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MHD Natural Convection Flow of Casson Nanofluid over Nonlinearly Stretching Sheet Through Porous Medium with Chemical Reaction and Thermal Radiation

MHD Natural Convection Flow of Casson Nanofluid over Nonlinearly Stretching Sheet Through Porous... In the present work, the effects of chemical reaction on hydromagnetic natural convection flow of Casson nanofluid induced due to nonlinearly stretching sheet immersed in a porous medium under the influence of thermal radiation and convective boundary condition are performed numerically. Moreover, the effects of velocity slip at stretching sheet wall are also examined in this study. The highly nonlinear-coupled governing equations are converted to nonlinear ordinary differential equations via similarity transformations. The transformed governing equations are then solved numerically using the Keller box method and graphical results for velocity, temperature, and nanoparticle concentration as well as wall shear stress, heat, and mass transfer rate are achieved through MATLAB software. Numerical results for the wall shear stress and heat transfer rate are presented in tabular form and compared with previously published work. Comparison reveals that the results are in good agreement. Findings of this work demonstrate that Casson fluids are better to control the temperature and nanoparticle concentration as compared to Newtonian fluid when the sheet is stretched in a nonlinear way. Also, the presence of suspended nanoparticles effectively promotes the heat transfer mechanism in the base fluid. Keywords: Casson fluid, Chemical reaction, Slip condition, Thermal radiation, Convective boundary condition Background conductivity as compared to conventional base fluids. Nanofluid is a new class of fluid consists of nanometer- Eastman et al. [2] further explored that the addition of sized particles suspended in a base fluid. Poor heat copper (10 nm) particles in ethylene glycol increases transfer fluids such as water, ethylene glycol, and engine the thermal conductivity up to 40%. Later on, many re- oil have low thermal conductivity, and are considered es- searchers [3–5] reported that addition of 1–5% by vol- sential for heat transfer coefficient between the heat ume of nanoparticles to ordinary heat transfer fluids transfer medium and the heat transfer surface. It has can enhancethethermalconductivitymorethan20%. been proven through experiments that the thermal Boungiorno [6] pointed two slip mechanisms, i.e., conductivity of nanofluid is appreciably higher than Brownian motion and thermophoresis out of seven slip thebasefluids. Theterm “nanofluid” was first coined mechanisms that effectively enhance the thermal con- by Choi and Eastman [1] and discovered that sus- ductivity of base fluid. Brownian motion is responsible for pended nanoparticles in the base fluid can enhance the the collision of nanoparticles moving in the base fluid. In thermal conductivity of base fluid efficiently. The fact, heat transfer due to the collision of two particles could nanoparticles are typically made of Al O , SiC, AlN, enhance the thermal conductivity of nanofluids. The com- 2 3 Cu, TiO and graphite, and have high thermal prehensive references and in-depth understanding on nano- fluid can be insightinmostrecentarticles[7–9]. The boundary layer flow caused by stretching a sheet * Correspondence: [email protected] Department of Mathematical Sciences, Faculty of Science, Universiti linearly or nonlinearly is an important engineering prob- Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia lem and has several industrial applications, including Full list of author information is available at the end of the article © The Author(s). 2016 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 2 of 15 extrusion of polymer sheets, melting spinning, the hot stretching sheet under the influence of first-order rolling, wire drawing, production of glass fiber, plastic chemical reaction was theoretically studied by Raptis and rubber sheets manufacturing, enhanced recovery of and Perdikis [22]. On the other hand, the numerical petroleum resources, and cooling of large plate in bath. and analytical solutions of steady-state boundary layer The heat transfer phenomenon in stretching sheet prob- flow of micropolar fluid induced due to nonlinearly lem is very important as cooling and heating are neces- stretching sheet were found by Damseh et al. [23] and sary factors for the quality of end product. The seminal Magyari and Chamkha [24], respectively. The combined work of Crane [10] was extended by Cortell [11, 12] and effects of slip and chemical reaction on electrically con- found numerical solutions for heat transfer flow of vis- ducting fluid over a nonlinearly porous stretching sheet cous fluid due to nonlinearly stretching sheet with and were analyzed by Yazdi et al. [25]. In the same year, without the effects of thermal radiation, respectively. In Bhattacharyya and Layek [26] investigated the velocity the same year, Abbas and Hayat [13] also explored the slip effects on boundary layer flow of viscous fluid past influence of thermal radiation on two-dimensional flow a permeable stretching sheet in the presence of chem- of viscous fluid towards nonlinearly stretching sheet ical reaction. The steady two-dimensional boundary saturated in a porous medium. The steady electrically layer flow of Newtonian fluid due to stretching sheet conducting flow of micropolar fluid caused by nonli- saturated in nanofluid in the presence of chemical reac- nearly stretching sheet was reported by Hayat et al. tion is explored by Kameswaran at al. [27]. Motivated [14]. Motivated by this, Anwar et al. [15] utilized the by this, Aurangzaib et al. [28] studied theoretically the Boungiorno model and investigated natural convection influence of thermal radiation on unsteady natural con- flow of viscous fluid induced by nonlinearly stretching vection flow caused by stretching surface in the pres- sheet saturated in a nanofluid. Mukhopadhyay [16] ence of chemical reaction and magnetic field. Shehzad studied two-dimensional boundary layer flow of Casson et al. [29] reported the effects of magnetic field on mass fluid past a nonlinearly stretching sheet and concluded transfer flow of Casson fluid past a permeable stretch- that fluid velocity is suppressed whereas temperature ing sheet in the presence of chemical reaction. Pal and enhanced by Casson parameter. The two-dimensional Mandal [30] explored the characteristics of mixed con- incompressible flow of viscous fluid caused by nonli- vection flow of nanofluid towards a stretching sheet nearly stretching sheet in a nanofluid is reported by under the influence of chemical reaction and thermal Zaimi et al. [17]. Motivate by this, Raju and Sandeep radiation. Similarity solutions for unsteady boundary [18] and Raju et al. [19] analyzed three-dimensional flow of Casson fluid induced due to stretching sheet electrically conducting flow of Casson-Carreau fluids embedded in a porous medium in the presence of first and nanofluids due to unsteady and steady stretching order chemical reaction were obtained by Makanda and sheet, respectively. Very recently, Pal et al. [20] investi- Shaw [31]. gated the influence of thermal radiation on mixed con- On the other hand, convective boundary condition vection flow of nanofluid caused by nonlinearly plays a vital role in many engineering processes and in- stretching/shrinking sheet. dustries such as gas turbines, material drying, textile However, the combined effects of heat and mass trans- drying, laser pulse heating, nuclear plants, transpiration fer and chemical reaction play a vital role in chemical cooling, and food process. It is because that the convect- and hydro-metallurgical industries. The chemical reac- ive boundary condition applied at the surface is more tion can be of any order, but the most simple of which is practical and realistic. The two-dimensional laminar the chemical reaction of first order where the reaction boundary layer flow of Newtonian fluid caused by por- rate and species concentration are directly proportional ous stretching surface in the presence of convective to each other. The formation of Smog is an example of boundary condition is investigated numerically by Ishak first order chemical reaction. In several chemical engin- [32]. Makinde and Aziz [33] analyzed steady incom- eering processes, chemical reaction between foreign pressible flow of nanofluids towards stretching sheet mass and working fluid often occurs because of stretch- with convective boundary condition using Boungiorno ing a sheet. The diffusive species can be absorbed or model. Moreover, RamReddy et al. [34] included the ef- generated due to different types of chemical reaction fects of Soret and investigated mixed convection flow with the ambient fluid which is greatly influenced by the due to vertical plate in a nanofluid under convective properties and quality of end product. Kandasamy and boundary condition. Das et al. [35] discussed the heat Periasamy [21] investigated heat and mass transfer free and mass transfer flow of hydromagnetic nanofluid past convection flow of Newtonian fluid past nonlinearly a stretching sheet place in a porous medium with con- stretching sheet in the presence of chemical reaction vective boundary condition. The three-dimensional and magnetic field. The laminar boundary layer flow laminar flow of Casson nanofluid due to