Statistical Criteria of Nanofluid Flows Over a Stretching Sheet with The

Home , Biot number, Eckert number, Knudsen number, Lewis number, Sherwood number

S S symmetry

Article Statistical Criteria of Nanoﬂuid Flows over a Stretching Sheet with the Eﬀects of Magnetic Field and Viscous Dissipation

Alias Jedi 1,* , Noorhelyna Razali 1, Wan Mohd Faizal Wan Mahmood 1 and Nor Ashikin Abu Bakar 2 1 Centre for Integrated Design for Advanced Mechanical Systems (PRISMA), Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia; [email protected] (N.R.); [email protected] (W.M.F.W.M.) 2 Institut Matematik Kejuruteraan, Universiti Malaysia Perlis, Arau 02600, Malaysia; [email protected] * Correspondence: [email protected]

Received: 28 August 2019; Accepted: 24 October 2019; Published: 4 November 2019

Abstract: In this study, the heat and mass transfer characteristics of nanofluid flow over a nonlinearly stretching sheet are investigated. The important effects of axisymmetric of thermal conductivity and viscous dissipation have been included in the model of nanofluids. The Buongiorno model is considered to solve the nanofluid boundary layer problem. The governing nonlinear partial differential equations have been transformed into a system of ordinary differential equations and are solved numerically via the shooting technique. The validity of this method was verified by comparison with previous work performed for nanofluids without the effects of the magnetic field and viscous dissipation. The analytical investigation is carried out for different governing parameters, namely, the Brownian motion parameter, thermophoresis parameter, magnetic parameter, Biot number, and Eckert number. The results indicate that the skin friction coefficient has a direct relationship with the Brownian motion number and thermophoresis number. Moreover, it can be seen that the Nusselt number decreases with the increase of the magnetic parameter and Eckert number.

Keywords: boundary layer flow; nanofluids; magnetic field; viscous dissipation; dual solutions

1. Introduction In recent decades, nanofluid technology has contributed to energy costs in terms of heat transfer control in energy systems. Nanoparticles have been used for diverse purposes; for instance, manufacturing and the automotive industry. The external magnetic field effect on nanofluids and fluid flow properties has attracted significant attention due to its applications [1]. Sheikhholeslami et al. [1] studied the radiation of the nanofluid effect in the presence of a magnetic field, and they discovered that the heat transfer rate was reduced with augmentation of Lorentz forces. The magnetic field effect on force convection heat transfer was investigated by Sheikhholeslami et al. [2], who found that the Kelvin force effects are more pronounced at high Reynolds numbers. Turkyilmazoglu [3] investigated the magnetohydrodynamic, electrically conductive slip flow of a non-Newtonian fluid past a shrinking sheet. Their results indicated that the magnetic field presence has considerable effects on temperature and velocity fields. Ishak [4] considered the effects of radiation on the steady two-dimensional magnetohydrodynamic (MHD) flow past a permeable stretching/shrinking sheet. They discovered that the surface rate of heat transfer decreases in the presence of radiation. Sheikholeslami and Kandelousi [5] investigated the spatially variable magnetic field effect on heat transfer and ferrofluid flow by examining the condition of a constant heat flux boundary. They discovered that, as heat

