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 SoretÐDufour effect and heat generationÐabsorption The impact of exponentially decaying heat generation effect are taken into account. The impact of SoretÐ 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 SoretÐDufour 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 [10Ð21]. The basic knowledge of thermal radiation is necessary τ =−pI + β A1. (1) for some physical and practical phenomena in engineering 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 . pI 0 1 A1 (4) the acceleration due to gravity,√σ is the electrical con- ductivity of the fluid, B = B0/ x is the applied variable Pantokratoras [29] explained this rheological model on magnetic field, T is the temperature, C is the concen- the basis of n and he fully described the nature of this tration, α is the thermal diffusivity, Dm is the diffusion model. The power-law index n characterised the fluid coefficient, k is the thermal diffusion ratio, C is the behaviour. The flow is said to be shear thinning when T s concentration susceptibility, cp is the specific heat, Tm n < 1 and the flow is shear thickening if n > 1. It is (= / ) = is the mean fluid temperature and R R0 x is the vari- noticed that the Newtonian fluid is obtained for n 1 able reaction rate. or = 0. The radiative heat flux qr is obtained from the Rosse- The velocity, temperature and concentration fields for land approximation two-dimensional laminar incompressible flow are considered as 4σ ∗ ∂T 4 qr =− , (9) 3k∗ ∂y V = u(x, y), v(x, y), 0 , T =T (x, y), C=C(x, y). where σ ∗ is the StefanÐBoltzmann constant and k∗ is The expression for rate of strain γ˙ and boundary layer the absorption coefficient. equation for linear momentum is followed by [30]. The highest difference of temperatures inside the flow is suitably small in such a manner that it can be exhibited as a linear function of temperature. So, expanding into 3. Mathematical formulation Tylor series about T∞ and ignoring higher-order terms, we get A steady laminar viscous incompressible two- T 4 =∼ 4T 3 T − 3T 4 . (10) dimensional MHD boundary layer flow of Carreau fluid ∞ ∞ with mixed convective heat and mass transfer over a The relative boundary conditions for the velocity com- vertical plate moving with constant velocity Uw is con- ponents, temperature and concentration for the consid- sidered. The free stream temperature is T∞ and free ered model are as follows: stream concentration is C∞. It is also assumed that the temperature and concentration of the plate are Tw and u = Uw,v= 0, T = Tw, C = Cw at y=0, Cw, respectively. The x-axis is taken along the plate and u → 0, T = T∞, C = C∞ at y →∞. (11) y-axis is normal to the plate. The velocity components v u and are supposed to be along x-andy-axes respec- The following transformations are introduced: tively.

The governing equations for the above considered Uw ψ(x, y) = νxUw · f (η), η = y , problem can be expressed as νx ∂u ∂v T = T∞+(Tw−T∞)θ(η), C =C∞ +(Cw −C∞)φ(η), + = 0, (5) ∂x ∂y where ψ is a stream function which is deﬁned in the ( − )/ ∂u ∂u ∂2u ∂u 2 n 1 2 usual form as u + v = ν 1 + 2 ∂ ∂ ∂ 2 ∂ ∂ψ ∂ψ x y y y u = ,v=− . ( − )/ ∂ ∂ ∂2u ∂u 2 ∂u 2 n 3 2 y x +ν(n − 1)2 1 + 2 ∂y2 ∂y ∂y Using the above transformations, eqs (6)Ð(8) are transformed as σ B2u +gβ (T − T∞) + gβ (C − C∞) − , 2 2 (n−3)/2 2 2 T C ρ (6) ff + 2{1 + We ( f ) } {1 + nWe ( f ) } f 108 Page 4 of 10 Pramana – J. Phys. (2020) 94:108