stretching of electrically conducted fluid towards nonlinearly sheet in the presence of convective boundary condition Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 3 of 15 is developed by Nadeem and Haq [36]. Motivated by here that the results are perceived in an excellent this, Malik et al. [37] investigated the influence of con- agreement. vective boundary condition past a stretching sheet in the presence of magnetic field. The two-dimensional Methods electrically conducting flow of Casson nanofluid caused Mathematical Formulation by stretching sheet with convective boundary condition is The steady incompressible natural convection flow of performed by Hussain et al. [38]. Very recently, Sulochana Casson fluid caused by nonlinearly stretching sheet et al. [39] established numerical solutions of three- through porous medium in the presence of chemical re- dimensional Casson nanofluid induced due to permeable action thermal radiation is considered. The x − axis is stretching sheet in the presence of convective boundary taken along the direction of stretching sheet and y − axis condition. is perpendicular to the surface (see Fig. 1). The sheet is Motivated by the above-cited literature survey and the stretched with the nonlinear velocity of the form u (x)= widespread engineering and industrial applications, it is cx ,where c is constant and n (> 0) represents the nonli- of prime importance to explore the effects of chemical nearly stretching sheet parameter (n = 1 corresponds to lin- reaction and thermal radiation on electrically conducting ear stretching sheet and n ≠ 1 represent nonlinear natural convection flow of Casson nanofluid caused by stretching sheet). Moreover, a variable magnetic field B(x)= (n − 1)/2 nonlinearly stretching sheet through porous medium in B x [22] is applied normally to the stretching sheet the presence of slip and convective boundary conditions. with constant B . Furthermore, it is also assumed that sheet The presence of momentum slip and convective bound- wall is heated by temperature T (x)= T + Ax (A being the f ∞ ary condition makes the present mathematical model of reference temperature and λ =2n − 1) and C (x)= C + Ex s ∞ a physical system to some extent difficult while inter- (E being the reference concentration). action of nanofluid with Casson fluid as base fluid. The The governing equations for Casson nanofluid along governing equations are converted to ordinary differen- with continuity equation are given as tial equation using similarity transformations and nu- merical solutions are obtained through the Keller box ∂u ∂υ þ ¼ 0; ð1Þ method [40]. To validate and examine the numerical al- ∂x ∂x gorithm developed in MATLAB software for the present problem, the results are compared with the ∂u ∂u μ 1 ∂ u σB ðÞ x μ ϕ B B existing literature results for pure Newtonian and u þ υ ¼ 1 þ − þ u ∂x ∂y ρ β ∂y ρ ρ k f f f Casson fluids as a limiting case. It is worth mentioning Fig. 1 Physical sketch and coordinate system Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 4 of 15 2 3 Table 1 Comparison of skin friction coefficient for different ρ −ρ p f f ∞ values of β and M when n =1, β→ ∞, Bi → ∞, Bi → ∞, and ∞ 1 2 4 5 þðÞ 1−C βðÞ T−T þ gβðÞ C−C g; ∞ ∞ ∞ T C ρ ρ M = K = Gr = Gm = δ = R =Le = N = N = R =0 d t b f f − 1 þ f ðÞ 0 ð2Þ β M Nadeem et al. [36] Ahmad and Nazar [42] Present results ∞ 0 1.0042 1.0042 1 ∂T ∂T ∂ T u þ υ ¼ α 5 1.0954 – 1.0955 ∂x ∂y ∂y "# 1 1.4142 – 1.4144 ∂C ∂T D ∂T 1 ∂q ∞ 10 3.3165 3.3165 3.3166 þτ D þ − ; ∂y ∂y T ∂y ðÞ ρc ∂y 5 3.6331 – 3.6332 ð3Þ 1 4.6904 – 4.6904 ∞ 100 10.049 10.0498 10.0499 2 2 ∂C ∂C ∂ C D ∂ T u þ υ ¼ D þ −kðÞ C−C ð4Þ B c ∞ 5 11.0091 – 11.0091 2 2 ∂x ∂y ∂y T ∂y 1 14.2127 – 14.2127 In theaboveexpressions, u and υ denote the velocity components in x − and y − directions, respectively, μ is the plastic dynamic viscosity, ρ is the fluid density, σ The radiative heat flux q described according to Ros- is the electrically conductivity, β is the Casson fluid seland approximation is given as parameter which has inverse relation with yield stress, pffiffiffiffi μ 2π 1 − n B  4 i.e., p ¼ [41], φ is the porosity, k (x)= k x is −4σ ∂T 1 0 q ¼ ð7Þ the variable permeability of porous medium, g is the 3k ∂y gravitational force due to acceleration, ρ is the density where σ is the Stefan-Boltzmann constant and k * is the of nanoparticle, β is the volumetric coefficient of ther- mean absorption coefficient. T can be expressed as lin- mal expansion, β is the coefficient of concentration ear function of temperature. By expanding T in a Taylor expansion, T is the fluid temperature, C is the nanopar- series about T and neglecting higher terms, we can ticle concentration, α ¼ is the thermal diffusivity ðÞ ρc write of the Casson fluid, k is the thermal conductivity of the 4 3 4 T ≅ 4T T−3T ð8Þ fluid, D is the Brownian diffusion coefficient, D is the B T ∞ ∞ ðÞ ρc thermophoretic diffusion coefficient, τ ¼ is the ra- ðÞ ρc Now putting Eqs. (7) and (8) in Eq. (3), we obtain tio of heat capacities in which (ρc) is the heat capacity 2  3 2 ∂T ∂T ∂ T 16σ T ∂ T of the fluid and (ρc) is the effective heat capacity of p u þ υ ¼ α þ ð9Þ 2 2 ∂x ∂y ∂y 3ðÞ ρc k ∂y nanoparticle material, c is the specific heat at constant f "# ak x pressure, q is the radiative heat flux and k ðÞ x ¼ is ∂C ∂T D ∂T r c þτ D þ the variable rate of chemical reaction, k is a constant ∂y ∂y T ∂y 2 ∞ reaction rate, and a is the reference length along the flow. The corresponding boundary conditions are written as follows: Table 2 Comparison of skin friction coefficient when β→ ∞, Bi → ∞, Bi → ∞, Pr = 6.8, and M = K = Gr = Gm = δ = R =Le = 1 2 d 1 ∂u ∂T > N = N = R =0 u ¼ u ðÞ x þ N ν 1 þ ; k ¼ −h T −T t b w 1 f f β ∂y ∂y 1 ″ − 1 þ f ðÞ 0 ∂C D ¼ −hðÞ C −C at y ¼ 0 ; B s s n Cortell [11] Pal et al. [20] Present results ∂y 0.0 0.6276 0.6275 0.6276 ð5Þ 0.2 0.7668 0.7668 0.7668 u→0; T→T ; C→C as y→∞: ð6Þ 0.5 0.8895 0.8895 0.8896 ∞ ∞ 1 1.0000 1.0000 1.0000 n−1 Here, N ðÞ x ¼ N x is the velocity slip, N being 1 0 0 3 1.1486 1.1486 1.1486 n−1 n−1 2 2 constant, h ðÞ x ¼ h x and h ðÞ x ¼ h x are the con- f 0 s 1 10 1.2349 1.2348 1.2349 vective heat and mass transfer with h , and h being 0 1 20 1.2574 1.2574 1.2574 constants. Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 5 of 15 Table 3 Comparison of − θ′(0) for different Pr with n =1, β→ ∞, Bi → ∞, Bi → ∞, and M = K = Gr = Gm = δ = R =Le = N = N = 1 21 d t b R =0 − θ′(0) Pr Yih [43] Aurangzaib et al. [28] Pal et al. [20] Present results 0.72 0.8086 0.8086 0.8086 0.8088 1 1.0000 1.0000 1.0000 1.0000 3 1.9237 1.9237 1.9237 1.9237 10 3.7207 3.7207 3.7206 3.7208 100 12.2940 12.3004 12.2939 12.3004 Now, introduce the stream function ψ defined in its usual notation in terms of velocity, a similar variable η, and the following similarity transformations; Fig. 3 Effect of β on temperature for various n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 2νc nþ1 ðÞ n þ 1 c n−1 2 2 ψ ¼ x fðÞ η ; η ¼ x y; ð10Þ n þ 1 2ν 22ðÞ n−1 N 1 00 0 0 00 T−T C−C φ þ Lef φ − Lef φ þ θ − RLeφ ¼ 0 ∞ ∞ θ ¼ ; φ ¼ n þ 1 N n þ 1 T −T C −C f ∞ s ∞ ð13Þ rffiffiffiffiffiffiffiffiffiffiffi 9 Finally, Eqs. (3–6) and Eq. (9) take the following form pffiffiffiffiffiffiffiffiffiffiffi 1 1 0 00 0 > f ðÞ 0 ¼ 1 þ δ n þ 1 1 þ f ðÞ 0 ; θ ðÞ 0 ¼ − Bi½ 1−θðÞ 0 ; 1 = β n þ 1 rffiffiffiffiffiffiffiffiffiffiffi 1 > 1 2n 1 φðÞ 0 ¼ − Bi½ 1−φðÞ 0 00 00 0 0 n þ 1 1 þ f þ ff − f − ðÞ M þ K f ð11Þ β n þ 1 n þ 1 ð14Þ þ ðÞ Grθ þ Gmφ¼ 0 n þ 1 f ð∞Þ¼ 0; θð∞Þ¼ 0; ϕð∞Þ¼ 0 ð15Þ 4 22ðÞ n−1 In the above expressions, M, K, Gr, Gm, δ,Pr, R , N , 00 0 0 0 0 d b 1 þ R θ þ Prf θ − Prf θ þ PrN φ θ d b N , Bi , Bi , Le, and R are the magnetic parameter, por- 3 n þ 1 t 1 2 osity parameter, Grashof number, mass Grashof number, þ PrN θ ¼ 0 slip parameter, Prandtl number, radiation parameter, Brownian motion parameter, thermophoresis parameter, ð12Þ Fig. 4 Effect of β on nanoparticle concentration for two different Fig. 2 Effect of β on velocity for two different values of n values of n Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 6 of 15 Fig. 7 Effect of n on nanoparticle concentration in the presence/ Fig. 5 Effect of n on velocity in the presence/absence of K absence of K Biot numbers, Lewis number, and chemical reaction par- ameter and are defined as The dimensionless skin friction coefficient Cf ¼ , ! ! 