Symmetry 2019, 11, 1367; doi:10.3390/sym11111367 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 1367 2 of 17 transfer increases, the Hartmann number increases and decreases with increases in the Rayleigh and magnetic numbers. The studies of ferrohydrodynamic and MHD effects on convective heat transfer and ferrofluid flow by Sheikholeslami and Ganji [6] demonstrated that the magnetic number results in a dissimilar effect to the Nusselt number. A magnetic field effect on heat transfer and unsteady nanofluid flow has been presented by Sheikholeslami et al. [7] using the Buongiorno model. They showed that there is direct relationship between the coefficient of skin friction with squeeze and the Hartmann number. The magnetic field plays an important role by controlling the cooling rate, which is desired in industrial production. Hence, researchers have studied the flow characteristics over stretching sheets in depth. Considering this situation, Pavlov [8] found that the MHD boundary layer equation gave a similarity solution in exact analytical form. Where in [8] the presence of a uniform transverse magnetic field is considered. The effect of a magnetic field on the viscous flow of an electrically conducting fluid was studied by Hayat et al. [9]. The increasing frictional drag from M = 0 to M = 3 is due to the Lorentz force. Thermal boundary layer becomes thick due to the effect of Lorentz force. Elbashbeshy et al. [10] on their study found that if the magnetic parameter M is increased to M = 2, temperature boundary layer is slightly increased. Results from Noghrehabadi et al. [11] show that the presence of a magnetic field at value M = 3 would significantly affect the boundary layer profiles. An increase in magnetic parameter from M = 0.1 to M = 3 would decrease the reduced Nusselt and Sherwood numbers. The magnetic field at M = 5 strongly affects the velocity and consequently temperature profiles; the effect of magnetic field is negligible on the thickness of the boundary layer. Elbashbeshy and Aldawody [12] also found that the local Nusselt number decreases when the magnetic parameter M = 0.4. Yasin et al. [13] considered the effects of radiation on the steady two-dimensional magnetohydrodynamic (MHD) flow past a permeable stretching/shrinking sheet. They discovered that the surface rate of heat transfer decreases in the presence of radiation. Pal et al. [14] analyzed the thermal radiation effect on hydromagnetic mixed convection flow over nonlinear shrinking/stretching sheets in nanofluids. They discovered that copper–water nanofluids have a higher rate of mass transfer in contrast with titanium dioxide—water and alumina—water nanofluids. Many researchers have concentrated on the improvement of heat transfer by altering nanoparticles in base fluids. This has been used in photonics, transportation, and electronics [15–19]. Nanoparticles in a base fluid are found to have better convective heat transfer coefficients and thermal conductivity than with the base fluid [20–28]. Based on these observations, the objective of this study is to analyze the effect of the thermal radiation, viscous dissipation, and thermal convective boundary condition on the flow of nanofluids over a nonlinear stretching sheet. From the examined literature, the problem of the effect of the magnetic field and viscous dissipation with a thermal convective boundary condition has not been discussed. We aim, in the present paper, to analyze the effect of the magnetic field and viscous dissipation over a stretching sheet using the Runge–Kutta–Fehlberg method with the shooting technique. The effects of governing parameters are analyzed based on fluid velocity, temperature, and particle concentration. The results are then shown graphically and also in tables. The available results in the literature are then compared with the results found in this study. Excellent agreement is found for both comparisons.

2. Problem Formulation Consider a steady axisymmetric thermal conductivity, laminar, incompressible, and one-dimensional flow in a nanofluid towards a permeable stretching sheet is placed on the plane y = 0, where y and x are Cartesian coordinates estimated along the stretching surface and typical of it, respectively. The flow is limited at y > 0. The sheet is extended with a speed Uw = ax along the x-direction while assuming the pressure gradient and external force to be zero, where a is a positive constant value. Further, we assume that the flat plate is heated from below by a hot fluid whose temperature is Tf, with a heat transfer coefficient hf. The scheme of the physical configuration is depicted in Figure1. Symmetry 20192018, 1110, 1367x FOR PEER REVIEW 33 of of 17