−2 f M2 + 2λθ+ 2δφ= 0, (12) where θ + ( + )θ + φ = , ⎧ Pr f 2 1 Nr 2Pr Df 0 (13) ⎪ ∂u ∂u 2 (n−1)/2 ⎪ 2 φ + φ + θ − φ = , ⎪ τw = β0 1 + , Scf 2 2 Sc Sr 2 Sc Rc 0 (14) ⎪ ∂y ∂y ⎨⎪ 16σ ∗T 3 ∂T =− + ∞ , (17) where ⎪ qw k 1 ∗ ⎪ 3k k ∂y ⎪ ⎪ ∂C 2 3 1/2 2 ⎩ =− . Uw σ B jw Dm We= , M2= 0 , ∂y νx ρUw From eqs (16)and(17), we get skin-friction coefficient, Gr Gr gβ (Tw − T∞)x3 λ= x ,δ= c , Gr = T , local Nusselt number and local Sherwood number as 2 2 x ν2 Rex Rex follows: 3 ⎧ gβ (Cw − C∞)x / Gr = C , ⎨⎪ 1 2 = ( )[ + 2( ( ))2](n−1)/2, c 2 Rex C fx f 0 1 We f 0 ν −1/2 ∗ 3 Rex Nux =−(1 + Nr)θ (0), (18) ν 16σ T∞ ⎩⎪ − / Pr = , Nr = , 1 2 =−φ( ), α ∗ Rex Shx 0 3k k ν DmkT Cw−C∞ where Re = Uwx/ν is the local Reynolds number. Sc= , Df= , x D C c ν Tw−T∞ m s p DmkT Tw−T∞ Sr= 4. Numerical method for solution Tmν Cw−C∞ Equations (12)Ð(14) along with boundary conditions and given by eq. (15) are solved by numerical technique specified as bvp4c, a MATLAB package. The initial R0 step in this numerical technique is to convert the bound- Rc= Uw ary value problem into an initial value problem and a first-order system of ordinary differential equations are η are the local Weissenberg number, the magnetic param- obtained. We check our solution for different ∞, until eter, the thermal buoyancy parameter, the solutal buoy- we get asymptotically convergent solution with satisfy- ancy parameter, the thermal Grashof number, the solutal ing specified tolerance level. In our complete analysis, η = Grashof number, the Prandtl number, the thermal radia- we take the value of ∞ 10 and integrate the first- tion parameter, the Schmidt number, the Dufour number, order system given by ⎧ the Soret number and the reaction rate parameter, ⎪ y = y2, respectively. ⎪ 1 ⎪ = , Using eq. (11), the corresponding boundary condi- ⎪ y2 y3 ⎪ 2 − λ − δ − tions become ⎪ 2y2 M 2 y4 2 y6 y1 y3 ⎪ y = , ⎪ 3 2{1 + We2(y )2}(n−3)/2{1 + nWe2(y )2} ⎪ 3 3 ⎨⎪ = , f (0) = 0, f (0) = 1,θ(0) = 1,φ(0) = 1, y4 y5 − + − f (∞) → 0,θ(∞) → 0,φ(∞) → 0. (15) ⎪ 2Df RcScPry6 Df PrScy7 y1 y1 y5 Pr ⎪ y = ⎪ 5 2(1 + Nr − PrScDf Sr) ⎪ = , ⎪ y6 y7 The physical quantities of interest are local skin-friction ⎪ ⎪ y = coefficient, local Nusselt number and local Sherwood ⎪ 7 number, which are given by ⎪ ( + ) − ( + ) + ⎩⎪ 2 1 Nr RcScy6 Sc 1 Nr y1 y7 SrScPry1 y5 . ⎧ 2(1+Nr−PrScDf Sr) (τ ) ⎪ w y=0 (19) ⎪ C f = , ⎪ x ρU 2 ⎨⎪ w with the boundary conditions x(qw)y=0 Nux = , (16) ( ) = , ( ) = , ( ) = , ( ) = , ⎪ k(Tw − T∞) y1 0 0 y2 0 1 y4 0 1 y6 0 1 ⎪ (∞) → , (∞) → , (∞) → . (20) ⎪ x( jw) = y2 0 y4 0 y6 0 ⎩⎪ Sh = y 0 , x ( − ) Dm Cw C∞ Here, y1 = f (η), y4 = θ(η) and y6 = φ(η). Pramana – J. Phys. (2020) 94:108 Page 5 of 10 108

