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Soret–Dufour effects on MHD heat and mass transfer of Walter’s-B viscoelastic fluid over a semi-infinite vertical plate: spectral relaxation analysis

Soret–Dufour effects on MHD heat and mass transfer of Walter’s-B viscoelastic fluid over a... JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 2019, VOL. 13, NO. 1, 49–62 https://doi.org/10.1080/16583655.2018.1523527 RESEARCH ARTICLE Soret–Dufour effects on MHD heat and mass transfer of Walter’s-B viscoelastic fluid over a semi-infinite vertical plate: spectral relaxation analysis A. S. Idowu and B. O. Falodun Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria ABSTRACT ARTICLE HISTORY Received 12 May 2018 Soret–Dufour effects on MHD heat and mass transfer of Walter’s-B viscoelastic fluid over a Revised 7 September 2018 semi-infinite vertical plate are considered. The equations of motion are set of partial differen- Accepted 10 September 2018 tial equations; these are non-dimensionalized by introducing an appropriate non-dimensional quantity. The dimensionless equations along with the boundary conditions are solved numer- KEYWORDS ically using the spectral relaxation method (SRM). All programs are coded in MATLAB R2012a. Soret–Dufour effect; heat Results are presented in graphs, and numerical computations of the local skin friction, local Nus- transfer; viscous dissipation; selt number and local Sherwood number are presented in a tabular form. The result revealed thermal radiation; chemical reaction; spectral relaxation that as the viscoelastic parameter increases, the velocity profile close to the plate decreases but method when far away from the plate, it increases slightly. The present results were found to be in good agreement with those of the existing literature. 1. Introduction increases the concentration boundary layer thickness. They reported that the behaviour of Dufour and Soret The physical effects occur in MHD basically when a on the temperature and concentration profiles is oppo- conductor migrates into a magnetic field, electric cur- site. Choudhury and Kumar Das [2] studied viscoelas- rent is induced and creates its own magnetic field tic MHD-free convective flow through porous media in (Lenz’s law). The conductor used in this paper is the the presence of radiation and chemical reaction with fluid with complex motions. Another important thing heat and mass transfer. Their governing partial differ- to note is that the moment currents are induced prob- ential equations were solved using the multiple per- ably by a motion of an electrically conducting fluid as turbation technique. They concluded that the velocity a result of a magnetic field; a resistive force acts on field possesses an accelerating trend with the grow- the fluid and decelerates its motion. MHD has gained ing effect of the viscoelastic parameter. Rashidi et al. considerable interest due to its fundamental impor- [3] did not consider the porous medium. The homo- tance in the industrial and technological applications topy analysis method with two auxiliary parameters was such as in coating of metals, crystal growth, electro- used and its results show that increasing Soret num- magnetic pumps, power generators, MHD accelerators ber or decreasing Dufour number leads to a decrease and reactor cooling. Many researchers in the field of in velocity and temperature profiles. Vedavathi et al. fluid dynamics have studied MHD viscoelastic fluid flow [4] considered radiation and mass transfer effects on by considering the effects of Soret and Dufour num- unsteady MHD convective flow over an infinite vertical ber. Soret–Dufour effects have been investigated by plate with Dufour and Soret effects. In their analysis, the numerous scholars in fluid mechanics due to their sig- effects of Soret and Dufour parameters on the veloc- nificance in sciences and engineering like Soret in iso- ity, temperature and concentration profiles are plotted tope separation. Gbadeyan et al. [1] considered the in Figures 12(a–c). It is noted that a decrease in Soret effects of Soret and Dufour in their analysis. Their equa- number and an increase in Dufour number produce a tions of motion were solved using the shooting method decrease in the velocity and concentration profiles. In with sixth-order of the Runge–Kutta technique. In their another investigation of Sharma and Aich [5], the effects analysis, the variation of Dufour in the velocity field is of Soret and Dufour were significant. They concluded depicted in Figure 1 and it shows that the increase in in their study that the rate of flow decreases with an Dufour number gives a slight increase in the velocity increase in both Soret and Dufour effects. Recently in of the fluid. Also, effect of Dufour number as plotted the work of Hayat et al. [6], the influence of both Soret in Figure 10 of their result show that increasing Dufour and Dufour was investigated on the peristaltic flow of number increases the fluid temperature. It was discov- Jeffery fluid. The report was made on the study that ered in their study that increasing the Soret number CONTACT B. O. Falodun [email protected] © 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group 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. 50 A. S. IDOWU AND B. O. FALODUN Non-Newtonian fluids are important in many tech- nological applications compared to the Newtonian flu- ids. The non-Newtonian fluid flows find applications in modern technology and in industries. The non-linearity of the mechanism of non-Newtonian fluids presents a challenge to mathematicians, engineers and physi- cists. This non-linearity manifests itself in fields such as food, drilling operations and bioengineering. Rashidi et al. [16] examined analytic approximate solutions for MHD boundary-layer viscoelastic fluid flow over a con- tinuously moving stretching surface by the homotopy analysis method with two auxiliary parameters. Graph- ical results of Figures 7 and 8 show that the veloc- ity decreases with increasing viscoelastic parameter. Sivaraj and Kumar [17] examined chemically reacting dusty viscoelastic fluid flow in an irregular channel with convective boundary. They reported that velocity pro- files of all physical parameters decrease at the wavy wall. Figure 1. Physical model of the problem. Manglesh and Gorla [18] studied the effects of thermal radiation, chemical reaction and rotation on unsteady MHD viscoelastic slip flow. They reported in their study the behaviour of Soret and Dufour number for tem- that increasing the viscoelastic parameter makes the perature and concentration is contradictory. Iqbal and hydrodynamic boundary layer to adhere strongly to the Khalid [7] studied the analysis of thermally developing surface and results in the retardation of the flow to the laminar convection in the finned double-pipe. Ahmed left half channel but accelerates in the right half when and Iqbal [8] examined MHD power-law fluid flow and there is no slip boundary condition. In 2013, Kumar heat transfer analysis through Darcy Brinkman porous and Sivaraj [19] examined heat and mass transfer in media in the annular sector. Their results revealed that MHD viscoelastic fluid flow over a vertical cone and flat an increase in the Hartman parameter increases the rate plate with variable viscosity. They reported in the study of heat transfer. Ahmed et al. [9] examined the study that magnetic field, buoyancy ratio parameter, viscosity of forced convective power-law fluid through an annu- variation parameter, Eckert number and chemical reac- lus sector duct numerically. Numerical study of heat tion parameter play an important role in viscoelastic transfer and fluid flow through an annular sector duct fluid flow through the porous medium. In the work of filled with porous media was carried out by Iqbal and Eswaramoorthi et al. [20], viscoelastic type of fluid was Afag Hamna [10]. Iqbal et al. [11] recently considered considered. The flow equations were solved using the the simulation of MHD-forced convection heat transfer homotopy analysis method. It was reported in the study through the annular sector. that increasing the viscoelastic parameter results in a The role of the magnetic field and surface corru- decrease in the velocity. Graphical result of Figure 10 gation on natural convection in a nanofluid-filled 3D shows that increasing the viscoelastic parameter leads trapezoidal cavity was carried out by Selimefendigil to a decrease in the temperature.Finite element anal- ysis of MHD viscoelastic nanofluid flow over a stretch- and Oztop [12]. Conjugate natural convection in a nanofluid-filled partitioned horizontal annulus formed ing sheet with radiation is examined by Madhu and by two isothermal cylinder surfaces under magnetic Kishan [21]. field was investigated by Selimefendigil and Oztop [13]. In 2013, Rao et al. [22] studied finite element anal- The work of Selimefendigil et al. [14] deals with the ysis of radiation and mass transfer flow pass over a flow of MHD in a lid-driven nanofluid-filled square cavity semi-infinite moving vertical plate with viscous dis- with a flexible side wall. Finite element formulation was sipation. In their study, they neglected the effects used as a method of solution and their findings revealed of Soret and Dufour parameters. In 2016, Alao et al. that the Brownian motion effect on the thermal conduc- [23] considered the effects of thermal radiation, Soret tivity of the nanofluid is significant. The finite element and Dufour on an unsteady heat and mass trans- method was used to solve the analysis of MHD mixed fer flow of a chemically reacting fluid pass over a convection in a flexible-walled and nanofluid- filled the semi-infinite vertical plate with viscous dissipation. lid-driven cavity with volumetric heat generation by The work of Alao et al. [23] was an extension of Selimefendigil and Oztop [15]. The study revealed that Rao et al’s. [22] work by considering the effects of decreasing values of Richardson number and increasing Soret and Dufour parameters in their governing equa- values of Hartmann number decreases the average heat tions. The two problems are centred on Newtonian transfer. fluid. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 51 The present study is a Non-Newtonian fluid model. Its Here, η is the limiting viscosity at the small rate of objective is to illustrate the use of the spectral relaxation shear and is given as method (SRM) for solving the equations of motion rep- ∞ ∞ resenting the physical model of Soret–Dufour effects η = N(τ )dτ and k = τN(τ )dτ (5) 0 0 0 0 on MHD heat and mass transfer of Walter’s B vis- coelastic fluid over a semi-infinite vertical plate and to N(τ ) is the relaxation spectrum as discussed by Walters explore the effects of different controlling flow param- [24]. This idealized model is a valid approximation of eters as encountered in the equations. SRM is an itera- Walters-B taking short relaxation time into account so tive method that employs the Gauss–Seidel approach that terms involving in solving both linear and non-linear different equa- tions. SRM is found to be effective and accurate. The t N(τ )dτ , n ≥2(6) governing equations are systems of partial differential equations which are transformed into a dimensionless have been neglected in the momentum equation and form by introducing suitable non-dimensional quanti- k is significant. ties. Numerical computations are carried out and graph- Under the above assumptions and the usual Boussi- ical results for the velocity, temperature, and concentra- nesq’s approximation, the governing equations and tion profiles as well as the local skin friction, local Nusselt boundary conditions are given as (see details in Alao number and local Sherwood number coefficients within et al. [23]) the boundary layer flow are discussed. ∂v =0(7) ∂y 2. Equations of motion ∂u ∂u ∂ u The unsteady free convective flow of a viscoelastic fluid + v = ν + gβ (T − T ) + gβ (C − C ) t ∞ c ∞ ∂t ∂y ∂y (Walters B’ model) over a semi-infinite vertical plate 2 3  3 with time-dependent oscillatory suction in the presence σ B u K ∂ u ∂ u − − + v (8) of a transfer magnetic field is considered. The plate is  2 3 ρ ρ ∂t ∂y ∂y considered infinite in the x -direction, thus the x -axis shall be taken along the vertical infinite plate in the ∂T ∂T ∂ T 1 ∂q μ ∂u upward direction and the y -axisnormaltothe plate ∂y + v = α − + ∂t ρc ∂y ρc ∂y ∂y p p (see Figure 1). The flow direction is vertically upward 2 ∗ and in the continuity equation, the term ∂u /∂x is DK ∂ C β u Q T 0 + + (T − T) + (T − T ) (9) ∞ ∞ neglected. It is assumed that initially at t ≤ 0, the plate 2 ρc ρc C c ∂y p p s p and fluid are at the same temperature. Soret, Dufour, heat generation or absorption is taken into account. 2 2 ∂C ∂C ∂ C DK ∂ T It is assumed that the magnetic Reynolds number is + v = D − K  (C − C ) + (10) 2 2 ∂t ∂y ∂y T ∂y small so that the induced magnetic field is neglected. subject to the conditions In the direction of they -axis, a magnetic field of uni- form strength B is applied. Walters-B viscoelastic type n t u = U , T = T + ψ(T − T )e , 0 w w ∞ of fluid is considered in this paper. n t Following Choudhury and Kumar Das [2], the consti- C = C + ψ(C − C )e at y = 0 (11) w w ∞ tutive equation for Walter’s-B viscoelastic fluid can be defined as u → 0, T → T , C → C as y →∞ (12) ∞ ∞ σ =−pg + σ (1) ∗ ik ik ik β (T − T) and Q (T − T ) are the heat generation ∞ 0 ∞ and absorption, respectively. ik ik ik σ = aη e − 2k e (2) 0 0 Both sides of the continuity equation (7) are inte- where η is the limiting viscosity at small rates of shear, grated to get v = constant. Obviously, the suction ik k is the elastic co-efficient, σ is the stress tensor, p is velocity normal to the plate is a constant function or the isotropic pressure, g is the metric tensor of a fixed assumed as a function of time. In this paper, it is con- ik co-ordinate system x and v is the velocity vector. The sidered as a case when it is both constant and time- ik contravariant form of e is given as dependent expressed as [23] ik n t ∂e ik m ik k im i mk v =−v (1 + εAe ) (13) e = + v e − V e − v e (3) ,m ,m ,m ∂t It is assumed in this paper that ∂q /∂y >> ∂q /∂x r r The convected derivative of the deformation rate and as a result the x -direction radiative flux ∂q /∂x is ik r tensor e is defined by neglected. However, the radiative heat flux that domi- ik 2e = v + v (4) nates the flow is ∂q /∂y . i,k k,i 52 A. S. IDOWU AND B. O. FALODUN Assuming that the temperature difference within the and the boundary conditions lead to flow regime is sufficiently small in such a way that T can ∂u ∂u ∂ u nt 2 be expressed as a linear function of the free stream tem- − (1 + εAe ) = + G ϑ + G ϕ − M u r m 4 ∂t ∂y ∂y perature T . Expanding T in the Taylor series about 3 3 T and neglecting the higher terms let us consider the ∂ u ∂ u nt − α − (1 + εe ) (23) 2 3 Taylor series expansion of the function m(x) about x 0 ∂t∂y ∂y (x − x ) m(x) = m(x ) + (x − x )m (x ) + m (x ) 2 0 0 0 0 ∂ϑ ∂ϑ 1 + Rr ∂ ϑ ∂u 2! nt − (1 + εAe ) = + E ∂t ∂y P ∂y ∂y (x − x ) + ··· + m (x ) (14) n! ∂ ϕ + D + δ ϑ − uϑ (24) u x ∂y 4 4 setting m(x) = T and m(x ) = m(T ) = T in the 0 ∞ equation above and neglecting higher order term lead 2 2 ∂ϕ ∂ϕ 1 ∂ ϕ ∂ ϑ nt 2 to: − (1 + εAe ) = − k ϕ + S (25) 2 2 ∂t ∂y S ∂y ∂y 4 3 4 T = 4T T − 3T (15) ∞ ∞ where G , G , P , Rr, E , S , k , D ,S , α,  and δ are r m r c c r u r x Using the Roseland approximation, the radiative the thermal Grashof number, mass Grashof number, heat flux is given by Prandtl number, radiation parameter, Eckert number, Schmidt number, chemical reaction parameter, Dufour 4σ ∂T number, Soret number, viscoelastic parameter, heat q =− (16) 3k ∂y generation/absorption coefficient and heat source/sink parameter, respectively. where σ is the Stefan–Boltzmann constant and k is e e The transformed initial and boundary conditions are the mean absorption coefficient. By using the Roseland approximation, the present study is limited to optically nt εe at y = 0 (26) thick fluids. If temperature differences within the flow are sufficiently small, then Equation (15) can be lin- u → 0, ϑ → 0, ϕ → 0, at y →∞ (27) earized and in view of Equations (15) and (16), Equation (9) reduces to The physical quantities of interest are the local skin friction coefficient (Cf ), local Nusselt number (Nu) and 2 2 ∂T ∂T ∂ T 16σ ∂ T μ ∂u 3 2 Sherwood number (Sh) of the flow in practical engi- + v = α + T + ( ) 2 2 ∂t ∂y ∂y 3ρc k ∂y ρc ∂y p e p neering. The skin friction coefficient, Nusselt and Sher- 2 ∗ D k ∂ C β u Q wood number are given as m T 0 + + (T − T) + (T − T ) ∞ ∞ C c ∂y ρc ρc s p p p ∂C (17) ∂y τ Kq w w y =0 Cf = , Nu =− , Sh = ρU V T − T C − C 0 0 w ∞ w ∞ In order to write the governing equations and the boundary conditions in dimensionless form, the follow- where ing non-dimensional quantities are introduced: 2  2 ∂u ∂ u ∂ u τ = η − K + 2 2 w 0 y=0 0 u v y v t υn ∂y  ∂t ∂y ∂y 0 0 y=0 u = , y = , t = , n = (18) u υ υ v ∂C 4σ ∂T q =−K − . T − T C − C υρc υ ∞ ∞ p ∂y 3k ∂y y=0 y=0 ϑ = , ϕ = , P = = (19) T − T C − C k α w ∞ w ∞ υ gβv(T − T ) gβ v(C − C ) 3. Method of solution w ∞ w ∞ S = , G = , G = c r m 2 2 u v u v 0 0 0 0 The non-dimensionless transformed system of partial (20) differential equations is solved in this section using the 2 2 ∗ u K V β u ν 0 0 spectral relaxation method (SRM). SRM is an iterative 0 0 E = , α = ,  = (21) 2 2 c (T − T ) ρν ρc v procedure that employs the Gauss–Siedel type of relax- p w ∞ ation approach to linearize and decouple the system of 2 3 2 k v 16σ T σ B v Q ν coupled differential equations. The resulting non-linear r e 0 2 ∞ 0 k = , Rr = , M = , δ = 2 2 2 differential equations are further discretized and solved 3k k v ρv ρc v e p 0 0 0 (22) with the Chebyshev pseudo-spectral method [25]. The The above non-dimensional quantities on the gov- linear terms in each equation are evaluated at the cur- erning momentum, energy, concentration equations rent iteration level (denoted by r + 1) and the non-linear JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 53 terms are assumed to be known from the previous iter- subject to ation level (denoted by r). The following are the basic nt u (0, t) = 1, ϑ (0, t) = 1 + εe , r+1 r+1 steps of the method: nt ϕ (0, t) = 1 + εe (36) r+1 (1) decoupling and rearrangement of the governing non-linear equations in a Gauss–Siedel manner. u (∞, t) = 0, ϑ (∞, t) = 0, ϕ (∞, t) = 0 r+1 r+1 r+1 (2) discretizing the linear differential equations. (37) nt (3) solving the discretized linear differential equations where β = 1 + εAe , setting iteratively using the Chebyshev pseudo-spectral nt nt a = β = 1 + εAe , a = β = 1 + εAe , 0,r 1,r method. a = Grϑ + Gmϕ , b = (1 + Rr), 2,r r r 0,r Applying the spectral relaxation method (SRM), we ∂u r+1 nt first re-arrange the transformed governing Equations b = Prβ = Pr(1 + εAe ), b = PrEc , 1,r 2,r ∂y (23)–(25) to yield; ∂ ϕ b = PrDu , b = Pru , 3 2 3,r 4,r r ∂u ∂ u ∂u ∂ u ∂y nt + α = (1 + εAe ) + + G ϑ 2 2 ∂t ∂t∂y ∂y ∂y ∂ ϑ r+1 nt c = Scβ = Sc(1 + εAe ), c = ScSr (38) 3 0,r 1,r ∂ u ∂y 2 nt + G ϕ − M u − α(1 + εe ) ∂y Substituting the above coefficient parameters into (28) (33)–(35) gives 3 2 ∂u ∂ u ∂u ∂ u r+1 r+1 r+1 r+1 2 + α = a + + a 1,r 2,r ∂ϑ ∂ϑ 1 + Rr ∂ ϑ ∂u 2 2 nt ∂t ∂t∂y ∂y ∂y = (1 + εAe ) + + E ∂t ∂y P ∂y ∂y r 3 ∂ u r+1 − M u + a (39) 2 r+1 0,r ∂ ϕ 3 ∂y + D + δ ϑ − uϑ (29) u x ∂y ∂ϑ ∂ϑ ∂ ϑ r+1 r+1 r+1 = b + b + b + b 1,r 0,r 2,r 3,r 2 2 2 ∂t ∂y ∂y ∂ϕ ∂ϕ 1 ∂ ϕ ∂ ϑ nt 2 = (1 + εAe ) + − k ϕ + S (30) 2 2 ∂t ∂y S ∂y ∂y c + Prδ ϑ − b ϑ (40) x r+1 4,r r+1 subject to ∂ϕ ∂ϕ ∂ ϕ r+1 r+1 r+1 = c + − k ϕ + c (41) 0,r r+1 1,r nt nt u = 1, ϑ = 1 + εe , ϕ = 1 + εe at y = 0 (31) ∂t ∂y ∂y subject to u → 0, ϑ → 0, ϕ → 0, at y →∞ (32) nt u (0, t) = 1, ϑ (0, t) = 1 + εe , r+1 r+1 nt Adopting the SRM on the non-linear coupled partial ϕ (0, t) = 1 + εe at y = 0 (42) r+1 differential equations (28)–(30) subject to (31) and (32), to obtain u (∞, t) = 0, ϑ (∞, t) = 0, r+1 r+1 2 3 2 ∂u ∂ u ∂ u ∂ u ∂u ϕ (∞, t) = 0, at y →∞ (43) r+1 r+1 r+1 r+1 r+1 r+1 + α = β + + β 2 3 2 ∂t ∂t ∂y ∂y ∂y The unknown functions are defined by the Gauss– + Grϑ + Gmϕ − M u (33) r r r+1 Lobatto points given as πj ξ = cos , j = 0, 1, 2, ... , N;1 ≤ ξ ≤−1 (44) 2 2 ∂ϑ ∂ ϑ ∂u ∂ ϕ r+1 r+1 r+1 r Pr = (1+Rr) +PrEc +PrDu 2 2 where N is the number of collocation points. The ∂t ∂y ∂y ∂y domain of the physical region [0, ∞) is transformed into ∂ϑ r+1 + Prδ ϑ + Pru ϑ + Prβ x r+1 r r+1 [−1, 1]. Thus, the problem is solved on the interval [0, L] ∂y instead of [0, ∞). The following transformation is used (34) to map the interval together η ξ + 1 = , − 1 ≤ ξ ≤ 1 (45) ∂ϕ ∂ ϕ ∂ϕ r+1 r+1 r+1 L 2 Sc = + Scβ − Sck ϕ r+1 ∂t ∂y ∂y where L is the scaling parameter used to implement the ∂ ϑ r+1 boundary conditions at infinity. The initial approxima- + ScSr (35) ∂y tions for solving Equations (23)–(25) are obtained at y = 54 A. S. IDOWU AND B. O. FALODUN 0 and are chosen with due consideration of the bound- before applying the finite differences. ary conditions (26) and (27). Hence, u (y, t), ϑ (y, t) and 0 0 du du r+1 r+1 2 3 2 + αD = a D u + D u 0,r r+1 r+1 ϕ (y, t) are chosen as follows: dt dt + a Du + a − M u −y −y nt 1,r r+1 2,r r+1 u (y, t) = e , ϑ (y, t) = ϕ (y, t) = e + εe (46) 0 0 0 (54) The system of Equations (39)–(41) can be solved iter- dϑ r+1 atively for