2 ρu . . p w f p − p ∞ p f xq 2gβ AðÞ 1−C 2gβ E T ∞ C w pf p the local Nusselt number Nu ¼ , and local 2σB 2νϕ f x M ¼ ; K ¼ ; Gr ¼ ; Gm ¼ ; αðÞ T −T f f ∞ 2 2 ρc k c c c rffiffiffiffiffi xq τD T −T c μcf 4σ T τDðÞ C −C T f ∞ ∞ B s ∞ Sherwood number Sh ¼ on the surface along δ ¼ N ; Pr ¼ ; R ¼ ; N ¼ ; N ¼ ; DðÞ C −C 0 d  b t B w ∞ 2ν k kk ν νT 1 ∞ x − direction, local Nusselt number Nu , and Sherwood 1 1 x h 2ν h 2ν ν 2νk 0 2 1 2 2 number Sh are given by Bi ¼ ; Bi ¼ ; Le ¼ ; R ¼ 1 2 x k c D c D c B B The wall skin friction, wall heat flux, and wall mass rffiffiffiffiffiffiffiffiffiffiffi n þ 1 1 flux, respectively, are defined by −1=2 1=2 Cf ¼ 1 þ f ðÞ 0 ; ðÞ Re Nu ðÞ Re x x x 2 β rffiffiffiffiffiffiffiffiffiffiffi 1 ∂u n þ 1 4 0 −1=2 τ ¼ μ 1 þ ; q B w ¼ − 1 þ R θ ðÞ 0 ;ðÞ Re Sh d x β ∂y y¼0 2 3 ! ! rffiffiffiffiffiffiffiffiffiffiffi 16σ T ∂T n þ 1 ¼ − α þ and q ¼ − φðÞ 0 3ðρcÞ k ∂y y¼0 ∂C ¼ −D nþ1 Re ¼cx ∂y where is the local Reynold number. y¼0 Fig. 6 Effect of n on temperature in the presence/absence of K Fig. 8 Effect of M on velocity for various values of n Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 7 of 15 Fig. 9 Effect of M on temperature for two values of n Fig. 11 Effect of K on velocity for various values of β Numerical Scheme Here, the step size η = 0.01 and boundary layer thick- The governing Eqs. (11)–(13) with associated boundary ness η = 10 is used. Further, the convergent criteria − 5 conditions (14) and (15) are solved numerically via the 10 is considered for all the cases. The numerical and Keller box method. The detail of this method is given in graphical results are generated through MATLAB soft- the book of Cebeci and Bradshaw [40]. This method is ware. In order to assess the accuracy and validate our unconditionally stable and has second-order accuracy. code, the comparison is made with previous results of The following four steps are involved in finding the nu- literature as a limiting case. merical solutions of the problem Results and Discussion (i) Initially, the transformed governing equations In the present study, the effects of slip and convective are converted to first-order system. boundary conditions on heat and mass transfer flow of (ii) Now, approximate the first-order system using nanofluid due to nonlinearly stretching surface saturated central difference formula about the mid-point. in a porous medium in the presence of chemical reac- (iii) The algebraic equations are then linearized via tion and thermal radiation were analyzed. Moreover, Newton’s method and write them in matrix-vector Casson fluid is used as base fluid. In order to analyze the notation. results, numerical calculations are carried out for various (iv) Finally, block tri-diagonal elimination technique is values of Casson fluid parameter β, nonlinear stretching used to solve the linear system. sheet parameter n, magnetic parameter M, porosity Fig. 10 Effect of M on nanoparticle concentration for two values of n Fig. 12 Effect of K on temperature for two values of β Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 8 of 15 Fig. 13 Effect of K on nanoparticle concentration for various β Fig. 15 Effect of Gr on temperature profile for two different values of n parameter K, Grashof number Gr, mass Grashof number M, K, Gr, Gm, δ,Pr, R , N , N , Bi , Bi , Le, and R on vel- d b t 1 2 Gm, Prandtl number Pr, radiation parameter R , Brown- ocity (f′(η)), temperature (θ(η)), and nanoparticle con- ian motion parameter N , thermophoresis parameter N , centration (φ(η)) profiles, respectively. Figures 2, 3, and b t Lewis number Le, slip parameter δ, and Biot numbers 4 exhibit the variation of β on velocity, temperature, and Bi , Bi . For validation of the present method, the results nanoparticle concentration, respectively, for the case of 1 2 are compared with previously reported results and dis- n = 1 and n ≠ 1. It is noteworthy here that the present played in Tables 1, 2, and 3. problem reduces to pure Newtonian nanofluid case Tables 1 and 2 present the comparison of skin friction when β→ ∞. Clearly, fluid velocity reduces as β in- coefficient for different values of β, M,and n, respect- creases. The reason behind this behavior is that increas- ively, with the results of Nadeem et al. [36], Ahmad and ing values of β implies rise in fluid viscosity, i.e., Nazar [42], Cortell [11], and Pal et al. [20]. The results reducing the yield stress. Consequently, the momentum showed an excellent agreement. Table 3 describes the boundary layer thickness reduces. It is also observed comparison of heat transfer rate for various values of Pr from this figure that fluid velocity decreases faster in the with the results of Yih [43], Aurangzaib et al. [28], and case of stretching a sheet in a nonlinear way. A similar Pal et al. [20] and revealed in a good agreement. trend of velocity profile was reported by Nadeem and Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, Haq [36] and Mukhopadhyay [16]. Conversely, both θ(η) 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, and and φ(η) enhance with increase in β (see Figs. 3 and 4). 31 are displayed to insight the physical behavior of β, n, Figures 5, 6, and 7 demonstrate the effect of n on fluid Fig. 14 Effect of Gr on velocity for two different values of n Fig. 16 Effect of Gr on nanoparticle concentration for various n Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 9 of 15 Fig. 17 Effect of Gm on velocity for two different values of n Fig. 19 Effect of Gm on nanoparticle concentration for different values of n velocity, temperature, and nanoparticle concentration, respectively, in non-porous (K = 0) and porous (K ≠ 0) A similar behavior for electrically conducting flow of medium. Interestingly, f′(η), θ(η), and φ(η) are all found Casson nanofluid due to stretching sheet was observed as decreasing functions of n. It is also noticed from Fig. 5 by Hussain et al. [38]. that momentum boundary layer thickness reduces rap- The effect of K on velocity, temperature, and nano- idly when K ≠ 0, whereas the case of thermal and con- particle concentration profiles for Newtonian and non- centration boundary layer thicknesses are quite opposite Newtonian fluids is portrayed in Figs. 11, 12, and 13, to this, i.e., both thicknesses decrease when K =0. respectively. It is interesting to note that the response Figures 8, 9, and 10 reveal the influence of M on fluid of K in these figures is completely the same as observed velocity, temperature, and nanoparticle concentration, for M. It is also found from these figures that momen- respectively, for both cases of n =1 and n ≠ 1. It is no- tum boundary layer become thinner in case of β→ ∞ ticeable that increasing values of M reduces the fluid and the opposite to this for thermal and concentration velocity whereas temperature and nanoparticle concen- boundary layer thicknesses. tration rises as M increases. As it is well-known fact Figures 14, 15, and 16 display the variation of Gr vel- that a resistive-type force produces as the current ocity, temperature, and nanoparticle concentration dis- passes through the moving fluid. This force is respon- tributions for n =1 and n ≠ 1. It is seen that fluid sible in slowing down fluid motion and increasing the velocity rises with increase in λ whereas temperature thermal and concentration boundary layer thicknesses. and nanoparticle concentration reduce as Gr increases. Fig. 18 Effect of Gm on temperature for two values of n Fig. 20 Effect of δ on velocity in the presence/absence of K Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 10 of 15 Fig. 21 Effect of δ on temperature in the presence/absence of K Fig. 23 Effect of Pr on temperature profile for various n Since the buoyancy force is dominant over viscous slip condition and δ ≠ 0 shows velocity slip at stretching force with increase in Gr. Consequently, Grashof num- sheet wall. It is interesting to see that increasing values of δ ber enhances the fluid flow which leads to increasing reduces the fluid velocity initially and then increases far velocity as well as thickness of momentum boundary from the sheet whereas dimensionless temperature and layer. In addition, since the buoyancy force tends to en- nanoparticle concentration increase with increase in δ. hance the temperature and concentration gradients, Physically, this shows that fluid velocity adjacent to the therefore, temperature and nanoparticle concentration sheet is less than the velocity of normal stretching sheet as fall. The same reason may be described for the behavior slip (δ ≠ 0) occurs. Increasing δ allowed more fluid slipping of Gm on velocity, temperature, and nanoparticle con- over the sheet and the flow decelerates near the sheet. centration distributions, as elucidated in Figs. 17, 18, Figure 23 reveals the influence of Pr on dimensionless and 19. It is also observed from these figures that influ- temperature profile when n = 1 and n ≠ 1. As expected, ence of Gr and Gm is more pronounced in case of lin- increasing Pr leads to reduction in dimensionless ear stretching sheet for all the three profiles. temperature. Based on the definition of Pr (the ratio of The variations of δ on dimensionless velocity, tem- momentum diffusivity to thermal diffusivity), therefore, perature, and nanoparticle concentration profiles for K =0 for large Pr, heat will diffuse more rapidly than the mo- and K ≠ 0 are depicted in Figs. 20, 21, and 22, respectively. mentum. Consequently, thickness of thermal boundary It is worth mentioning here that δ = 0 corresponds to no layer reduces as Pr increases. It is also noticed that Fig. 22 Effect of δ on nanoparticle concentration in the presence/ Fig. 24 Effect of R on temperature profile in the presence/absence absence of K of K Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 11 of 15 Fig. 27 Effect of N on temperature profile in the presence/absence Fig. 25 Effect of N on temperature profile for different n of R higher values of Pr reduces the temperature more dras- Figures 25 and 26 illustrate the variation of N on di- tically. It is because of the fact that low thermal conduct- mensionless temperature and nanoparticle concentration ivity of fluid associated with larger Pr, which decreases distributions. It is noted that dimensionless temperature conduction and results a temperature fall. enhances with increase in N while nanoparticle concen- The effect of R on dimensionless temperature for K tration is found decreasing as N increases. It is well d b =0 and K ≠ 0 is exhibited in Fig. 24. It is noteworthy known that Brownian motion is a diffusive process. The here that R = 0 denotes no radiation and R ≠ 0shows higher diffusivity implies higher