Table 1. Comparison of the values of −θ (0) for different values of Pr when n = 1.0, We = λ = δ = Nr = Df = Sr = Sc = Rc = 0. Pr Chen [31] Present results 0.7 0.3509 0.350893 2.0 0.6832 0.683218 10.0 1.6802 1.680241

Table 2. Computed values of skin-friction coefﬁcient, local Nusselt number and Sherwood number when λ = 0.2, δ = 0.1, Pr = 3.0, Nr = 0.5, Df = Sr = 0.2, Sc = 1.0, Rc = 0.5. 1/2 nWeMRex C fx −θ (0) −φ (0) 0.5 1.0 0.0 −0.2104787 0.5067148 0.8249067 0.5 −0.4287817 0.4571329 0.8173638 1.0 −0.8505231 0.3489916 0.8056785 1.5 0.0 −0.2195566 0.5085762 0.8251498 0.5 −0.4507626 0.4626061 0.8180942 1.0 −0.9416982 0.3701997 0.8181475 0.5 2.0 0.0 −0.1985787 0.5041799 0.8245851 0.5 −0.4019266 0.4501439 0.8164468 1.0 −0.7608425 0.3267760 0.8030944 1.5 0.0 −0.2306655 0.5107889 0.8254522 0.5 −0.4746709 0.4680944 0.8188184 1.0 −1.0097915 0.3842609 0.8096093 0.5 3.0 0.0 −0.1832577 0.5007419 0.8241623 0.5 −0.3712248 0.4418296 0.8153896 1.0 −0.6860153 0.3074941 0.8009439 1.5 0.0 −0.2446037 0.5134636 0.8258354 0.5 −0.5015577 0.4737370 0.8195613 1.0 −1.0725233 0.3959170 0.8107568

Table 3. Computed values of skin-friction coefﬁcient, local Nusselt number and local Sherwood number when We = 1.0, M = 0.5, λ = 0.2, δ = 0.1, Pr = 3.0, Nr = 0.5, Sc = 1.0, Rc = 0.5. 1/2 nSrDfRex C fx −θ (0) −φ (0) 0.5 0.0 0.0 −0.4517761 0.6694436 0.8395939 0.2 −0.4311951 0.4505399 0.8414417 0.5 −0.4015248 0.1256070 0.8440289 1.0 0.0 −0.4642420 0.6723395 0.8404371 0.2 −0.4425850 0.4533569 0.8421602 0.5 −0.4114864 0.1282524 0.8445923 1.5 0.0 −0.4761290 0.6749946 0.8412119 0.2 −0.4535289 0.4559831 0.8428269 0.5 −0.4211533 0.1307656 0.8451221 0.5 0.5 0.0 −0.4420314 0.6727937 0.6975095 0.2 −0.4250706 0.4677817 0.7782086 0.5 −0.3978975 −0.0153271 0.9802433 1.0 0.0 −0.4538593 0.6754716 0.6975364 0.2 −0.4360041 0.4706482 0.7780493 0.5 −0.4075901 −0.0120079 0.9797303 1.5 0.0 −0.4651722 0.6779351 0.6975706 0.2 −0.4465241 0.4733218 0.7779045 0.5 −0.4169967 −0.0088644 0.9792459 108 Page 6 of 10 Pramana – J. Phys. (2020) 94:108

Figure 1. Effects of M and n on velocity, temperature and concentration proﬁles for We = 1.0, 2.0 and 3.0.

