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Soret and Dufour Effects on MHD Boundary Layer Flow of Non Pramana – J. Phys. (2020) 94:108 © Indian Academy of Sciences https://doi.org/10.1007/s12043-020-01984-z Soret and Dufour effects on MHD boundary layer flow of non-Newtonian Carreau fluid with mixed convective heat and mass transfer over a moving vertical plate ANIL KUMAR GAUTAM1, AJEET KUMAR VERMA1, KRISHNENDU BHATTACHARYYA1 ,∗ and ASTICK BANERJEE2 1Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221 005, India 2Department of Mathematics, Sidho-Kanho-Birsha University, Purulia 723 104, India ∗Corresponding author. E-mail: [email protected], [email protected] MS received 1 August 2019; revised 24 April 2020; accepted 3 May 2020 Abstract. In this analysis, the mixed convection boundary layer MHD flow of non-Newtonian Carreau fluid subjected to Soret and Dufour effects over a moving vertical plate is studied. The governing flow equations are converted into a set of non-linear ordinary differential equations using suitable transformations. For numerical computations, bvp4c in MATLAB package is used to solve the resulting equations. Impacts of various involved parameters, such as Weissenberg number, power-law index, magnetic parameter, thermal buoyancy parameter, solutal buoyancy parameter, thermal radiation, Dufour number, Soret number and reaction rate parameter, on velocity, temperature and concentration are shown through figures. Also, the local skin-friction coefficient, local Nusselt number and local Sherwood number are calculated and shown graphically and in tabular form for different parameters. Some important facts are revealed during the investigation. The temperature and concentration show decreasing trends with increasing values of power-law index, whereas velocity shows reverse trend and these trends are more prominent for larger values of Weissenberg number. For stronger magnetic field, velocity decreases, while the temperature and concentration increase. It was also found that for shear thinning fluid the drag coefficient exhibits an increasing character when Weissenberg number increases, but for shear thickening fluid the drag coefficient shows the contrary nature. For small values of Dufour number, heat transfer rate enhances with increasing Soret number, but for higher values of Dufour number it slightly dies down with Soret number and the mass transfer rate reacts oppositely. In addition, due to increasing chemical reaction rate, the concentration and velocity decrease. Keywords. Soret and Dufour effects; magnetohydrodynamics boundary layer; Carreau fluid; mixed convection; moving vertical plate. PACS Nos 47.65.−d; 52.65.Kj 1. Introduction ids is known as magnetohydrodynamics (MHD). MHD effect has many applications in several engineering The study of fluid flow past a moving flat plate or mov- fields, geophysics and many other branches of science. ing vertical plate is very important in different types of In medical sciences, health-related issues, such as blood practical applications. Sakiadis [1] was the first person pressure, can be reduced by using external magnetic to initiate and develop the flow field caused by a flat field and cancer therapy needs the help of magnetic surface which is moving with constant velocity. Tsou field. In the last few years, many researchers and sci- et al [2] investigated both analytically and experimen- entists have realised that MHD is very important in tally the flow in boundary layer on an eternal moving the field of boundary layer theory and hence the study surface and showed that these types of problems are of MHD flows attracts their attention. On the other physically possible. hand, simultaneous impact of heat and mass transfer on The area of science in which we study the behaviour MHD-related problems are useful in several engineer- and magnetic properties of electrically conducting flu- ing problems related to electrically conducting fluids. In 0123456789().: V,-vol 108 Page 2 of 10 Pramana – J. Phys. (2020) 94:108 addition, the MHD flow over a moving surface has more a moving vertical plate with Newtonian heating was useful applications in different industrial processes, illustrated by Narahari and Ishak [23]. such as geothermal energy extractions, petroleum and Soret effect is the effect when the temperature gradi- chemical engineering problems and many metalworking ent is responsible for creating a mass flux, while Dufour processes. effect is the effect when concentration differences are In fluid mechanics, mixed convection is the phe- responsible for energy flux. The impacts of Soret and nomenon in which both forced and free convection are Dufour effects play vital roles in the area of geoscience, involved and these act simultaneously. In manufactur- in the context of density differences in the flow, in chem- ing industry, such as in manufacturing nuclear plants, ical engineering and in many other fields. Uwanta et al aircrafts, missiles etc., the role of mixed