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Effects of thermophoresis, Soret-Dufour on heat and mass transfer flow of magnetohydrodynamics non-Newtonian nanofluid over an inclined plate

Effects of thermophoresis, Soret-Dufour on heat and mass transfer flow of magnetohydrodynamics... ARAB JOURNAL OF BASIC AND APPLIED SCIENCES University of Bahrain 2020, VOL. 27, NO. 1, 149–165 https://doi.org/10.1080/25765299.2020.1746017 Effects of thermophoresis, Soret-Dufour on heat and mass transfer flow of magnetohydrodynamics non-Newtonian nanofluid over an inclined plate A. S. Idowu and B. O. Falodun Department of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Kwara State, Nigeria ABSTRACT ARTICLE HISTORY Received 5 November 2019 Heat together with mass transfer of magnetohydrodynamics (MHD) non-Newtonian nano- Revised 15 February 2020 fluid flow over an inclined plate embedded in a porous medium with influence of thermo- Accepted 11 March 2020 phoresis and Soret-Dufour is studied. The novelty of this study is the combined effects of Soret, Dufour and thermophoresis with nanofluid flow on heat together with mass transfer. KEYWORDS The flow is considered over an inclined plate embedded in a porous medium. Appropriate Soret-Dufour; mixed similarity transformations were used to simplify the governing coupled nonlinear partial dif- convective; inclined plate; ferential equations into coupled nonlinear ordinary differential equations. A novel and accur- Chebyshev pseudospectral ate numerical method called spectral homotopy analysis method (SHAM) was used in method; SHAM; MHD; solving the modelled equations. SHAM is the numerical version of the well-known homotopy chemical reaction analysis method (HAM). It involves the decomposition of the nonlinear equations into linear and nonlinear equations. The decomposed linear equations were solved using Chebyshev pseudospectral method. The findings revealed that the applied magnetic field gives rise to an opposing force which slows the motion of an electrically conducting fluid. Increase in the non-Newtonian Casson fluid parameter increases the skin friction factor and reduces the rate of heat and mass transfer. The present results are compared with existing work and found to be in good agreement. 1. Introduction stretchable disk. Their flow equations were solved using Runge-Kutta-Fehlberg method. Heat and mass Combined effects of heat together with mass trans- transfer on mixed convection flow of chemically fer problem have received considerable attention by reacting nanofluid have been considered by many researchers because of their applications in sci- Mahanthesh, Gireesha, and Gorla (2016). They used ence and chemical engineering processes. Heat Laplace transform method to obtain a close form transfer fluids such as oil, water, and ethylene glycol solutions. It is noticed in their study that increase in mixtures are poor thermal conductivity and poor chemical reaction in the presence of nanofluid heat transfer fluid. As a result of their poor thermal brings a rapid decrease to the dimensionless concen- conductivity, they are used as a cooling tool in tration profiles. Ullah, Shafie, Khan, and Hsiao (2018) enhancing manufacturing and costs of operation. presented Brownian diffusion and thermophoresis Many researchers have attempted the enhancement mechanism in Casson fluid over a moving wedge. of these fluids thermal conductivity by suspending They solved their flow equations numerically using nanoparticles in liquids (Abu-Nada, Hakan, and Pop the Keller box method. The study concluded that 2012). Nanofluids contains Ultrafine nanoparticles thermal radiation assisted the heat transfer rate. suspended in a base fluid, it can be an organic solv- Rafique et al. (2019) elucidate solution of Casson ent or water (Choi, 2009). Rajesh, Chamkha, and nanofluid numerically. It was concluded in the study Mallesh (2016) presented transient MHD free convec- that thermophoresis factor increases the temperature tion flow and heat transfer of nanofluid using impli- and decreases concentration profile. Waqas (2020) cit finite difference numerical method. The study examined heat transfer analysis of ferromagnetic concluded that Cu-water nanofluid achieved an non-Newtonian liquid with heterogeneous and improved heat transfer rate compared with the other homogeneous reactions. Their flow equations were nanofluid for all values of t. Latiff, Uddin, and Md. solved using bvp4c scheme and their result shows Ismail (2016) examined Stefan blowing effect on bio- that velocity and thermal fields are having opposite convective flow of nanofluid over a solid rotating behaviour because of the presence of CONTACT B. O. Falodun [email protected] Department of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Kwara State, Nigeria 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 150 A. S. IDOWU AND B. O. FALODUN ferrohydrodynamics interaction parameter. Recently, method. They found out that the chemical reaction Ullah, Nadeem, Khan, Ul Haq, and Tlili (2020) studied acts destructively by reducing the concentration influence of metallic nanoparticles in water. Their fields. Hayat et al. (2016a) explored mixed convec- flow equations were solved using shooting method tion flow of a Burgers nanofluid. Another study of and concluded that increase in the ratio of nanopar- Hayat et al. (2016b) examined 2D stratified flow of ticles has a significant increase in temperature. Oldroyd-B fluid with chemical reaction. Hayat, Mondal, Mishra, Kundu, and Sibanda (2020) exam- Zubair, Waqas, Alsaedi, and Ayub (2017) considered ined entropy generation of variable viscosity and double stratified chemically reactive flow of Powell- thermal radiation on magneto nanofluid flow. Their Erying liquid. It was concluded in the study that skin flow equations were solved using spectral quasi-lin- friction enhances due to the presence of wall thick- earization method. They concluded that local Nusselt ness parameter. Khan, Waqas, Hayat, and Alsaedi number decreases with increase in Brownian motion. (2017) considered Soret-Dufour effects on Jeffrey Naz, Noor, Hayat, Javed, and Alsaedi (2020) solved stretching fluid flow. Their flow equations were the dynamism of magnetohydrodynamic cross nano- solved analytically and they concluded that higher fluid using optimal homotopy analysis method. They Prandtl number leads to reduction in temperature. concluded in the study that Brownian motion has Mondal, Pal, Chatterjee, and Sibanda (2018) exam- great effect on the fluid concentration ined MHD mixed convection mass transfer over an The study of non-Newtonian fluid flow with heat inclined plate. It was concluded in the study that as transport processes are widely considered in recent thermophoretic parameter increases, the concentra- times due to their applications in engineering. tion profile decreases. Fagbade, Falodun, and Examples of such fluids include biological fluids Omowaye (2018) considered MHD natural convection (blood, Salvia, etc.), foodstuffs (jellies, jams, soups, flow of viscoelastic fluid using spectral homotopy etc.) are non-Newtonian fluid because of their analysis approach. behaviour as explained by Xu and Liao (2009). Non- Double diffusive flow (heat and mass transfer) Newtonian fluid finds application in polymer indus- finds applications in many chemical engineering tries, electronic cooling system, heat exchangers and processes. Heat and mass transfer flow is driven by so on. The non-Newtonian fluid of nonlinearity buoyancy due to both temperature and concentra- between shear rate and shear stress reforms the tion gradients. The simultaneous occurrence of heat behaviour of the fluid flow and hereby affect the and mass transfer in a fluid on motion results to ability of the fluid transporting heat. A comparative complications in the relations between the energy study of non-Newtonian fluids flow past a stretching fluxes and the driving potentials. Dufour or diffu- sheet was investigated by Ramana Reddy, Anantha sion-thermal effect is the energy flux caused by com- Kumar, Sugunamma, and Sandeep (2018). Their flow position gradient while Soret or thermal-diffusion equations were solved using Runge-Kutta Fehlberg effect is the mass fluxes created by temperature gra- technique and the analysis shows that Casson fluid dient. The effect of both Soret and Dufour are attains highest velocity when it is compared with mostly neglected in the past due to their smaller Maxwell fluid. Animasaun and Pop (2017) studied order of magnitude as presented by Fick’s laws. non-Newtonian Carreau fluid flow driven by catalytic Soret effect has been utilized for isotope separation. surface reactions numerically using shooting method. Alao et al. (2016) explained that the effects of Soret They concluded that the temperature distribution in and Dufour on the velocity, temperature and con- the flow of viscoelastic Carreau fluid is greater than centration boundary layers are opposite. Omowaye, that of a Newtonian fluid. Gireesha, Ganesh Kumar, Fagbade, and Ajayi (2015) presented Dufour and Ramesh, and Prasannakumara (2018) studied heat Soret effects on steady MHD convective flow. The and mass transfer of Oldroyd-B nanofluid over a study concluded that an increase in the Dufour num- stretching sheet. Their reduced equations were ber reduces the skin friction coefficient and rate of solved using RKF-45 method and result obtained heat transfer. Amanulla, Saleem, Wakif, and AlQarni revealed that nonlinear radiation is more effective (2019) studied MHD Prandtl fluid flow past an iso- than linear radiation. thermal permeable sphere with slip effects. Ahmed Animasaun (2015) examined effects of thermo- and Rashed (2019) examined MHD natural convec- phoresis, variable viscosity and thermal conductivity tion in a heat generating porous medium-filled wavy on free convective flow using shooting method. It enclosures using Buongiorno’s nanofluid model. was concluded in the study that variable viscosity Hayat et al. presented Soret and Dufour effects on and thermal conductivity has a significant effect on MHD peristaltic flow of Prandtl fluid in a rotat- the flow layers. Alao, Fagbade, and