Dufour and Soret effects on pulsatile hydromagnetic ﬂow of Casson ﬂuid in a vertical non-Darcian porous space

. This article aims to inspect the pulsating hydromagnetic slip ﬂow of Casson ﬂuid in a vertical porous channel with heat and mass transfer. The ﬂuid is injected into the channel from the left wall and removed at the opposite wall with the same velocity. The impact of non-Darcy, Soret, and Dufour effects are taken under consideration. The governing partial differential equations (PDEs) are converted to ordinary differential equations (ODEs) using perturbation method and solved by utilizing 4th-order Runge–Kutta (R–K) technique together with shooting method. The impact of dissimilar parameters on ﬂow, heat and mass transfer characteristics are displayed and discussed.


Introduction
Studies pertaining to the MHD flows of non-Newtonian fluids in a porous medium are important because of its applications in irrigation problems, process of petroleum, heatstorage beds, paper, textile, and polymer composite industries. The most famous among these fluids is Casson fluid. Casson fluid model was introduced by Casson [4] for prediction of the flow behaviour of pigment-oil suspensions in lithographic varnishes. We can characterize Casson fluid as a shear thinning liquid which is considered to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear. The examples of Casson fluid are concentrated fruit juices, honey, tomato sauce, jelly, and human blood [8,10,14,17]. Chamkha [7] investigated the hydromagnetic fully developed laminar mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions in the presence or absence of heat generation or absorption effects. Loganathan and Deepa [21] researched the EMHD flow of the Casson fluid on a permeable Riga-plate. Pantokratoras [25] investigated the natural convection of non-Newtonian power-law fluids over a vertical plate. Walawender et al. [33] employed Casson model for portraying blood flow curves.
Pulsatile flow in a porous pipe or a channel is an important study due to its application in both engineering systems (microelectromechanical systems or MEMS, pulse combustors, IC engines, and reciprocating pumps) and natural systems (vascular diseases, respiratory systems, circulatory systems). Especially, the pulsatile flow in a porous channel is significant in the dialysis of blood in artificial kidneys [1,3,16,19,26]. Chamkha [6] examined the problem of flow and heat transfer of two electrically conducting and heat generating or absorbing immiscible fluids in a vertical infinitely long channel in the presence or absence of a porous medium and applied magnetic field. Kumar et al. [20] analyzed the problem of fully-developed convective flow of a micropolar and viscous fluid between parallel plate vertical channels with asymmetric wall temperature distribution. Celli and Kuznetsov [5] investigated the pulsatile viscous flow inside a horizontal infinitely wide channel. Haddad et al. [12] examined the pulsating laminar and incompressible fully developed pipe flows. Srinivas et al. [30] studied the pulsatile flow of hydromagnetic Casson fluid in a porous channel. Wang [34] illustrated the pulsating flow in a porous channel.
The fluids displaying boundary slip are significant in technological applications like inertial cavities and polishing of artificial heart valves [22,27,35]. The study of convective boundary condition is of extraordinary significance due to its application in engineering and industrial processes like material drying and transpiration cooling process [9,31]. Malathy et al. [23] have elucidated the influences of slip and thermal radiation on MHD pulsatile flow of an Oldroyd-B fluid in a porous space with convective boundary condition. Sayed et al. [28] studied the peristaltic transport of nanofluid in an inclined asymmetric channel in the presence of slip and convective boundary conditions. The energy flux caused by a concentration gradient was found in 1873 by Dufour and was correspondingly named the Dufour effect. It is additionally known as the diffusionthermo effect. Then again, mass flux is able to be made by a temperature gradient as was recognized by Soret. This is the thermal-diffusion effect. Dufour and Soret effects assume a significant role when large density contrasts exist in the flow regime. Radiation, Dufour, and Soret effects on MHD flows emerge in applied physics and numerous areas of engineering like catalytic reactors, heat insulation, geothermal systems, MHD generators, and drying technology [13,24,29]. Dzulkifli et al. [11] numerically discussed the Dufour and Soret parameters on the boundary layer flow in nanofluid through shrinking/stretching sheet. Khan et al. [15] discussed the cross diffusion effects on Carreau-Yasuda fluid flow over a porous stretchable surface. Umavathi and Chamkha [32] examined the stability analysis of cross diffusion when a nanofluid saturated with porous space was filled in a horizontal channel. Recently, Kumar and Srinivas [18] performed the simulation for pulsation flow of hydromagnetic Casson fluid in a vertical channel with Dufour and Soret effects.