5. Computational results and discussion fluids). More specifically, in the analysis the values are considered as n = 0.5, 1.0and1.5. Figure 1 displays The numerical solutions are obtained for several values the effect of magnetic parameter M and Weissenberg of different parameters and those solutions, in the form number We on velocity, temperature and concentration of velocity, temperature and concentration, for change for the aforesaid three values of n. From the figure, in one or more dimensionless flow controlling param- it is evident that with increasing values of M,there eters are exhibited through graphs. If any parameter is is diminution in velocity profiles for shear thinning, not varying in any figure, then this means that it is fixed. Newtonian and shear thickening fluids. It is addition- The fixed values of parameters are taken as We = 1.0, ally ascertained that for shear thinning fluid the velocity M = 0.5, λ = 0.2, δ = 0.1, Nr = 0.5, Pr = 3.0, decreases with growth of We, while for shear thicken- Sc = 1.0, Sr = 0.2, Df = 0.2andRc = 0.5. These ing fluids opposite trend in velocity profile is observed. values are fixed in the whole numerical computation. When the values of power-law index n increases, the To confirm the accuracy of the above-described numer- velocity enhances. It means that for shear thinning fluid ical scheme, the values of the computed local Nusselt the velocity is minimum, whereas, for shear thickening numbers are compared with those of Chen [31]forvar- fluid it is maximum for fixed value of Weand this char- ious values of Pr with n = 1.0, We = 0, M = 0, acter is more prominent when We increases, because λ = 0, δ = 0, Nr = 0, Sc = 0, Sr = 0, Df = 0 We controls the non-Newtonian behaviour of fluids. and Rc = 0intable1 and it seems that those values An increment in temperature is observed with We for are in good agreement. In this article, the influence of shear thinning and shear thickening fluids, and oppo- different parameters are discussed for all possible cases site behaviour in temperature profile is found. It is also of power-law index n, i.e., n < 1 (shear thinning fluids), noticed that the temperature decreases with rising val- n = 1.0 (Newtonian fluids) and n > 1 (shear thickening ues of power-law index n. It is additionally noticed that Pramana – J. Phys. (2020) 94:108 Page 7 of 10 108

Figure 2. Effects of Df and Sr on velocity, temperature and concentration proﬁles for n = 0.5, 1.0 and 1.5.

Figure 3. Effect of Rc on velocity, temperature and concentration proﬁles for n = 0.5, 1.0 and 1.5.

Figure 4. Effect of Nr on velocity, temperature and concentartion proﬁles for n = 0.5, 1.0 and 1.5. 108 Page 8 of 10 Pramana – J. Phys. (2020) 94:108

Figure 5. Effects of δ and λ on velocity, temperature and concentration proﬁles for n = 0.5, 1.0 and 1.5.

Figure 6. Variations in skin friction coefficient, local Nusselt number and local Sherwood number with M for different values of n and We. by increasing the magnetic parameter, temperature rises trend in concentration is noticed. In figure 2, the impacts for both shear thinning and shear thickening fluids. Fig- of Dufour (Df)andSoret(Sr) numbers on velocity, ure 1 also shows that with stronger magnetic field the temperature and concentration profiles for three distinct concentration rises for shear thinning as well as shear values of n are demonstrated. This figure depicts that for thickening fluids. It also indicates that the concentration all values of n, when the value of Df increases, veloc- decreases with increasing values of n. From the figure, ity also increases. Similar trend is noticed in velocity one can observe that for shear thinning fluid, the con- for increment in the value of Soret number (Sr). But, centration shows enhancing behaviour with increasing the influences of Df and Sr on temperature for different values of We, while for shear thickening fluid the reverse values of n reveal that temperature increases with larger Pramana – J. Phys. (2020) 94:108 Page 9 of 10 108

Figure 7. Variations in skin friction coefﬁcient, local Nusselt number and local Sherwood number with Df for various values of Sr and n.