the unknown functions starting from the ini- Pr = b D ϑ + b Dϑ + b + b 0,r r+1 1,r r+1 2,r 3,r dt tial approximations in (46). The iteration schemes (39), + Prδ ϑ + b ϑ (55) (40) and (41) are solved iteratively for u (y, t), ϑ x r+1 4,r r+1 r+1 r+1 (y, t) and ϕ (y, t) when r = 0, 1, 2. In order to solve r+1 dϕ r+1 2 2 the system of Equations (39)–(41), the equations are dis- Sc = D ϕ + c Dϕ − Sck ϕ + c (56) r+1 0,r r+1 r+1 1,r dt cretized with the help of the Chebyshev spectral collo- subject to cation method in the y−direction and the implicit finite difference method in the t−direction. The finite differ- u (x , t) = 1, u (xNx, t) = 0, r+1 0 r+1 ence scheme is used with centring about a mid-point nt ϑ (x , t) = 1 + εe (57) r+1 0 n+1 n between t and t . The mid-point is expressed as nt ϑ (xNx, t) = 0, ϕ (x , t) = 1 + εe , n+1 n r+1 r+1 0 1 t + t n+ t = (47) ϕ (xNx, t) = 0 (58) 2 r+1 1 Simplifying Equations (54)–(56) further leads to n+ Thus using the centring about t to the unknown du du r+1 r+1 2 3 2 functions, say u(y, t), ϑ(y, t) and ϕ(y, t) and its associ- + αD = (a D + D 0,r dt dt ated derivative yield +a D − M )u + a (59) 1,r r+1 2,r n+1 1 u + u n+ j j n+ u y , t = u = , dϑ r+1 Pr = (b + b + Prδ + b )ϑ + b 0,r 1,r x 4,r r+1 2,r n+1 n dt n+ 2 u − u ∂u j j = (48) + b + b (60) 3,1 4,r ∂t t dϕ r+1 2 2 Pr = (D + c D − Sck )ϕ + c (61) 0,r r+1 1,r dt n+1 n ϑ + ϑ n+ j j n+ 2 2 subject to (57) and (58) where ϑ y , t = ϑ = , ⎡ ⎤ u (x , t) n+1 r+1 0 n+ ϑ − ϑ ∂ϑ j j ⎢ ⎥ u (x , t) r+1 1 = (49) ⎢ ⎥ ∂t t ⎢ ⎥ u = ⎢ ⎥ , r+1 ⎢ ⎥ ⎣ ⎦ u (x , t) r+1 N x−1 n+1 1 ϕ + ϕ u (x , t) r+1 N n+ j j x n+ 2 ϕ y , t = ϕ = , ⎡ ⎤ ϑ (x , t) r+1 0 n+1 n n+ ⎢ ⎥ 2 ϕ − ϕ ϑ (x , t) r+1 1 ∂ϕ j j ⎢ ⎥ = (50) ⎢ ⎥ ∂t t ϑ = ⎢ ⎥ , r+1 ⎢ ⎥ ⎣ ⎦ ϑ (x , t) r+1 N x−1 The concept behind spectral collocation method is ϑ (x , t) r+1 N the use of differentiation matrix D to approximate the ⎡ ⎤ derivatives of unknown variables defined as ϕ (x , t) r+1 0 ⎢ ⎥ ϕ (x , t) r+1 1 ⎢ ⎥ d u r r ⎢ ⎥ = D u(ξ ) = D u, i = 0, 1, ... N (51) k ϕ = ⎢ ⎥ (62) ik r+1 k=0 dy ⎢ ⎥ ⎣ ⎦ ϕ (x , t) r+1 N x−1 ϕ (x , t) r+1 N r x d ϑ N r r = D ϑ(ξ ) = D ϑ, i = 0, 1, ... N (52) ik k=0 dy ⎡ ⎤ a (x , t) 0,r 0 ⎢ ⎥ a (x , t) 0,r 1 ⎢ ⎥ d ϕ r r ⎢ ⎥ = D ϕ(ξ ) = D ϕ, i = 0, 1, ... N (53) ik ⎢ ⎥ r . a = k=0 dy 0,r ⎢ ⎥ ⎢ ⎥ ⎣ . ⎦ where r is the order of differentiation. Chebyshev spec- a (x , t) tral collocation method is first applied on (39)–(41) 0,r N x JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 55 ⎡ ⎤ 2 2 b (x , t) Sc (D + c D − Srk ) 0,r 0,r 1 r n+1 − ϕ r+1 ⎢ ⎥ b (x , t) t 2 0,r 2 ⎢ ⎥ ⎢ ⎥ 2 2 ⎢ ⎥ . Sc (D + c D − Srk ) b = 0,r 0,r r n ⎢ ⎥ = − ϕ + c (69) 1,r r+1 ⎢ ⎥ t 2 ⎣ ⎦ b (x , t) 0,r N Upon further simplification leads to ⎡ ⎤ c (x , t) 0,r 0 n+1 n ⎢ ⎥ c (x , t) N u = H u + G (70) 1 1 1 0,r 1 r+1 r+1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ c = 0,r ⎢ ⎥ ⎢ ⎥ n+1 n ⎣ ⎦ . N ϑ = H ϑ + G (71) 2 2 2 r+1 r+1 c (x , t) 0,r N (63) n+1 n N ϕ = H ϕ + G (72) 3 3 3 r+1 r+1 The same diagonal matrix goes for a , a , b , b , 1,r 2,r 1,r 2,r subject to b , b , c and c . Applying the forward finite dif- 3,r 4,r 0,r 1,r ference scheme defined in (48)–(50) on Equations n n u (xN , t ) = ϑ (xN , t ) = 0 r+1 x r+1 x (59)–(61) (73) ϕ (xN , t ) = 0 r+1 x n+1 u − u 2 r+1 r+1 (1 + αD ) n n nt u (x , t ) = 1, ϑ (x , t ) = 1 + ξe r+1 0 r+1 0 (74) n+1 n nt u + u ϕ (x , t ) = 1 + ξe , n = 1, 2, ... r+1 0 3 2 2 r+1 r+1 = (a D + D + a D − M ) 0,r 1,r (64) −y −y nt j j u (y ,0) = e , ϑ (y ,0) = e + ξe r+1 j r+1 j (75) −y nt ϕ (y ,0) = e + ξe r+1 j n+1 n ϑ − ϑ r+1 r+1 The above matrices are defined as Pr = (b D + b D + b + Prδ ) 0,r 1,r 4,r x 1 1 n+ n+ n+1 n 2 3 2 2 2 ϑ + ϑ a D + D + a D − M I r+1 r+1 2 0,r 1,r (1 + αD ) × + b + b 2,r 3,r N = − 2 1 t 2 (65) 1 1 n+ n+ 2 3 2 2 2 n+1 n a D + D + a D − M I 2 0,r 1,r ϕ − ϕ r+1 r+1 (1 + αD ) Sc H = + t 2 n+1 ϕ + ϕ 2 2 r+1 r+1 1 1 1 = (D + c D − Sck ) + c (66) 0,r 1,r n+ n+ n+ 2 2 2 2 b D + b D + b + Prδ 0,r 1,r 4,r Pr N = − t 2 Upon simplification leads to 2 3 2 2 (1 + αD ) (a D + D + a D − M ) 0,r 1,r n+1 1 1 1 − u n+ n+ n+ r+1 2 2 2 2 b D + b D + b + Prδ t 2 0,r 1,r 4,r Pr ⎡ ⎤ 3 2 H = + (a D + D 0,r t 2 ⎢ 2 ⎥ +a D − M ) (1 + αD ) 1,r ⎢ ⎥ = + u + a (67) ⎢ ⎥ 2,r r+1 ⎣ t 2 ⎦ 1 n+ 2 2 2 D + c D − Sck 0,r r Sc N = − , t 2 Pr (b D + b D + b + Prδ ) 0,r 1,r 4,r x n+1 − ϑ r+1 t 2 n+ 2 2 2 D + c D − Sck ⎡ ⎤ r 0,r Sc (b D + b D 0,r 1,r H = + ⎢ ⎥ t 2 +b + Prδ ) Pr 4,r x ⎢ ⎥ = + ϑ + (b + b ) ⎢ ⎥ 2,r 3,r r+1 ⎣ t 2 ⎦ 1 1 1 1 n+ n+ n+ n+ 2 2 2 2 G = a , G = b + b , G = c 1 2 3 1,r 2,r 3,r 1,r (68) 56 A. S. IDOWU AND B. O. FALODUN the physics of the problem. Hence, all numerical 4. Results and discussion computations correspond to the above-stated values Equations (23)–(25) subject to (26) and (27) have been unless or otherwise stated. The effects of Soret param- solved using the spectral relaxation method (SRM). SRM eter Sr and Dufour parameter Du were investigated employs the idea of Gauss-seidel relaxation approach separately in this study. Remarkably, the results were to linearize and decoupled system of non-linear dif- compared with those of the existing literature and were ferential equations [26]. Using the SRM, numerical found to be in good agreement. It is worth mention- computations are carried out for the velocity, tem- ing that, the moment when α = 0 is set in this study, perature, concentration, local skin friction, local Nus- the model is categorized as a Newtonian fluid phe- selt number and Sherwood number. Results are pre- nomenon. sented in the tabular and graphical form. All pro- grams are coded in MATLAB R2012a. The results were generated using the scaling parameter L = 15 and it 4.1. Velocity profiles is observed that increasing the value of L does not The effect of the Prandtl number Pr on the velocity is change the result to a reasonable extent. The num- depicted in Figure 2 (a). It is observed that the fluid ber of collocation point used in generating the results velocity decreases with the increase in Pr. This result was N = 120. The accuracy is seen to improve with is in agreement with that of Alao et al. [23], but it an increase in the number of collocation points while is observed that the presence of parameters such as it was noted that there are errors using a few col- α, δ ,  tends to influence the profile more. It is obvi- location points. The range of parameters are cho- ous that in Figure 2(a) as the value of Pr is increas- sen to Gr = Gm = 2.0, Rr = A = kr = 0.5, t = 1.0, n = ing, it causes a reduction in the velocity as higher Pr 0.5, Du = 0.2, Sr = 0.5, Ec = 0.01, α = 0.2, δ = 0.04, is expected to reduce both the velocity and the local = 0.01, ε = 0.0001 and the value for magnetic param- skin friction. In fact, higher values of Pr aremoreorless eter between 0.1 and 1 to obtain a clear insight into Figure 2. Effects of (a) Prandtl number and (b) radiation param- Figure 3. Effects of (a) Soret number and (b) Dufour number on eter on velocity profiles. velocity profiles. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 57 increased in the thermal conductivity of the fluid and for mass fraction of solute concentration. This is called ther- that reason heat is able to migrate away from the heated mal gradient and cooperative solutal, the solute diffuses surface more speedily for lower values of Pr. Physically, to cold regions but for the case when Soret number < higher viscosity and lesser thermal conductivity result in 0, an increase in temperature brings about an increase larger Pr.Thus, Pr is useful in increasing the rate of cool- in the temperature and it leads to an increase in den- ing in conducting flows. Figure 2(b) shows the variation sity. It is called thermal gradient and competitive solutal of the radiation parameter Rr on velocity. It is noted from and the solute diffuses to warmer regions. Figure 3(b) Figure 2(b) that increasing Rr intensifies the thermal illustrates the variation of Dufour parameter Du on the condition of the fluid environment. Increase in the fluid velocity profile. It is observed that there is an increase temperature incites more flow in the boundary layer in the fluid velocity by increasing Du. Figure 4(a) illus- making the velocity of the fluid to increase through trates the effect of the viscoelastic fluid parameter α the buoyancy effect. Also, the hydrodynamic bound- on the velocity profile. The viscoelastic fluid parame- ary layer thickness increases when the thermal radiation ter α connotes the effect of normal stress coefficient parameter is increased. This is true because the increase on the flow. It is noted that at a point on the flow in radiation parameter releases heat energy to the flow. domain that increasing the viscoelastic parameter has Figure 3(a) represents the effect of Soret parameter Sr the tendency of decreasing the boundary layer thick- on the velocity profile. It is found out that increasing the ness. Interestingly, very close to the wall, the fluid veloc- values of Sr increases the velocity profile. This is due to ity decreases and increases far away from the plate. The the fact that, when Sr is raised, there will be greater ther- effect of heat generation/absorption coefficient param- mal diffusion and this results to increase in the velocity eter  on the velocity profile is depicted in Figure 4(b). of the fluid. The positive Soret parameter has a stabi- Increasing the values of heat generation/absorption lizing effect. When Soret number > 0, an increase in coefficient parameter  increases the velocity profile. temperature will cause a decrease in both density and This is expected because when > 0, the behaviour of the fluid velocity changes and it drastically causes an Figure 4. Effects of (a) Viscoelastic fluid parameter and heat Figure 5. Effects of (a) Schmidt number and (b) chemical reac- generation on velocity profiles. tion parameter on the velocity profiles. 