The literature reveals that no study related to hydromagnetic pulsatile Casson fluid in a vertical permeable channel has been explored so far. Inspired by the past investigations [1,23,26,34] and keeping in the perspective on wide applications, we made an endeavour to portray the cross diffusion impacts on pulsating hydromagnetic slip flow of Casson fluid in a vertical porous channel with convective boundary. The coupled PDEs are converted to ODEs using perturbation method and solved by utilizing 4th-order R-K technique together with shooting method.

Formulation of the model
Consider the laminar and incompressible pulsating flow of Casson fluid between two vertical parallel walls at a distance h. The strength B 0 of a uniform magnetic field is applied opposite to the flow direction. We assume that the plates are very wide and very long, so that the flow is essentially axial. So that onlyx-component ofû of the velocity does not vanish. The condition of fully developed flow implies that ∂û/∂x = 0. Since the velocity is solenoidal, we obtain ∂v/∂ŷ = 0. As a consequence, the velocity component v is constant in any channel section and is equal to zero at the channel walls, sov must be vanishing at any position. Theŷ-momentum balance equation can be expressed as ∂p/∂ŷ = 0 (see [2,12,19]). The slip parameter, Joule heating, convective boundary, Dufour and Soret effects are considered. A Cartesian coordinate system is taken so that thex-axis is taken along the flow direction (vertical), andŷ-axis is orthogonal to the walls (see Fig. 1). The channel walls possess the characteristics of convective-type boundary condition. The left and right walls maintain temperatures are T 0 , T 1 ( > T 0 ), and concentrations are C 0 , C 1 ( > C 0 ), respectively. The convective boundary conditions at the left wall and the right wall are −κ∂T /∂ŷ = h f (T − T 0 ) and −κ∂T /∂ŷ = h f (T − T 1 ), respectively. The fluid is injected into the channel from the left wall with a velocity v 0 and removed at the opposite wall with the same velocity. The stress and strain relationship is designed as (Kumar et al. [19], Loganathan and Deepa [21]): The (a, b) th component of the deformation rate and shear stress tensor are e ab and τ ab , respectively. Pŷ is yield stress, π ( = e 2 ab ) is the product of shear rate components, π c is the critical value of π, µ B is the plastic dynamic viscisity. The governing equations are (Adesanya et al. [1], Radhakrishnamacharya and Maiti [26]) where κ is the thermal conductivity, µ is the dynamic viscosity, σ is the electrical conductivity, ω is the frequency, ν is the kinematic viscosity, ρ is the density of the fluid,T ,Ĉ are the temperature and concentration of the fluid,p is pressure, Φ and k are the porosity and permeability of porous medium, βĈ is the coefficient of concentration expansion, D is the coefficient of mass diffusivity, k 1 is the 1st-order chemical reaction rate,t is time, βT is the coefficient of thermal expansion, g is the acceleration due to gravity, β = µ B √ 2π c /Pŷ is the Casson fluid parameter, T m is the mean temperature of the fluid, k T is the thermal diffusion ratio, C s is the concentration susceptibility, q r is the radiative heat flux, Q 0 is the coefficient of heat source/sink, C b is the form of drag coefficient, C p is the specific heat at constant pressure. Thermal radiation is simulated utilizing the Rosseland approximation (Makinde et al. [22]), and as per this, q r is specified by where χ andσ are the Rosseland mean absorption coefficient and Stefan-Boltzmann constant. Assuming an adequately small temperature difference in the flow and expandinĝ T 4 by Taylor's series about T 0 , we getT 4 ∼ = 4T 3 0T − 3T 4 0 (higher-order terms are neglected). In view of q r andT 4 , Eq. (2) becomes The boundary conditions are (Malathy et al. [23], Xinhui et al. [35]) where α and h f are slip coefficient at the surface of the porous walls and heat coefficient, respectively.