Figure 1. Schematic diagram of the problem. Figure 1. Schematic diagram of the problem. Using boundary layer approximations, the final steady momentum, energy, continuity, and nanoparticleUsing boundary concentration layer equations approximations, are defined the asfinal below steady (see momentum, Zaimi et al. [ 23energy,]): continuity, and nanoparticle concentration equations are defined as below (see Zaimi et al. [23]): ∂u ∂v ∂∂uv+ = 0, (1) ∂x +=∂y 0, (1) ∂∂xy 2 2 ∂u ∂u ∂ u γB (2x) u∂∂uu+ v = ν ∂2 uγ Bx()u, (2) uv∂x+=∂y ν ∂y2 − −ρ · ⋅ u, ∂∂ ∂2 ρf (2) xy y f ∂v ∂v 1 ∂ρ ∂2v γB2(x) u + v = + ν 2 u, (3) ∂ ∂ ∂ρ ∂22 γ B ()x v∂x+ ∂vy=−−1ρ ∂y +ν∂y v−− ρ f · ⋅ u v 2 u, (3) ∂x2 ∂y ρ ∂y ∂y !2 ρ 2 !2 ∂T ∂T ∂ T  ∂C ∂T D ∂T  Df K ∂ C V ∂u u + v = α + τD + T  + m T + , (4) 2  B  2 ∂x ∂y ∂y ∂y ∂y T ∂y 22cscp ∂y cp ∂y ∂∂TT ∂22 T ∂∂ CTTD∞  ∂DK ∂ CVu  ∂ +=ατ +2 +T 2 +mT2 + uv∂C22∂C DB ∂ C DT∂ T DmKT ∂ T , (4) ∂∂xyu ∂ y+ v = ∂∂D yyTyB + ∞ ∂+ ccycy ∂, ∂ (5) ∂x ∂y ∂y2 T ∂y2 Tm sp∂y2 p  ∞ referring to the initial boundary∂∂CC conditions ∂22 CD ∂ TDK ∂ 2 T uv+= D +T +mT , (5) ∂∂ B ∂22 ∂ ∂ 2 xy∂u yTyTy∂T ∞ m u = U + Nρν , v = vw, k = h T T , C = Cw at y = 0, ∂y − ∂y f f − (6) referring to the initial boundary conditionsu 0, T T , C C as y → → ∞ → ∞ → ∞ ∂ ∂ where v and u are= the+ correspondingρν  u  = velocity−  componentsT  = () in− the y and= x directions,= respectively. u U N   ,v vw , k   hf T f T ,C Cw at y 0, In Equation (3) by using the scaling ∂y analysis, the ∂ pressurey  gradient ∂p/∂y can be ignored because(6) the pressure change across the thickness of boundary layer is negligible. Therefore, the Equation (3) is u → 0,T → T∞ ,C → C∞ as y →∞ neglected so that the momentum Equation (2) is considered as one-dimensional. For the next part, wewhere identify v and the u are temperature the corresponding in the boundary velocity layercomponents (T) and in ambient the y and temperature x directions,T ,respectively. where C is the In ∞ Equationnanoparticle (3) concentrationby using the scaling in the boundaryanalysis, the layer, pressureCw is the gradient nanoparticle ∂p/∂y volumecan be ignored fraction because at the plate, the Tpressurew is the surfacechange temperature,across the thicknessC is the of nanoparticleboundary layer volume is negligible. fraction farTherefore, from the the plate, Equationν represents (3) is ∞ neglectedthe coefficient so that of kinematicthe momentum viscosity, EquationDB is the (2) coeis coffinsideredcient of Brownian as one-dimensional. diffusion, D TForis thethe coenextffi cientpart, weof thermophoresis identify the temperature diffusion, inα =thek /boundary(ρc) f is the layer fluid (T) thermal and ambient diffusivity, temperatureγ is electrical T∞, where conductivity, C is the nanoparticleτ = (ρc)p/(ρ concentrationc) f is the ratio in between the boundary the eff ectivelayer, heatCw is capacitythe nanoparticle of the nanoparticle volume fraction material at the and plate, the Theatw is capacity the surface of the temperature, fluid,ρp is the C particle∞ is the density, nanoparticleρ f is the fluidvolume density, fractionDB isfar the from Brownian the plate, diffusion ν representscoefficient, thec is coefficient the volumetric of kinematic volume expansion viscosity, coeDB ffiis cient,the coefficientKT is the thermalof Brownian diffusion diffusion, ratio, DT isis the thermophoresis diffusion coefficient, DB is theα = Brownian()ρ diffusion coefficient, Dm is the coefficientγ of coefficient of thermophoresis diffusion, k c f is the fluid thermal diffusivity, is mass diffusivity, Kn is the Knudsen number, T is the mean fluid temperature, c is the concentration τ = ()ρ ()ρ m s electricalsusceptibility, conductivity,cp is the specific heatc p at constantc f is pressure, the ratioN betweenρν is the velocitythe effective slip factor, heat vcapacityw is the suctionof the or injection velocity with vw > 0 for suction and vw < 0 forρ injection, and ε is the stretchingρ /shrinking nanoparticle material and the heat capacity of the fluid, p is the particle density, f is the fluid parameter with ε > 0 for a stretching case and ε < 0 for a shrinking case. The constant n is the density, DB is the Brownian diffusion coefficient, c is the volumetric volume expansion coefficient, nonlinearity parameter with n = 1 for the linear case and n , 1 for the nonlinear case. We assume that KT is the thermal diffusion ratio, DT is the thermophoresis diffusion coefficient, DB is the Brownian

Symmetry 2019, 11, 1367 4 of 17

(n 1)/2 the variable magnetic field B (x) = B0 x − . A flow with a coefficient of convective heat transfer hf as well as temperature of Tf moves through the stretching surface. The plate and the fluid have a slippery surface, upon which the length of slip will be created; this length is important for enhancing the heat transfer rate and widening the range of possible dual solutions. In a recent study, we investigated the relations between the effects of the partial slip and the thermal convective boundary condition towards the performance of the heat and mass transfer rate. Moreover, in finding dual solutions, this study needs to consider a permeable surface where the suction is presented to allow the movement of nanoparticles and fluid. By considering the nanofluid model of Buongiorno [25], we assume that the nanofluid’s relative motion is caused by Brownian motion and thermophoresis but only in the case of general nanoparticles and fluid base; i.e., size, nature, concentration. We now introduce the following dimensionless variables:

1/2 n av(n+1) h n 1 i u = ax f (η), v = f (η) + − η f (η) , 0 − 2 0 n+1 0 1/2 (7) av(n+1) (n 1)/2 T T C C η = y x , θ(η) = − ∞ , φ(η) = − ∞ , 2v − T f T Cw C − ∞ − ∞ where η is the similarity variable. Moreover, φ(η), f (η), and θ(η) are the dimensionless nanoﬂuid concentration, velocity, and temperature in the boundary layer region, respectively. A nonlinear equation is written as axn, where n is the number of degrees, and a is constant. By introducing the relation (7) into the Equations (2) to (5), we obtain the following nonlinear ordinary diﬀerential equations.