58 A. S. IDOWU AND B. O. FALODUN increase. Figure 5(a) depicts the effect of the Schmidt that increasing the magnetic parameter causes a reduc- number Sc on the velocity profile. It is obvious from tion in the velocity profile. The magnetic field induces Figure 5(a) that increasing Sc causes retardation on currents in a moving conductive fluid. The induced cur- the velocity profile. This causes a decrease in the con- rents polarize the fluid and change the magnetic field. centration buoyancy effects leading to a reduction in Figure 7(a) depicts the effect of thermal Grashof num- the fluid velocity. Decrease in the velocity and concen- ber Gr on the velocity profile. Gr is the ratio of buoyancy tration is accompanied by a simultaneous decrease in to the viscous acting on the fluid. The velocity is cou- both velocity and concentration boundary layer. This pled with temperature and concentration via thermal behaviour is shown in Figure 5(a) and 10(b). Figure 5(b) Grashof number and mass Grashof number as seen in shows the effect of the chemical reaction parameter Equation (8). Enhancement of thermal buoyancy force kr on the velocity profile. It is observed that there is a gives rise to the velocity. When the values of Gr increase, reduction in the velocity profile with increasing value of it increases rapidly close to the plate and decreases kr. Figure 6(a) depicts the influence of Eckert number the free stream velocity. This behaviour is shown in Ec on the velocity profile. Ec connotes the relationship Figure 7(a). Figure 7(b) illustrates the influence of mass between the kinetic energy in the flow and enthalpy. Grashof number Gm on the velocity profile. It is evident Greater Eckert number causes a rise in velocity. It is from Figure 7(b) that the fluid velocity increases and the clearly seen from Figure 6(a) that the Eckert number ele- peak value is more distinctive because of the increase in vates the velocity profile because heat energy is stored the species buoyancy force. in the liquid due to frictional heating. The variation of different values of the magnetic parameter M on the 4.2. Temperature profiles velocity is depicted in Figure 6(b). The applied mag- netic field strength B gives rise to a resistive force called 0 From Figure 8(a) it is observed that the increase in Lorentz force and it causes the motion of an electri- Pr decreases the temperature profile. It is noted that cally conducting fluid. It is clearly seen in Figure 6(b) when Pr < 1, the fluid in the hydrodynamic, thermal and concentration boundary layer is highly conducive. Figure 6. Effects of (a) Eckert number and (b) magnetic param- Figure 7. Effects of (a) thermal Grashof number and (b) mass eter on the velocity profiles. Grashof number on velocity profiles. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 59 Figure 8. Effects of (a) Prandtl number and (b) radiation param- Figure 9. Effects of (a) Dufour number and (b) heat generation eter on temperature profiles. parameter on the temperature profiles. Also, small values of Pr are equivalent to the enrich- heat generation/absorption coefficient parameter  on ment of thermal conductivities. Prandtl number can be the temperature profile is depicted in Figure 9(b). defined as the ratio of momentum diffusion to thermal increases the temperature profile as seen in Figure 9(b) diffusion. With the increase in Pr , the thermal diffusion because the boundary layer gets thicker and the parti- decreases and therefore the thermal boundary layer cles of the fluid become warm. becomes thinner. However, fluids with a very large Pr and higher heat capacity tend to increase the rate of 4.3. Concentration profile heat transfer and decreases the non-dimensional tem- perature. The effect of Rr on the temperature profile The variation of the Soret parameter Sr on the is presented in Figure 8(b) shows that the tempera- concentration profile is depicted in Figure 10(a). The ture profile increases with an increase in Rr. When Rr is concentration profile rises when increasing the Soret raised the temperature of the fluid will increase caus- parameter as shown in Figure 10(a). Figure 10(b) illus- ing an increase in the profile. Physically, increase in the trates the variation of the Schmidt number Sc on the radiation parameter brings heat energy to the thermal concentration profile. It is obvious from Figure 10(b) boundary layer. As a result of this, the heat energy gives that increasing the values of Sc drastically decreases the room for more temperature and hereby increases the concentration profile. The effect of the chemical reac- non-dimensional temperature. Figure 9(a) depicts the tion parameter kr on the concentration profile is plotted effect of Dufour parameter Du on the temperature pro- in Figure 11. It is observed that the concentration profile file. The Dufour term explains the effect of concentra- decreases with increasing values of kr. The implication tion gradients as seen in Equation (9) that plays a signif- of this is that the buoyancy effects due to concentra- icant role in aiding the fluid flow and has the tendency tion and temperature difference are significant in the to elevate the thermal energy in the boundary layer. plate. The fluid motion is retarded on account of a chem- From Figure 9(a), it is observed that as Du increases it ical reaction. This implies that the destructive reaction gives a rise in the temperature profile. The variation of kr > 0 decreases the concentration field which weakens 60 A. S. IDOWU AND B. O. FALODUN Table 1. Computational values for skin friction coefficient Cf and Nusselt number for various values of Schmidt number com- pared with [22]when α = δ =  = Du = Sr = 0. Present study [22] Sc Cf Sh Cf Sh 0.22 3.1067 0.4512 3.1068 0.4515 0.60 2.4546 0.8429 2.4548 0.8431 0.78 2.2766 1.0212 2.2767 1.0214 0.94 2.1539 1.1744 2.1540 1.1745 Table 2. Computational values for skin friction coefficient Cf and Nusselt number for various values of thermal radiation parameter compared with [23]when α = δ =  = 0. Present study [23] Rr Cf ϑ (0) Cf ϑ (0) 0.0 1.7939 0.7452 1.7940 0.7455 0.4 1.9164 0.6532 1.9166 0.6533 0.8 2.0108 0.6049 2.0111 0.6051 1.0 2.0509 0.5889 2.0512 0.5892 Table 3. Computational values for skin friction coefficient Cf, Nusselt number and Sherwood number with variation of Dufour parameter. Du Cf Nh Sh 0.0 0.9572 0.4174 0.6993 0.5 1.3468 0.5551 0.6993 1.0 1.7364 0.6928 0.6993 Table 4. Computational values for skin friction coefficient Cf, Nusselt number and Sherwood number with variation of Soret parameter. Figure 10. Effects of (a) Soret number and (b) Schmidt number Sr Cf Nh Sh on the concentration profiles. 0.0 1.2097 0.6377 0.5353 0.5 1.5806 0.6377 0.6993 1.0 1.9514 0.6377 0.8633 5. Concluding remarks In this paper, spectral relaxation method (SRM) is employed to solve a third-order-coupled partial differ- ential equations that govern the effects of Soret and Dufour on MHD heat and mass transfer of a viscoelas- tic fluid pass over a semi-infinite vertical plate. Detailed explanation on SRM is discussed in the previous sec- tions. Comparison of the present results is done with previously published work and was found to be in excellent agreement. The present results will be help- ful in understanding the complex problems of MHD viscoelastic fluid over a semi-infinite vertical plate. The SRM is efficient and gives accurate results compared Figure 11. Effect of chemical reaction parameter on the con- centration profile. to other numerical methods used in the reviewed lit- erature. Physically, the magnetic term in Equation (8) gives rise to the Lorentz force which interacts with the buoyancy force in the governing velocity and temper- the buoyancy effects due to concentration gradients. ature fields. It slows down the motion of the fluid. The Thus, the chemical reaction reduces the concentration, presence of the radiation term gives rise to the fluid thereby increasing its concentration gradient and con- temperature. When the radiation is further increased, centration flux (Tables 1–4). there is a higher temperature which adds to the velocity JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 61 of the fluid. The graphical representation is in Figure 2b medium over a non-isothermal stretching sheet. IOSR J Math. 2016;12(1):53–60. and 8b of this study. The presence of thermal con- [6] Hayat T, Hina Z, Anum T, Ahmad A. Soret and Dufour ductivity is very important in the study of the fluid effects on MHD peristaltic transport of Jeffrey fluid in phenomenon. An increase in the thermal conductiv- a curved channel with convective boundary conditions. ity reduces the velocity and the temperature field. PLOS ONE. 2017;12(2) e0164854. DOI:10.1371/journal. From our numerical computations, we deduced the pone.0164854. [7] Iqbal M, Khalid S. Analysis of thermally developing lami- following: nar convection in the fined double-pipe. Heat Transf Res. 2013; 45(1): DOI:10.1615/Heat TransRes.2013006719. (i) When Dufour parameter is increased, the veloc- [8] Ahmed F, Iqbal M. MHD power law fluid flow and heat ity profile as well as the temperature profiles transfer analysis through Darcy Brinkamn porous media increases. in annular sector. Int J Mech Sci. 130:508–517. (ii) Increasing the Soret parameter increases both the [9] Ahmed F, Iqbal M, Sher Akbar. Numerical study of forced convective power law fluid flow through an annulus sec- velocity and concentration profiles. tor duct. Eur Phys J Plus. 131:341. https://doi.org/10.1140 (iii) Effects of Soret and Dufour have opposite effect on /epjp/i.16341-x the Nusselt and Sherwood number. [10] Iqbal M, Afag Hamna. Fluid flow and heat transfer (iv) It was discovered that as the viscoelastic parame- through an annular sector duct filled with porous media. ter increases, the velocity profile close to the plate J Porous Media. 18(7):679–687. DOI:10.1615/JPorMedia. v18.i7.30. decreases while far away from the plate it increases [11] Iqbal M, Ahmed F, Rashidi MM. Simulation of MHD forced slightly. convection heat transfer through annular sector duct. J (v) The thermal Grashof number increases the hydro- Thermophys Heat Transfer. 32(2):469–474. dynamic boundary layer when it is increased. [12] Selimefendigil F, Oztop H.F. Role of magnetic field and surface corrugation on natural convection in a nanofluid The effects of Soret–Dufour, thermal radiation and filled 3D trapezoidal cavity. Int Commun Heat and Mass. 2018; 95: 182–196. Walters’-B viscoelastic fluid as well as chemical reaction [13] Selimefendigil F, Oztop HF. Conjugate natural convec- parameter on the flow profiles are significant in indus- tion in a nanofluid filled partitioned horizontal annu- trial and technological applications such as in coating of lus formed by two isothermal cylinder surfaces under metals, crystal growth, electromagnetic pumps, power magnetic field. Int J Heat Mass Transf. 2017; 108: generators, MHD accelerators and reactor cooling. 156–171. [14] Selimefendigil F, Oztop HF, Chamkha AJ. Fluid structure- magnetic field lid-driven cavity with flexible side wall. Eur J Mech-B/Fluids. 