Falodun (2016) ing channel. presented unsteady heat and mass transfer flow of a The objective of this paper is to investigate mech- chemically reacting fluid using spectral relaxation anism of Soret-Dufour and thermophoresis on heat ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 151 and mass transfer flow of MHD non-Newtonian Table 1. Numerical values of nanoparticles and water. Fluid phase nanofluid over and inclined plate. To the very best Thermophysical properties (water) Al O Cu Ni 2 3 of my knowledge, no studies available in the litera- 4179 765 385 444 C k ture discussing Soret-Dufour and thermophoresis kg kg 997.1 3970 8933 8900 effects on heat and mass transfer flow of MHD non- q 0.613 40 400 90.9 Newtonian nanofluid. The aim of this paper is to elu- mk cidate the flow behaviour of reservoir fluids. The flow behaviour based on the characteristics of the reservoir. Hence, the type of fluid considered in this paper is non-Newtonian Casson fluid and the viscos- ity and thermal conductivity of the fluid varies. The physical situation that is modelled is the flow of a Casson non-Newtonian model embedded in a por- ous medium in an inclined plate. The fluid viscosity and thermal conductivity is considered to vary within the boundary layer. The present paper considered the viscosity and thermal conductivity to vary because as the fluid moves in the layers, their viscos- ity and temperature changes. Hence, it is not realistic to consider the viscosity and thermal conductivity to be constant. The reduced system of ordinary differ- ential equations is solved numerically by using spec- tral homotopy analysis method (SHAM). The numerical outcomes are obtained for the physical Figure 1. Physical model of the problem. parameters the skin friction, local Nusselt number and local Sherwood number against different values of flow parameters. induced magnetic field is neglected. Based on the work of Fredrickson (1964) and the definition of 2. Flow analysis @u viscosity s ¼ l j ¼ 0 , the rheological equation @y y Consider a two-dimensional, laminar flow of an of a Casson fluid can be written as: incompressible fluid with nanoparticles in an inclined plane. The plate is inclined at an acute s ¼ l þ pffiffiffiffiffiffi 2e when p>p ij ij c 2p o o angle U ð0  U  90 Þ to the horizontal. The (1) s ¼ l þ pffiffiffiffiffiffiffi 2e when p<p temperature and concentration of the nanoparticle ij ij c 2p volume fraction of the plate surface are h and where P is the yield stress of the fluid expressed as / respectively. Also, the free stream temperature y pffiffiffiffiffiffiffiffiffi and concentration of the nanoparticle volume frac- l ð2pÞ P ¼ (2) tion are denoted by h and / respectively (see 1 1 Figure 1). In this paper, we ignored convective l ¼ plastic dynamic viscosity, p ¼ e e ¼ product of ij ij acceleration and diffusion so that the porous l the component of deformation rate with itself and medium can reduce to the Darcy’s law i:e u : e ¼ deformation rate and p ¼ critical value based ij c The flow direction is towards x-axis at constant on Casson non-Newtonian model. The Casson fluid wall temperature h . The fluid considered is water- flow where p>p , it is convenient to say that based which involves solid particles such as Al O , Cu and Ni. The fluid properties are constant 2 2 l ¼ l þ pffiffiffiffiffiffi (3) 0 b 2p except the viscosity and thermal conductivity of the fluid. See Table 1 for thermophysical properties putting Equations (2) into (3), hence kinematic vis- of modified nanofluid. The porous medium is cosity hereby depends on plastic dynamic viscosity assumed to be homogeneous and saturated with l , the density q and Casson parameter b which fluid in local thermodynamic equilibrium. All the leads to properties of the fluid are constant except density l 1 in the buoyancy term of momentum equation. A l ¼ 1 þ (4) q b magnetic field of uniform strength B is applied in ydirection normal to the plate and the magnetic Under the assumptions above, the simplified gov- Reynolds number is assumed to be small so that erning equations becomes 152 A. S. IDOWU AND B. O. FALODUN @w @n Applying the Roseland approximation, we have þ ¼ 0 (5) @x @y the radiative heat flux as @w @w 1 1 @ @w 4 4r @h w þ n ¼ 1 þ l ðTÞ q ¼ (14) @x @y b q @y @y 3k @y þ gb cos ðUÞðhh Þ t 1 where r -Stefan-Boltmann constant and k -mean e e rB absorption coefficient. The stream function WðgÞ is þ gb cos ðUÞð// Þ w c 1 q @W @W defined as w ¼ and n ¼ , thus the stream @y @x l ðTÞ 1 function automatically satisfied the continuity equa- 1 þ w kq b tion (5). The similarity transformation used in this (6) study are given by @h @h 1 @ @h 1 @q l ðTÞ b rffiffiffi w þ n ¼ kðTÞ  þ 1 c hh @x @y qc @y @y qc @y qc p p p 2 W ¼ðcÞ fðgÞ, g ¼ y, TðgÞ¼ , CðgÞ h  h 2 w 1 1 @w 1 þ // b @y ¼ (15) "# /  / w 1 @/ @h D @h þ s D þ Using Equations (11)–(15) on the governing @y @y T @y Equations (6)–(8) and the boundary conditions (9) Q Dk @ / 0 T and (10), the transformed momentum, energy and þ ðhh Þþ qc c c @y p s p concentration equations with the boundary condi- (7) tions are @/ @/ @ / @ðV /Þ 1 1 w þ n ¼ D k ð// Þ l 0 00 00 00 1 þ  T f þ 1 þ ð1 þ  TÞf 0þ ff @x @y @y @y a a b b Dk D @ h T T 0 2 ðf Þ þ  cos ðUÞT þ  cos ðUÞC þ þ (8) a b T T @y m 1 1 1 2 0 M f0 1 þ ð1 þ  TÞf ¼ 0 together with the boundary conditions a P b w ¼ cx, n ¼ðxÞ, h ¼ h , / ¼ / at y ¼ 0 (9) (16) w ! 0, h ! 0, / ! 0 as y ! 0 (10) ð1 þ d TÞþ R d y p y 00 0 0 2 T þ fT þ ðT Þ þ HT Pr Pr In the concentration equation (8), the thermopho- (17) retic velocity V can be written as Alam, Rahman, þ E ð1 þ  TÞ 1 þ ðf00Þ n a and Sattar (2009) 00 0 0 0 þ D C þ N C T þ N ðT Þ ¼ 0 f b t T k @T V ¼k ¼ (11) 1 N T T @y t 00 0 00 00 0 0 ref ref C þ fC C C þ S þ T þ s CT þ T C ¼ 0 p o Sc L N n b th th where k -thermophoretic coefficient. The k values (18) as reported by Batchelor and Shen (1985) and dis- together with the boundary conditions cussed by Animasaun (2015) are taking between 0.2 to 1.2 and it is defined from the theory of Talbot, f ¼ 1, f ¼ S , T ¼ 1, C ¼ 1 at g ¼ 0 (19) Cheng, Schefer, and Willis (1980) as: 0 f ! 0, T ! 0, C ! 0 as g ! 0 (20) hi g  C gb ðh h Þ 2c þ C k 1 þ k C þ C exp w 1 s t n n 1 2 k k Note that:  ¼ bðh h Þ, ¼ ,  ¼ p n a w 1 a 2 b a x k ¼ (12) rB l k gb ð/ / Þ c w 1 0 b ð1 þ 3C k Þ 1 þ 2 þ 2C k , M ¼ , P ¼ , d ¼ nðh h Þ, R ¼ m n t n 2 s y w 1 p k a x qa kqa l ðaxÞ 16r h qc e p Q 1 0 b , Pr ¼ , H ¼ , E ¼ , D ¼ n f 3k k k qc a qc ðh h Þ e p p w 1 where C , C , C , C , c , C are constant, k and k are 1 2 3 m s t g p Dk ð/ / Þ sD ð/ / Þ sD ðh h Þ T B T w 1 w 1 w 1 , Nb ¼ , N ¼ , Sc ¼ c c ðh h Þ  h the thermal conductivity of both fluid and diffused s p w 1 1 k Dk ðh h Þ kðh h Þ l T w 1  w 1 particles respectively, k is the Knudsen number. , C ¼ , S ¼ , Ln ¼ , s ¼ are p o D a T ð/ / Þ D T m w 1 B ref Suppose that the temperature difference within the variable viscosity parameter, thermal Grashof the flow regime is sufficiently small such that h can number, mass Grashof number, magnetic parameter, be expressed as a linear function of h : With the permeability parameter, variable thermal conductiv- expansion of h in Taylor’s series about T and ity parameter, radiation parameter, Prandtl number, neglecting higher order terms to obtain (Idowu and heat generation parameter, Eckert number, Dufour Falodun, 2019). number, Brownian motion parameter, thermophore- sis parameter, Schmidt number, chemical reaction 4 3 4 h ¼ 4h hh (13) 1 1 parameter, Soret number, Lewis number and ther- mophoretic parameter. The physical quantities of ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 153 Figure 2. Effect of Casson parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, Sc ¼ a b n b p t o p f 0:61, S ¼ 3:0: interest are the skin friction, Nusselt number and The rate of mass transfer gives rise to the Sherwood Sherwood number. The skin friction due to viscous number given by drag in the vicinity of the walls is given as Sh ¼ ¼ C ð0Þ s D ð/  / Þ C ¼ w 1 qw where where @/ D d ¼ D ¼ ð/ / ÞC ð0Þ P @u y w 1 @y b s ¼ l þ pffiffiffiffiffiffi ¼ 0 w B y¼0 @y 2p ðReÞ C ¼ 1 þ f ð0Þ 3. Solution techniques The rate of heat transfer (Nusselt number) due to the In this section, the SHAM is applied on the trans- heat transfer between the fluid and wall is given as formed equations (16)–(18) subject to (19) and (20). SHAM is the discrete version of the traditional homo- Nu ¼ ¼ T ð0Þ topy analysis method (HAM). HAM is useful in ðh  h Þ w 1 decomposing system of nonlinear differential equa- tion to linear differential equations. By applying where SHAM on the decomposed linear ordinary differential equations, the Chebyshev spectral collocation @h c q ¼ k ¼ ðh h ÞT ð0Þ w w 1 method is used to solve the resulting equations. The @y b y¼0 physical region is first transformed from the physical 154 A. S. IDOWU AND B. O. FALODUN Figure 3. Effect of Dufour parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, Sc ¼ a b n b p t o p f 0:61, S ¼ 3:0: region ½0, 1Þ to the ½1, 1 by using the technique of 1 1 00 000 00 00 þ 1 þ T f 0þ  1 þ T f þ ff þ ff a 0 a 0 0 0 domain truncation (Sibanda, Motsa, and Makukula, b b 00 00 0 2 0 0 0 2 2012). Hence, the problem solution is obtained in the þf f þ f f ðf Þ 2f f ðf Þ þ  cos ðUÞT 0 0 a 0 0 0 interval ½0, g  and not ½0, 1Þ again. It resulted to the 1 2 þ cos ðUÞT þ  cos ðUÞC þ  cos ðUÞC M f0 a 0 b b 0 use of the following algebraic mapping 1 1 1 1 2 0 0 M f0  1 þ f  1 þ f 2g P b P b f ¼ 1, n2½ 1, 1 (21) s s 1 1 1 1 0 0 1 þ  Tf  1 þ  ff a a For convenience we make the boundary conditions P b P b s s homogeneous by applying the transformations 1 1 1 1 0 0 1 þ  T f  1 þ  T f ¼ 0 a 0 a 0 fðgÞ¼ fðnÞþ f ðgÞ, TðgÞ¼ TðnÞþ T ðgÞ, CðgÞ P b P b 0 0 s s (23) ¼ CðnÞþ C ðgÞ (22) 00 00 00 00 ð1 þ R ÞT þð1 þ R ÞT þ d TT þ d TT þ d T T p p y y 0 y 0 substituting Equation (22) into (16)–(18) to obtain 00 0 0 0 0 0 2 þ d T T þ PrfT þ PrfT þ Prf T þ Prf T þ d ðT Þ y 0 0 0 y 0 0 0 1 1 1 0 00 0 00 0 00 0 0 0 2 1 þ  T f þ 1 þ  T f þ 1 þ  T f a a a þ 2d T T þ d ðT Þ þ PrHT þPrHT 0 0 y y 0 0 0 b b b 1 1 00 00 1 1 1 0 00 00 000 þPrE 1 þ ðf00Þ þ 2PrE 1 þ f f n n þ 1 þ  T f þ 1 þ f 0þ 1 þ f 0 0 0 b b b b b 00 00 00 1 1 00 000 þPrE 1 þ ðf Þ þ D C þ D C f f 0 0 þ 1 þ Tf 0þ 