2n 2 f 000 + f f 00 f 0 M f 0 = 0, (8) − n + 1 − ·

1 2 2 θ00 + f θ0 + Nbθ0φ0 + Ntθ0 + Duφ00 + Ec f 00 = 0, (9) Pr Nt φ00 + Le f φ + θ00 + Sr Le θ00 = 0, (10) 0 Nb where prime denotes the diﬀerentiation with respect to η. Further, Nb is the Brownian motion parameter, M is the magnetic parameter, Pr is the Prandtl number, Nt is the thermophoresis parameter, Ec is the Eckert number, Du is the Dufour number, Sr is the Soret number, and Le is the Lewis number. The boundary conditions are now transformed to

f (0) = S, f (0) = 1 + σ f 00 (0), θ (0) = Bi(θ 1), φ(0) = 1, 0 0 − (11) f (η) 0, θ(η) 0, φ(0) 0as η , 0 → → → → ∞ With !1/2 av(n + 1) (n 1)/2 vw = x − S, (12) − 2v where S is known as the constant mass transfer parameter, with S < 0 corresponding to mass injection and S > 0 corresponding to mass suction. Next, σ is the velocity slip parameter, and Bi is the second-order velocity slip parameter or Biot number. The parameters are composed as follows:

!1/2 h 1/2 (2n 1)/2 a(n + 1) f u σ = Nρνax − , Bi = (13) 2v k vw Symmetry 2019, 11, 1367 5 of 17

The other parameters are written as

(ρc)pDB(Cw C ) (ρc)pDT(T f T ) Pr = ν , Le = ν , Nb = − ∞ , Nt = − ∞ α DB (ρc) f ν (ρc) f T ν ∞ DmKT(Cw C ) DmKT(T f T ) Du = − ∞ , Sr = − ∞ , (14) cscpν(T T ) νTm(Cw C ) f ∞ −2 ∞ 2 − Ec = u , M = 2σBi cp(Tw T ) αρ f (n+1) − ∞ where Le is the Lewis number, Pr is the Prandtl number, Nt is the thermophoresis parameter, Nb is the Brownian motion parameter, Sr is the Soret number, Du is the Dufour number, M is the magnetic parameter, and Ec is the Eckert number. The physical quantities of interest are the local Nusselt number Nux, skin friction coeﬃcient C f , and local Sherwood number Shx, known as

τw xqw xqm C f = , Nux = , Shx = , (15) 2 D (C C ) ρU k T f T B w ∞ − ∞ − ∞ where the local heat flux qw, stress of wall shear τw, as well as the local mass flux qm are defined as follows: ! ! ! ∂u ∂T ∂T τw = µ , qw = k , qm = DB , (16) ∂y y=0 − ∂y y=0 − ∂y y=0 where µ is the dynamic viscosity of the nanofluids, and k is the thermal conductivity of the nanofluids, and k is taken to be a constant nonisotropic axisymmetric tensor of thermal conductivity. Using the similarity variables (7), the local Nusselt number, local Sherwood number, and reduced skin friction coefficient are

1/2 1/2 1/2 C Rex = f 00 (0), NuxRex− = θ0(0), ShxRex− = φ0(0), (17) f − − where Rex = U x/ν is the local Reynolds number. ∞ Using dimensional analysis, the function of the Reynolds and Schmidt numbers is expressed by the Froessling equation: 1/2 1/3 Sh = Sh0 + CRe Sc (18) where Sh0 is the Sherwood number due only to natural convection and not forced convection while C is a constant. Regarding Equation (15), the Sherwood numbers are calculated with several parameters dependent on the concentration. Sherwood numbers are shown for diﬀerent Ec, Bi, Nb, Nt, and M values according to Equation (18).

3. Results and Discussion The flow, mass, and heat transfer on a stretching sheet with magnetic, partial slip, Eckert, and Biot effects were analyzed. Using the shooting method, we solved ordinary differential system Equations (8) to (10) subject to boundary conditions (11). This method is stated by Torrance and Jaluria in their book [24] and has been applied extensively in the present work; see Salleh et al. [25]. Using this method, dual solutions were obtained by employing various initial guesses for unknown values of the reduced local Nusselt number θ (0), reduced local Sherwood number φ (0), and reduced skin friction − 0 − 0 coefficient f 00 (0), where the infinity boundary conditions are satisfied by all the temperature, velocity, and nanoparticle concentrations (11) asymptotically but with different boundary layer thicknesses and shapes. Table1 shows a comparison of previous results with the local Nusselt number [ 23] in the absence of partial slip and magnetic, Eckert, and Biot effects as a benchmark, showing that the numerical results are in perfect agreement. We assume M = 0, Ec = 0, and Bi = 0 so that the values of the reduced local Nusselt number are similar with those in [23] by using similar shooting method. Comparison are also made using M = 0.1, Ec = 0.1, and Bi = 0.1, and it is found that the present results Symmetry 2019, 11, 1367 6 of 17 in Table1 show the values of local Nusselt number is higher than [ 23]. The numerical values of some important physical quantities are presented in Table2. Di fferent governing nondimensional parameters effect are discussed, namely, the Brownian motion parameter Nb, Prandtl number Pr, thermophoresis parameter Nt, Dufour number Du, Soret number Sr, magnetic number M, Eckert number Ec, Biot number Bi, Lewis number Le, mass suction parameter S, velocity slip parameter σ, stretching parameter ε, and nonlinearity parameter n. We assumed constant values for several parameters, which are Pr = 1, Nb = 0.5, Nt = 0.5, Sr = 0.1, Du = 0.1, S = 2.5, σ = 0.1, n = 2, and Le = 2. The values of parameters are based on the behavior of velocity, temperature, and nanoparticle concentration profiles to ensure the figures move asymptotically in the x-axis. Besides his, the chosen of the value of suction S > 2 is important to obtain dual solutions.

Table 1. The reduced local Nusselt number θ (0) with eﬀects of σ and ε.(Nb = 0.5, Nt = 0.5, Le = 2, − 0 S = 2.5, n = 2, Pr = 1, σ = 0.1, Sr = 0.1, Du = 0.5).

θ0 (0) − σ ε Present Results Present Results Zaimi et al. [23] M = 0, Ec = 0, and Bi = 0 M = 0.1, Ec = 0.1, and Bi = 0.1 1 1.4298738 [1.5108520] 1.4298738 [1.5108520] 1.5132575 [1.3789473] − 0.5 - 1.4586293 [1.5716212] 1.5839247 [1.4452501] 0.1 − 0 1.4850957 [1.6068106] 1.4850957 [1.6068106] 1.5957988 [1.4850957] 0.5 - 1.5096806 [1.6326134] 1.5630076 [1.4951495] 1 1.4657975 [1.5215138] 1.4657975 [1.5215138] 1.5206108 [1.4588067] − 0.5 - 1.4757430 [1.5499348] 1.5577777 [1.4740737] − 1 0 1.4850957 [1.5716768] 1.4850957 [1.5716768] 1.5817981 [1.4850957] 0.5 - 1.4939579 [1.5879273] 1.5940551 [1.4921329] 1 1.5024036 [1.60007110] 1.5024036 [1.60007110] 1.5973980 [1.4953789] [ ] second solution.

Table 2. Values f 00 (0), θ (0), and φ (0) with Bi, M, Ec, and ε when Nb = 0.5, Nt = 0.5, Sr = 0.1, − 0 − 0 Du = 0.1, Pr = 1, S = 2.5, σ = 0.1, n = 2, and Le = 2.

Bi M Ec ε f”(0) θ0 (0) φ0 (0) − − 1 2.2391822 0.0815095 2.4485787 − [ 3.7001711] [0.0706781] [1.7947827] − 1.0756013 0.0838298 2.4521358 0.2 0.1 − [ 2.5565807] [0.0759208] [1.8890286] − 0 1.2279546 0.0844259 2.3986889 [ 1.5382880] [0.0790001] [1.8908379] 0.1 − 1 2.2208376 0.0781744 2.3545960 − [ 3.6428921] [0.0597493] [1.4806476] − 0.5 1.0652139 0.0830049 2.4294086 0.1 0.2 − [ 2.5002446] [0.07057078] [1.7555915] − 0 0 0.0844259 2.3986889 [ 1.4795582] [0.0771729] [1.8760263] − Symmetry 2018, 10, x FOR PEER REVIEW 7 of 17

[−3.6428921] [0.0597493] [1.4806476] 0.5 −1.0652139 0.0830049 2.4294086 [−2.5002446] [0.07057078] [1.7555915] Symmetry 2019, 11, 1367 7 of 17 0 0 0.0844259 2.3986889