2017; 61(1):77–85. Disclosure statement [15] Selimefendigil F, Oztop HF. Analysis of MHD mixed con- No potential conflict of interest was reported by the authors. vection in a flexible walled and nanofluids filled lid- driven cavity with volumetric heat generation. Int J Mech Sci 2016; 118, 113–124. ORCID [16] Rashidi MM, Momoniat E, Rostani B. Analytic approx- imate solutions for MHD boundary-layer viscoelastic B. O. Falodun http://orcid.org/0000-0003-1020-1677 fluid flow over continuously moving stretching sur- face by homotopy analysis method with two auxiliary parameters. J Appl Math. Article ID 780415, 2012;1–19. References DOI:10.1155/2012/780415. [17] Sivaraj R, Kumar BR. Chemically reacting dusty viscoelas- [1] Gbadeyan JA, Idowu AS, Ogunsola AW, et al. Heat and tic fluid flow in an irregular channel with convective mass transfer for Soret and Dufour effect on mixed con- boundary. Ain Shams Eng J. 2013;4:93–101. vection boundary layer flow over a stretching vertical sur- [18] Manglesh A, Gorla. The effects of thermal radiation, face in a porous medium filled with a viscoelastic fluid in chemical reaction and rotation on unsteady MHD vis- the presence of magnetic field. Global Journal of Science coelastic slip flow. Global Journal of Science Frontier Frontier Research. 2011;11(8):96–114. Research Mathematics and Decision Sciences. 2012; [2] Choudhury R., Kumar Das S. Viscoelastic MHD free con- 12(14):1–15. vective flow through porous media in presence of radia- [19] Kumar BR, Sivaraj. Heat and mass transfer in MHD tion and chemical reaction with heat and mass transfer. J viscoelastic fluid flow over a vertical cone and flat Appl Fluid Mech. 2014;7(4):603–609. plate with variable viscosity. Int J Heat Mass Transf. [3] Rashidi MM, Ali M, Rostani B, Rostani P, et al. Heat and 2013;56(1):370–379. mass transfer for MHD viscoelastic fluid flow over a verti- [20] Eswaramoorthi S, Bhuvaneswari M, Sivasankaran S, et al. cal stretching sheet with considering Soret and Dufour Effect of radiation on MHD convective flow and heat effects. Math Probl Eng. Article ID 861065, 2015;1–12. transfer of a viscoelastic fluid over a stretching surface. http://dx.doi.org/10.1155//2015/861065. 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Effects of thermal radi- computational applied mathematics and mathematical ation, Soret and Dufour on an unsteady heat and mass modeling in fluid flow. School of mathematics, statis- transfer flow of a chemically reacting fluid past a semi- tics and computer science, Pietermaritzburg Campus infinite vertical plate with viscous dissipation. J Nigerian 2012;9–13. Math Soc. 2016;35:142–158. [26] Motsa SS, Dlamini PG, Khumalo M. Spectral relaxation [24] Walters K. Non-Newtonian effects in some elastico- method and spectral quasilinearization method for solv- viscous liquids whose behaviour at small rates of shear ing unsteady boundary layer flow problems. Adv Math is characterized by a general linear equations of state. Phys. Article ID 341964, 2014;1–12. http://dx.doi.org/ Quantum J Mech Appl Math. 1962;15:63–76. 10.1155/2014/341964. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Taibah University of Science Taylor & Francis

Soret–Dufour effects on MHD heat and mass transfer of Walter’s-B viscoelastic fluid over a semi-infinite vertical plate: spectral relaxation analysis

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JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 2019, VOL. 13, NO. 1, 49–62 https://doi.org/10.1080/16583655.2018.1523527 RESEARCH ARTICLE Soret–Dufour effects on MHD heat and mass transfer of Walter’s-B viscoelastic fluid over a semi-infinite vertical plate: spectral relaxation analysis A. S. Idowu and B. O. Falodun Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria ABSTRACT ARTICLE HISTORY Received 12 May 2018 Soret–Dufour effects on MHD heat and mass transfer of Walter’s-B viscoelastic fluid over a Revised 7 September 2018 semi-infinite vertical plate are considered. The equations of motion are set of partial differen- Accepted 10 September 2018 tial equations; these are non-dimensionalized by introducing an appropriate non-dimensional quantity. The dimensionless equations along with the boundary conditions are solved numer- KEYWORDS ically using the spectral relaxation method (SRM). All programs are coded in MATLAB R2012a. Soret–Dufour effect; heat Results are presented in graphs, and numerical computations of the local skin friction, local Nus- transfer; viscous dissipation; selt number and local Sherwood number are presented in a tabular form. The result revealed thermal radiation; chemical reaction; spectral relaxation that as the viscoelastic parameter increases, the velocity profile close to the plate decreases but method when far away from the plate, it increases slightly. The present results were found to be in good agreement with those of the existing literature. 1. Introduction increases the concentration boundary layer thickness. They reported that the behaviour of Dufour and Soret The physical effects occur in MHD basically when a on the temperature and concentration profiles is oppo- conductor migrates into a magnetic field, electric cur- site. Choudhury and Kumar Das [2] studied viscoelas- rent is induced and creates its own magnetic field tic MHD-free convective flow through porous media in (Lenz’s law). The conductor used in this paper is the the presence of radiation and chemical reaction with fluid with complex motions. Another important thing heat and mass transfer. Their governing partial differ- to note is that the moment currents are induced prob- ential equations were solved using the multiple per- ably by a motion of an electrically conducting fluid as turbation technique. They concluded that the velocity a result of a magnetic field; a resistive force acts on field possesses an accelerating trend with the grow- the fluid and decelerates its motion. MHD has gained ing effect of the viscoelastic parameter. Rashidi et al. considerable interest due to its fundamental impor- [3] did not consider the porous medium. The homo- tance in the industrial and technological applications topy analysis method with two auxiliary parameters was such as in coating of metals, crystal growth, electro- used and its results show that increasing Soret num- magnetic pumps, power generators, MHD accelerators ber or decreasing Dufour number leads to a decrease and reactor cooling. Many researchers in the field of in velocity and temperature profiles. Vedavathi et al. fluid dynamics have studied MHD viscoelastic fluid flow [4] considered radiation and mass transfer effects on by considering the effects of Soret and Dufour num- unsteady MHD convective flow over an infinite vertical ber. Soret–Dufour effects have been investigated by plate with Dufour and Soret effects. In their analysis, the numerous scholars in fluid mechanics due to their sig- effects of Soret and Dufour parameters on the veloc- nificance in sciences and engineering like Soret in iso- ity, temperature and concentration profiles are plotted tope separation. Gbadeyan et al. [1] considered the in Figures 12(a–c). It is noted that a decrease in Soret effects of Soret and Dufour in their analysis. Their equa- number and an increase in Dufour number produce a tions of motion were solved using the shooting method decrease in the velocity and concentration profiles. In with sixth-order of the Runge–Kutta technique. In their another investigation of Sharma and Aich [5], the effects analysis, the variation of Dufour in the velocity field is of Soret and Dufour were significant. They concluded depicted in Figure 1 and it shows that the increase in in their study that the rate of flow decreases with an Dufour number gives a slight increase in the velocity increase in both Soret and Dufour effects. Recently in of the fluid. Also, effect of Dufour number as plotted the work of Hayat et al. [6], the influence of both Soret in Figure 10 of their result show that increasing Dufour and Dufour was investigated on the peristaltic flow of number increases the fluid temperature. It was discov- Jeffery fluid. The report was made on the study that ered in their study that increasing the Soret number CONTACT B. O. Falodun [email protected] © 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group 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. 50 A. S. IDOWU AND B. O. FALODUN Non-Newtonian fluids are important in many tech- nological applications compared to the Newtonian flu- ids. The non-Newtonian fluid flows find applications in modern technology and in industries. The non-linearity of the mechanism of non-Newtonian fluids presents a challenge to mathematicians, engineers and physi- cists. This non-linearity manifests itself in fields such as food, drilling operations and bioengineering. Rashidi et al. [16] examined analytic approximate solutions for MHD boundary-layer viscoelastic fluid flow over a con- tinuously moving stretching surface by the homotopy analysis method with two auxiliary parameters. Graph- ical results of Figures 7 and 8 show that the veloc- ity decreases with increasing viscoelastic parameter. Sivaraj and Kumar [17] examined chemically reacting dusty viscoelastic fluid flow in an irregular channel with convective boundary. They reported that velocity pro- files of all physical parameters decrease at the wavy wall. Figure 1. Physical model of the problem. Manglesh and Gorla [18] studied the effects of thermal radiation, chemical reaction and rotation on unsteady MHD viscoelastic slip flow. They reported in their study the behaviour of Soret and Dufour number for tem- that increasing the viscoelastic parameter makes the perature and concentration is contradictory. Iqbal and hydrodynamic boundary layer to adhere strongly to the Khalid [7] studied the analysis of thermally developing surface and results in the retardation of the flow to the laminar convection in the finned double-pipe. Ahmed left half channel but accelerates in the right half when and Iqbal [8] examined MHD power-law fluid flow and there is no slip boundary condition. In 2013, Kumar heat transfer analysis through Darcy Brinkman porous and Sivaraj [19] examined heat and mass transfer in media in the annular sector. Their results revealed that MHD viscoelastic fluid flow over a vertical cone and flat an increase in the Hartman parameter increases the rate plate with variable viscosity. They reported in the study of heat transfer. Ahmed et al. [9] examined the study that magnetic field, buoyancy ratio parameter, viscosity of forced convective power-law fluid through an annu- variation parameter, Eckert number and chemical reac- lus sector duct numerically. Numerical study of heat tion parameter play an important role in viscoelastic transfer and fluid flow through an annular sector duct fluid flow through the porous medium. In the work of filled with porous media was carried out by Iqbal and Eswaramoorthi et al. [20], viscoelastic type of fluid was Afag Hamna [10]. Iqbal et al. [11] recently considered considered. The flow equations were solved using the the