temperature, and as d d the presence of radiation. Clearly, dimensionless consequences, the thermal conductivity becomes higher. temperature is higher as R increases. The reason be- Also, Brownian motion in nanofluid occurs only due to hind this fact is that heat energy released to the fluid nanometer size of nanoparticles. In addition to this, the as R increases and this results rise in temperature. It kinetic energy of nanoparticles enhance mainly due to is also evident from this figure that thermal boundary the increase in N and resulting higher temperature of layer thickness increases faster with increase in R nanofluids. The influence of N on dimensionless d t when K ≠ 0. This shows that influence of radiation in a temperature and nanoparticle concentration distribu- porous medium is more effective when high tions for R = 0 and R ≠ 0 are shown in Figs. 27 and 28, d d temperature is required for the desired thickness of respectively. It is evident from these figures that both end product. θ(η) and φ(η) are increasing functions of N . According Fig. 26 Effect of N on nanoparticle concentration for two different Fig. 28 Effect of N on nanoparticle concentration in the presence/ b t values of M absence of R d Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 12 of 15 Fig. 29 Effect of Bi on temperature profile for various n Fig. 31 Effect of Le on nanoparticle concentration for various values of n to the definition of N , i.e., higher values of N implies wall temperature and constant wall concentration case t t higher temperature differences and shear gradient. when Bi → ∞ and Bi → ∞. Apparently, dimensionless 1 2 Therefore, increasing values of N tends to higher temperature increases with increase in Bi . It is also evi- t 1 temperature difference across the boundary layer. On dent from this figure that increasing values of Bi lead to the other hand, nanoparticle concentration is a strong rise in sheet convective heating. Furthermore, smaller function of N ; for this reason, it is significantly influ- values of Bi (< < 1) rapidly enhance the temperature as t 1 enced by increasing values of N . These conclusions are well as the corresponding boundary layer thickness in agreement with Hussain et al. [38] and Nadeem et al. across the boundary region. Since Bi is inversely pro- [36]. It is also interesting to note from Fig. 28 that con- portional to Brownian diffusivity coefficient. Therefore, centration peak values reveal that stronger N intensifies thermal diffusivity reduces whereas momentum diffusiv- the thermal conductivity of the nanofluids near the wall. ity rises, and as a consequence, nanoparticle concentra- Further, it is also observed from Fig. 28 that nanoparticle tion enhances and related boundary layer becomes concentration falls with increase in R . thinner. Figures 29 and 30 demonstrate the effect of Bi and Figure 31 exhibits the variation of Le on nanoparticle Bi on dimensionless temperature and nanoparticle con- concentration profile for linear and nonlinear stretching centration profiles for n = 1 and n ≠ 1. It is worth men- sheet. It is perceived that in both cases, nanoparticle tioning that the present study will reduce to constant concentration is lower for higher Le. It is a fact that Fig. 32 Effect of R on nanoparticle concentration for two different Fig. 30 Effect of Bi on nanoparticle concentration for various n values of n 2 Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 13 of 15 Fig. 33 Variation of skin friction coefficient for various values of β, Fig. 35 variation of Nusselt number for various values of R , β, and δ Gr, and K lower D corresponds to larger Le and the fluids having B shear stress for increasing values of β, Gr,and K. qffiffiffiffiffiffiffi smaller Le have higher D . In other words, mass transfer B ″ nþ1 1 Clearly, 1 þ f ðÞ 0 decreases with increase rate higher for large Le. The influence of R on nanoparti- in β and Gr, whereas it increases with increase in K.It cle concentration distribution for n = 1 and n ≠1is is also evident from this figure that the skin friction co- depicted in Fig. 32. It is noteworthy here that R = 0 de- efficient is negative for all values of β, Gr,and K which notes no chemical reaction and R ≠ 0 corresponds to the indicates that fluid experiences a resistive force at the presence of chemical reaction. It is evident that stronger boundary. This is also in agreement with the results of R leads to reduce nanoparticle concentration. The ex- Table 1. The variation of skin friction coefficient Cf for planation for this behavior is that destructive chemical different values of δ, n,and Gm is portrayed in Fig. 34. rate (R > 0) enhances the mass transfer rate and results a qffiffiffiffiffiffiffi nþ1 1 decrease in nanoparticle concentration. It is found that 1 þ f ðÞ 0 increases with n The effect of skin friction coefficient Cf , local Nus- while reduces as δ and Gm increases. The influence of selt number Nu , and Sherwood number Sh for some hqffiffiffiffiffiffiffi x x nþ1 4 heat transfer rate 1 þ = R Þθ ðÞ 0 for R , β,and physical parameters β, n, K, Gr, Gm, δ,Pr, R , N , N , 3 d d d b t Le,and R are displayed in Figs. 33, 34, 35, 36, 37, and δ is perceived in Fig. 35. It is observed that heat 38, respectively. Figure 33 reveals the variation of wall Fig. 34 Variation of skin friction coefficient for different values of δ, Fig. 36 Variation of Nusselt number for various values of N , N , t b n, and Gm and Pr Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 14 of 15 Conclusions Two-dimensional electrically conducting natural convec- tion flow of Casson nanofluid towards nonlinearly stretching sheet in the presence of chemical reaction and thermal radiation was numerically discussed in this study. Moreover, the effect of slip and convective bound- ary conditions were also considered. Similarity transfor- mations are employed for the conversion of nonlinear partial differential equations to nonlinear ordinary differ- ential equations. Numerical solutions are found by the Keller box method, and graphical results are obtained through MATLAB software. Results are compared with previous work as a limiting case, and excellent accuracy is achieved with those results. It is found that β reduces the fluid velocity whereas dimensionless temperature and nanoparticle concentration increase with an increase Fig. 37 Variation of Sherwood number for various values of N , N , t b and R in β. Increasing values of n diminish the fluid velocity, temperature, and nanoparticle concentration. Velocity is observed to be enhanced as Gr and Gm increased. It is transfer rate higher for higher values of R , β whereas also noted that momentum boundary layer thickness de- increasing values of δ diminish the heat transfer rate. creases as δ increases. The dimensionless temperature Figure 36 elucidates the effect of heat transfer rate and nanoparticle concentration profiles increase with in- for various values of N , N ,and Pr.Itis noted that crease in Bi and Bi , respectively. Furthermore, it is also t b 1 2 mass transfer rate reduces as N and N increases noticed that dimensionless temperature and nanoparticle t b while increasing values of Pr enhance the mass trans- concentration distributions are increasing function of N . fer rate. It is also observed that higher value of Pr enhances the heat transfer rate significantly. Figure 37 Acknowledgements The authors would like to acknowledge Ministry of Higher Education (MOHE) illustrates the influence of Sherwood number for sev- and Research Management Centre Universiti Teknologi Malaysia (UTM) for eral values of N , N ,and R. It is noticed that stron- t b the financial support through vote numbers 4F713, 4F538 and 06H67 for this ger N promotes a progressive increase in mass research. transfer rate φ′(0) whereas increasing values of N and R decrease the mass transfer rate. Finally, Fig. 38 Authors’ Contributions IU and SS modeled the problem. IU solved the problem and generated the demonstrates the influence of mass transfer rate for graphs. IK contributed in drafting and revision of the manuscript. All authors increasing values of β, Le,and R . This figure clears read and approved the manuscript. that mass transfer rate higher for higher values of β, Le,and R . Competing Interests The authors declare that they have no competing interests. Author details Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia. Basic Sciences Department, College of Engineering, Majmaah University, Majmaah 11952, Saudi Arabia. Received: 3 October 2016 Accepted: 21 November 2016 References 1. Choi SUS, Eastman JA (1995) Enhancing thermal conductivity of fluids with nanoparticles. In: Proc 1995 ASME Int Mech Engineering Congr Expo. ASME, FED231/MD66, San Francisco, pp 99–105 2. Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ (2001) Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett 78:718–720. doi: 10.1063/1.1341218 3. Xuan Y, Li Q (2003) Investigation on convective heat transfer and flow features of nanofluids. J Heat Transfer 125:151. doi:10.1115/1.1532008 Fig. 38 variation of Sherwood number for various values of β, Le, 4. 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MHD Natural Convection Flow of Casson Nanofluid over Nonlinearly Stretching Sheet Through Porous Medium with Chemical Reaction and Thermal Radiation

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Springer Journals
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Copyright © 2016 by The Author(s).