Table[−1.4795582] 2. Cont. [0.0771729] [1.8760263] 1 −2.2391822 0.5819357 1.8647797 Bi M Ec ε f”(0) θ0 (0) φ0 (0) [−3.7001711] [0.4988329] − [1.1832779]− 1 2.2391822 0.5819357 1.8647797 0.5 −1.0756013 − 0.5950489 1.8480621 [ 3.7001711] [0.4988329] [1.1832779] 0.2 0.1 − [−2.5565807]0.5 1.0756013[0.5315346] 0.5950489 [1.2200802] 1.8480621 0.2 0.1 − 0 1.2279461 [ 2.5565807] 0.5950033 [0.5315346] 1.7859959 [1.2200802] − [−1.5382880]0 1.2279461[0.5474027] 0.5950033 [1.1766139] 1.7859959 1 1 −2.2208376 [ 1.5382880]0.5577470 [0.5474027] 1.7953111 [1.1766139] 1 − [−3.6428921]1 2.2208376[0.4218679] 0.5577470 [0.9766738] 1.7953111 − [ 3.6428921] [0.4218679] [0.9766738] 0.5 −1.0652139 − 0.5891219 1.8315048 0.1 0.2 0.5 1.0652139 0.5891219 1.8315048 0.1 0.2 [−2.5002447] − [0.4946099] [1.1522619] [ 2.5002447] [0.4946099] [1.1522619] 0 0 − 0.5950033 1.7859959 0 0 0.5950033 1.7859959 [−1.4795582] [0.5356944] [1.2030547] [ 1.4795582] [0.5356944] [1.2030547] [ ] second solution. − [ ] second solution. The results of the reduced local Nusselt number −𝜃(0) as well as the reduced local Sherwood numberThe −𝜙′ results(0) variation of the reduced with the local parameter Nusselt of number mass suctionθ (0) asS for well Bi as= 0.1, the 0.3, reduced and 0.5 local when Sherwood Le = 2, − 0 Prnumber = 1, Nb =φ 0.5,(0) Ntvariation = 0.5, 𝜎with = 0.1, the n = parameter 2, Du = 0.2, of Sr mass = 0.1, suction M = 0.1,S for and Bi Ec= =0.1, 0.1 0.3, for anda stretching 0.5 when surfaceLe = 2, − 0 (Pr𝜀= =1)1, areNb shown= 0.5, Ntin Figure= 0.5, σ 2a,b,= 0.1, respectively.n = 2, Du = 0.2,FromSr th=ese0.1, figures, M = 0.1, when and Ecthe= Bi0.1 increases, for a stretching the values surface of −𝜃′(ε =(01)) areincreases; shown inmeanwhile, Figure2a,b, respectively.the value of From −𝜙′ these(0) ﬁgures,decreases. when Figure the Bi 2a increases, indicates the valuesthat the of dimensionlessθ (0) increases; heat meanwhile, transfer rate the increases value of withφ (0 mass) decreases. suction Figure S and2 athe indicates Biot number. that the Physically, dimensionless this − 0 − 0 canheat be transfer attributed rate increasesto the increase with massof temperature suction S and gradient the Biot with number. respect Physically, to mass suction this can and be attributed the Biot number.to the increase However, of temperature the perplexing gradient results with obtained respect to in mass Figure suction 2b show and thethat Biot the number. dimensionless However, mass the transferperplexing rate results decreases obtained when in the Figure Biot2b number show that increa the dimensionlessses. The impact mass of suction transfer on rate the decreases rate of whenmass transfer,the Biot numberas shown increases. in Figure The 2b, impactis evidence of suction that th one rate the rateof mass of mass transfer transfer, reduces as shown with inan Figureincrease2b, in is theevidence Biot and that suction the rate values. of mass transfer reduces with an increase in the Biot and suction values.

(a) (b)

FigureFigure 2. 2. ((a)) ReducedReduced locallocal Nusselt Nusselt number number with with the the parameter parameter of mass of mass suction suctionS for S di forﬀerent different Bi values. Bi values.(b) Reduced (b) Reduced local Sherwood local Sherwood number number with the with parameter the parameter of mass suction of massS forsuction diﬀerent S for Bi different values. Bi values.

