via a vertical plate. It was reported that rotation parameter enhanced velocity profiles. Oldroyd-B Rotating MHD radiative fluid through a vertical porous channel was carried out by Garg et al. (2014b). Guria and Jana (2013) examined rotatory viscoelastic fluid past a porous plate under a uniform suction. It was discovered that the presence of viscoelastic parameter contributed to the increase in the plate heat transfer. Abdulmaleque (2017) investigated the effects of temperature dependent suction/injection on non-Newtonian casson radiative fluid flow with viscous dissipation. Also, Garg et al. (2014a) presented oscillatory viscoelastic fluid flow through a porous rotating vertical channel with an assumption of an optically thin radiation and constant suction. The result showed that as the rotation parameter increased, the velocity decreased. Even though, the above investigations had contributed to the studies of fluid flow but the effects of chemical reaction was neglected in the studies and chemical reactions have tremendous impacts in changing the rate of mass diffusion.
In fluid flow that involves both heat and mass transfer, driving po-tentials and the fluxes relation are significantly noticed. The energy flux that is generated due to concentration gradient is referred to as diffusionthermo, while mass flux resulting from temperature gradients is thermaldiffusion. Mostly, the effects of diffusion-thermo and thermal-diffusion are often neglected in most studies on the bases that they are of low magnitude in relation to the rest chemical species. The effects of Dufour and Soret become significant phenomena in areas like petrology, hydrology, geosciences, etc. The effect of thermal-diffusion is relevant, for example, in the separation of isotope and mixture of gases that has light molecular weight. Therefore, Sarma and Govardhan (2016) reported on the effects of thermal-diffusion and diffusion-thermo on natural convection heat and mass transfer with thermal radiation in the presence of viscous dissipation in a porous medium. A Newtonian fluid was examined in the study and finite difference method was used in the computations of the results. It was reported that velocity profiles was accelerated by increase in viscous dissipation. The effects of thermal-diffusion and diffusion-thermo on free convection MHD flow of Rivlin-Ericksen fluid was examined by Reddy et al. (2016). Rotatory and thermal radiation effects were considered to be insignificant, the result shown that an increase in diffusion-thermo and thermal-diffusion speedup the skin-friction. Gbadeyan et al. (2011) examined the influence of Soret and Dufour with heat and mass transfer on mixed convective viscoelastic fluid flow past a porous medium. It was observed from the result that Soret enhanced both concentration and temperature profiles. Furthermore, Dada and Salawu (2017) presented heat and mass transfer of pressure-driven flow with inclined magnetic field. The result revealed that an increase in chemical reaction reduced both pressure and concentration profiles. Ibrahim and Suneetha (2015) studied effects of Soret and chemical reaction on MHD unsteady viscoelastic fluid past an infinite vertical plate. The study concluded that both concentration and velocity profiles increased as thermal-diffusion increased. Hayat et al. (2017) investigated Dufour and Soret effects on MHD Jeffrey fluid of peristaltic transport in a curved channel. It was observed that Dufour and Soret have opposite behaviour for concentration and temperature. Babu et al. (2017) considered diffusion-thermo and thermal-diffusion effects on heat and mass transfer MHD Jeffery fluid flow in a stretching sheet. The result revealed that temperature profiles was reduced by an increase in either Prandtl number or Soret parameter. Influence of thermal-diffusion on Kurshinshiki fluid in the presence of heat and mass transfer past a vertical porous plate was investigated by Jimoh et al. (2014). At the boundary layer, the result shown that increase in the heat sources parameter improved both velocity and temperature profiles. However, as impressive as the above studies were, rotatory Rivlin-Ericksen fluid flows have received no significant attention.
A careful examination of all the above studies on heat and mass transfer showed that combined effects of time dependence suction, pressure gradient and heat absorption in Rivlin-Ericksen convective fluid flow in a rotating medium with diffusion-thermo and thermal-diffusion have received little or no attention. Considering various phenomena, combined effects of all these parameters come into consideration in a practical flows of fluid and are of practical applications in the field of engineering, chemical processing industry, rotating machinery, paper and food processing industry, petroleum industry and other areas that involve viscoelastic fluid flow. Hence, this present study analyses the effects of diffusion-thermo, thermal-diffusion and radiation effects on convective Rivlin-Ericksen fluid in a rotating system with chemical reaction.

MATHEMATICAL ANALYSIS
Consider a non-Newtonian, two-dimensional incompressible free convective Rivlin-Ericksen flow of an electrically conducting fluid through a rotating vertical channel with a periodic suction. The following assumptions are made in the formulation of this problem: (i) an unsteady and laminar flow is considered; (ii) induced magnetic field and Hall effects are ignored due to the fact that magnetic Reynolds number and transversely applied magnetic field is considered to be very small; (iii) a magnetic field (B0) of uniform strength is perpendicularly applied to the plates; (iv) there is a rotation of the entire system through the perpendicular axis to the plates; (v) thermal-diffusion and diffusion-thermo are assumed to be of substantial magnitude, hence, they are not negligible; (vi) the plates are considered to be infinite in x * -direction, hence all physical quantities excluding pressure are functions of coordinate z * and time t * ; (vii) in the flow field, pressure is taken to be constant; and (viii) the fluid is finitely conducting with constant physical properties.
With the above assumptions, the flow chart and governing equations are as follows: Frontiers in Heat and Mass Transfer (FHMT), 11, 31 (2018) DOI: 10.5098/hmt.11.31 Global Digital Central ISSN: 2151-8629 The boundary conditions for the problem are: The time dependent suction velocity is expressed in exponential form as: Das et al. (2011)), and A is small values less than unity. qR is the radiative heat flux and is defined base on Rosseland approximation (Brewster (1972)) as: This present analysis is limited to optically thick fluid, hence Rosseland approximation is used. Considering the temperature differences within the flow to be sufficiently small, T * 4 (quartic temperature function) can be expanded using Taylor series expansion and neglecting higher order terms gives; This is substituted into radiative heat flux term that was used in Eq. (4). The pressure gradient for the fluid is considered in the form; where H is a constant and it oscillates only in x-axis direction. The accompanying non-dimensional variables are utilized to reduce the governing equations to non-dimensional form.