1 þ Tf a a 0 b b b ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 155 Figure 4. Effect of magnetic parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ deg d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, y a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: 00 0 1 1 a ¼ f , a ¼ f , a ¼2f , a 2 00 00 5 6 0 7 8 0 0 þ E 1 þ  Tðf00Þ þ 2E 1 þ  Tf f n a n a b b 1 1 ¼ 1 þ  f , 1 1 2 2 P b þE 1 þ  Tðf Þ þ E 1 þ  T ðf00Þ n a n a 0 b b 1 1 a ¼ 1 þ  T 9 a 0 1 1 2 P b 00 00 00 þ 2E 1 þ  T f f þ E 1 þ  T ðf Þ n a 0 n a 0 0 0 b b 1 1 2 0 00 000 0 0 0 0 0 0 0 0 0 G ðgÞ¼ 1 þ  T f  1 þ f þ N C T þ N C T þ N C T þ N C T þ N ðT Þ 1 a b b b b t 0 0 0 0 0 0 0 b b 0 0 0 þ 2N T T þ N ðT Þ ¼ 0 t t 0 0 000 00 0 2 (24) 1 þ T f f f þðf Þ a 0 0 0 0 0 00 00 0 0 0 C þ C S C CS C C þ S fC þ S fC þ S f C n p n p 0 n n n 0 0 0 S N S N n t n t  cos ðUÞT  cos ðUÞC þ M f0 0 00 00 a 0 b 0 0 þ S f C þ S S þ T þ S S þ T n 0 n 0 n 0 0 0 L N L N n b n b 1 1 1 1 00 00 00 00 0 0 0 0 0 0 þ sCT þ sCT þ sC T þ sC T þ sT C þ sT C þ 1 þ f þ 1 þ  T f a 0 0 0 0 0 0 0 0 P b P b s s 0 0 0 0 þ sT C þ sT C ¼ 0 0 0 0 0 0 b ¼ d T , b ¼ PrT , b ¼ Prf , b ¼ 2d T , y 0 0 y (25) 1 2 3 4 0 0 Simplifying the above equations by setting 1 1 00 00 b ¼ 2PrE 1 þ f b ¼ 2E 1 þ  f , n n a 5 0 6 0 b b 1 1 00 0 a ¼ 1 þ  f , a ¼ 1 þ  T , 1 a 2 a 0 0 b b 1 1 00 2 b ¼ En 1 þ  ðf Þ , b ¼ E 1 þ  T , 1 1 a n a 0 7 0 8 b b a ¼  1 þ f , a ¼  1 þ T , 3 a 4 a 0 b b 156 A. S. IDOWU AND B. O. FALODUN Figure 5. Effect of Brownian motion parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼ deg ¼ d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ a y a b n b p t o p f 2:0, Sc ¼ 0:61, S ¼ 3:0: Substituting the above coefficient parameters into 00 0 b ¼ 2E 1 þ  T f , b ¼ N T , n a 0 b 9 10 0 0 Equations (23)–(25) to obtain 1 1 0 0 0 00 0 00 00 b ¼ N C , b ¼ 2N T 1 þ  T f þ a T þ a f þ 1 þ f 0 b t a 1 2 11 12 0 0 b b 00 00 0 2 G ðgÞ¼ð1 þ R ÞT d T T d ðT Þ PrHT 2 p y 0 y 0 00 00 0 0 0 þ  1 þ Tf 0þ a T þ a f 0 a 3 4 00 2 00 00 0 0 PrE 1 þ ðf Þ n þ ff þ a f þ a f ðf Þ þ a f þ  cos ðaÞT 5 6 7 a þ  cos ðaÞCM f0 0 00 00 1 1 Prf T D C E 1 þ  T ðf Þ 0 n a 0 0 0 0 0 0 1 þ  Tf þ a T þ a f ¼ G ðgÞ a 8 9 1 P b 0 0 0 (26) N C T N ðT Þ b t 0 0 0 00 00 00 0 0 ð1 þ R ÞT þ d TT þ b T þ b T þ PrfT b f þ b T p y 1 1 2 3 0 00 c ¼ S C , c ¼ S f , c ¼ sT , c ¼ sC , c n n 0 0 1 0 2 3 0 4 5 2 2 0 0 00 þ d ðT Þ þ b T þ PrHT þ PrE 1 þ ðf00Þ þ b f y n 4 5 0 0 ¼ sC , c ¼ sT 0 6 0 00 00 þ D C þ E 1 þ  Tðf00Þ þ b f f n a S N 6 n t 00 0 00 G ðgÞ¼ C þ S C C S f C  S S þ T 3 n p 0 n 0 n 0 0 0 0 L N n b 00 0 0 0 þb T þ b ðf00Þ þ b f þ N C T þ b C 7 8 9 a 10 00 0 0 sC T sT C 0 0 0 0 0 2 þb T þ N ðT Þ þ b T ¼ G ðgÞ (27) t 2 11 12 ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 157 Figure 6. Effect of porosity parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ deg d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, y a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: S N 1 n t 00 0 0 00 0 00 000 000 00 C S C C þ S fC þ c f þ c C þ S S þ T a T þ a f þ 1 þ f þ a T þ a f þ a f þ a f n p n n 0 1 2 3 l 4 5 l 6 1 2 l l l l l L N b n b 00 00 0 0 0 0 þsCT þ c C þ c T þ sT C þ c T þ c C ¼ G ðgÞ 0 2 3 4 5 6 þ a f þ  cos ðUÞT þ  cos ðUÞC M f0 7 a l b l l (28) 1 1 0 0 1 þ f þ a T þ a f ¼ G ðgÞ 8 l 9 1 l l Equations (26)–(28) are nonlinear equations. In P b the application of SHAM, the nonlinear equations are (31) 00 00 0 0 decomposed into linear and nonlinear parts. It ð1 þ R ÞT þ b T þ b T þ b f þ b T þ b T þ PrHT p l l l l 1 1 l 2 3 l 4 l should be noted that the derivatives of f, T and C 00 00 00 00 0 þ b f þ D C þ b f þ b T þ b f þ b C f l 5 l l 6 l 7 9 l 10 l are with respect to n defined by: þ b T þ b T ¼ G ðgÞ l 2 11 l 12 d 2 d (32) ¼ (29) dn L dg S N n t 00 0 00 C S C C þ c f þ c C þ S S þ T n p l l n 0 l 1 2 l l An initial guess is chosen with reference to the L N n b boundary conditions (15) and (16) as the following þ c C þ c T 3 4 l functions 0 0 þ c T þ c C ¼ G ðgÞ 5 l 6 l g g f ðgÞ¼ S þ e þ 1, T ðgÞ¼ C ¼ e (30) 0 w 0 ðgÞ (33) subject to: The non-homogeneous linear part of Equations (26)–(28) are decomposed from the nonlinear part as 0 0 f ð1Þ¼ f ð1Þ¼ f ð1Þ¼ 0, Tð1Þ¼ T ð1Þ l l l l the following equations ¼ 0, C ð1Þ¼ C ð1Þ¼ 0 (34) l l 158 A. S. IDOWU AND B. O. FALODUN Figure 7. Effect of radiation parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ deg d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, y a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: The boundary conditions (34) is chosen to be all collocation points (Trefethen, 2000) given by equals to zero with reference to the transformed pj domain ½1, 1, to be able to implement the linear n ¼ cos (38) part SHAM solution. The Chebyshev pseudospectral method is applied on Equations (26)–(28). The provided j ¼ 0, 1, :::N and N þ 1 is the number of col- unknown functions f ðnÞ, T ðnÞ and T in the l l l location points. The unknown functions f ðnÞ, T ðnÞ l l Equations (26)–(28) are approximated as a truncated and C ðnÞ are approximated by the use of Lagrange series of Chebyshev polynomials given by Fagbade form of interpolating polynomial which interpolates et al. (2018) as: the unknown functions f ðnÞ, T ðnÞ and C ðnÞ at the l l l Gauss-Lobatto collocation points as defined in f ðnÞuf ðn Þþ f T ðn Þ, j ¼ 0, :::, N (35) l j k 1k j l Equation (38). k¼0 T ðnÞuT ðn Þþ T T ðn Þ, j ¼ 0, :::, N (36) l j k 2k j 4. Results and discussion k¼0 The transformed governing equations (16)–(18) sub- C ðnÞuC ðn Þþ C T ðn Þ, j ¼ 0, :::, N (37) ject to the boundary conditions (19) and (20) are set l k 3k j j k¼0 of coupled highly non-linear ordinary differential equations. These set of equations were solved where T , T and T are the kth Chebyshev polyno- 1k 2k 3k mial and n , n , :::, n are Gauss-Lobatto collocation numerically using spectral homotopy analysis 0 1 N point. The Chebyshev nodes in the transformed method. To study the behaviour of various flow domain ½1, 1 are defined by the Gauss-Lobatto parameters such as Casson, heat generation ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 159 Figure 8. Effect of Soret parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, Sc ¼ a b n b p t o p f 0:61, S ¼ 3:0: parameter, Soret-Dufour parameter, chemical reac- effect on the concentration profile. The effect of D tion parameter, radiation parameter, etc, a compre- as presented in Figure 3 is in good agreement with hensive numerical computation is carried out. that of Alao et al. (2016). The diffusion-thermal Figure 2 represents the effect of Casson non- added to the thermal boundary layer influences the Newtonian fluid parameter ðbÞ on the velocity, tem- temperature field. perature and concentration fields. With increase in b, The effect of magnetic parameter (M) on velocity, it is noticed in Figure 2 that transport rate reduces temperature and concentration fields is shown in within the thermal boundary layer. A slight decrease Figure 4.In Figure 4, increase in the values of mag- in the temperature field is observed as the values of netic parameter causes a damping effect on the vel- b increases. It worth mentioning that as Casson fluid ocity field by producing a drag-like force called parameter approaches infinity, the fluid behaves as Lorentz force. This force acts in the opposite direc- Newtonian fluid. Because of increase in the elasticity tion and thereby reduces the motion of an electric- stress parameter, there is thickening of the thermal ally conducting fluid. With increasing value of the boundary layer. The graphical results in Figure 2 is in magnetic parameter, the momentum boundary layer excellent agreement with the previous work of thickness decreases while the thermal boundary Zaigham Zia, Ullah, Waqas, Alsaedi, and Hayat layer slightly increases far away from the plate as (2018). Figure 3 illustrates the effect of Dufour par- shown in Figure 4. This result was found to be in ameter ðD Þ on the velocity, temperature and con- agreement with the previous work of Fagbade et al. centration fields. It is observed that increase in the (2018). In addition, increase in the values of M has values of D increases the velocity, temperature no effect on the concentration field. Figure 5 illus- fields, hydrodynamics and thermal boundary layer. trates the effect of Brownian motion parameter ðN Þ Increase in the values of D is noticed to has no on the velocity, temperature and concentration f 160 A. S. IDOWU AND B. O. FALODUN Figure 9. Effect of Schmidt number on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: fields. Increase in the values of Brownian motion par- a result of increase in the porosity term, the thermal ameter within the hydrodynamic boundary layer boundary layer thickness increases. This result is as brings more erratic movement of nanoparticles in shown in Figure 6. Effect of P is negligible on the the porous medium surroundings. Increase in the concentration field. Figure 7 illustrates the effect of values of N in Figure 5 is observed to decrease vel- radiation parameter ðR Þ on the velocity, tempera- ocity and concentration fields of the fluid nanopar- ture and concentration fields. When the radiation ticles. This is as a result of the random collision of parameter is increased, the fluid temperature, hydro- the fluid particles which causes reduction on the dynamic and thermal boundary layer increases. fluid velocity. It is obvious from Figure 8 that Physically, increase in R added more heat energy to increase in N slightly increases temperature and the the thermal boundary layer. This gives room for thermal boundary layer because of the nanofluid more temperature and thereby increase temperature porosity in the hydrodynamic and thermal boundary field. The temperature field as shown in Figure 7 layer. In addition, increase in the values of N in the increases the entire thermal boundary layer. Increase boundary layer decreases the solutal concentration. in R is seen to decrease the nanofluid concentration Figure 6 depicts the effect of porosity term ðP Þ close to the plate. on the velocity, temperature and concentration Figure 8 depicts the effect of Soret parameter ðS Þ fields. Increasing the porosity expands holes and on the velocity, temperature