Subject
Materials Science; Nanotechnology; Nanotechnology and Microengineering; Nanoscale Science and Technology; Nanochemistry; Molecular Medicine
ISSN
1931-7573
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1556-276X
DOI
10.1186/s11671-016-1745-6
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Abstract

In the present work, the effects of chemical reaction on hydromagnetic natural convection flow of Casson nanofluid induced due to nonlinearly stretching sheet immersed in a porous medium under the influence of thermal radiation and convective boundary condition are performed numerically. Moreover, the effects of velocity slip at stretching sheet wall are also examined in this study. The highly nonlinear-coupled governing equations are converted to nonlinear ordinary differential equations via similarity transformations. The transformed governing equations are then solved numerically using the Keller box method and graphical results for velocity, temperature, and nanoparticle concentration as well as wall shear stress, heat, and mass transfer rate are achieved through MATLAB software. Numerical results for the wall shear stress and heat transfer rate are presented in tabular form and compared with previously published work. Comparison reveals that the results are in good agreement. Findings of this work demonstrate that Casson fluids are better to control the temperature and nanoparticle concentration as compared to Newtonian fluid when the sheet is stretched in a nonlinear way. Also, the presence of suspended nanoparticles effectively promotes the heat transfer mechanism in the base fluid. Keywords: Casson fluid, Chemical reaction, Slip condition, Thermal radiation, Convective boundary condition Background conductivity as compared to conventional base fluids. Nanofluid is a new class of fluid consists of nanometer- Eastman et al. [2] further explored that the addition of sized particles suspended in a base fluid. Poor heat copper (10 nm) particles in ethylene glycol increases transfer fluids such as water, ethylene glycol, and engine the thermal conductivity up to 40%. Later on, many re- oil have low thermal conductivity, and are considered es- searchers [3–5] reported that addition of 1–5% by vol- sential for heat transfer coefficient between the heat ume of nanoparticles to ordinary heat transfer fluids transfer medium and the heat transfer surface. It has can enhancethethermalconductivitymorethan20%. been proven through experiments that the thermal Boungiorno [6] pointed two slip mechanisms, i.e., conductivity of nanofluid is appreciably higher than Brownian motion and thermophoresis out of seven slip thebasefluids. Theterm “nanofluid” was first coined mechanisms that effectively enhance the thermal con- by Choi and Eastman [1] and discovered that sus- ductivity of base fluid. Brownian motion is responsible for pended nanoparticles in the base fluid can enhance the the collision of nanoparticles moving in the base fluid. In thermal conductivity of base fluid efficiently. The fact, heat transfer due to the collision of two particles could nanoparticles are typically made of Al O , SiC, AlN, enhance the thermal conductivity of nanofluids. The com- 2 3 Cu, TiO and graphite, and have high thermal prehensive references and in-depth understanding on nano- fluid can be insightinmostrecentarticles[7–9]. The boundary layer flow caused by stretching a sheet * Correspondence: [email protected] Department of Mathematical Sciences, Faculty of Science, Universiti linearly or nonlinearly is an important engineering prob- Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia lem and has several industrial applications, including Full list of author information is available at the end of the article © The Author(s). 2016 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 2 of 15 extrusion of polymer sheets, melting spinning, the hot stretching sheet under the influence of first-order rolling, wire drawing, production of glass fiber, plastic chemical reaction was theoretically studied by Raptis and rubber sheets manufacturing, enhanced recovery of and Perdikis [22]. On the other hand, the numerical petroleum resources, and cooling of large plate in bath. and analytical solutions of steady-state boundary layer The heat transfer phenomenon in stretching sheet prob- flow of micropolar fluid induced due to nonlinearly lem is very important as cooling and heating are neces- stretching sheet were found by Damseh et al. [23] and sary factors for the quality of end product. The seminal Magyari and Chamkha [24], respectively. The combined work of Crane [10] was extended by Cortell [11, 12] and effects of slip and chemical reaction on electrically con- found numerical solutions for heat transfer flow of vis- ducting fluid over a nonlinearly porous stretching sheet cous fluid due to nonlinearly stretching sheet with and were analyzed by Yazdi et al. [25]. In the same year, without the effects of thermal radiation, respectively. In Bhattacharyya and Layek [26] investigated the velocity the same year, Abbas and Hayat [13] also explored the slip effects on boundary layer flow of viscous fluid past influence of thermal radiation on two-dimensional flow a permeable stretching sheet in the presence of chem- of viscous fluid towards nonlinearly stretching sheet ical reaction. The steady two-dimensional boundary saturated in a porous medium. The steady electrically layer flow of Newtonian fluid due to stretching sheet conducting flow of micropolar fluid caused by nonli- saturated in nanofluid in the presence of chemical reac- nearly stretching sheet was reported by Hayat et al. tion is explored by Kameswaran at al. [27]. Motivated [14]. Motivated by this, Anwar et al. [15] utilized the by this, Aurangzaib et al. [28] studied theoretically the Boungiorno model and investigated natural convection influence of thermal radiation on unsteady natural con- flow of viscous fluid induced by nonlinearly stretching vection flow caused by stretching surface in the pres- sheet saturated in a nanofluid. Mukhopadhyay [16] ence of chemical reaction and magnetic field. Shehzad studied two-dimensional boundary layer flow of Casson et al. [29] reported the effects of magnetic field on mass fluid past a nonlinearly stretching sheet and concluded transfer flow of Casson fluid past a permeable stretch- that fluid velocity is suppressed whereas temperature ing sheet in the presence of chemical reaction. Pal and enhanced by Casson parameter. The two-dimensional Mandal [30] explored the characteristics of mixed con- incompressible flow of viscous fluid caused by nonli- vection flow of nanofluid towards a stretching sheet nearly stretching sheet in a nanofluid is reported by under the influence of chemical reaction and thermal Zaimi et al. [17]. Motivate by this, Raju and Sandeep radiation. Similarity solutions for unsteady boundary [18] and Raju et al. [19] analyzed three-dimensional flow of Casson fluid induced due to stretching sheet electrically conducting flow of Casson-Carreau fluids embedded in a porous medium in the presence of first and nanofluids due to unsteady and steady stretching order chemical reaction were obtained by Makanda and sheet, respectively. Very recently, Pal et al. [20] investi- Shaw [31]. gated the influence of thermal radiation on mixed con- On the other hand, convective boundary condition vection flow of nanofluid caused by nonlinearly plays a vital role in many engineering processes and in- stretching/shrinking sheet. dustries such as gas turbines, material drying, textile However, the combined effects of heat and mass trans- drying, laser pulse heating, nuclear plants, transpiration fer and chemical reaction play a vital role in chemical cooling, and food process. It is because that the convect- and hydro-metallurgical industries. The chemical reac- ive boundary condition applied at the surface is more tion can be of any order, but the most simple of which is practical and realistic. The two-dimensional laminar the chemical reaction of first order where the reaction boundary layer flow of Newtonian fluid caused by por- rate and species concentration are directly proportional ous stretching surface in the presence of convective to each other. The formation of Smog is an example of boundary condition is investigated numerically by Ishak first order chemical reaction. In several chemical engin- [32]. Makinde and Aziz [33] analyzed steady incom- eering processes, chemical reaction between foreign pressible flow of nanofluids towards stretching sheet mass and working fluid often occurs because of stretch- with convective boundary condition using Boungiorno ing a sheet. The diffusive species can be absorbed or model. Moreover, RamReddy et al. [34] included the ef- generated due to different types of chemical reaction fects of Soret and investigated mixed convection flow with the ambient fluid which is greatly influenced by the due to vertical plate in a nanofluid under convective properties and quality of end product. Kandasamy and boundary condition. Das et al. [35] discussed the heat Periasamy [21] investigated heat and mass transfer free and mass transfer flow of hydromagnetic nanofluid past convection flow of Newtonian fluid past nonlinearly a stretching sheet place in a porous medium with con- stretching sheet in the presence of chemical reaction vective boundary condition. The three-dimensional and magnetic field. The