Df and identical nature in temperature is observed for show enhancing character with increasing We and the larger values of Sr near the plate and far away from the effect of Weon skin-friction coefficient is exactly oppo- plate. For all cases of n, it is ascertained that the con- site. One can also notice that the values of Nusselt and centration decreases with amplifying values of Df.For Sherwood numbers are superior for shear thickening flu- higher values of Df without Soret effect (Sr = 0), it ids than for shear thinning fluids, while skin-friction seems that the concentration reduces slightly, whereas coefficient shows contrary nature. Finally, for stronger in the presence of Soret effect (Sr = 0.5), the effects magnetic field, i.e., for increasing M all three reduce. are different near the plate and far away from it. In fig- Figure 7 presents the influences of Df and Sr on ure 3, the influence of reaction rate parameter (Rc)on skin-friction coefficient, heat transfer rate and the mass velocity, temperature and concentration profiles for all transfer rate for three values of n. From the figure, it three values of n are demonstrated. In general, when Rc is determined that for increasing Df, the skin-friction is larger then the concentration boundary layer becomes coefficient increases and it is also so with increasing thinner and the same is obtained in the present analysis. Sr. But, the heat transfer rate reduces with increasing Due to this reason, diminution in concentration profile values of Df. It is also observed that for smaller val- is observed for increase in Rc. Decrement in velocity is ues of Df, heat transfer rate increases with increasing also detected for larger values of Rc. It is observed that values of Sr but for larger values of Df, heat transfer with higher reaction rate, the temperature increases ini- rate decreases with increasing Sr. The figure also reveals tially but for larger η, it shows opposite behaviour with that for small values of Df, the local Sherwood number, increasing values of Rc. The impact of thermal radia- i.e., the mass transfer rate decreases with increment in tion on velocity, temperature and concentration profile is the value of Sr, but for larger values of Df, it increases exhibited in figure 4. Larger values of radiation param- with increasing Sr. The variations in local Sherwood eter (Nr) offer additional heat to operating fluid which number for changes in n are different with and without produces associated growth in the temperature field. It Soret effect. The evidences of the above-described vari- is also clear that with rise in Nr, velocity increases ations in skin-friction coefficient, local Nusselt number while concentration profile shows opposite behaviour. and local Sherwood number can also be confirmed from Figure 5 displays how velocity, temperature and concen- tables 2 and 3. tration are affected by the variation in thermal buoyancy parameter λ and solutal buoyancy parameter δ for different values of n. Observations from the figure reveal 6. Conclusions that by increasing the magnitude of solutal buoyancy parameter, for buoyancy-assisting flow (δ>0) the con- The present study examines the Soret and Dufour effects centration and temperature profile decrease but velocity on mixed convective boundary layer MHD flow of non- profile shows increasing behaviour and for buoyancy- Newtonian Carreau fluid over a moving vertical plate. opposing flow (δ<0), opposite graphical structures are MATLAB package bvp4c is used to solve the final witnessed. Identical nature is observed for λ for both transformed ODEs. The notable characteristics of this buoyancy-assisting and opposing cases. Figure 6 rep- investigation can be concluded as: resents skin-friction coefficient, heat transfer rate and mass transfer rate with M for different values of We • The temperature and concentration increase with and n. From the figure, it is clear that for shear thinning increasing values of M, i.e., for stronger magnetic fluid, heat transfer rate and mass transfer rate decrease field and the velocity decreases with M. with increasing values of We, whereas for shear thick- • The concentration and temperature decrease with ening fluid heat transfer rate and mass transfer rate increasing values of n, while velocity profile shows 108 Page 10 of 10 Pramana – J. Phys. (2020) 94:108