Adomian Decomposition method
The ordinary differential Eqs. (29)-(46), though linear but are highly coupled, hence Adomian decomposition methods is applied in solving the problem. A differential equation can be written in a general form as; where F represents an operator of nonlinear ordinary differential equation containing both linear and nonlinear terms. Lψ represents the linear term, and the invertible linear operator is L. Taking the highest-ordered derivative as L, L −1 is n-fold integration operator from 0 to η for L = d n dη n . For the linear operator L, the remainder is R and N ψ is the nonlinear term. Hence, Since L is invertible, thus The highest-order in Equations (29)-(46) is two, therefore, substituting for L −1 Lψ in Equation (51), the equation becomes; Hence, ψ can be written in series form as: also, the nonlinear term as: An (57) where Substituting Equations (56) and (57) into equation (55) gives; An dηds (59) The first three terms are identified as ψ0 which is the initial approximation, that is and An dηds (61) is the recurrence relation. All the components can be determined since A0 depends on ψ0 only, A1 depends on ψ0 and ψ1 and so on. The solution then is the n-term approximation or approximant to ψ. From Eqs. (60) and (61), the approximate solutions for Eqs (29)-(46), which converges at n = 5, can be written as: Series solutions (62) are substituted in Eqs. (17) and (28)

Skin-friction, Nusselt and Sherwood number in term of Amplitude
With reference to the boundary conditions, the amplitude is defined in terms of primary and secondary velocities for steady and unsteady flow. Therefore, total resultant velocity can be written as; where velocity is defined as The Skin-friction is given as; Nusselt number (Heat transfer coefficient) is defined as; Sherwood Number(Mass transfer coefficient) is expressed as:

DISCUSSION OF RESULTS
The solutions for the partial differential equations (13), (14) and (16) with the corresponding boundary conditions (15)        The effect of Ω on resultant velocity is seen in Fig. 15. The result revealed that, higher values of rotation parameter enhanced resultant velocity profiles, which showed an overwhelming effect of rotation. A diminishing in Rv due to a decrease in Ω is because of the presence of gravitational and Lorentz force rotating at very low speeds. This indicates that a friction factor is noticed, hence Rv decreases. The same trend is apparent in Figs. 16 and 17, which represented velocity and temperature profiles for different values of Pr. Prandtl number can be defined as the ratio of momentum diffusivity to thermal diffusivity. It is, therefore, obvious that a lower thermal conductivity material leads to high velocity and a different trend is seen for higher thermal conductivity. Hence, in Fig. 17, it is seen that an increase in Prandtl number accelerates the resultant velocity profiles. Likewise, in Figs. 16, an increase in Pr reduces the thermal boundary layer thickness and average temperature within the boundary. This implies that, an increase in Pr makes the thermal conductivity of the fluid to increase. Thus, resulting in rapid diffusivity of the heated surface. Furthermore, effect of Sc on concentration and resultant velocity profiles is revealed in Figs. 18 and 19. Here, it is observed that higher Sc leads to a decline in concentration profiles, while the resultant velocity is enhanced. . It is seen that with a rise in φ, the temperature diminishes. Thus, when heat is absorbed, the buoyancy force decreases the temperature profile. Effect of VR on resultant velocity is displayed in Fig. 21. It is evident in Fig. 21 that resultant velocity is accelerated by an increase in VR.  Effect of Dp on resultant velocity, temperature and concentration profiles is presented in Figs. 25 -27. Resultant velocity profiles diminishes as Dp is increased, while temperature profiles increases. This is as a result of the generation of energy flux that enhances the temperature. A rise in Dp makes concentration profiles to fall within 0 ≤ η ≤ 0.7. and within 0.7 ≤ η ≤ 1, a rise in concentration profile is observed. Tables 1 and 2 display the variation of fluid parameters (Kr, E, Ω, Sp, Dp and VR) on Skin-friction, Nusselt number and Sherwood Number at η = 0 and η = 1. It is seen in Table 1 that the Skin-friction is diminished with the presence of Kr, Ω, Dp and VR, while it is strengthened by E and Sp. Nusselt number is reduced with an increase in Kr, N , and Dp. On the other hand, increasing the values of Sp enhances the Nusselt number. In like manner, Sherwood number increases with an increase in chemical reaction and Dufour parameter. The mass transfer coefficient value is reduced with an increase in E and Sp. Consequently, Table 2 shows that skin friction is quickened by an increase in Kr, E, Ω, Sp and VR, while higher values of Dufour parameter decreases the Skin-friction. Nusselt number is risen with an increase in Kr and Dp but diminishes with increment in the values of E and Sp. Increasing E, Sp and Dp make Sherwood number to rise and it decelerates by increasing the values of Kr.