simulation of MHD-forced convection heat transfer homotopy analysis method. It was reported in the study through the annular sector. that increasing the viscoelastic parameter results in a The role of the magnetic field and surface corru- decrease in the velocity. Graphical result of Figure 10 gation on natural convection in a nanofluid-filled 3D shows that increasing the viscoelastic parameter leads trapezoidal cavity was carried out by Selimefendigil to a decrease in the temperature.Finite element anal- ysis of MHD viscoelastic nanofluid flow over a stretch- and Oztop [12]. Conjugate natural convection in a nanofluid-filled partitioned horizontal annulus formed ing sheet with radiation is examined by Madhu and by two isothermal cylinder surfaces under magnetic Kishan [21]. field was investigated by Selimefendigil and Oztop [13]. In 2013, Rao et al. [22] studied finite element anal- The work of Selimefendigil et al. [14] deals with the ysis of radiation and mass transfer flow pass over a flow of MHD in a lid-driven nanofluid-filled square cavity semi-infinite moving vertical plate with viscous dis- with a flexible side wall. Finite element formulation was sipation. In their study, they neglected the effects used as a method of solution and their findings revealed of Soret and Dufour parameters. In 2016, Alao et al. that the Brownian motion effect on the thermal conduc- [23] considered the effects of thermal radiation, Soret tivity of the nanofluid is significant. The finite element and Dufour on an unsteady heat and mass trans- method was used to solve the analysis of MHD mixed fer flow of a chemically reacting fluid pass over a convection in a flexible-walled and nanofluid- filled the semi-infinite vertical plate with viscous dissipation. lid-driven cavity with volumetric heat generation by The work of Alao et al. [23] was an extension of Selimefendigil and Oztop [15]. The study revealed that Rao et al’s. [22] work by considering the effects of decreasing values of Richardson number and increasing Soret and Dufour parameters in their governing equa- values of Hartmann number decreases the average heat tions. The two problems are centred on Newtonian transfer. fluid. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 51 The present study is a Non-Newtonian fluid model. Its Here, η is the limiting viscosity at the small rate of objective is to illustrate the use of the spectral relaxation shear and is given as method (SRM) for solving the equations of motion rep- ∞ ∞ resenting the physical model of Soret–Dufour effects η = N(τ )dτ and k = τN(τ )dτ (5) 0 0 0 0 on MHD heat and mass transfer of Walter’s B vis- coelastic fluid over a semi-infinite vertical plate and to N(τ ) is the relaxation spectrum as discussed by Walters explore the effects of different controlling flow param- [24]. This idealized model is a valid approximation of eters as encountered in the equations. SRM is an itera- Walters-B taking short relaxation time into account so tive method that employs the Gauss–Seidel approach that terms involving in solving both linear and non-linear different equa- tions. SRM is found to be effective and accurate. The t N(τ )dτ , n ≥2(6) governing equations are systems of partial differential equations which are transformed into a dimensionless have been neglected in the momentum equation and form by introducing suitable non-dimensional quanti- k is significant. ties. Numerical computations are carried out and graph- Under the above assumptions and the usual Boussi- ical results for the velocity, temperature, and concentra- nesq’s approximation, the governing equations and tion profiles as well as the local skin friction, local Nusselt boundary conditions are given as (see details in Alao number and local Sherwood number coefficients within et al. [23]) the boundary layer flow are discussed. ∂v =0(7) ∂y 2. Equations of motion ∂u ∂u ∂ u The unsteady free convective flow of a viscoelastic fluid + v = ν + gβ (T − T ) + gβ (C − C ) t ∞ c ∞ ∂t ∂y ∂y (Walters B’ model) over a semi-infinite vertical plate 2 3  3 with time-dependent oscillatory suction in the presence σ B u K ∂ u ∂ u − − + v (8) of a transfer magnetic field is considered. The plate is  2 3 ρ ρ ∂t ∂y ∂y considered infinite in the x -direction, thus the x -axis shall be taken along the vertical infinite plate in the ∂T ∂T ∂ T 1 ∂q μ ∂u upward direction and the y -axisnormaltothe plate ∂y + v = α − + ∂t ρc ∂y ρc ∂y ∂y p p (see Figure 1). The flow direction is vertically upward 2 ∗ and in the continuity equation, the term ∂u /∂x is DK ∂ C β u Q T 0 + + (T − T) + (T − T ) (9) ∞ ∞ neglected. It is assumed that initially at t ≤ 0, the plate 2 ρc ρc C c ∂y p p s p and fluid are at the same temperature. Soret, Dufour, heat generation or absorption is taken into account. 2 2 ∂C ∂C ∂ C DK ∂ T It is assumed that the magnetic Reynolds number is + v = D − K  (C − C ) + (10) 2 2 ∂t ∂y ∂y T ∂y small so that the induced magnetic field is neglected. subject to the conditions In the direction of they -axis, a magnetic field of uni- form strength B is applied. Walters-B viscoelastic type n t u = U , T = T + ψ(T − T )e , 0 w w ∞ of fluid is considered in this paper. n t Following Choudhury and Kumar Das [2], the consti- C = C + ψ(C − C )e at y = 0 (11) w w ∞ tutive equation for Walter’s-B viscoelastic fluid can be defined as u → 0, T → T , C → C as y →∞ (12) ∞ ∞ σ =−pg + σ (1) ∗ ik ik ik β (T − T) and Q (T − T ) are the heat generation ∞ 0 ∞ and absorption, respectively. ik ik ik σ = aη e − 2k e (2) 0 0 Both sides of the continuity equation (7) are inte- where η is the limiting viscosity at small rates of shear, grated to get v = constant. Obviously, the suction ik k is the elastic co-efficient, σ is the stress tensor, p is velocity normal to the plate is a constant function or the isotropic pressure, g is the metric tensor of a fixed assumed as a function of time. In this paper, it is con- ik co-ordinate system x and v is the velocity vector. The sidered as a case when it is both constant and time- ik contravariant form of e is given as dependent expressed as [23] ik n t ∂e ik m ik k im i mk v =−v (1 + εAe ) (13) e = + v e − V e − v e (3) ,m ,m ,m ∂t It is assumed in this paper that ∂q /∂y >> ∂q /∂x r r The convected derivative of the deformation rate and as a result the x -direction radiative flux ∂q /∂x is ik r tensor e is defined by neglected. However, the radiative heat flux that domi- ik 2e = v + v (4) nates the flow is ∂q /∂y . i,k k,i 52 A. S. IDOWU AND B. O. FALODUN Assuming that the temperature difference within the and the boundary conditions lead to flow regime is sufficiently small in such a way that T can ∂u ∂u ∂ u nt 2 be expressed as a linear function of the free stream tem- − (1 + εAe ) = + G ϑ + G ϕ − M u r m 4 ∂t ∂y ∂y perature T . Expanding T in the Taylor series about 3 3 T and neglecting the higher terms let us consider the ∂ u ∂ u nt − α − (1 + εe ) (23) 2 3 Taylor series expansion of the function m(x) about x 0 ∂t∂y ∂y (x − x ) m(x) = m(x ) + (x − x )m (x ) + m (x ) 2 0 0 0 0 ∂ϑ ∂ϑ 1 + Rr ∂ ϑ ∂u 2! nt − (1 + εAe ) = + E ∂t ∂y P ∂y ∂y (x − x ) + ··· + m (x ) (14) n! ∂ ϕ + D + δ ϑ − uϑ (24) u x ∂y 4 4 setting m(x) = T and m(x ) = m(T ) = T in the 0 ∞ equation above and neglecting higher order term lead 2 2 ∂ϕ ∂ϕ 1 ∂ ϕ ∂ ϑ nt 2 to: − (1 + εAe ) = − k ϕ + S (25) 2 2 ∂t ∂y S ∂y ∂y 4 3 4 T = 4T T − 3T (15) ∞ ∞ where G , G , P , Rr, E , S , k , D ,S , α,  and δ are r m r c c r u r x Using the Roseland approximation, the radiative the thermal Grashof number, mass Grashof number, heat flux is given by Prandtl number, radiation parameter, Eckert number, Schmidt number, chemical reaction parameter, Dufour 4σ ∂T number, Soret number, viscoelastic parameter, heat q =− (16) 3k ∂y generation/absorption coefficient and heat source/sink parameter, respectively. where σ is the Stefan–Boltzmann constant and k is e e The transformed initial and boundary conditions are the mean absorption coefficient. By using the Roseland approximation, the present study is limited to optically nt εe at y = 0 (26) thick fluids. If temperature differences within the flow are sufficiently small, then Equation (15) can be lin- u → 0, ϑ → 0, ϕ → 0, at y →∞ (27) earized and in view of Equations (15) and (16), Equation (9) reduces to The physical quantities of interest are the local skin friction coefficient (Cf ), local Nusselt number (Nu) and 2 2 ∂T ∂T ∂ T 16σ ∂ T μ ∂u 3 2 Sherwood number (Sh) of the flow in practical engi- + v = α + T + ( ) 2 2 ∂t ∂y ∂y 3ρc k ∂y ρc ∂y p e p neering. The skin friction coefficient, Nusselt and Sher- 2 ∗ D k ∂ C β u Q wood number are given as m T 0 + + (T − T) + (T − T ) ∞ ∞ C c ∂y ρc ρc s p p p ∂C (17) ∂y τ Kq w w y =0 Cf = , Nu =− , Sh = ρU V T − T C − C 0 0 w ∞ w ∞ In order to write the governing equations and the boundary conditions in dimensionless form, the follow- where ing non-dimensional quantities are introduced: 2  2 ∂u ∂ u ∂ u τ = η − K + 2 2 w 0 y=0 0 u v y v t υn ∂y  ∂t ∂y ∂y 0 0 y=0 u = , y = , t = , n = (18) u υ υ v ∂C 4σ ∂T q =−K − . T − T C − C υρc υ ∞ ∞ p ∂y 3k ∂y y=0 y=0 ϑ = , ϕ = , P = = (19) T − T C − C k α w ∞ w ∞ υ gβv(T − T ) gβ v(C − C ) 3. Method of solution w ∞ w ∞ S = , G = , G = c r m 2 2 u v u v 0 0 0 0 The non-dimensionless transformed system of partial (20) differential equations is solved in this section using the 2 2 ∗ u K V β u ν 0 0 spectral relaxation method (SRM). SRM is an iterative 0 0 E = , α = ,  = (21) 2 2 c (T − T ) ρν ρc v procedure that employs the Gauss–Siedel type of relax- p w ∞ ation approach to linearize and decouple the system of 2 3 2 k v 16σ T σ B v Q ν coupled differential equations. The resulting non-linear r e 0 2 ∞ 0 k = , Rr = , M = , δ = 2 2 2 differential equations are further discretized and solved 3k k v ρv ρc v e p 0 0 0 (22) with the Chebyshev pseudo-spectral method [25]. The The above non-dimensional quantities on the gov- linear terms in each equation are evaluated at the cur- erning momentum, energy, concentration equations rent iteration level (denoted by r + 1) and the non-linear JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 53 terms are assumed to be known from the previous iter- subject to ation level (denoted by r). The following are the basic nt u (0, t) = 1, ϑ (0, t) = 1 + εe , r+1 r+1 steps of the method: nt ϕ (0, t) = 1 + εe (36) r+1 (1) decoupling and rearrangement of the governing non-linear equations in a Gauss–Siedel manner. u (∞, t) = 0, ϑ (∞, t) = 0, ϕ (∞, t) = 0 r+1 r+1 r+1 (2) discretizing the linear differential equations. (37) nt (3) solving the discretized linear differential equations where β = 1 + εAe , setting iteratively using the Chebyshev pseudo-spectral nt nt a = β = 1 + εAe , a = β = 1 + εAe , 0,r 1,r method. a = Grϑ + Gmϕ , b = (1 + Rr), 2,r r r 0,r Applying the spectral relaxation method (SRM), we ∂u r+1 nt first re-arrange the transformed governing Equations b = Prβ = Pr(1 + εAe ), b = PrEc , 1,r 2,r ∂y (23)–(25) to yield; ∂ ϕ b = PrDu , b = Pru , 3 2 3,r 4,r r ∂u ∂ u ∂u ∂ u ∂y nt + α = (1 + εAe ) + + G ϑ 2 2 ∂t ∂t∂y ∂y ∂y ∂ ϑ r+1 nt c = Scβ = Sc(1 + εAe ), c = ScSr (38) 3 0,r 1,r ∂ u ∂y 2 nt + G ϕ − M u − α(1 + εe ) ∂y Substituting the above coefficient parameters into (28) (33)–(35) gives 3 2 ∂u ∂ u ∂u ∂ u r+1 r+1 r+1 r+1 2 + α = a + + a 1,r 2,r ∂ϑ ∂ϑ 1 + Rr ∂ ϑ ∂u 2 2 nt ∂t ∂t∂y ∂y ∂y = (1 + εAe ) + + E ∂t ∂y P ∂y ∂y r 3 ∂ u r+1 − M u + a (39) 2 r+1 0,r ∂ ϕ 3 ∂y + D + δ ϑ − uϑ (29) u x ∂y ∂ϑ ∂ϑ ∂ ϑ r+1 r+1 r+1 = b + b + b + b 1,r 0,r 2,r 3,r 2 2 2 ∂t ∂y ∂y ∂ϕ ∂ϕ 1 ∂ ϕ ∂ ϑ nt 2 = (1 + εAe ) + − k ϕ + S (30) 2 2 ∂t ∂y S ∂y ∂y c + Prδ ϑ − b ϑ (40) x r+1 4,r r+1 subject to ∂ϕ ∂ϕ ∂ ϕ r+1 r+1 r+1 = c + − k ϕ + c (41) 0,r r+1 1,r nt nt u = 1, ϑ = 1 + εe , ϕ = 1 + εe at y = 0 (31) ∂t ∂y ∂y subject to u → 0, ϑ → 0, ϕ → 0, at y →∞ (32) nt u (0, t) = 1, ϑ (0, t) = 1 + εe , r+1 r+1 nt Adopting the SRM on the non-linear coupled partial ϕ (0, t) = 1 + εe at y = 0 (42) r+1 differential equations (28)–(30) subject to (31) and (32), to obtain u (∞, t) = 0, ϑ (∞, t) = 0, r+1 r+1 2 3 2 ∂u ∂ u ∂ u ∂ u ∂u ϕ (∞, t) = 0, at y →∞ (43) r+1 r+1 r+1 r+1 r+1 r+1 + α = β + + β 2 3 2 ∂t ∂t ∂y ∂y ∂y The unknown functions are defined by the Gauss– + Grϑ + Gmϕ − M u (33) r r r+1 Lobatto points given as πj ξ = cos , j = 0, 1, 2, ... , N;1 ≤ ξ ≤−1 (44) 2 2 ∂ϑ ∂ ϑ ∂u ∂ ϕ r+1 r+1 r+1 r Pr = (1+Rr) +PrEc +PrDu 2 2 where N is the number of collocation points. The ∂t ∂y ∂y ∂y domain of the physical region [0, ∞) is transformed into ∂ϑ r+1 + Prδ ϑ + Pru ϑ + Prβ x r+1 r r+1 [−1, 1]. Thus, the problem is solved on the interval [0, L] ∂y instead of [0, ∞). The following transformation is used (34) to map the interval together η ξ + 1 = , − 1 ≤ ξ ≤ 1 (45) ∂ϕ ∂ ϕ ∂ϕ r+1 r+1 r+1 L 2 Sc = + Scβ − Sck ϕ r+1 ∂t ∂y ∂y where L is the scaling parameter used to implement the ∂ ϑ r+1 boundary conditions at infinity. The initial approxima- + ScSr (35) ∂y tions for solving Equations (23)–(25) are obtained at y = 54 A. S. IDOWU AND B. O. FALODUN 0 and are chosen with due consideration of the bound- before applying the finite differences. ary conditions (26) and (27). Hence, u (y, t), ϑ (y, t) and 0 0 du du r+1 r+1 2 3 2 + αD = a D u + D u 0,r r+1 r+1 ϕ (y, t) are chosen as follows: dt dt + a Du + a − M u −y −y nt 1,r r+1 2,r r+1 u (y, t) = e , ϑ (y, t) = ϕ (y, t) = e + εe (46) 0 0 0 (54) The system of Equations (39)–(41) can be solved iter- dϑ r+1 atively for the unknown functions starting from the ini- Pr = b D ϑ + b Dϑ + b + b 0,r r+1 1,r r+1 2,r 3,r dt tial approximations in (46). The iteration schemes (39), + Prδ ϑ + b ϑ (55) (40) and (41) are solved iteratively for u (y, t), ϑ x r+1 4,r r+1 r+1 r+1 (y, t) and ϕ (y, t) when r = 0, 1, 2. In order to solve r+1 dϕ r+1 2 2 the system of Equations (39)–(41), the equations are dis- Sc = D ϕ + c Dϕ − Sck ϕ + c (56) r+1 0,r r+1 r+1 1,r dt cretized with the help of the Chebyshev spectral collo- subject to cation method in the y−direction and the implicit finite difference method in the t−direction. The finite differ- u (x , t) = 1, u (xNx, t) = 0, r+1 0 r+1 ence scheme is used with centring about a mid-point nt ϑ (x , t) = 1 + εe (57) r+1 0 n+1 n between t and t . The mid-point is expressed as nt ϑ (xNx, t) = 0, ϕ (x , t) = 1 + εe , n+1 n r+1 r+1 0 1 t + t n+ t = (47) ϕ (xNx, t) = 0 (58) 2 r+1 1 Simplifying Equations (54)–(56) further leads to n+ Thus using the centring about t to the unknown du du r+1 r+1 2 3 2 functions, say u(y, t), ϑ(y, t) and ϕ(y, t) and its associ- + αD = (a D + D 0,r dt dt ated derivative yield +a D − M )u + a (59) 1,r r+1 2,r n+1 1 u + u n+ j j n+ u y , t = u = , dϑ r+1 Pr = (b + b + Prδ + b )ϑ + b 0,r 1,r x 4,r r+1 2,r n+1 n dt n+ 2 u − u ∂u j j = (48) + b + b (60) 3,1 4,r ∂t t dϕ r+1 2 2 Pr = (D + c D − Sck )ϕ + c (61) 0,r r+1 1,r dt n+1 n ϑ + ϑ n+ j j n+ 2 2 subject to (57) and (58) where ϑ y , t = ϑ = , ⎡ ⎤ u (x , t) n+1 r+1 0 n+ ϑ − ϑ ∂ϑ j j ⎢ ⎥ u (x , t) r+1 1 = (49) ⎢ ⎥ ∂t t ⎢ ⎥ u = ⎢ ⎥ , r+1 ⎢ ⎥ ⎣ ⎦ u (x , t) r+1 N x−1 n+1 1 ϕ + ϕ u (x , t) r+1 N n+ j j x n+ 2 ϕ y , t = ϕ = , ⎡ ⎤ ϑ (x , t) r+1 0 n+1 n n+ ⎢ ⎥ 2 ϕ − ϕ ϑ (x , t) r+1 1 ∂ϕ j j ⎢ ⎥ = (50) ⎢ ⎥ ∂t t ϑ = ⎢ ⎥ , r+1 ⎢ ⎥ ⎣ ⎦ ϑ (x , t) r+1 N x−1 The concept behind spectral collocation method is ϑ (x , t) r+1 N the use of differentiation matrix D to approximate the ⎡ ⎤ derivatives of unknown variables defined as ϕ (x , t) r+1 0 ⎢ ⎥ ϕ (x , t) r+1 1 ⎢ ⎥ d u r r ⎢ ⎥ = D u(ξ ) = D u, i = 0, 1, ... N (51) k ϕ = ⎢ ⎥ (62) ik r+1 k=0 dy ⎢ ⎥ ⎣ ⎦ ϕ (x , t) r+1 N x−1 ϕ (x , t) r+1 N r x d ϑ N r r = D ϑ(ξ ) = D ϑ, i = 0, 1, ... N (52) ik k=0 dy ⎡ ⎤ a (x , t) 0,r 0 ⎢ ⎥ a (x , t) 0,r 1 ⎢ ⎥ d ϕ r r ⎢ ⎥ = D ϕ(ξ ) = D ϕ, i = 0, 1, ... N (53) ik ⎢ ⎥ r . a = k=0 dy 0,r ⎢ ⎥ ⎢ ⎥ ⎣ . ⎦ where r is the order of differentiation. Chebyshev spec- a (x , t) tral collocation method is first applied on (39)–(41) 0,r N x JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 55 ⎡ ⎤ 2 2 b (x , t) Sc (D + c D − Srk ) 0,r 0,r 1 r n+1 − ϕ r+1 ⎢ ⎥ b (x , t) t 2 0,r 2 ⎢ ⎥ ⎢ ⎥ 2 2 ⎢ ⎥ . Sc (D + c D − Srk ) b = 0,r 0,r r n ⎢ ⎥ = − ϕ + c (69) 1,r r+1 ⎢ ⎥ t 2 ⎣ ⎦ b (x , t) 0,r N Upon further simplification leads to ⎡ ⎤ c (x , t) 0,r 0 n+1 n ⎢ ⎥ c (x , t) N u = H u + G (70) 1 1 1 0,r 1 r+1 r+1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ c = 0,r ⎢ ⎥ ⎢ ⎥ n+1 n ⎣ ⎦ . N ϑ = H ϑ + G (71) 2 2 2 r+1 r+1 c (x , t) 0,r N (63) n+1 n N ϕ = H ϕ + G (72) 3 3 3 r+1 r+1 The same diagonal matrix goes for a , a , b , b , 1,r 2,r 1,r 2,r subject to b , b , c and c . Applying the forward finite dif- 3,r 4,r 0,r 1,r ference scheme defined in (48)–(50) on Equations n n u (xN , t ) = ϑ (xN , t ) = 0 r+1 x r+1 x (59)–(61) (73) ϕ (xN , t ) = 0 r+1 x n+1 u − u 2 r+1 r+1 (1 + αD ) n n nt u (x , t ) = 1, ϑ (x , t ) = 1 + ξe r+1 0 r+1 0 (74) n+1 n nt u + u ϕ (x , t ) = 1 + ξe , n = 1, 2, ... r+1 0 3 2 2 r+1 r+1 = (a D + D + a D − M ) 0,r 1,r (64) −y −y nt j j u (y ,0) = e , ϑ (y ,0) = e + ξe r+1 j r+1 j (75) −y nt ϕ (y ,0) = e + ξe r+1 j n+1 n ϑ − ϑ r+1 r+1 The above matrices are defined as Pr = (b D + b D + b + Prδ ) 0,r 1,r 4,r x 1 1 n+ n+ n+1 n 2 3 2 2 2 ϑ + ϑ a D + D + a D − M I r+1 r+1 2 0,r 1,r (1 + αD ) × + b + b 2,r 3,r N = − 2 1 t 2 (65) 1 1 n+ n+ 2 3 2 2 2 n+1 n a D + D + a D − M I 2 0,r 1,r ϕ − ϕ r+1 r+1 (1 + αD ) Sc H = + t 2 n+1 ϕ + ϕ 2 2 r+1 r+1 1 1 1 = (D + c D − Sck ) + c (66) 0,r 1,r n+ n+ n+ 2 2 2 2 b D + b D + b + Prδ 0,r 1,r 4,r Pr N = − t 2 Upon simplification leads to 2 3 2 2 (1 + αD ) (a D + D + a D − M ) 0,r 1,r n+1 1 1 1 − u n+ n+ n+ r+1 2 2 2 2 b D + b D + b + Prδ t 2 0,r 1,r 4,r Pr ⎡ ⎤ 3 2 H = + (a D + D 0,r t 2 ⎢ 2 ⎥ +a D − M ) (1 + αD ) 1,r ⎢ ⎥ = + u + a (67) ⎢ ⎥ 2,r r+1 ⎣ t 2 ⎦ 1 n+ 2 2 2 D + c D − Sck 0,r r Sc N = − , t 2 Pr (b D + b D + b + Prδ ) 0,r 1,r 4,r x n+1 − ϑ r+1 t 2 n+ 2 2 2 D + c D − Sck ⎡ ⎤ r 0,r Sc (b D + b D 0,r 1,r H = + ⎢ ⎥ t 2 +b + Prδ ) Pr 4,r x ⎢ ⎥ = + ϑ + (b + b ) ⎢ ⎥ 2,r 3,r r+1 ⎣ t 2 ⎦ 1 1 1 1 n+ n+ n+ n+ 2 2 2 2 G = a , G = b + b , G = c 1 2 3 1,r 2,r 3,r 1,r (68) 56 A. S. IDOWU AND B. O. FALODUN the physics of the problem. Hence, all numerical 4. Results and discussion computations correspond to the above-stated values Equations (23)–(25) subject to (26) and (27) have been unless or otherwise stated. The effects of Soret param- solved using the spectral relaxation method (SRM). SRM eter Sr and Dufour parameter Du were investigated employs the idea of Gauss-seidel relaxation approach separately in this study. Remarkably, the results were to linearize and decoupled system of non-linear dif- compared with those of the existing literature and were ferential equations [26]. Using the SRM, numerical found to be in good agreement. It is worth mention- computations are carried out for the velocity, tem- ing that, the moment when α = 0 is set in this study, perature, concentration, local skin friction, local Nus- the model is categorized as a Newtonian fluid phe- selt number and Sherwood number. Results are pre- nomenon. sented in the tabular and graphical form. All pro- grams are coded in MATLAB R2012a. The results were generated using the scaling parameter L = 15 and it 4.1. Velocity profiles is observed that increasing the value of L does not The effect of the Prandtl number Pr on the velocity is change the result to a reasonable extent. The num- depicted in Figure 2 (a). It is observed that the fluid ber of collocation point used in generating the results velocity decreases with the increase in Pr. This result was N = 120. The accuracy is seen to improve with is in agreement with that of Alao et al. [23], but it an increase in the number of collocation points while is observed that the presence of parameters such as it was noted that there are errors using a few col- α, δ ,  tends to influence the profile more. It is obvi- location points. The range of parameters are cho- ous that in Figure 2(a) as the value of Pr is increas- sen to Gr = Gm = 2.0, Rr = A = kr = 0.5, t = 1.0, n = ing, it causes a reduction in the velocity as higher Pr 0.5, Du = 0.2, Sr = 0.5, Ec = 0.01, α = 0.2, δ = 0.04, is expected to reduce both the velocity and the local = 0.01, ε = 0.0001 and the value for magnetic param- skin friction. In fact, higher values of Pr aremoreorless eter between 0.1 and 1 to obtain a clear insight into Figure 2. Effects of (a) Prandtl number and (b) radiation param- Figure 3. Effects of (a) Soret number and (b) Dufour number on eter on velocity profiles. velocity profiles. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 57 increased in the thermal conductivity of the fluid and for mass fraction of solute concentration. This is called ther- that reason heat is able to migrate away from the heated mal gradient and cooperative solutal, the solute diffuses surface more speedily for lower values of Pr. Physically, to cold regions but for the case when Soret number < higher viscosity and lesser thermal conductivity result in 0, an increase in temperature brings about an increase larger Pr.Thus, Pr is useful in increasing the rate of cool- in the temperature and it leads to an increase in den- ing in conducting flows. Figure 2(b) shows the variation sity. It is called thermal gradient and competitive solutal of the radiation parameter Rr on velocity. It is noted from and the solute diffuses to warmer regions. Figure 3(b) Figure 2(b) that increasing Rr intensifies the thermal illustrates the variation of Dufour parameter Du on the condition of the fluid environment. Increase in the fluid velocity profile. It is observed that there is an increase temperature incites more flow in the boundary layer in the fluid velocity by increasing Du. Figure 4(a) illus- making the velocity of the fluid to increase through trates the effect of the viscoelastic fluid parameter α the buoyancy effect. Also, the hydrodynamic bound- on the velocity profile. The viscoelastic fluid parame- ary layer thickness increases when the thermal radiation ter α connotes the effect of normal stress coefficient parameter is increased. This is true because the increase on the flow. It is noted that at a point on the flow in radiation parameter releases heat energy to the flow. domain that increasing the viscoelastic parameter has Figure 3(a) represents the effect of Soret parameter Sr the tendency of decreasing the boundary layer thick- on the velocity profile. It is found out that increasing the ness. Interestingly, very close to the wall, the fluid veloc- values of Sr increases the velocity profile. This is due to ity decreases and increases far away from the plate. The the fact that, when Sr is raised, there will be greater ther- effect of heat generation/absorption coefficient param- mal diffusion and this results to increase in the velocity eter  on the velocity profile is depicted in Figure 4(b). of the fluid. The positive Soret parameter has a stabi- Increasing the values of heat generation/absorption lizing effect. When Soret number > 0, an increase in coefficient parameter  increases the velocity profile. temperature will cause a decrease in both density and This is expected because when > 0, the behaviour of the fluid velocity changes and it drastically causes an Figure 4. Effects of (a) Viscoelastic fluid parameter and heat Figure 5. Effects of (a) Schmidt number and (b) chemical reac- generation on velocity profiles. tion parameter on the velocity profiles. 58 A. S. IDOWU AND B. O. FALODUN increase. Figure 5(a) depicts the effect of the Schmidt that increasing the magnetic parameter causes a reduc- number Sc on the velocity profile. It is obvious from tion in the velocity profile. The magnetic field induces Figure 5(a) that increasing Sc causes retardation on currents in a moving conductive fluid. The induced cur- the velocity profile. This causes a decrease in the con- rents polarize the fluid and change the magnetic field. centration buoyancy effects leading to a reduction in Figure 7(a) depicts the effect of thermal Grashof num- the fluid velocity. Decrease in the velocity and concen- ber Gr on the velocity profile. Gr is the ratio of buoyancy tration is accompanied by a simultaneous decrease in to the viscous acting on the fluid. The velocity is cou- both velocity and concentration boundary layer. This pled with temperature and concentration via thermal behaviour is shown in Figure 5(a) and 10(b). Figure 5(b) Grashof number and mass Grashof number as seen in shows the effect of the chemical reaction parameter Equation (8). Enhancement of thermal buoyancy force kr on the velocity profile. It is observed that there is a gives rise to the velocity. When the values of Gr increase, reduction in the velocity profile with increasing value of it increases rapidly close to the plate and decreases kr. Figure 6(a) depicts the influence of Eckert number the free stream velocity. This behaviour is shown in Ec on the velocity profile. Ec connotes the relationship Figure 7(a). Figure 7(b) illustrates the influence of mass between the kinetic energy in the flow and enthalpy. Grashof number Gm on the velocity profile. It is evident Greater Eckert number causes a rise in velocity. It is from Figure 7(b) that the fluid velocity increases and the clearly seen from Figure 6(a) that the Eckert number ele- peak value is more distinctive because of the increase in vates the velocity profile because heat energy is stored the species buoyancy force. in the liquid due to frictional heating. The variation of different values of the magnetic parameter M on the 4.2. Temperature profiles velocity is depicted in Figure 6(b). The applied mag- netic field strength B gives rise to a resistive force called 0 From Figure 8(a) it is observed that the increase in Lorentz force and it causes the motion of an electri- Pr decreases the temperature profile. It is noted that cally conducting fluid. It is clearly seen in Figure 6(b) when Pr < 1, the fluid in the hydrodynamic, thermal and concentration boundary layer is highly conducive. Figure 6. Effects of (a) Eckert number and (b) magnetic param- Figure 7. Effects of (a) thermal Grashof number and (b) mass eter on the velocity profiles. Grashof number on velocity profiles. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 59 Figure 8. Effects of (a) Prandtl number and (b) radiation param- Figure 9. Effects of (a) Dufour number and (b) heat generation eter on temperature profiles. parameter on the temperature profiles. Also, small values of Pr are equivalent to the enrich- heat generation/absorption coefficient parameter  on ment of thermal conductivities. Prandtl number can be the temperature profile is depicted in Figure 9(b). defined as the ratio of momentum diffusion to thermal increases the temperature profile as seen in Figure 9(b) diffusion. With the increase in Pr , the thermal diffusion because the boundary layer gets thicker and the parti- decreases and therefore the thermal boundary layer cles of the fluid become warm. becomes thinner. However, fluids with a very large Pr and higher heat capacity tend to increase the rate of 4.3. Concentration profile heat transfer and decreases the non-dimensional tem- perature. The effect of Rr on the temperature profile The variation of the Soret parameter Sr on the is presented in Figure 8(b) shows that the tempera- concentration profile is depicted in Figure 10(a). The ture profile increases with an increase in Rr. When Rr is concentration profile rises when increasing the Soret raised the temperature of the fluid will increase caus- parameter as shown in Figure 10(a). Figure 10(b) illus- ing an increase in the profile. Physically, increase in the trates the variation of the Schmidt number Sc on the radiation parameter brings heat energy to the thermal concentration profile. It is obvious from Figure 10(b) boundary layer. As a result of this, the heat energy gives that increasing the values of Sc drastically decreases the room for more temperature and hereby increases the concentration profile. The effect of the chemical reac- non-dimensional temperature. Figure 9(a) depicts the tion parameter kr on the concentration profile is plotted effect of Dufour parameter Du on the temperature pro- in Figure 11. It is observed that the concentration profile file. The Dufour term explains the effect of concentra- decreases with increasing values of kr. The implication tion gradients as seen in Equation (9) that plays a signif- of this is that the buoyancy effects due to concentra- icant role in aiding the fluid flow and has the tendency tion and temperature difference are significant in the to elevate the thermal energy in the boundary layer. plate. The fluid motion is retarded on account of a chem- From Figure 9(a), it is observed that as Du increases it ical reaction. This implies that the destructive reaction gives a rise in the temperature profile. The variation of kr > 0 decreases the concentration field which weakens 60 A. S. IDOWU AND B. O. FALODUN Table 1. Computational values for skin friction coefficient Cf and Nusselt number for various values of Schmidt number com- pared with [22]when α = δ =  = Du = Sr = 0. Present study [22] Sc Cf Sh Cf Sh 0.22 3.1067 0.4512 3.1068 0.4515 0.60 2.4546 0.8429 2.4548 0.8431 0.78 2.2766 1.0212 2.2767 1.0214 0.94 2.1539 1.1744 2.1540 1.1745 Table 2. Computational values for skin friction coefficient Cf and Nusselt number for various values of thermal radiation parameter compared with [23]when α = δ =  = 0. Present study [23] Rr Cf ϑ (0) Cf ϑ (0) 0.0 1.7939 0.7452 1.7940 0.7455 0.4 1.9164 0.6532 1.9166 0.6533 0.8 2.0108 0.6049 2.0111 0.6051 1.0 2.0509 0.5889 2.0512 0.5892 Table 3. Computational values for skin friction coefficient Cf, Nusselt number and Sherwood number with variation of Dufour parameter. Du Cf Nh Sh 0.0 0.9572 0.4174 0.6993 0.5 1.3468 0.5551 0.6993 1.0 1.7364 0.6928 0.6993 Table 4. Computational values for skin friction coefficient Cf, Nusselt number and Sherwood number with variation of Soret parameter. Figure 10. Effects of (a) Soret number and (b) Schmidt number Sr Cf Nh Sh on the concentration profiles. 0.0 1.2097 0.6377 0.5353 0.5 1.5806 0.6377 0.6993 1.0 1.9514 0.6377 0.8633 5. Concluding remarks In this paper, spectral relaxation method (SRM) is employed to solve a third-order-coupled partial differ- ential equations that govern the effects of Soret and Dufour on MHD heat and mass transfer of a viscoelas- tic fluid pass over a semi-infinite vertical plate. Detailed explanation on SRM is discussed in the previous sec- tions. Comparison of the present results is done with previously published work and was found to be in excellent agreement. The present results will be help- ful in understanding the complex problems of MHD viscoelastic fluid over a semi-infinite vertical plate. The SRM is efficient and gives accurate results compared Figure 11. Effect of chemical reaction parameter on the con- centration profile. to other numerical methods used in the reviewed lit- erature. Physically, the magnetic term in Equation (8) gives rise to the Lorentz force which interacts with the buoyancy force in the governing velocity and temper- the buoyancy effects due to concentration gradients. ature fields. It slows down the motion of the fluid. The Thus, the chemical reaction reduces the concentration, presence of the radiation term gives rise to the fluid thereby increasing its concentration gradient and con- temperature. When the radiation is further increased, centration flux (Tables 1–4). there is a higher temperature which adds to the velocity JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 61 of the fluid. The graphical representation is in Figure 2b medium over a non-isothermal stretching sheet. IOSR J Math. 2016;12(1):53–60. and 8b of this study. The presence of thermal con- [6] Hayat T, Hina Z, Anum T, Ahmad A. Soret and Dufour ductivity is very important in the study of the fluid effects on MHD peristaltic transport of Jeffrey fluid in phenomenon. 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Journal

Journal of Taibah University of ScienceTaylor & Francis

Published: Dec 11, 2019

Keywords: Soret–Dufour effect; heat transfer; viscous dissipation; thermal radiation; chemical reaction; spectral relaxation method

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