laminar boundary layer flow laminar flow of Casson nanofluid due to stretching of electrically conducted fluid towards nonlinearly sheet in the presence of convective boundary condition Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 3 of 15 is developed by Nadeem and Haq [36]. Motivated by here that the results are perceived in an excellent this, Malik et al. [37] investigated the influence of con- agreement. vective boundary condition past a stretching sheet in the presence of magnetic field. The two-dimensional Methods electrically conducting flow of Casson nanofluid caused Mathematical Formulation by stretching sheet with convective boundary condition is The steady incompressible natural convection flow of performed by Hussain et al. [38]. Very recently, Sulochana Casson fluid caused by nonlinearly stretching sheet et al. [39] established numerical solutions of three- through porous medium in the presence of chemical re- dimensional Casson nanofluid induced due to permeable action thermal radiation is considered. The x − axis is stretching sheet in the presence of convective boundary taken along the direction of stretching sheet and y − axis condition. is perpendicular to the surface (see Fig. 1). The sheet is Motivated by the above-cited literature survey and the stretched with the nonlinear velocity of the form u (x)= widespread engineering and industrial applications, it is cx ,where c is constant and n (> 0) represents the nonli- of prime importance to explore the effects of chemical nearly stretching sheet parameter (n = 1 corresponds to lin- reaction and thermal radiation on electrically conducting ear stretching sheet and n ≠ 1 represent nonlinear natural convection flow of Casson nanofluid caused by stretching sheet). Moreover, a variable magnetic field B(x)= (n − 1)/2 nonlinearly stretching sheet through porous medium in B x [22] is applied normally to the stretching sheet the presence of slip and convective boundary conditions. with constant B . Furthermore, it is also assumed that sheet The presence of momentum slip and convective bound- wall is heated by temperature T (x)= T + Ax (A being the f ∞ ary condition makes the present mathematical model of reference temperature and λ =2n − 1) and C (x)= C + Ex s ∞ a physical system to some extent difficult while inter- (E being the reference concentration). action of nanofluid with Casson fluid as base fluid. The The governing equations for Casson nanofluid along governing equations are converted to ordinary differen- with continuity equation are given as tial equation using similarity transformations and nu- merical solutions are obtained through the Keller box ∂u ∂υ þ ¼ 0; ð1Þ method [40]. To validate and examine the numerical al- ∂x ∂x gorithm developed in MATLAB software for the present problem, the results are compared with the ∂u ∂u μ 1 ∂ u σB ðÞ x μ ϕ B B existing literature results for pure Newtonian and u þ υ ¼ 1 þ − þ u ∂x ∂y ρ β ∂y ρ ρ k f f f Casson fluids as a limiting case. It is worth mentioning Fig. 1 Physical sketch and coordinate system Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 4 of 15 2 3 Table 1 Comparison of skin friction coefficient for different ρ −ρ p f f ∞ values of β and M when n =1, β→ ∞, Bi → ∞, Bi → ∞, and ∞ 1 2 4 5 þðÞ 1−C βðÞ T−T þ gβðÞ C−C g; ∞ ∞ ∞ T C ρ ρ M = K = Gr = Gm = δ = R =Le = N = N = R =0 d t b f f − 1 þ f ðÞ 0 ð2Þ β M Nadeem et al. [36] Ahmad and Nazar [42] Present results ∞ 0 1.0042 1.0042 1 ∂T ∂T ∂ T u þ υ ¼ α 5 1.0954 – 1.0955 ∂x ∂y ∂y "# 1 1.4142 – 1.4144 ∂C ∂T D ∂T 1 ∂q ∞ 10 3.3165 3.3165 3.3166 þτ D þ − ; ∂y ∂y T ∂y ðÞ ρc ∂y 5 3.6331 – 3.6332 ð3Þ 1 4.6904 – 4.6904 ∞ 100 10.049 10.0498 10.0499 2 2 ∂C ∂C ∂ C D ∂ T u þ υ ¼ D þ −kðÞ C−C ð4Þ B c ∞ 5 11.0091 – 11.0091 2 2 ∂x ∂y ∂y T ∂y 1 14.2127 – 14.2127 In theaboveexpressions, u and υ denote the velocity components in x − and y − directions, respectively, μ is the plastic dynamic viscosity, ρ is the fluid density, σ The radiative heat flux q described according to Ros- is the electrically conductivity, β is the Casson fluid seland approximation is given as parameter which has inverse relation with yield stress, pffiffiffiffi μ 2π 1 − n B  4 i.e., p ¼ [41], φ is the porosity, k (x)= k x is −4σ ∂T 1 0 q ¼ ð7Þ the variable permeability of porous medium, g is the 3k ∂y gravitational force due to acceleration, ρ is the density where σ is the Stefan-Boltzmann constant and k * is the of nanoparticle, β is the volumetric coefficient of ther- mean absorption coefficient. T can be expressed as lin- mal expansion, β is the coefficient of concentration ear function of temperature. By expanding T in a Taylor expansion, T is the fluid temperature, C is the nanopar- series about T and neglecting higher terms, we can ticle concentration, α ¼ is the thermal diffusivity ðÞ ρc write of the Casson fluid, k is the thermal conductivity of the 4 3 4 T ≅ 4T T−3T ð8Þ fluid, D is the Brownian diffusion coefficient, D is the B T ∞ ∞ ðÞ ρc thermophoretic diffusion coefficient, τ ¼ is the ra- ðÞ ρc Now putting Eqs. (7) and (8) in Eq. (3), we obtain tio of heat capacities in which (ρc) is the heat capacity 2  3 2 ∂T ∂T ∂ T 16σ T ∂ T of the fluid and (ρc) is the effective heat capacity of p u þ υ ¼ α þ ð9Þ 2 2 ∂x ∂y ∂y 3ðÞ ρc k ∂y nanoparticle material, c is the specific heat at constant f "# ak x pressure, q is the radiative heat flux and k ðÞ x ¼ is ∂C ∂T D ∂T r c þτ D þ the variable rate of chemical reaction, k is a constant ∂y ∂y T ∂y 2 ∞ reaction rate, and a is the reference length along the flow. The corresponding boundary conditions are written as follows: Table 2 Comparison of skin friction coefficient when β→ ∞, Bi → ∞, Bi → ∞, Pr = 6.8, and M = K = Gr = Gm = δ = R =Le = 1 2 d 1 ∂u ∂T > N = N = R =0 u ¼ u ðÞ x þ N ν 1 þ ; k ¼ −h T −T t b w 1 f f β ∂y ∂y 1 ″ − 1 þ f ðÞ 0 ∂C D ¼ −hðÞ C −C at y ¼ 0 ; B s s n Cortell [11] Pal et al. [20] Present results ∂y 0.0 0.6276 0.6275 0.6276 ð5Þ 0.2 0.7668 0.7668 0.7668 u→0; T→T ; C→C as y→∞: ð6Þ 0.5 0.8895 0.8895 0.8896 ∞ ∞ 1 1.0000 1.0000 1.0000 n−1 Here, N ðÞ x ¼ N x is the velocity slip, N being 1 0 0 3 1.1486 1.1486 1.1486 n−1 n−1 2 2 constant, h ðÞ x ¼ h x and h ðÞ x ¼ h x are the con- f 0 s 1 10 1.2349 1.2348 1.2349 vective heat and mass transfer with h , and h being 0 1 20 1.2574 1.2574 1.2574 constants. Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 5 of 15 Table 3 Comparison of − θ′(0) for different Pr with n =1, β→ ∞, Bi → ∞, Bi → ∞, and M = K = Gr = Gm = δ = R =Le = N = N = 1 21 d t b R =0 − θ′(0) Pr Yih [43] Aurangzaib et al. [28] Pal et al. [20] Present results 0.72 0.8086 0.8086 0.8086 0.8088 1 1.0000 1.0000 1.0000 1.0000 3 1.9237 1.9237 1.9237 1.9237 10 3.7207 3.7207 3.7206 3.7208 100 12.2940 12.3004 12.2939 12.3004 Now, introduce the stream function ψ defined in its usual notation in terms of velocity, a similar variable η, and the following similarity transformations; Fig. 3 Effect of β on temperature for various n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 2νc nþ1 ðÞ n þ 1 c n−1 2 2 ψ ¼ x fðÞ η ; η ¼ x y; ð10Þ n þ 1 2ν 22ðÞ n−1 N 1 00 0 0 00 T−T C−C φ þ Lef φ − Lef φ þ θ − RLeφ ¼ 0 ∞ ∞ θ ¼ ; φ ¼ n þ 1 N n þ 1 T −T C −C f ∞ s ∞ ð13Þ rffiffiffiffiffiffiffiffiffiffiffi 9 Finally, Eqs. (3–6) and Eq. (9) take the following form pffiffiffiffiffiffiffiffiffiffiffi 1 1 0 00 0 > f ðÞ 0 ¼ 1 þ δ n þ 1 1 þ f ðÞ 0 ; θ ðÞ 0 ¼ − Bi½ 1−θðÞ 0 ; 1 = β n þ 1 rffiffiffiffiffiffiffiffiffiffiffi 1 > 1 2n 1 φðÞ 0 ¼ − Bi½ 1−φðÞ 0 00 00 0 0 n þ 1 1 þ f þ ff − f − ðÞ M þ K f ð11Þ β n þ 1 n þ 1 ð14Þ þ ðÞ Grθ þ Gmφ¼ 0 n þ 1 f ð∞Þ¼ 0; θð∞Þ¼ 0; ϕð∞Þ¼ 0 ð15Þ 4 22ðÞ n−1 In the above expressions, M, K, Gr, Gm, δ,Pr, R , N , 00 0 0 0 0 d b 1 þ R θ þ Prf θ − Prf θ þ PrN φ θ d b N , Bi , Bi , Le, and R are the magnetic parameter, por- 3 n þ 1 t 1 2 osity parameter, Grashof number, mass Grashof number, þ PrN θ ¼ 0 slip parameter, Prandtl number, radiation parameter, Brownian motion parameter, thermophoresis parameter, ð12Þ Fig. 4 Effect of β on nanoparticle concentration for two different Fig. 2 Effect of β on velocity for two different values of n values of n Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 6 of 15 Fig. 7 Effect of n on nanoparticle concentration in the presence/ Fig. 5 Effect of n on velocity in the presence/absence of K absence of K Biot numbers, Lewis number, and chemical reaction par- ameter and are defined as The dimensionless skin friction coefficient Cf ¼ , ! ! 