and concentration allows more movement of nanoparticles within the fields. Increase in the values of S is noticed to hydrodynamics and thermal boundary layer. Increase increase the fluid velocity and the momentum in the porosity term P brings decrease to the boundary layer thickness as shown in Figure 8. Also, momentum boundary layer thickness and thereby increase in the values of Soret parameter accelerates decreases the fluid velocity as shown in Figure 6.As the concentration field close to the plate and ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 161 Figure 10. Effect of thermal Grashof number on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼ deg ¼ d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ a y a b n b p t o p f 2:0, Sc ¼ 0:61, S ¼ 3:0: negligible at the free stream. Effect of the Schmidt Figure 11 depicts the effect of Prandtl number number ðS Þ on the velocity, temperature and con- (Pr) on the velocity, temperature and concentration centration fields is illustrated in Figure 9. The profiles. The velocity profile is observed to decrease Schmidt is a dimensionless number which is defined with increase in the values of Prandtl number (Pr). as the ratio of fluid viscosity to mass diffusivity. This is because fluids that have higher Pr possess Hence, if the viscosity of fluid is greater than mass greater viscosities and thereby reduces the fluid diffusivity, there is going to be more Schmidt num- velocities and lower the wall skin friction coefficient. ber within the entire boundary layer. It is observed In addition, increase in the values of Pr correspond that increase in S drastically decrease the velocity to a decrease in the fluid temperature and the ther- and concentration field due to the nanofluid porosity mal boundary layer thickness. For small values of Pr, within the hydrodynamic and solutal boundary layer. that is Pr < 1 the fluid becomes very conductive. Increase in the values of S , increase the solutal Increase in the values of Pr is noticed to increase boundary layer. The effect of thermal Grashof num- fluid concentration at the wall. ber ð Þ on the velocity, temperature and concen- From Table 2, increase in the values of Lewis tration field is illustrated in Figure 10. The number increases the skin friction coefficient and buoyancy force acts like as favourable pressure gra- Sherwood number whereas it decreases the Nusselt dient on the fluid flow and increase nanofluid par- number. From Table 2, increase in the magnetic par- ticles within the boundary layer. This is shown in ameter (M) decreases the three physical quantities of Figure 10 as increase in the thermal Grashof num- engineering (that is, skin friction coefficient, Nusselt ber ð Þ increases the momentum boundary layer number and Sherwood number). In Table 3, increase and slightly reduces the thermal and mass bound- in the values of Lewis number (Ln) increases the ary layer. Nusselt number and decreases the skin friction 162 A. S. IDOWU AND B. O. FALODUN Figure 11. Effect of Prandtl number on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: coefficient and the Sherwood number. In Table 3, thermal conductivity is considered. A parametric increase in the values of Brownian motion parameter study on the velocity, temperature, and concentra- (Nb) decreases the skin friction coefficient, Nusselt tion. The flow equations were solved numerically number and Sherwood number. From Table 3, using spectral homotopy analysis method. From the increase in the values of Schmidt number decreases results obtained, increase in the non-Newtonian Casson fluid parameter brings increase to the skin the wall skin friction coefficient and the Nusselt number whereas increase in the Sherwood number friction coefficient and reduces the rate of heat and mass transfer. The results in the present study is noticeable. In Table 3, it is noticed that increase in the non-Newtonian fluid parameter ðbÞ increases the revealed that the behaviour of the non-Newtonian fluid parameter changes to Newtonian fluid as the Nusselt and Sherwood number but decreases the Casson fluid parameter approaches infinity. Increase skin friction coefficient. Tables 4 and 5 shows the in the magnetic parameter (M) brings a decrease to comparison of the present results with previous pub- velocity of the fluid. It is observed that increase in lished works and was found to be in the magnetic parameter brings to the fluid concen- good agreement. tration close to the plate and reduces the free stream at the porous medium. The present result 5. Concluding remarks gives account to the effect of radiation parameter This study examined the effects of thermophoresis and it is noticed that increase in radiation parameter and Soret-Dufour on heat and mass transfer mixed gives rise to the fluid temperature and has no effect convective flow of MHD non-Newtonian nanofluid on the velocity. It is observed that increase in the over an inclined plate embedded in a porous Schmidt number gives rise to the concentration field medium. The analysis for variable viscosity and at the free stream whereas it increase the fluid ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 163 0 0 Table 2. Computational values for skin friction coefficient ðC Þ, Nusselt number ðT ð0ÞÞ, and sherwood number ðC ð0ÞÞ for different values of Ln and M. Ln M C Nh Sh 0.0 1.0 1.08233781 0.72334184 1.02490066 0.5 1.0 1.14285394 0.35872612 1.08228644 1.0 1.0 1.20739181 0.22940864 1.14323113 2.0 1.0 1.34923800 0.18349278 1.27658671 0.1 0.4 1.45926666 0.83996184 1.09810897 0.1 0.6 1.31269481 0.79692501 1.05550050 0.1 0.8 1.20198770 0.75406987 1.04802955 0.1 1.0 1.06855863 0.72694283 1.00977521 0 0 Table 3. Computational values for skin friction coefficient ðC Þ, Nusselt number ðT ð0ÞÞ, and sherwood number ðC ð0ÞÞ for different values of Ln, Nb, Sc and b. Parameters Present work Ln Nb Sc b Cf Nh Sh 0.2 1.0 0.61 3.0 0.13265182 0.88477929 0.78652409 0.4 1.0 0.61 3.00 0.12774386 0.90996351 0.58398366 0.6 1.0 0.61 3.0 0.12488548 0.93522964 0.38152589 0.8 1.0 0.61 3.0 0.12289942 0.96147715 0.17200602 1.0 0.0 0.61 3.0 0.08967359 0.94746978 0.34074944 1.0 1.0 0.61 3.0 0.12519871 0.92975180 0.42525615 1.0 2.0 0.61 3.0 0.12488548 0.93522964 0.38152589 1.0 3.0 0.61 3.0 0.12448055 0.94174484 0.32969808 1.0 1.0 0.3 3.0 1.04462222 0.78476059 0.67090855 1.0 1.0 0.6 3.0 0.97607905 0.73329484 0.94697241 1.0 1.0 0.9 3.0 0.93182279 0.69528927 1.15619714 1.0 1.0 1.2 3.0 0.89901583 0.66435691 1.33199172 1.0 1.0 0.61 0.0 0.95154181 0.73022820 0.95324841 1.0 1.0 0.61 0.2 1.77694769 0.78778324 1.00573400 1.0 1.0 0.61 0.3 1.57443332 0.88549205 1.09642636 1.0 1.0 0.61 0.4 1.55314440 0.77899759 1.07697499 Table 4. Comparison of the present result with that of Mondal et al. (2018) when P ¼ H ¼ N ¼ N ¼ 0 and at constant s b t viscosity and thermal conductivity. Mondal et al. (2018) Present result D So C Nu Sh C Nu Sh f x x f x x 0.030 2.0 6.238707 1.151932 0.1554706 6.237707 1.142931 0.1554601 0.037 1.6 6.160867 1.144651 0.2306910 6.151877 1.143551 0.2306811 0.050 1.2 6.087934 1.135752 0.3057275 6.077864 1.125750 0.3057266 0.075 0.8 6.023187 1.123219 0.3808005 6.012188 1.123218 0.3808004 0.150 0.4 5.996934 1.096560 0.4574460 5.99694 1.096561 0.4574461 Table 5. Comparison of the present result with that of Rafique et al. (2019) when P ¼ Ec ¼ H ¼ C ¼ s ¼ 0 and at con- s p stant viscosity and thermal conductivity. Rafique et al. (2019) Present result N N b t C Nu C Nu f x f x 0.1 0.1 0.9524 2.1294 0.9532 2.1296 0.2 0.2 0.3654 2.5152 0.3666 2.5155 0.3 0.3 0.1355 2.6088 0.1359 2.6090 0.4 0.4 0.0495 2.6038 0.0499 2.6039 0.5 0.5 0.0179 2.5731 0.0177 2.5736 temperature close to the plate and reduces it at the The present study is useful in glass blowing pro- free stream. The novelty of this paper the investiga- cess. In glass blowing process, maintaining the tem- tion of varying viscosity and thermal conductivity on perature for glass blowing is very important. This is the flow of Casson non-Newtonian fluid in a porous because it requires optimal temperature to make it medium of an inclined plate. The result of this study flexible for blow so as to obtain the required shape. will be of help in chemical engineering processes Hence parameters such as thermal radiation and and the production of food. heat generation considered in this study are very 164 A. S. IDOWU AND B. O. FALODUN important in glass blowing process. Also, at very TðgÞ dimensionless temperature CðgÞ dimensionless concentration high temperature the properties of fluid such as vis- cosity and thermal conductivity may vary. It is observed from this study that increase in viscosity Disclosure statement brings decrease to the fluid flow. This allows us to No potential conflict of interest was reported by be able to control the flow of the fluid. On the other the authors. hand, an increase in thermal conductivity gives rise to the fluid temperature in such a way that it con- ORCID ducts more heat. The use of thermal conductivity in regulating the conducting nature of heat of the fluid B. O. Falodun http://orcid.org/0000-0003-1020-1677 also helps in controlling the blowing glass substance temperature. The present study is very useful in References many chemical engineering processes such as metal- Abu-Nada, E., Hakan, F. O., & Pop, I. (2012). Buoyancy lurgical and extrusion of polymer which involves induced flow in a nanofluid filled enclosure partially cooling of molten liquid. The polyethylene oxide and exposed to forced convection. 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Effects of thermophoresis, Soret-Dufour on heat and mass transfer flow of magnetohydrodynamics non-Newtonian nanofluid over an inclined plate

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Abstract