opposite behaviour and these characters are promi- References nent for larger We. • For shear thinning fluid, temperature and [1] B C Sakiadis, AICHE J. 7, 26 (1961) concentration profiles show increasing nature with [2] F K Tsou, E M Sparrow and R J Goldstein, Int. J. Heat increasing values of We, but velocity profile shows Mass Transfer 10, 219 (1967) decreasing behaviour and for shear thickening fluid [3] N Rilley, J. Fluid Mech. 18(4), 577 (1964) the whole effect is exactly the opposite. [4] M Ali and F Al-Yousef, Heat Mass Transfer 33(4), 301 • Due to Dufour effect, velocity and temperature (1998) [5] O D Makinde, Therm. Sci. 15, 137 (2011) increase, while concentration decreases. • [6] O D Makinde, The Can. J. Chem. Eng. 88(6), 983 Velocity increases for increasing values of Sr and (2010) temperature also increases near the plate and far [7] L S Yao, J. Heat Transfer 109(2), 440 (1987) away from it. [8] Hashim, M Khan and A S Alshomrani, Eur. Phys. J. E • Concentration and velocity decrease with Rc.How- 40, 8 (2017) ever, temperature initially increases with increas- [9] B I Olajuwon, Therm. Sci. 15(2), 241 (2011) ing values of Rc, but far away from the plate, it [10] M Khan, M Irfan, W A Khan and A S Alshomrani, decreases. Results Phys. 7, 2692 (2017) • For shear thinning fluid, local Nusselt number and [11] N S Akbar and S Nadeem, Ain Shams Eng. J. 5(4), 1307 local Sherwood number decrease with increasing (2014) We, while for shear thickening fluid, reverse trend is [12] G R Machireddy and S Naramgari, Ain Shams Eng. J. 9(4), 1189 (2018) obtained and contrary effect of We on skin-friction [13] T Hayat, N Saleem, S Mesloub and N Ali, Z. Natur- coefficient is seen. • forsch. A 66, 215 (2011) The skin-friction coefficient decreases with power- [14] I Khan, Shafquatullah, M Y Malik, A Hussain and M law index n and with increasing values of Df and Khan, Results Phys. 7, 4001 (2017) Sr skin-friction increases. [15] T Hayat, M Waqas, S A Shehzad and A Alsaedi, • For small Df, local Sherwood number, i.e., the Pramana—J. Phys. 86, 3 (2016) mass transfer rate reduces with increasing Sr but for [16] M Khan and H Sardar, Results Phys. 8, 516 (2018) slightly larger values of Df it becomes stronger with [17] M Khan, M Y Malik, T Salahuddin and I Khan, Results Sr and heat transfer rate exhibits opposite nature. Phys. 7, 2384 (2017) • Temperature and concentration decrease/increase [18] M Khan, H Sardar, M M Gulzar and A S Alshomrani, with increasing magnitudes of λ for assisting/ Results Phys. 8, 926 (2018) opposing flow, while velocity demonstrates opposite [19] M Khan, M Irfan, W A Khan and M Ayaz, Pramana – J. Phys. 91: 14 (2018) trend. • δ [20] B Ramadevi, K A Kumar, V Sugunamma and N Variation of shows similar impact on concentration, Sandeep, Pramana – J. Phys. 93: 86 (2019) λ temperature and velocity profiles like in buoyancy [21] M Khan and M Azam, Results Phys. 6, 1168 (2016) assisting and opposing flows. [22] R Muthucumaraswamy and G S Kumar, Theoret. Appl. Mech. 31(1), 35 (2004) [23] M Narahari and A Ishak, J. Appl. Sci. 11(7), 1096 Acknowledgements (2011) [24] I J Uwanta, K K Asogwa and U A Ali, Int. J. Comput. The research of A K Gautam is supported by the Appl. 45(2), 8 (2008) University Grants Commission, New Delhi, Ministry [25] M J Subhakar, K Gangadhar and N B Reddy, Adv. Appl. of Human Resource Development, Government of Sci. Res. 3(5), 3165 (2012) [26] R G Srinivasa, B Ramana, R B Reddy and G Vidyasagar, India Grant [1220/(CSIR-UGC NET DEC. 2016)]. The Int. J. Emer. Trends Eng. Dev. 2(4), 215 (2014) research of A K Verma is funded by the Council of [27] K G Kumar, M Archana, B J Gireesha, M R Krishana- Scientific and Industrial Research, New Delhi, Min- murthy and N G Rudraswamy, Results Phys. 8, 694 istry of Human Resources, Government of India Grant (2018) [09/013(0724)/2017-EMR-I] and the author K Bhat- [28] P J Carreau, Trans. Soc. Rheol. 16, 99 (1972) tacharyya is thankful to UGC for financial support [29] A Pantokratoras, J. Taiwan Inst. Chem. Eng. 56, 1 (2015) by UGC-BSR Research Start-up-Grant project [F.30- [30] M Khan and Hashim, AIP Adv. 5, 1 (2015) 370/2017(BSR)]. [31] C H Chen, Acta Mech. 142, 195 (2000)