2 ρu . . p w f p − p ∞ p f xq 2gβ AðÞ 1−C 2gβ E T ∞ C w pf p the local Nusselt number Nu ¼ , and local 2σB 2νϕ f x M ¼ ; K ¼ ; Gr ¼ ; Gm ¼ ; αðÞ T −T f f ∞ 2 2 ρc k c c c rffiffiffiffiffi xq τD T −T c μcf 4σ T τDðÞ C −C T f ∞ ∞ B s ∞ Sherwood number Sh ¼ on the surface along δ ¼ N ; Pr ¼ ; R ¼ ; N ¼ ; N ¼ ; DðÞ C −C 0 d  b t B w ∞ 2ν k kk ν νT 1 ∞ x − direction, local Nusselt number Nu , and Sherwood 1 1 x h 2ν h 2ν ν 2νk 0 2 1 2 2 number Sh are given by Bi ¼ ; Bi ¼ ; Le ¼ ; R ¼ 1 2 x k c D c D c B B The wall skin friction, wall heat flux, and wall mass rffiffiffiffiffiffiffiffiffiffiffi n þ 1 1 flux, respectively, are defined by −1=2 1=2 Cf ¼ 1 þ f ðÞ 0 ; ðÞ Re Nu ðÞ Re x x x 2 β rffiffiffiffiffiffiffiffiffiffiffi 1 ∂u n þ 1 4 0 −1=2 τ ¼ μ 1 þ ; q B w ¼ − 1 þ R θ ðÞ 0 ;ðÞ Re Sh d x β ∂y y¼0 2 3 ! ! rffiffiffiffiffiffiffiffiffiffiffi 16σ T ∂T n þ 1 ¼ − α þ and q ¼ − φðÞ 0 3ðρcÞ k ∂y y¼0 ∂C ¼ −D nþ1 Re ¼cx ∂y where is the local Reynold number. y¼0 Fig. 6 Effect of n on temperature in the presence/absence of K Fig. 8 Effect of M on velocity for various values of n Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 7 of 15 Fig. 9 Effect of M on temperature for two values of n Fig. 11 Effect of K on velocity for various values of β Numerical Scheme Here, the step size η = 0.01 and boundary layer thick- The governing Eqs. (11)–(13) with associated boundary ness η = 10 is used. Further, the convergent criteria − 5 conditions (14) and (15) are solved numerically via the 10 is considered for all the cases. The numerical and Keller box method. The detail of this method is given in graphical results are generated through MATLAB soft- the book of Cebeci and Bradshaw [40]. This method is ware. In order to assess the accuracy and validate our unconditionally stable and has second-order accuracy. code, the comparison is made with previous results of The following four steps are involved in finding the nu- literature as a limiting case. merical solutions of the problem Results and Discussion (i) Initially, the transformed governing equations In the present study, the effects of slip and convective are converted to first-order system. boundary conditions on heat and mass transfer flow of (ii) Now, approximate the first-order system using nanofluid due to nonlinearly stretching surface saturated central difference formula about the mid-point. in a porous medium in the presence of chemical reac- (iii) The algebraic equations are then linearized via tion and thermal radiation were analyzed. Moreover, Newton’s method and write them in matrix-vector Casson fluid is used as base fluid. In order to analyze the notation. results, numerical calculations are carried out for various (iv) Finally, block tri-diagonal elimination technique is values of Casson fluid parameter β, nonlinear stretching used to solve the linear system. sheet parameter n, magnetic parameter M, porosity Fig. 10 Effect of M on nanoparticle concentration for two values of n Fig. 12 Effect of K on temperature for two values of β Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 8 of 15 Fig. 13 Effect of K on nanoparticle concentration for various β Fig. 15 Effect of Gr on temperature profile for two different values of n parameter K, Grashof number Gr, mass Grashof number M, K, Gr, Gm, δ,Pr, R , N , N , Bi , Bi , Le, and R on vel- d b t 1 2 Gm, Prandtl number Pr, radiation parameter R , Brown- ocity (f′(η)), temperature (θ(η)), and nanoparticle con- ian motion parameter N , thermophoresis parameter N , centration (φ(η)) profiles, respectively. Figures 2, 3, and b t Lewis number Le, slip parameter δ, and Biot numbers 4 exhibit the variation of β on velocity, temperature, and Bi , Bi . For validation of the present method, the results nanoparticle concentration, respectively, for the case of 1 2 are compared with previously reported results and dis- n = 1 and n ≠ 1. It is noteworthy here that the present played in Tables 1, 2, and 3. problem reduces to pure Newtonian nanofluid case Tables 1 and 2 present the comparison of skin friction when β→ ∞. Clearly, fluid velocity reduces as β in- coefficient for different values of β, M,and n, respect- creases. The reason behind this behavior is that increas- ively, with the results of Nadeem et al. [36], Ahmad and ing values of β implies rise in fluid viscosity, i.e., Nazar [42], Cortell [11], and Pal et al. [20]. The results reducing the yield stress. Consequently, the momentum showed an excellent agreement. Table 3 describes the boundary layer thickness reduces. It is also observed comparison of heat transfer rate for various values of Pr from this figure that fluid velocity decreases faster in the with the results of Yih [43], Aurangzaib et al. [28], and case of stretching a sheet in a nonlinear way. A similar Pal et al. [20] and revealed in a good agreement. trend of velocity profile was reported by Nadeem and Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, Haq [36] and Mukhopadhyay [16]. Conversely, both θ(η) 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, and and φ(η) enhance with increase in β (see Figs. 3 and 4). 31 are displayed to insight the physical behavior of β, n, Figures 5, 6, and 7 demonstrate the effect of n on fluid Fig. 14 Effect of Gr on velocity for two different values of n Fig. 16 Effect of Gr on nanoparticle concentration for various n Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 9 of 15 Fig. 17 Effect of Gm on velocity for two different values of n Fig. 19 Effect of Gm on nanoparticle concentration for different values of n velocity, temperature, and nanoparticle concentration, respectively, in non-porous (K = 0) and porous (K ≠ 0) A similar behavior for electrically conducting flow of medium. Interestingly, f′(η), θ(η), and φ(η) are all found Casson nanofluid due to stretching sheet was observed as decreasing functions of n. It is also noticed from Fig. 5 by Hussain et al. [38]. that momentum boundary layer thickness reduces rap- The effect of K on velocity, temperature, and nano- idly when K ≠ 0, whereas the case of thermal and con- particle concentration profiles for Newtonian and non- centration boundary layer thicknesses are quite opposite Newtonian fluids is portrayed in Figs. 11, 12, and 13, to this, i.e., both thicknesses decrease when K =0. respectively. It is interesting to note that the response Figures 8, 9, and 10 reveal the influence of M on fluid of K in these figures is completely the same as observed velocity, temperature, and nanoparticle concentration, for M. It is also found from these figures that momen- respectively, for both cases of n =1 and n ≠ 1. It is no- tum boundary layer become thinner in case of β→ ∞ ticeable that increasing values of M reduces the fluid and the opposite to this for thermal and concentration velocity whereas temperature and nanoparticle concen- boundary layer thicknesses. tration rises as M increases. As it is well-known fact Figures 14, 15, and 16 display the variation of Gr vel- that a resistive-type force produces as the current ocity, temperature, and nanoparticle concentration dis- passes through the moving fluid. This force is respon- tributions for n =1 and n ≠ 1. It is seen that fluid sible in slowing down fluid motion and increasing the velocity rises with increase in λ whereas temperature thermal and concentration boundary layer thicknesses. and nanoparticle concentration reduce as Gr increases. Fig. 18 Effect of Gm on temperature for two values of n Fig. 20 Effect of δ on velocity in the presence/absence of K Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 10 of 15 Fig. 21 Effect of δ on temperature in the presence/absence of K Fig. 23 Effect of Pr on temperature profile for various n Since the buoyancy force is dominant over viscous slip condition and δ ≠ 0 shows velocity slip at stretching force with increase in Gr. Consequently, Grashof num- sheet wall. It is interesting to see that increasing values of δ ber enhances the fluid flow which leads to increasing reduces the fluid velocity initially and then increases far velocity as well as thickness of momentum boundary from the sheet whereas dimensionless temperature and layer. In addition, since the buoyancy force tends to en- nanoparticle concentration increase with increase in δ. hance the temperature and concentration gradients, Physically, this shows that fluid velocity adjacent to the therefore, temperature and nanoparticle concentration sheet is less than the velocity of normal stretching sheet as fall. The same reason may be described for the behavior slip (δ ≠ 0) occurs. Increasing δ allowed more fluid slipping of Gm on velocity, temperature, and nanoparticle con- over the sheet and the flow decelerates near the sheet. centration distributions, as elucidated in Figs. 17, 18, Figure 23 reveals the influence of Pr on dimensionless and 19. It is also observed from these figures that influ- temperature profile when n = 1 and n ≠ 1. As expected, ence of Gr and Gm is more pronounced in case of lin- increasing Pr leads to reduction in dimensionless ear stretching sheet for all the three profiles. temperature. Based on the definition of Pr (the ratio of The variations of δ on dimensionless velocity, tem- momentum diffusivity to thermal diffusivity), therefore, perature, and nanoparticle concentration profiles for K =0 for large Pr, heat will diffuse more rapidly than the mo- and K ≠ 0 are depicted in Figs. 20, 21, and 22, respectively. mentum. Consequently, thickness of thermal boundary It is worth mentioning here that δ = 0 corresponds to no layer reduces as Pr increases. It is also noticed that Fig. 22 Effect of δ on nanoparticle concentration in the presence/ Fig. 24 Effect of R on temperature profile in the presence/absence absence of K of K Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 11 of 15 Fig. 27 Effect of N on temperature profile in the presence/absence Fig. 25 Effect of N on temperature profile for different n of R higher values of Pr reduces the temperature more dras- Figures 25 and 26 illustrate the variation of N on di- tically. It is because of the fact that low thermal conduct- mensionless temperature and nanoparticle concentration ivity of fluid associated with larger Pr, which decreases distributions. It is noted that dimensionless temperature conduction and results a temperature fall. enhances with increase in N while nanoparticle concen- The effect of R on dimensionless temperature for K tration is found decreasing as N increases. It is well d b =0 and K ≠ 0 is exhibited