ARAB JOURNAL OF BASIC AND APPLIED SCIENCES University of Bahrain 2020, VOL. 27, NO. 1, 149–165 https://doi.org/10.1080/25765299.2020.1746017 Effects of thermophoresis, Soret-Dufour on heat and mass transfer flow of magnetohydrodynamics non-Newtonian nanofluid over an inclined plate A. S. Idowu and B. O. Falodun Department of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Kwara State, Nigeria ABSTRACT ARTICLE HISTORY Received 5 November 2019 Heat together with mass transfer of magnetohydrodynamics (MHD) non-Newtonian nano- Revised 15 February 2020 fluid flow over an inclined plate embedded in a porous medium with influence of thermo- Accepted 11 March 2020 phoresis and Soret-Dufour is studied. The novelty of this study is the combined effects of Soret, Dufour and thermophoresis with nanofluid flow on heat together with mass transfer. KEYWORDS The flow is considered over an inclined plate embedded in a porous medium. Appropriate Soret-Dufour; mixed similarity transformations were used to simplify the governing coupled nonlinear partial dif- convective; inclined plate; ferential equations into coupled nonlinear ordinary differential equations. A novel and accur- Chebyshev pseudospectral ate numerical method called spectral homotopy analysis method (SHAM) was used in method; SHAM; MHD; solving the modelled equations. SHAM is the numerical version of the well-known homotopy chemical reaction analysis method (HAM). It involves the decomposition of the nonlinear equations into linear and nonlinear equations. The decomposed linear equations were solved using Chebyshev pseudospectral method. The findings revealed that the applied magnetic field gives rise to an opposing force which slows the motion of an electrically conducting fluid. Increase in the non-Newtonian Casson fluid parameter increases the skin friction factor and reduces the rate of heat and mass transfer. The present results are compared with existing work and found to be in good agreement. 1. Introduction stretchable disk. Their flow equations were solved using Runge-Kutta-Fehlberg method. Heat and mass Combined effects of heat together with mass trans- transfer on mixed convection flow of chemically fer problem have received considerable attention by reacting nanofluid have been considered by many researchers because of their applications in sci- Mahanthesh, Gireesha, and Gorla (2016). They used ence and chemical engineering processes. Heat Laplace transform method to obtain a close form transfer fluids such as oil, water, and ethylene glycol solutions. It is noticed in their study that increase in mixtures are poor thermal conductivity and poor chemical reaction in the presence of nanofluid heat transfer fluid. As a result of their poor thermal brings a rapid decrease to the dimensionless concen- conductivity, they are used as a cooling tool in tration profiles. Ullah, Shafie, Khan, and Hsiao (2018) enhancing manufacturing and costs of operation. presented Brownian diffusion and thermophoresis Many researchers have attempted the enhancement mechanism in Casson fluid over a moving wedge. of these fluids thermal conductivity by suspending They solved their flow equations numerically using nanoparticles in liquids (Abu-Nada, Hakan, and Pop the Keller box method. The study concluded that 2012). Nanofluids contains Ultrafine nanoparticles thermal radiation assisted the heat transfer rate. suspended in a base fluid, it can be an organic solv- Rafique et al. (2019) elucidate solution of Casson ent or water (Choi, 2009). Rajesh, Chamkha, and nanofluid numerically. It was concluded in the study Mallesh (2016) presented transient MHD free convec- that thermophoresis factor increases the temperature tion flow and heat transfer of nanofluid using impli- and decreases concentration profile. Waqas (2020) cit finite difference numerical method. The study examined heat transfer analysis of ferromagnetic concluded that Cu-water nanofluid achieved an non-Newtonian liquid with heterogeneous and improved heat transfer rate compared with the other homogeneous reactions. Their flow equations were nanofluid for all values of t. Latiff, Uddin, and Md. solved using bvp4c scheme and their result shows Ismail (2016) examined Stefan blowing effect on bio- that velocity and thermal fields are having opposite convective flow of nanofluid over a solid rotating behaviour because of the presence of CONTACT B. O. Falodun [email protected] Department of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Kwara State, Nigeria 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 150 A. S. IDOWU AND B. O. FALODUN ferrohydrodynamics interaction parameter. Recently, method. They found out that the chemical reaction Ullah, Nadeem, Khan, Ul Haq, and Tlili (2020) studied acts destructively by reducing the concentration influence of metallic nanoparticles in water. Their fields. Hayat et al. (2016a) explored mixed convec- flow equations were solved using shooting method tion flow of a Burgers nanofluid. Another study of and concluded that increase in the ratio of nanopar- Hayat et al. (2016b) examined 2D stratified flow of ticles has a significant increase in temperature. Oldroyd-B fluid with chemical reaction. Hayat, Mondal, Mishra, Kundu, and Sibanda (2020) exam- Zubair, Waqas, Alsaedi, and Ayub (2017) considered ined entropy generation of variable viscosity and double stratified chemically reactive flow of Powell- thermal radiation on magneto nanofluid flow. Their Erying liquid. It was concluded in the study that skin flow equations were solved using spectral quasi-lin- friction enhances due to the presence of wall thick- earization method. They concluded that local Nusselt ness parameter. Khan, Waqas, Hayat, and Alsaedi number decreases with increase in Brownian motion. (2017) considered Soret-Dufour effects on Jeffrey Naz, Noor, Hayat, Javed, and Alsaedi (2020) solved stretching fluid flow. Their flow equations were the dynamism of magnetohydrodynamic cross nano- solved analytically and they concluded that higher fluid using optimal homotopy analysis method. They Prandtl number leads to reduction in temperature. concluded in the study that Brownian motion has Mondal, Pal, Chatterjee, and Sibanda (2018) exam- great effect on the fluid concentration ined MHD mixed convection mass transfer over an The study of non-Newtonian fluid flow with heat inclined plate. It was concluded in the study that as transport processes are widely considered in recent thermophoretic parameter increases, the concentra- times due to their applications in engineering. tion profile decreases. Fagbade, Falodun, and Examples of such fluids include biological fluids Omowaye (2018) considered MHD natural convection (blood, Salvia, etc.), foodstuffs (jellies, jams, soups, flow of viscoelastic fluid using spectral homotopy etc.) are non-Newtonian fluid because of their analysis approach. behaviour as explained by Xu and Liao (2009). Non- Double diffusive flow (heat and mass transfer) Newtonian fluid finds application in polymer indus- finds applications in many chemical engineering tries, electronic cooling system, heat exchangers and processes. Heat and mass transfer flow is driven by so on. The non-Newtonian fluid of nonlinearity buoyancy due to both temperature and concentra- between shear rate and shear stress reforms the tion gradients. The simultaneous occurrence of heat behaviour of the fluid flow and hereby affect the and mass transfer in a fluid on motion results to ability of the fluid transporting heat. A comparative complications in the relations between the energy study of non-Newtonian fluids flow past a stretching fluxes and the driving potentials. Dufour or diffu- sheet was investigated by Ramana Reddy, Anantha sion-thermal effect is the energy flux caused by com- Kumar, Sugunamma, and Sandeep (2018). Their flow position gradient while Soret or thermal-diffusion equations were solved using Runge-Kutta Fehlberg effect is the mass fluxes created by temperature gra- technique and the analysis shows that Casson fluid dient. The effect of both Soret and Dufour are attains highest velocity when it is compared with mostly neglected in the past due to their smaller Maxwell fluid. Animasaun and Pop (2017) studied order of magnitude as presented by Fick’s laws. non-Newtonian Carreau fluid flow driven by catalytic Soret effect has been utilized for isotope separation. surface reactions numerically using shooting method. Alao et al. (2016) explained that the effects of Soret They concluded that the temperature distribution in and Dufour on the velocity, temperature and con- the flow of viscoelastic Carreau fluid is greater than centration boundary layers are opposite. Omowaye, that of a Newtonian fluid. Gireesha, Ganesh Kumar, Fagbade, and Ajayi (2015) presented Dufour and Ramesh, and Prasannakumara (2018) studied heat Soret effects on steady MHD convective flow. The and mass transfer of Oldroyd-B nanofluid over a study concluded that an increase in the Dufour num- stretching sheet. Their reduced equations were ber reduces the skin friction coefficient and rate of solved using RKF-45 method and result obtained heat transfer. Amanulla, Saleem, Wakif, and AlQarni revealed that nonlinear radiation is more effective (2019) studied MHD Prandtl fluid flow past an iso- than linear radiation. thermal permeable sphere with slip effects. Ahmed Animasaun (2015) examined effects of thermo- and Rashed (2019) examined MHD natural convec- phoresis, variable viscosity and thermal conductivity tion in a heat generating porous medium-filled wavy on free convective flow using shooting method. It enclosures using Buongiorno’s nanofluid model. was concluded in the study that variable viscosity Hayat et al. presented Soret and Dufour effects on and thermal conductivity has a significant effect on MHD peristaltic flow of Prandtl fluid in a rotat- the flow layers. Alao, Fagbade, and Falodun (2016) ing channel. presented unsteady heat and mass transfer flow of a The objective of this paper is to investigate mech- chemically reacting fluid using spectral relaxation anism of Soret-Dufour and thermophoresis on heat ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 151 and mass transfer flow of MHD non-Newtonian Table 1. Numerical values of nanoparticles and water. Fluid phase nanofluid over and inclined plate. To the very best Thermophysical properties (water) Al O Cu Ni 2 3 of my knowledge, no studies available in the litera- 4179 765 385 444 C k ture discussing Soret-Dufour and thermophoresis kg kg 997.1 3970 8933 8900 effects on heat and mass transfer flow of MHD non- q 0.613 40 400 90.9 Newtonian