Symmetry 2019, 11, 1367 8 of 17 Symmetry 2018, 10, x FOR PEER REVIEW 8 of 17

Figure 33a,ba,b demonstratesdemonstrates thethe valuesvalues ofof thethe reducedreduced locallocal NusseltNusselt number −𝜃θ((0)) and reduced − 0 local SherwoodSherwood number numberφ −𝜙′(0),(0 respectively,) , respectively, for the for stretching the stretching case (ε = 1)case for several( 𝜀 = thermophoresis1) for several − 0 parameterthermophoresis values, parameter Nt = 0.1, values, Nt = 0.3, Nt = and 0.1, Nt Nt= =0.5 0.3, when and NtLe = 0.52, Prwhen= 1, LeNb = =2, 0.5,Pr =Bi 1, =Nb0.1, = 0.5,σ = Bi0.1, = 0.1,n = 2,𝜎 Du= 0.1,= 0.2, n = Sr2, Du= 0.1, = 0.2, M Sr= 0.1,= 0.1, and M Ec= 0.1,= 0.1.andFrom Ec = 0.1. Figure From3a, Figure it is seen 3a, thatit is seen as Nt thatincreases, as Nt increases, the heat thetransfer heat ratestransfer increase rates and,increase thus, and, the thermalthus, the boundary thermal boundary layer thickness layer decreases.thickness decreases. As the boundary As the boundarylayer thickness layer decreases, thickness the decreases, temperature the gradienttemperatur at thee gradient surface increases.at the surface However, increases. Figure However,3b shows Figurethat a smaller 3b shows Nt that is enough a smaller to increase Nt is enough transfer to rates. increase This transfer is due torates. the factThis that is due the to thermal the fact boundary that the thermallayer thickness boundary will layer increase thickness once Nt will is increase intensiﬁed. once As Nt the is thermalintensified. boundary As the layerthermal thickness boundary increases, layer wethickness can expect increases, that the we temperature can expect that gradient the temper at theature surface gradient grows at smaller. the surface Thus, grows the local smaller. Sherwood Thus, thenumber local decreases.Sherwood number Increasing decreases.S values Increasing will increase S values the mass will andincrease heat the transfer mass rate,and heat as shown transfer in rate,Figure as3 a,b,shown respectively. in Figure 3a,b, respectively.

(a) (b)

Figure 3.3. ((aa)) ReducedReduced local local Nusselt Nusselt number number with with the the parameter parameter of mass of mass suction suctionS for diS ffforerent different Nt values. Nt (values.b) Reduced (b) Reduced local Sherwood local Sherwood number withnumber the parameterwith the parameter of mass suction of massS forsuction different S for Nt different values. Nt values. Figure4a,b depicts the reduced local Nusselt number θ (0) as well as reduced local Sherwood − 0 number φ0(0) variation, respectively, for a stretching case−𝜃′ for( various0) values of Eckert Ec with other Figure− 4a,b depicts the reduced local Nusselt number as well as reduced local Sherwood numberfixed value −𝜙′ parameters(0) variation, (Pr respectively,= 1, Le = 2, Ntfor =a stretchi0.5, Nbng= 0.5,case Bi for= various0.1, σ = values0.1, n =of2, EckertDu = Ec0.2, withSr =other0.1, fixedM = 0.1, value and parameters S = 2.5). From (Pr = Figure 1, Le =4 a,b,2, Nt it = is 0.5, clearly Nb = seen 0.5, thatBi = 0.1, the decreasing𝜎 = 0.1, n = Eckert 2, Du number= 0.2, Sr increases= 0.1, M = 0.1,the rateand ofS = mass 2.5). andFrom heat Figure transfer, 4a,b, respectively. it is clearly Theseen e ffthatect the of the decreasing Eckert number Eckert (Ec)number is negligible increases when the ratethere of is mass greater and suction. heat transfer, It is also respectively. shown that The as theeffe Eckertct of the number Eckert (Ec)number increases, (Ec) is thenegligible local Nusselt when number θ (0) and local Sherwood number φ (0) will decrease; hence, it is prone to increasing the there is greater− 0 suction. It is also shown that− as0 the Eckert number (Ec) increases, the local Nusselt numbernanoparticle −𝜃′( and0) and thermal local Sherwood boundary layernumber thicknesses. −𝜙′(0) will decrease; hence, it is prone to increasing the nanoparticle and thermal boundary layer thicknesses.

Symmetry 2018, 10, x FOR PEER REVIEW 9 of 17 Symmetry 2019, 11, 1367 9 of 17 Symmetry 2018, 10, x FOR PEER REVIEW 9 of 17

(a) (b)