in Fig. 24. It is noteworthy known that Brownian motion is a diffusive process. The here that R = 0 denotes no radiation and R ≠ 0shows higher diffusivity implies higher temperature, and as d d the presence of radiation. Clearly, dimensionless consequences, the thermal conductivity becomes higher. temperature is higher as R increases. The reason be- Also, Brownian motion in nanofluid occurs only due to hind this fact is that heat energy released to the fluid nanometer size of nanoparticles. In addition to this, the as R increases and this results rise in temperature. It kinetic energy of nanoparticles enhance mainly due to is also evident from this figure that thermal boundary the increase in N and resulting higher temperature of layer thickness increases faster with increase in R nanofluids. The influence of N on dimensionless d t when K ≠ 0. This shows that influence of radiation in a temperature and nanoparticle concentration distribu- porous medium is more effective when high tions for R = 0 and R ≠ 0 are shown in Figs. 27 and 28, d d temperature is required for the desired thickness of respectively. It is evident from these figures that both end product. θ(η) and φ(η) are increasing functions of N . According Fig. 26 Effect of N on nanoparticle concentration for two different Fig. 28 Effect of N on nanoparticle concentration in the presence/ b t values of M absence of R d Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 12 of 15 Fig. 29 Effect of Bi on temperature profile for various n Fig. 31 Effect of Le on nanoparticle concentration for various values of n to the definition of N , i.e., higher values of N implies wall temperature and constant wall concentration case t t higher temperature differences and shear gradient. when Bi → ∞ and Bi → ∞. Apparently, dimensionless 1 2 Therefore, increasing values of N tends to higher temperature increases with increase in Bi . It is also evi- t 1 temperature difference across the boundary layer. On dent from this figure that increasing values of Bi lead to the other hand, nanoparticle concentration is a strong rise in sheet convective heating. Furthermore, smaller function of N ; for this reason, it is significantly influ- values of Bi (< < 1) rapidly enhance the temperature as t 1 enced by increasing values of N . These conclusions are well as the corresponding boundary layer thickness in agreement with Hussain et al. [38] and Nadeem et al. across the boundary region. Since Bi is inversely pro- [36]. It is also interesting to note from Fig. 28 that con- portional to Brownian diffusivity coefficient. Therefore, centration peak values reveal that stronger N intensifies thermal diffusivity reduces whereas momentum diffusiv- the thermal conductivity of the nanofluids near the wall. ity rises, and as a consequence, nanoparticle concentra- Further, it is also observed from Fig. 28 that nanoparticle tion enhances and related boundary layer becomes concentration falls with increase in R . thinner. Figures 29 and 30 demonstrate the effect of Bi and Figure 31 exhibits the variation of Le on nanoparticle Bi on dimensionless temperature and nanoparticle con- concentration profile for linear and nonlinear stretching centration profiles for n = 1 and n ≠ 1. It is worth men- sheet. It is perceived that in both cases, nanoparticle tioning that the present study will reduce to constant concentration is lower for higher Le. It is a fact that Fig. 32 Effect of R on nanoparticle concentration for two different Fig. 30 Effect of Bi on nanoparticle concentration for various n values of n 2 Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 13 of 15 Fig. 33 Variation of skin friction coefficient for various values of β, Fig. 35 variation of Nusselt number for various values of R , β, and δ Gr, and K lower D corresponds to larger Le and the fluids having B shear stress for increasing values of β, Gr,and K. qffiffiffiffiffiffiffi smaller Le have higher D . In other words, mass transfer B ″ nþ1 1 Clearly, 1 þ f ðÞ 0 decreases with increase rate higher for large Le. The influence of R on nanoparti- in β and Gr, whereas it increases with increase in K.It cle concentration distribution for n = 1 and n ≠1is is also evident from this figure that the skin friction co- depicted in Fig. 32. It is noteworthy here that R = 0 de- efficient is negative for all values of β, Gr,and K which notes no chemical reaction and R ≠ 0 corresponds to the indicates that fluid experiences a resistive force at the presence of chemical reaction. It is evident that stronger boundary. This is also in agreement with the results of R leads to reduce nanoparticle concentration. The ex- Table 1. The variation of skin friction coefficient Cf for planation for this behavior is that destructive chemical different values of δ, n,and Gm is portrayed in Fig. 34. rate (R > 0) enhances the mass transfer rate and results a qffiffiffiffiffiffiffi nþ1 1 decrease in nanoparticle concentration. It is found that 1 þ f ðÞ 0 increases with n The effect of skin friction coefficient Cf , local Nus- while reduces as δ and Gm increases. The influence of selt number Nu , and Sherwood number Sh for some hqffiffiffiffiffiffiffi x x nþ1 4 heat transfer rate 1 þ = R Þθ ðÞ 0 for R , β,and physical parameters β, n, K, Gr, Gm, δ,Pr, R , N , N , 3 d d d b t Le,and R are displayed in Figs. 33, 34, 35, 36, 37, and δ is perceived in Fig. 35. It is observed that heat 38, respectively. Figure 33 reveals the variation of wall Fig. 34 Variation of skin friction coefficient for different values of δ, Fig. 36 Variation of Nusselt number for various values of N , N , t b n, and Gm and Pr Ullah et al. Nanoscale Research Letters (2016) 11:527 Page 14 of 15 Conclusions Two-dimensional electrically conducting natural convec- tion flow of Casson nanofluid towards nonlinearly stretching sheet in the presence of chemical reaction and thermal radiation was numerically discussed in this study. Moreover, the effect of slip and convective bound- ary conditions were also considered. Similarity transfor- mations are employed for the conversion of nonlinear partial differential equations to nonlinear ordinary differ- ential equations. Numerical solutions are found by the Keller box method, and graphical results are obtained through MATLAB software. Results are compared with previous work as a limiting case, and excellent accuracy is achieved with those results. It is found that β reduces the fluid velocity whereas dimensionless temperature and nanoparticle concentration increase with an increase Fig. 37 Variation of Sherwood number for various values of N , N , t b and R in β. Increasing values of n diminish the fluid velocity, temperature, and nanoparticle concentration. Velocity is observed to be enhanced as Gr and Gm increased. It is transfer rate higher for higher values of R , β whereas also noted that momentum boundary layer thickness de- increasing values of δ diminish the heat transfer rate. creases as δ increases. The dimensionless temperature Figure 36 elucidates the effect of heat transfer rate and nanoparticle concentration profiles increase with in- for various values of N , N ,and Pr.Itis noted that crease in Bi and Bi , respectively. Furthermore, it is also t b 1 2 mass transfer rate reduces as N and N increases noticed that dimensionless temperature and nanoparticle t b while increasing values of Pr enhance the mass trans- concentration distributions are increasing function of N . fer rate. It is also observed that higher value of Pr enhances the heat transfer rate significantly. Figure 37 Acknowledgements The authors would like to acknowledge Ministry of Higher Education (MOHE) illustrates the influence of Sherwood number for sev- and Research Management Centre Universiti Teknologi Malaysia (UTM) for eral values of N , N ,and R. It is noticed that stron- t b the financial support through vote numbers 4F713, 4F538 and 06H67 for this ger N promotes a progressive increase in mass research. transfer rate φ′(0) whereas increasing values of N and R decrease the mass transfer rate. Finally, Fig. 38 Authors’ Contributions IU and SS modeled the problem. IU solved the problem and generated the demonstrates the influence of mass transfer rate for graphs. IK contributed in drafting and revision of the manuscript. All authors increasing values of β, Le,and R . This figure clears read and approved the manuscript. that mass transfer rate higher for higher values of β, Le,and R . Competing Interests The authors declare that they have no competing interests. Author details Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia. Basic Sciences Department, College of Engineering, Majmaah University, Majmaah 11952, Saudi Arabia. Received: 3 October 2016 Accepted: 21 November 2016 References 1. Choi SUS, Eastman JA (1995) Enhancing thermal conductivity of fluids with nanoparticles. In: Proc 1995 ASME Int Mech Engineering Congr Expo. ASME, FED231/MD66, San Francisco, pp 99–105 2. Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ (2001) Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett 78:718–720. doi: 10.1063/1.1341218 3. Xuan Y, Li Q (2003) Investigation on convective heat transfer and flow features of nanofluids. J Heat Transfer 125:151. doi:10.1115/1.1532008 Fig. 38 variation of Sherwood number for various values of β, Le, 4. 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Published: Nov 28, 2016

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