The following dimensionless quantities are invoked: Using Eq. (5), Eqs. (1), (4), and (3) transformed to where U is the characteristic velocity, θ, φ are dimensionless temperature and concentration, is the Soret number, γ = k 1 h 2 /ν is the chemical reaction parameter, Sc = ν/D is the Schmidt number, Du = Dk T (C 1 − C 0 )/[C s C p ν(T 1 − T 0 )] is the Dufour number, and K 1 = The corresponding boundary conditions are where B i = h f h/κ is the heat transfer Biot number, and L = √ k/(αh) is the slip parameter.
It is noted that the system of ODEs (10) of B i includes a higher degree of convective cooling at the channel walls, subsequently inflicting lower temperature at the channel walls and additionally within the bulk fluid. It is predictable that as B i → ∞, the convective boundary conditions will turn into the prescribed wall temperatures. Figure 3(b) depicts the effect of thermal radiation parameter on θ. It is observed that θ increases with an enhancement of Rd . This phenomenon can be ascribed to the physical fact that the thermal boundary layer thickness rises with an enhancing Rd . The variation of temperature with respect to the Dufour number is shown in Fig. 3(c). It is seen that there is an enhancement in temperature with an enhancing of Du. Actually, Du is associated in energy flux caused by a concentration gradient.  Subsequently, bigger concentration gradient cause to rise the temperature. Figure 3(d) reveals that an increase in Sr results in decrease of temperature distribution. Figure 4 illustrates the influences of Sc, Sr , and γ on concentration distribution. Figure 4(a) exhibits that the concentration profile decreases for given rise in Schmidt number. This is observed due to a rise in Sc that it turn makes the concentration boundary layer thinner than momentum boundary layer. Figure 4(b) depicts the influence of Soret number on φ. One can infer that φ is enhanced with increasing Sr . Figure 4(c) depicts γ effect on concentration. It is seen that the concentration falls with a rise in the destructive chemical reaction (γ > 0). The contrary pattern can be noticed for the case of generative chemical reaction (γ < 0). The effect of Du and Sr on the temperature and concentration distributions is shown in Figs. 5(a)-5(b). Figure 5(a) elucidates that a rise in Du with a decrease in Sr rises the thermal boundary layer growth. It is noticed from Fig. 5(b) that a decrease in Sr with a rise in Du has the tendency to decrease the concentration distribution. Physically, Soret effect reports that mass flux is made once a system is underneath a temperature gradient. Further, a rise in Du improves the convention velocity over combined influences of thermal and solutal buoyancy forces that leads to enhance the heat transfer but fall the mass transfer of the fluid.
The present numerical values corresponding to the Nusselt number compared with the previously published numerical results of Kumar et al. [19] are shown in Table 1. This comparison shows that the present results in limiting case are in good agrement with the  published results. Table 2 compares the findings obtained by an analytical method (double perturbation) with the results obtained by a numerical method (4th-order R-K method together with shooting technique) to verify the validity of the current model. Table 3 shows the variations in Nu and Sh for different values of Fs, β, B i , Du, Sr , and Sc. The Nusselt and Sherwood numbers at the left and right walls are denoted by Nu 0 , Nu 1 , and Sh 0 , Sh 1 , respectively. It is noticed that Nu falls with a rise in Forchheimer number at the left wall, while it rises at the right wall. The Nusselt number rises at both walls by rising Sr , whereas it falls at both walls by rising Du. One can observe that at the left wall, Nu enhances with a rise in the β, B i , and Sc, while it falls at the right wall. One can infer that at both walls, Sh rises by rising β and Sr , while it falls at both walls by increasing Fs. It can be noticed that Sh decreases with an increase in B i and Sc at the left wall, whereas it rises at the right wall. The reverse trend can be seen for the case of Dufour number.

Conclusions
An analysis is made on pulsating MHD slip flow of Casson fluid in a vertical non-Darcian porous space with convective boundary condition. The considered investigation is significant as flow of Casson fluids (drilling muds, greases, clay coating, certain oils, numerous impulsions, and blood) in porous channel are utilized in modelling biological and science research. The analytical and numerical solutions are constructed for flow variables. The salient points of this investigation are as follows: • The velocity increases with an increasing slip and Casson fluid parameter, while it falls with an enhancing Forchheimer number. • It is noticed that the temperature rises with an enhancing Dufour number and radiation parameter, while it falls with an enhancing Soret number.