nanofluid. The aim of this paper is to elu- mk cidate the flow behaviour of reservoir fluids. The flow behaviour based on the characteristics of the reservoir. Hence, the type of fluid considered in this paper is non-Newtonian Casson fluid and the viscos- ity and thermal conductivity of the fluid varies. The physical situation that is modelled is the flow of a Casson non-Newtonian model embedded in a por- ous medium in an inclined plate. The fluid viscosity and thermal conductivity is considered to vary within the boundary layer. The present paper considered the viscosity and thermal conductivity to vary because as the fluid moves in the layers, their viscos- ity and temperature changes. Hence, it is not realistic to consider the viscosity and thermal conductivity to be constant. The reduced system of ordinary differ- ential equations is solved numerically by using spec- tral homotopy analysis method (SHAM). The numerical outcomes are obtained for the physical Figure 1. Physical model of the problem. parameters the skin friction, local Nusselt number and local Sherwood number against different values of flow parameters. induced magnetic field is neglected. Based on the work of Fredrickson (1964) and the definition of 2. Flow analysis @u viscosity s ¼ l j ¼ 0 , the rheological equation @y y Consider a two-dimensional, laminar flow of an of a Casson fluid can be written as: incompressible fluid with nanoparticles in an inclined plane. The plate is inclined at an acute s ¼ l þ pffiffiffiffiffiffi 2e when p>p ij ij c 2p o o angle U ð0  U  90 Þ to the horizontal. The (1) s ¼ l þ pffiffiffiffiffiffiffi 2e when p<p temperature and concentration of the nanoparticle ij ij c 2p volume fraction of the plate surface are h and where P is the yield stress of the fluid expressed as / respectively. Also, the free stream temperature y pffiffiffiffiffiffiffiffiffi and concentration of the nanoparticle volume frac- l ð2pÞ P ¼ (2) tion are denoted by h and / respectively (see 1 1 Figure 1). In this paper, we ignored convective l ¼ plastic dynamic viscosity, p ¼ e e ¼ product of ij ij acceleration and diffusion so that the porous l the component of deformation rate with itself and medium can reduce to the Darcy’s law i:e u : e ¼ deformation rate and p ¼ critical value based ij c The flow direction is towards x-axis at constant on Casson non-Newtonian model. The Casson fluid wall temperature h . The fluid considered is water- flow where p>p , it is convenient to say that based which involves solid particles such as Al O , Cu and Ni. The fluid properties are constant 2 2 l ¼ l þ pffiffiffiffiffiffi (3) 0 b 2p except the viscosity and thermal conductivity of the fluid. See Table 1 for thermophysical properties putting Equations (2) into (3), hence kinematic vis- of modified nanofluid. The porous medium is cosity hereby depends on plastic dynamic viscosity assumed to be homogeneous and saturated with l , the density q and Casson parameter b which fluid in local thermodynamic equilibrium. All the leads to properties of the fluid are constant except density l 1 in the buoyancy term of momentum equation. A l ¼ 1 þ (4) q b magnetic field of uniform strength B is applied in ydirection normal to the plate and the magnetic Under the assumptions above, the simplified gov- Reynolds number is assumed to be small so that erning equations becomes 152 A. S. IDOWU AND B. O. FALODUN @w @n Applying the Roseland approximation, we have þ ¼ 0 (5) @x @y the radiative heat flux as @w @w 1 1 @ @w 4 4r @h w þ n ¼ 1 þ l ðTÞ q ¼ (14) @x @y b q @y @y 3k @y þ gb cos ðUÞðhh Þ t 1 where r -Stefan-Boltmann constant and k -mean e e rB absorption coefficient. The stream function WðgÞ is þ gb cos ðUÞð// Þ w c 1 q @W @W defined as w ¼ and n ¼ , thus the stream @y @x l ðTÞ 1 function automatically satisfied the continuity equa- 1 þ w kq b tion (5). The similarity transformation used in this (6) study are given by @h @h 1 @ @h 1 @q l ðTÞ b rffiffiffi w þ n ¼ kðTÞ  þ 1 c hh @x @y qc @y @y qc @y qc p p p 2 W ¼ðcÞ fðgÞ, g ¼ y, TðgÞ¼ , CðgÞ h  h 2 w 1 1 @w 1 þ // b @y ¼ (15) "# /  / w 1 @/ @h D @h þ s D þ Using Equations (11)–(15) on the governing @y @y T @y Equations (6)–(8) and the boundary conditions (9) Q Dk @ / 0 T and (10), the transformed momentum, energy and þ ðhh Þþ qc c c @y p s p concentration equations with the boundary condi- (7) tions are @/ @/ @ / @ðV /Þ 1 1 w þ n ¼ D k ð// Þ l 0 00 00 00 1 þ  T f þ 1 þ ð1 þ  TÞf 0þ ff @x @y @y @y a a b b Dk D @ h T T 0 2 ðf Þ þ  cos ðUÞT þ  cos ðUÞC þ þ (8) a b T T @y m 1 1 1 2 0 M f0 1 þ ð1 þ  TÞf ¼ 0 together with the boundary conditions a P b w ¼ cx, n ¼ðxÞ, h ¼ h , / ¼ / at y ¼ 0 (9) (16) w ! 0, h ! 0, / ! 0 as y ! 0 (10) ð1 þ d TÞþ R d y p y 00 0 0 2 T þ fT þ ðT Þ þ HT Pr Pr In the concentration equation (8), the thermopho- (17) retic velocity V can be written as Alam, Rahman, þ E ð1 þ  TÞ 1 þ ðf00Þ n a and Sattar (2009) 00 0 0 0 þ D C þ N C T þ N ðT Þ ¼ 0 f b t T k @T V ¼k ¼ (11) 1 N T T @y t 00 0 00 00 0 0 ref ref C þ fC C C þ S þ T þ s CT þ T C ¼ 0 p o Sc L N n b th th where k -thermophoretic coefficient. The k values (18) as reported by Batchelor and Shen (1985) and dis- together with the boundary conditions cussed by Animasaun (2015) are taking between 0.2 to 1.2 and it is defined from the theory of Talbot, f ¼ 1, f ¼ S , T ¼ 1, C ¼ 1 at g ¼ 0 (19) Cheng, Schefer, and Willis (1980) as: 0 f ! 0, T ! 0, C ! 0 as g ! 0 (20) hi g  C gb ðh h Þ 2c þ C k 1 þ k C þ C exp w 1 s t n n 1 2 k k Note that:  ¼ bðh h Þ, ¼ ,  ¼ p n a w 1 a 2 b a x k ¼ (12) rB l k gb ð/ / Þ c w 1 0 b ð1 þ 3C k Þ 1 þ 2 þ 2C k , M ¼ , P ¼ , d ¼ nðh h Þ, R ¼ m n t n 2 s y w 1 p k a x qa kqa l ðaxÞ 16r h qc e p Q 1 0 b , Pr ¼ , H ¼ , E ¼ , D ¼ n f 3k k k qc a qc ðh h Þ e p p w 1 where C , C , C , C , c , C are constant, k and k are 1 2 3 m s t g p Dk ð/ / Þ sD ð/ / Þ sD ðh h Þ T B T w 1 w 1 w 1 , Nb ¼ , N ¼ , Sc ¼ c c ðh h Þ  h the thermal conductivity of both fluid and diffused s p w 1 1 k Dk ðh h Þ kðh h Þ l T w 1  w 1 particles respectively, k is the Knudsen number. , C ¼ , S ¼ , Ln ¼ , s ¼ are p o D a T ð/ / Þ D T m w 1 B ref Suppose that the temperature difference within the variable viscosity parameter, thermal Grashof the flow regime is sufficiently small such that h can number, mass Grashof number, magnetic parameter, be expressed as a linear function of h : With the permeability parameter, variable thermal conductiv- expansion of h in Taylor’s series about T and ity parameter, radiation parameter, Prandtl number, neglecting higher order terms to obtain (Idowu and heat generation parameter, Eckert number, Dufour Falodun, 2019). number, Brownian motion parameter, thermophore- sis parameter, Schmidt number, chemical reaction 4 3 4 h ¼ 4h hh (13) 1 1 parameter, Soret number, Lewis number and ther- mophoretic parameter. The physical quantities of ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 153 Figure 2. Effect of Casson parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, Sc ¼ a b n b p t o p f 0:61, S ¼ 3:0: interest are the skin friction, Nusselt number and The rate of mass transfer gives rise to the Sherwood Sherwood number. The skin friction due to viscous number given by drag in the vicinity of the walls is given as Sh ¼ ¼ C ð0Þ s D ð/  / Þ C ¼ w 1 qw where where @/ D d ¼ D ¼ ð/ / ÞC ð0Þ P @u y w 1 @y b s ¼ l þ pffiffiffiffiffiffi ¼ 0 w B y¼0 @y 2p ðReÞ C ¼ 1 þ f ð0Þ 3. Solution techniques The rate of heat transfer (Nusselt number) due to the In this section, the SHAM is applied on the trans- heat transfer between the fluid and wall is given as formed equations (16)–(18) subject to (19) and (20). SHAM is the discrete version of the traditional homo- Nu ¼ ¼ T ð0Þ topy analysis method (HAM). HAM is useful in ðh  h Þ w 1 decomposing system of nonlinear differential equa- tion to linear differential equations. By applying where SHAM on the decomposed linear ordinary differential equations, the Chebyshev spectral collocation @h c q ¼ k ¼ ðh h ÞT ð0Þ w w 1 method is used to solve the resulting equations. The @y b y¼0 physical region is first transformed from the physical 154 A. S. IDOWU AND B. O. FALODUN Figure 3. Effect of Dufour parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, Sc ¼ a b n b p t o p f 0:61, S ¼ 3:0: region ½0, 1Þ to the ½1, 1 by using the technique of 1 1 00 000 00 00 þ 1 þ T f 0þ  1 þ T f þ ff þ ff a 0 a 0 0 0 domain truncation (Sibanda, Motsa, and Makukula, b b 00 00 0 2 0 0 0 2 2012). Hence, the problem solution is obtained in the þf f þ f f ðf Þ 2f f ðf Þ þ  cos ðUÞT 0 0 a 0 0 0 interval ½0, g  and not ½0, 1Þ again. It resulted to the 1 2 þ cos ðUÞT þ  cos ðUÞC þ  cos ðUÞC M f0 a 0 b b 0 use of the following algebraic mapping 1 1 1 1 2 0 0 M f0  1 þ f  1 þ f 2g P b P b f ¼ 1, n2½ 1, 1 (21) s s 1 1 1 1 0 0 1 þ  Tf  1 þ  ff a a For convenience we make the boundary conditions P b P b s s homogeneous by applying the transformations 1 1 1 1 0 0 1 þ  T f  1 þ  T f ¼ 0 a 0 a 0 fðgÞ¼ fðnÞþ f ðgÞ, TðgÞ¼ TðnÞþ T ðgÞ, CðgÞ P b P b 0 0 s s (23) ¼ CðnÞþ C ðgÞ (22) 00 00 00 00 ð1 þ R ÞT þð1 þ R ÞT þ d TT þ d TT þ d T T p p y y 0 y 0 substituting Equation (22) into (16)–(18) to obtain 00 0 0 0 0 0 2 þ d T T þ PrfT þ PrfT þ Prf T þ Prf T þ d ðT Þ y 0 0 0 y 0 0 0 1 1 1 0 00 0 00 0 00 0 0 0 2 1 þ  T f þ 1 þ  T f þ 1 þ  T f a a a þ 2d T T þ d ðT Þ þ PrHT þPrHT 0 0 y y 0 0 0 b b b 1 1 00 00 1 1 1 0 00 00 000 þPrE 1 þ ðf00Þ þ 2PrE 1 þ f f n n þ 1 þ  T f þ 1 þ f 0þ 1 þ f 0 0 0 b b b b b 00 00 00 1 1 00 000 þPrE 1 þ ðf Þ þ D C þ D C f f 0 0 þ 1 þ Tf 0þ 1 þ Tf a a 0 b b b ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 155 Figure 4. Effect of magnetic parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ deg d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, y a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: 00 0 1 1 a ¼ f , a ¼ f , a ¼2f , a 2 00 00 5 6 0 7 8 0 0 þ E 1 þ  Tðf00Þ þ 2E 1 þ  Tf f n a n a b b 1 1 ¼ 1 þ  f , 1 1 2 2 P b þE 1 þ  Tðf Þ þ E 1 þ  T ðf00Þ n a n a 0 b b 1 1 a ¼ 1 þ  T 9 a 0 1 1 2 P b 00 00 00 þ 2E 1 þ  T f f þ E 1 þ  T ðf Þ n a 0 n a 0 0 0 b b 1 1 2 0 00 000 0 0 0 0 0 0 0 0 0 G ðgÞ¼ 1 þ  T f  1 þ f þ N C T þ N C T þ N C T þ N C T þ N ðT Þ 1 a b b b b t 0 0 0 0 0 0 0 b b 0 0 0 þ 2N T T þ N ðT Þ ¼ 0 t t 0 0 000 00 0 2 (24) 1 þ T f f f þðf Þ a 0 0 0 0 0 00 00 0 0 0 C þ C S C CS C C þ S fC þ S fC þ S f C n p n p 0 n n n 0 0 0 S N S N n t n t  cos ðUÞT  cos ðUÞC þ M f0 0 00 00 a 0 b 0 0 þ S f C þ S S þ T þ S S þ T n 0 n 0 n 0 0 0 L N L N n b n b 1 1 1 1 00 00 00 00 0 0 0 0 0 0 þ sCT þ sCT þ sC T þ sC T þ sT C þ sT C þ 1 þ f þ 1 þ  T f a 0 0 0 0 0 0 0 0 P b P b s s 0 0 0 0 þ sT C þ sT C ¼ 0 0 0 0 0 0 b ¼ d T , b ¼ PrT , b ¼ Prf , b ¼ 2d T , y 0 0 y (25) 1 2 3 4 0 0 Simplifying the above equations by setting 1 1 00 00 b ¼ 2PrE 1 þ f b ¼ 2E 1 þ  f , n n a 5 0 6 0 b b 1 1 00 0 a ¼ 1 þ  f , a ¼ 1 þ  T , 1 a 2 a 0 0 b b 1 1 00 2 b ¼ En 1 þ  ðf Þ , b ¼ E 1 þ  T , 1 1 a n a 0 7 0 8 b b a ¼  1 þ f , a ¼  1 þ T , 3 a 4 a 0 b b 156 A. S. IDOWU AND B. O. FALODUN Figure 5. Effect of Brownian motion parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼ deg ¼ d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ a y a b n b p t o p f 2:0, Sc ¼ 0:61, S ¼ 3:0: Substituting the above coefficient parameters into 00 0 b ¼ 2E 1 þ  T f , b ¼ N T , n a 0 b 9 10 0 0 Equations (23)–(25) to obtain 1 1 0 0 0 00 0 00 00 b ¼ N C , b ¼ 2N T 1 þ  T f þ a T þ a f þ 1 þ f 0 b t a 1 2 11 12 0 0 b b 00 00 0 2 G ðgÞ¼ð1 þ R ÞT d T T d ðT Þ PrHT 2 p y 0 y 0 00 00 0 0 0 þ  1 þ Tf 0þ a T þ a f 0 a 3 4 00 2 00 00 0 0 PrE 1 þ ðf Þ n þ ff þ a f þ a f ðf Þ þ a f þ  cos ðaÞT 5 6 7 a þ  cos ðaÞCM f0 0 00 00 1 1 Prf T D C E 1 þ  T ðf Þ 0 n a 0 0 0 0 0 0 1 þ  Tf þ a T þ a f ¼ G ðgÞ a 8 9 1 P b 0 0 0 (26) N C T N ðT Þ b t 0 0 0 00 00 00 0 0 ð1 þ R ÞT þ d TT þ b T þ b T þ PrfT b f þ b T p y 1 1 2 3 0 00 c ¼ S C , c ¼ S f , c ¼ sT , c ¼ sC , c n n 0 0 1 0 2 3 0 4 5 2 2 0 0 00 þ d ðT Þ þ b T þ PrHT þ PrE 1 þ ðf00Þ þ b f y n 4 5 0 0 ¼ sC , c ¼ sT 0 6 0 00 00 þ D C þ E 1 þ  Tðf00Þ þ b f f n a S N 6 n t 00 0 00 G ðgÞ¼ C þ S C C S f C  S S þ T 3 n p 0 n 0 n 0 0 0 0 L N n b 00 0 0 0 þb T þ b ðf00Þ þ b f þ N C T þ b C 7 8 9 a 10 00 0 0 sC T sT C 0 0 0 0 0 2 þb T þ N ðT Þ þ b T ¼ G ðgÞ (27) t 2 11 12 ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 157 Figure 6. Effect of porosity parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ deg d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, y a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: S N 1 n t 00 0 0 00 0 00 000 000 00 C S C C þ S fC þ c f þ c C þ S S þ T a T þ a f þ 1 þ f þ a T þ a f þ a f þ a f n p n n 0 1 2 3 l 4 5 l 6 1 2 l l l l l L N b n b 00 00 0 0 0 0 þsCT þ c C þ c T þ sT C þ c T þ c C ¼ G ðgÞ 0 2 3 4 5 6 þ a f þ  cos ðUÞT þ  cos ðUÞC M f0 7 a l b l l (28) 1 1 0 0 1 þ f þ a T þ a f ¼ G ðgÞ 8 l 9 1 l l Equations (26)–(28) are nonlinear equations. In P b the application of SHAM, the nonlinear equations are (31) 00 00 0 0 decomposed into linear and nonlinear parts. It ð1 þ R ÞT þ b T þ b T þ b f þ b T þ b T þ PrHT p l l l l 1 1 l 2 3 l 4 l should be noted that the derivatives of f, T and C 00 00 00 00 0 þ b f þ D C þ b f þ b T þ b f þ b C f l 5 l l 6 l 7 9 l 10 l are with respect to n defined by: þ b T þ b T ¼ G ðgÞ l 2 11 l 12 d 2 d (32) ¼ (29) dn L dg S N n t 00 0 00 C S C C þ c f þ c C þ S S þ T n p l l n 0 l 1 2 l l An initial guess is chosen with reference to the L N n b boundary conditions (15) and (16) as the following þ c C þ c T 3 4 l functions 0 0 þ c T þ c C ¼ G ðgÞ 5 l 6 l g g f ðgÞ¼ S þ e þ 1, T ðgÞ¼ C ¼ e (30) 0 w 0 ðgÞ (33) subject to: The non-homogeneous linear part of Equations (26)–(28) are decomposed from the nonlinear part as 0 0 f ð1Þ¼ f ð1Þ¼ f ð1Þ¼ 0, Tð1Þ¼ T ð1Þ l l l l the following equations ¼ 0, C ð1Þ¼ C ð1Þ¼ 0 (34) l l 158 A. S. IDOWU AND B. O. FALODUN Figure 7. Effect of radiation parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ deg d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, y a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: The boundary conditions (34) is chosen to be all collocation points (Trefethen, 2000) given by equals to zero with reference to the transformed pj domain ½1, 1, to be able to implement the linear n ¼ cos (38) part SHAM solution. The Chebyshev pseudospectral method is applied on Equations (26)–(28). The provided j ¼ 0, 1, :::N and N þ 1 is the number of col- unknown functions f ðnÞ, T ðnÞ and T in the l l l location points. The unknown functions f ðnÞ, T ðnÞ l l Equations (26)–(28) are approximated as a truncated and C ðnÞ are approximated by the use of Lagrange series of Chebyshev polynomials given by Fagbade form of interpolating polynomial which interpolates et al. (2018) as: the unknown functions f ðnÞ, T ðnÞ and C ðnÞ at the l l l Gauss-Lobatto collocation points as defined in f ðnÞuf ðn Þþ f T ðn Þ, j ¼ 0, :::, N (35) l j k 1k j l Equation (38). k¼0 T ðnÞuT ðn Þþ T T ðn Þ, j ¼ 0, :::, N (36) l j k 2k j 4. Results and discussion k¼0 The transformed governing equations (16)–(18) sub- C ðnÞuC ðn Þþ C T ðn Þ, j ¼ 0, :::, N (37) ject to the boundary conditions (19) and (20) are set l k 3k j j k¼0 of coupled highly non-linear ordinary differential equations. These set of equations were solved where T , T and T are the kth Chebyshev polyno- 1k 2k 3k mial and n , n , :::, n are Gauss-Lobatto collocation numerically using spectral homotopy analysis 0 1 N point. The Chebyshev nodes in the transformed method. To study the behaviour of various flow domain ½1, 1 are defined by the Gauss-Lobatto parameters such as Casson, heat generation ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 159 Figure 8. Effect of Soret parameter on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, Sc ¼ a b n b p t o p f 0:61, S ¼ 3:0: parameter, Soret-Dufour parameter, chemical reac- effect on the concentration profile. The effect of D tion parameter, radiation parameter, etc, a compre- as presented in Figure 3 is in good agreement with hensive numerical computation is carried out. that of Alao et al. (2016). The diffusion-thermal Figure 2 represents the effect of Casson non- added to the thermal boundary layer influences the Newtonian fluid parameter ðbÞ on the velocity, tem- temperature field. perature and concentration fields. With increase in b, The effect of magnetic parameter (M) on velocity, it is noticed in Figure 2 that transport rate reduces temperature and concentration fields is shown in within the thermal boundary layer. A slight decrease Figure 4.In Figure 4, increase in the values of mag- in the temperature field is observed as the values of netic parameter causes a damping effect on the vel- b increases. It worth mentioning that as Casson fluid ocity field by producing a drag-like force called parameter approaches infinity, the fluid behaves as Lorentz force. This force acts in the opposite direc- Newtonian fluid. Because of increase in the elasticity tion and thereby reduces the motion of an electric- stress parameter, there is thickening of the thermal ally conducting fluid. With increasing value of the boundary layer. The graphical results in Figure 2 is in magnetic parameter, the momentum boundary layer excellent agreement with the previous work of thickness decreases while the thermal boundary Zaigham Zia, Ullah, Waqas, Alsaedi, and Hayat layer slightly increases far away from the plate as (2018). Figure 3 illustrates the effect of Dufour par- shown in Figure 4. This result was found to be in ameter ðD Þ on the velocity, temperature and con- agreement with the previous work of Fagbade et al. centration fields. It is observed that increase in the (2018). In addition, increase in the values of M has values of D increases the velocity, temperature no effect on the concentration field. Figure 5 illus- fields, hydrodynamics and thermal boundary layer. trates the effect of Brownian motion parameter ðN Þ Increase in the values of D is noticed to has no on the velocity, temperature and concentration f 160 A. S. IDOWU AND B. O. FALODUN Figure 9. Effect of Schmidt number on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: fields. Increase in the values of Brownian motion par- a result of increase in the porosity term, the thermal ameter within the hydrodynamic boundary layer boundary layer thickness increases. This result is as brings more erratic movement of nanoparticles in shown in Figure 6. Effect of P is negligible on the the porous medium surroundings. Increase in the concentration field. Figure 7 illustrates the effect of values of N in Figure 5 is observed to decrease vel- radiation parameter ðR Þ on the velocity, tempera- ocity and concentration fields of the fluid nanopar- ture and concentration fields. When the radiation ticles. This is as a result of the random collision of parameter is increased, the fluid temperature, hydro- the fluid particles which causes reduction on the dynamic and thermal boundary layer increases. fluid velocity. It is obvious from Figure 8 that Physically, increase in R added more heat energy to increase in N slightly increases temperature and the the thermal boundary layer. This gives room for thermal boundary layer because of the nanofluid more temperature and thereby increase temperature porosity in the hydrodynamic and thermal boundary field. The temperature field as shown in Figure 7 layer. In addition, increase in the values of N in the increases the entire thermal boundary layer. Increase boundary layer decreases the solutal concentration. in R is seen to decrease the nanofluid concentration Figure 6 depicts the effect of porosity term ðP Þ close to the plate. on the velocity, temperature and concentration Figure 8 depicts the effect of Soret parameter ðS Þ fields. Increasing the porosity expands holes and on the velocity, temperature and concentration allows more movement of nanoparticles within the fields. Increase in the values of S is noticed to hydrodynamics and thermal boundary layer. Increase increase the fluid velocity and the momentum in the porosity term P brings decrease to the boundary layer thickness as shown in Figure 8. Also, momentum boundary layer thickness and thereby increase in the values of Soret parameter accelerates decreases the fluid velocity as shown in Figure 6.As the concentration field close to the plate and ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 161 Figure 10. Effect of thermal Grashof number on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼ deg ¼ d ¼ 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ a y a b n b p t o p f 2:0, Sc ¼ 0:61, S ¼ 3:0: negligible at the free stream. Effect of the Schmidt Figure 11 depicts the effect of Prandtl number number ðS Þ on the velocity, temperature and con- (Pr) on the velocity, temperature and concentration centration fields is illustrated in Figure 9. The profiles. The velocity profile is observed to decrease Schmidt is a dimensionless number which is defined with increase in the values of Prandtl number (Pr). as the ratio of fluid viscosity to mass diffusivity. This is because fluids that have higher Pr possess Hence, if the viscosity of fluid is greater than mass greater viscosities and thereby reduces the fluid diffusivity, there is going to be more Schmidt num- velocities and lower the wall skin friction coefficient. ber within the entire boundary layer. It is observed In addition, increase in the values of Pr correspond that increase in S drastically decrease the velocity to a decrease in the fluid temperature and the ther- and concentration field due to the nanofluid porosity mal boundary layer thickness. For small values of Pr, within the hydrodynamic and solutal boundary layer. that is Pr < 1 the fluid becomes very conductive. Increase in the values of S , increase the solutal Increase in the values of Pr is noticed to increase boundary layer. The effect of thermal Grashof num- fluid concentration at the wall. ber ð Þ on the velocity, temperature and concen- From Table 2, increase in the values of Lewis tration field is illustrated in Figure 10. The number increases the skin friction coefficient and buoyancy force acts like as favourable pressure gra- Sherwood number whereas it decreases the Nusselt dient on the fluid flow and increase nanofluid par- number. From Table 2, increase in the magnetic par- ticles within the boundary layer. This is shown in ameter (M) decreases the three physical quantities of Figure 10 as increase in the thermal Grashof num- engineering (that is, skin friction coefficient, Nusselt ber ð Þ increases the momentum boundary layer number and Sherwood number). In Table 3, increase and slightly reduces the thermal and mass bound- in the values of Lewis number (Ln) increases the ary layer. Nusselt number and decreases the skin friction 162 A. S. IDOWU AND B. O. FALODUN Figure 11. Effect of Prandtl number on the (a) velocity, (b) temperature and (c) concentration profiles when b ¼  ¼ d ¼ a y deg 3:0, ¼  ¼ H ¼ 2:0,U ¼ 30 , M ¼ E ¼ N ¼ C ¼ N ¼ Ln ¼ s ¼ 1:0, P ¼ 0:5, Pr ¼ 0:71, R ¼ 0:6, D ¼ 2:0, a b n b p t o p f Sc ¼ 0:61, S ¼ 3:0: coefficient and the Sherwood number. In Table 3, thermal conductivity is considered. A parametric increase in the values of Brownian motion parameter study on the velocity, temperature, and concentra- (Nb) decreases the skin friction coefficient, Nusselt tion. The flow equations were solved numerically number and Sherwood number. From Table 3, using spectral homotopy analysis method. From the increase in the values of Schmidt number decreases results obtained, increase in the non-Newtonian Casson fluid parameter brings increase to the skin the wall skin friction coefficient and the Nusselt number whereas increase in the Sherwood number friction coefficient and reduces the rate of heat and mass transfer. The results in the present study is noticeable. In Table 3, it is noticed that increase in the non-Newtonian fluid parameter ðbÞ increases the revealed that the behaviour of the non-Newtonian fluid parameter changes to Newtonian fluid as the Nusselt and Sherwood number but decreases the Casson fluid parameter approaches infinity. Increase skin friction coefficient. Tables 4 and 5 shows the in the magnetic parameter (M) brings a decrease to comparison of the present results with previous pub- velocity of the fluid. It is observed that increase in lished works and was found to be in the magnetic parameter brings to the fluid concen- good agreement. tration close to the plate and reduces the free stream at the porous medium. The present result 5. Concluding remarks gives account to the effect of radiation parameter This study examined the effects of thermophoresis and it is noticed that increase in radiation parameter and Soret-Dufour on heat and mass transfer mixed gives rise to the fluid temperature and has no effect convective flow of MHD non-Newtonian nanofluid on the velocity. It is observed that increase in the over an inclined plate embedded in a porous Schmidt number gives rise to the concentration field medium. The analysis for variable viscosity and at the free stream whereas it increase the fluid ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 163 0 0 Table 2. Computational values for skin friction coefficient ðC Þ, Nusselt number ðT ð0ÞÞ, and sherwood number ðC ð0ÞÞ for different values of Ln and M. Ln M C Nh Sh 0.0 1.0 1.08233781 0.72334184 1.02490066 0.5 1.0 1.14285394 0.35872612 1.08228644 1.0 1.0 1.20739181 0.22940864 1.14323113 2.0 1.0 1.34923800 0.18349278 1.27658671 0.1 0.4 1.45926666 0.83996184 1.09810897 0.1 0.6 1.31269481 0.79692501 1.05550050 0.1 0.8 1.20198770 0.75406987 1.04802955 0.1 1.0 1.06855863 0.72694283 1.00977521 0 0 Table 3. Computational values for skin friction coefficient ðC Þ, Nusselt number ðT ð0ÞÞ, and sherwood number ðC ð0ÞÞ for different values of Ln, Nb, Sc and b. Parameters Present work Ln Nb Sc b Cf Nh Sh 0.2 1.0 0.61 3.0 0.13265182 0.88477929 0.78652409 0.4 1.0 0.61 3.00 0.12774386 0.90996351 0.58398366 0.6 1.0 0.61 3.0 0.12488548 0.93522964 0.38152589 0.8 1.0 0.61 3.0 0.12289942 0.96147715 0.17200602 1.0 0.0 0.61 3.0 0.08967359 0.94746978 0.34074944 1.0 1.0 0.61 3.0 0.12519871 0.92975180 0.42525615 1.0 2.0 0.61 3.0 0.12488548 0.93522964 0.38152589 1.0 3.0 0.61 3.0 0.12448055 0.94174484 0.32969808 1.0 1.0 0.3 3.0 1.04462222 0.78476059 0.67090855 1.0 1.0 0.6 3.0 0.97607905 0.73329484 0.94697241 1.0 1.0 0.9 3.0 0.93182279 0.69528927 1.15619714 1.0 1.0 1.2 3.0 0.89901583 0.66435691 1.33199172 1.0 1.0 0.61 0.0 0.95154181 0.73022820 0.95324841 1.0 1.0 0.61 0.2 1.77694769 0.78778324 1.00573400 1.0 1.0 0.61 0.3 1.57443332 0.88549205 1.09642636 1.0 1.0 0.61 0.4 1.55314440 0.77899759 1.07697499 Table 4. Comparison of the present result with that of Mondal et al. (2018) when P ¼ H ¼ N ¼ N ¼ 0 and at constant s b t viscosity and thermal conductivity. Mondal et al. (2018) Present result D So C Nu Sh C Nu Sh f x x f x x 0.030 2.0 6.238707 1.151932 0.1554706 6.237707 1.142931 0.1554601 0.037 1.6 6.160867 1.144651 0.2306910 6.151877 1.143551 0.2306811 0.050 1.2 6.087934 1.135752 0.3057275 6.077864 1.125750 0.3057266 0.075 0.8 6.023187 1.123219 0.3808005 6.012188 1.123218 0.3808004 0.150 0.4 5.996934 1.096560 0.4574460 5.99694 1.096561 0.4574461 Table 5. Comparison of the present result with that of Rafique et al. (2019) when P ¼ Ec ¼ H ¼ C ¼ s ¼ 0 and at con- s p stant viscosity and thermal conductivity. Rafique et al. (2019) Present result N N b t C Nu C Nu f x f x 0.1 0.1 0.9524 2.1294 0.9532 2.1296 0.2 0.2 0.3654 2.5152 0.3666 2.5155 0.3 0.3 0.1355 2.6088 0.1359 2.6090 0.4 0.4 0.0495 2.6038 0.0499 2.6039 0.5 0.5 0.0179 2.5731 0.0177 2.5736 temperature close to the plate and reduces it at the The present study is useful in glass blowing pro- free stream. The novelty of this paper the investiga- cess. In glass blowing process, maintaining the tem- tion of varying viscosity and thermal conductivity on perature for glass blowing is very important. This is the flow of Casson non-Newtonian fluid in a porous because it requires optimal temperature to make it medium of an inclined plate. The result of this study flexible for blow so as to obtain the required shape. will be of help in chemical engineering processes Hence parameters such as thermal radiation and and the production of food. heat generation considered in this study are very 164 A. S. IDOWU AND B. O. FALODUN important in glass blowing process. Also, at very TðgÞ dimensionless temperature CðgÞ dimensionless concentration high temperature the properties of fluid such as vis- cosity and thermal conductivity may vary. It is observed from this study that increase in viscosity Disclosure statement brings decrease to the fluid flow. This allows us to No potential conflict of interest was reported by be able to control the flow of the fluid. On the other the authors. hand, an increase in thermal conductivity gives rise to the fluid temperature in such a way that it con- ORCID ducts more heat. The use of thermal conductivity in regulating the conducting nature of heat of the fluid B. O. Falodun http://orcid.org/0000-0003-1020-1677 also helps in controlling the blowing glass substance temperature. The present study is very useful in References many chemical engineering processes such as metal- Abu-Nada, E., Hakan, F. O., & Pop, I. (2012). Buoyancy lurgical and extrusion of polymer which involves induced flow in a nanofluid filled enclosure partially cooling of molten liquid. The polyethylene oxide and exposed to forced convection. Superlattices and polyisobutylene solution in cetane are having elec- Microstructures, 51(3), 381–395. doi:10.1016/j.spmi.2012. tromagnetic properties which are used as cooling 01.002 liquids because they are regulated by external mag- Ahmed, S. E., & Rashed, Z. Z. (2019). MHD natural convec- tion in a heat generating porous medium-filled wavy netic fields. Hence, controlling parameters such as enclosures using Buongiorno’s nanofluid model. Case thermal radiation, viscous dissipation parameter, Studies in Thermal Engineering, 14, 100430. doi:10.1016/j. magnetic field parameter, chemical reaction param- csite.2019.100430 eter etc. considered in this study finds application in Alam, M. S., Rahman, M. M., & Sattar, M. A. (2009). chemical engineering processes. Transient magnetohydrodynamic free convective heat and mass transfer flow with thermophoresis past a radi- ate inclined permeable plate dependent viscosity. 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Journal

Arab Journal of basic and Applied SciencesTaylor & Francis

Published: Jan 1, 2020

Keywords: Soret-Dufour; mixed convective; inclined plate; Chebyshev pseudospectral method; SHAM; MHD; chemical reaction

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