convective heat [24] considered Soret and Dufour effects on MHD fluid and mass transfer flow is seriously significant. In 1964, flow over a vertical plate moving with constant velocity. Rilley [3] studied MHD free convection. The effect of Subhakar et al [25] demonstrated the MHD convective mixed convection on a moving vertical plate with suc- flow along moving vertical plate and in the investigation tion and injection was studied by Ali and Al-Yousef [4]. the Soret–Dufour effect and heat generation–absorption The impact of exponentially decaying heat generation effect are taken into account. The impact of Soret– on boundary layer flow with convective boundary condi- Dufour number and suction parameter on heat and mass tion over a vertical plate which is moving continuously transfer for MHD boundary layer flow past a moving was studied by Makinde [5]. In addition, Makinde [6] vertical plate was investigated by Srinivasa et al [26]. studied the MHD flow of boundary layer considering Recently, Kumar et al [27] examined the MHD bound- heat and mass transfer over a moving vertical plate with ary layer flow of Casson fluid with Soret–Dufour effect a convective surface boundary condition. Yao [7] stud- over a moving vertical plate considering nonlinear ther- ied the two-dimensional, mixed convection along a flat mal radiation. plate. From the review of the previously published research In fluid mechanics, modern-day researchers and sci- articles, it is observed that considering Carreau fluid entists have focussed their attention mainly on non- model, the mixed convection problem on moving ver- Newtonian fluid flows. The fluids which satisfy New- tical plate is not discussed yet. It is also found that ton’s law of viscosity are known as Newtonian fluids, in Carreau fluid flow on a moving vertical plate, the while the fluids which do not obey this law are called heat and mass transfer investigations with the Soret and non-Newtonian fluids. Carreau fluid is one type of non- Dufour effects are unavailable in the literature. Also, no Newtonian fluid. This model shows interesting features previous study has been done to analyse the simulta- for the low shear rate as well as high shear rate. When the neous effects of Soret and Dufour numbers on mixed shear rate is low, Carreau fluid behaves like Newtonian convection flow for Carreau fluid model in the presence fluid, but for high shear rate Carreau fluid shows charac- of magnetic field over a moving vertical plate. The main ters of power law fluid. As the relation and dependency objective of this investigation is to study a mathemati- of shear stress and shear rate is highly nonlinear in Car- cal model of mixed convection heat and mass transfer reau fluid model, the analysis of the behaviour of this in Carreau fluid over a moving vertical plate with Soret Carreau fluid become more complicated with respect to and Dufour effects in the presence of thermal radiation other non-Newtonian fluids. Hashim et al [8] explored and magnetic field. the characterisation of heat transfer analysis of Carreau fluid over a streching cylinder. Olajuwon [9] illustrated the unsteady mixed convection flow of Carreau fluid 2. Carreau rheological model with heat and mass transfer over a moving porous plate in the presence of thermal diffusion and thermal radi- Carreau fluid model was introduced and fully explained ation. In recent years, many researchers and scientists by Carreau [28] in 1972. The stress tensor for Carreau investigated and explained different flow problems of model is Carreau fluid model [10–21]. The basic knowledge of thermal radiation is necessary τ =−pI + β A1. (1) for some physical and practical phenomena in engi- neering problems. In the energy conversion system at The relation between zero shear rate viscosity β0,infi- high temperature, thermal radiation plays a significant nite shear rate viscosity β∞ and apparent viscosity β is role. The effect of thermal radiation with heat and mass given by transfer on moving vertical plate was studied by Muthu- (β − β∞) ( − )/ cumaraswamy and Kumar [22]. The theoretical study =[1 + (γ)˙ 2] n 1 2. (2) of the effect of radiation on free convection flow past (β0 − β∞) Pramana – J. Phys. (2020) 94:108 Page 3 of 10 108 In eqs (1)and(2), p is the pressure, I is the identity ∂T ∂T ∂2T D k ∂2C 1 ∂q u + v = α + m T − r , tensor, A1 is the first Rivlin Ericksen tensor, n denotes ∂x ∂y ∂y2 C c ∂y2 ρc ∂y s p p the power-law index and is the material time constant. (7) 2 2 ∂C ∂C ∂ C DmkT ∂ T γ˙ = ,= ( 2). u +v =Dm + −R(C − C∞), tr A1 (3) ∂ ∂ ∂ 2 ∂ 2 2 x y y Tm y (8) Weconsider the case of infinite shear rate viscosity being very small compared to zero shear rate viscosity. Then where ν = β0/ρ is the kinematic viscosity, βT is the from eq. (2), the resulting stress tensor in eq. (1) becomes volumetric coefficient of thermal expansion, βC is the volumetric coefficient of concentration expansion, g is τ =− + β [ + (γ)˙ 2](n−1)/2 .
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