Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness

Abstract In the study, we inspect the impact of cross diffusion and aligned magnetic field on Casson fluid flow along a stretched surface of variable thickness. The differential equations explaining the flow situation have been transitioned with the succor of suited transfigurations. The solution of the problem is achieved by using bvp5c Matlab package. From the solution, it is perceived that the flow, temperature and concentration fields are affected by the sundry physical quantities. Results explored for the flow over a uniform and a non-uniform thickness surfaces. The influence of emerging parameters on the flow, energy and mass transport are discussed with graphical and tabular results. Results show that the thermal, flow and species boundary layers are uneven for the flow over a uniform and non-uniform thickness stretched surfaces.


Introduction
The convective mass and heat transfer past a stretched surface plays an essential part in modern industries for intends of reliable apparatus. The researchers showing distinct fascination on mass and heat transfer in non-Newtonian ows because of its signi cance in the recent applications and technology in thermal engineering and in addition other astrophysical and geophysical studies. Rashidi et al. [1] analytically studied the thermal radiation e ect on micropolar uid ow between porous medium. Bhattacharya et al. [2] extended this work by considering the ow towards a porous shrinking surface. The mass and heat transfer in magnetohydrodynamic ow past a at plate with heat source/sink was presented by Chamkha et al. [3]. MHD viscous ow past an in nite vertical plate with constant mass ux has been reported by Saravana et al. [4]. Alam et al. [5] illustrated the impacts of thermophoresis and variable suction on MHD mass and heat transfer ow towards an inclined plate with thermal radiation.
The e ects of cross di usion on chemically reacting MHD ow past a permeable stretched surface with Brownian motion and thermophoresis was numerically analyzed by Kandasamy et al. [6]. Unsteady liquid lm ow of pseudo-plastic nanoliquid with viscous dissipation and variable thermal conductivity was studied by Lin et al. [7]. The analytical investigation of multi and singlephase models used for the reduction of nano uid ow was studied by Turkyilmazoglu [8]. A chemical reaction and transpiration e ect on magnetohydrodynamic ow over a wedge was theoretically investigated by Kandasamy et al. [9]. An analytical investigation for chemically reacting MHD ow towards a surface was proposed by Ouaf [10]. A variable temperature e ect on mixed convection ow over a wedge was presented by Hossain et al. [11]. MHD ow and heat transfer over an isothermal sheet with chemical reaction e ect was proposed by Kabeir et al. [12]. Chemical reaction and thermal radiation e ects on MHD ow past a permeable stretched surface by considering suction was discussed by Mohankrishna et al. [13]. MHD viscous ow past an expanding surface was analytically studied by Turkyilmazoglu [14]. Sandeep and Sulochana [15] numerically studied the mixed convection micropolar uid ow towards an expanding/contracting surface with nonuniform heat source/sink. MHD heat transfer ow of a non-Newtonian uid past a shrinking surface was numerically explained by Akbar et al. [16]. A theoretical investigation on heat transfer and Carreau liquid ow was done by Jenny [17]. Mixed convection ow over a rotating cone was numerically studied by Anilkumar and Roy [18]. A new buoyancy induced model of Al-water nano uid over a parabolic region was numer-ically studied by Sandeep and Animasaun [19]. Further, they extended their work by considering the ow over a stagnation region [20]. Chankha et al. [21] discussed the e ect of thermal radiation on the ow over a wedge lled with porous medium. Cross di usion e ects on the MHD non-Newtonian uid ows over a parabolic region was presented by Kumaran and Sandeep [22]. Koriko et al. [23] studied the ow over upper at surface of a paraboloid of revolution with of Brownian motion and thermophoresis. Very recently, the reserchers [24][25][26] investigated the MHD ow over various ow geometries by considering the thermal radiation and Cattenao-Christov heat ux.
By keeping the above references in view, In this paper, we inspect the impact of cross di usion and aligned magnetic eld on magneto hydrodynamic Casson uid. The ow is considered beside a stretched surface of variable thickness. The governing partial di erential equations of the ow, heat and mass transfer are transformed into ODE's equations solved numerically by using bvp5c Matlab package. From the solution, it is perceived that the ow, concentration and temperature elds are a ected by the sundry physical quantities.

Formulation of the problem
A steady 2D ow of magnetohydrodynamic Casson uid over a slendering stretched sheet is considered. The x-axis is considered along the sheet and the y-axis is perpendicular to it. It is supposed that This study induced magnetic eld is neglected. Combined in uence of Soret and Dufour impacts are considered. An aligned magnetic eld of strength B is employed as depicted in Fig. 1 at di erent angles. In this study, m ≠ deals with the slendering sheet and m = deals with the uniform thickness sheet.
With the above assumptions, the governing equations can be expressed as (refer [27]) we now suggest the following similarity transformations: If stream function ψ be described as u = ∂ψ ∂y andv = − ∂ψ with the help of (12), (13), equations (2)-(4) converted as and the corresponding conditions are The physical quantities of engineering interest, the friction factor, local Nusselt and Sherwood numbers are Tw(x)−T∞ y using (5), (19) becomes

Discussion of the results
The set of ODEs (14)

Numerical Procedure (bvp5c)
Bvp5c is a one of the boundary value problem solver in Matlab package. The bvp5c function is used exactly like bvp4c, with the exception of the meaning of error tolerances between the two solvers. If S(x) approximates the solution y(x), bvp4c controls the residual |S (x)-f(x,S(x))|. This controls indirectly the true error |y(x)-S(x)|. bvp5c controls the true error directly.

Conclusions
The in uence of cross di usion and aligned magnetic eld on magnetohydrodynamic Casson uid is investigated theoretically along a stretched surface of variable thickness. The di erential equations explaining the ow situation have been transitioned with the succor of suited trans gurations. The solution of the problem is achieved by using bvp5c Matlab package. From the solution, it is perceived that the ow, temperature and concentration elds are af-    The heat and mass transfer rate is high in the ow over a variable thickness surface when compared to the uniform thickness surface. Casson parameter has tended to decline the heat and mass transfer rate. • Cross di usion regulates the temperature and concentration elds. • Slip parameters monitor the heat and mass transfer performance.