Figure 4. (a) Reduced(a )local Nusselt number with the parameter of mass suction(b) S for different Ec values. (b) Reduced local Sherwood number with the parameter of mass suction S for different Ec Figure 4.4. ((aa)) ReducedReduced local local Nusselt Nusselt number number with with the the parameter parameter of mass of mass suction suctionS for diS ffforerent different Ec values. Ec values. (values.b) Reduced (b) Reduced local Sherwood local Sherwood number withnumber the parameterwith the parameter of mass suction of massS forsuction different S for Ec different values. Ec values. Figure 55aa portrays the the local local Nusselt Nusselt number number variation variation with with the the Brownian Brownian motion motion parameter parameter (Nb) for(Nb )someFigure for some thermophoresis 5a thermophoresisportrays the localparameter parameter Nusselt Nt number Ntvaluesvalues variation when when stretching with stretching the Brownian is is considered; considered; motion meanwhile, meanwhile,parameter (Nbthe ) correspondingfor some thermophoresis local Sherwood parameter number Nt is illustrated values when in Figure stretching 55b.b. ForFor is Figure Figureconsidered;5 5a,b,a,b, other other meanwhile, parameters parameters the 𝜎 arecorresponding fixed fixed ( Le = 2,local2, Pr Pr =Sherwood= 1,1, NbNb == 0.5, 0.5,number Bi Bi = =0.1, 0.1,is illustrated σ== 0.1,0.1, n n =in =2, Figure 2,DuDu = 0.2,=5b.0.2, SrFor Sr= Figure0.1,= 0.1, Ec 5a,b,Ec= 0.1,= other0.1, M = M 0.1,parameters= 0.1,andand S = 2.5).Sare= 2.5).fixed From From ( LeFigure = Figure2, Pr5a, =5 it a,1, is itNb clearly is clearly= 0.5, shown Bi shown = 0.1, that that𝜎 when= when0.1, Nbn Nb= increases,2,increases, Du = 0.2, the theSr local = local 0.1, Nusselt NusseltEc = 0.1, number number M = 0.1, gradually gradually and S = decreases,2.5). From whileFigurewhile it it5a, increases increases it is clearly along along shown with with increasing that increasing whenNt Nb; thus, Nt increases,; itthus, will it cause the will local a cause decrease Nusselt a decrease in number thermal in gradually boundary thermal boundarylayerdecreases, thickness. layerwhile thickness. Figure it increases5b demonstratesFigure along 5b demonstrwith that increasing theates increasing that Nt the; thus,increasi of thermophoresis it ngwill of causethermophoresis aNt decreasedecreases Nt in decreases thethermal mass thetransferboundary mass rate; transfer layer in otherthickness. rate; words, in other Figure the words, ratio5b demonstr ofthe the ratio convectiveates of the that convective the mass increasi transfer massng of totransfer thermophoresis the mass to the diff massusivity Nt diffusivity decreases becomes becomessmaller.the mass Thissmaller.transfer is due rate;This to inis the otherdue lager to words, the values lager the of valuesratio Nt where of of the Nt theconvective where viscosity the mass viscosity of the transfer base of the fluidto thebase is mass strongfluid diffusivity is and strong has andtendencybecomes has tendency smaller. for nanoparticles Thisfor nanoparticle is due to to move thes to amonglager move values themselves among of Nt themselves where less easily. the less viscosity This easily. phenomenon ofThis the phenomenon base makes fluid is the makesstrong fluid thetoand cool fluid has slower tendencyto cool and slower for therefore nanoparticle and therefor the heats toe transferthe move heat among rate transfer decreases. themselves rate decreases. less easily. This phenomenon makes the fluid to cool slower and therefore the heat transfer rate decreases.

(a) (b)

Figure 5. ((aa)) Reduced Reduced(a ) local local Nusselt Nusselt number number with with Nb for didifferentﬀerent NtNt values.values.(b) ( (b)) Reduced local Sherwood number with Nb for didifferentﬀerent Nt values. Figure 5. (a) Reduced local Nusselt number with Nb for different Nt values. (b) Reduced local Sherwood number with Nb for different Nt values.

Symmetry 2018, 10, x FOR PEER REVIEW 10 of 17

The local Nusselt number and local Sherwood number variations with the parameter of Brownian motion Nb for various influences of the magnetic field parameter M are plotted in Figure 5a,b, respectively, while other parameters are fixed in the stretching case (Pr = 1, Le = 2, Nt = 0.5, Nb = 0.5, Bi = 0.1, 𝜎 = 0.1, n = 2, Du = 0.2, Sr = 0.1, Ec = 0.1, and S = 2.5). Figure 6a shows that the local Nusselt number values decrease when Nb increases; however, as M increases, the local Nusselt number decreases. Figure 6b shows the local Sherwood number increases when Nb increases but has a lower value when M increases. These graphs illustrate the decreasing trend of the local Nusselt number and local Sherwood number against the Brownian motion parameter as the magnetic Symmetry 2019, 11, 1367 10 of 17 parameter increases. This figure also demonstrates that there are nonunique solutions at some level of the magnetic parameter, where the presence of those solutions depends on linear heat and velocity stability.The localThis Nusseltshows that number fluid and and local thermal Sherwood motion number on the variations wall of the with sheet the parameterdecelerated of Brownianwhen we strengthenedmotion Nb for the various effects of influences the magnetic of the parameter. magnetic field parameter M are plotted in Figure5a,b, respectively,Based on while the results other parametersin Figures 2–4, are fixedthe Sh in values the stretching for Bi, Nt, case and (Pr Ec= against1, Le = 2, suctionNt = 0.5, areNb aligned= 0.5, toBi the= 0.1, Froesslingσ = 0.1, curven = 2, inDu equation= 0.2, Sr18.= Since0.1, PrEc == 1,0.1, the and Sherwood S = 2.5). number Figure is6 acalculated shows that with the several local parametersNusselt number as stated values in decreaseequation when18. TheNb effectincreases; of the however, suction parameter as M increases, caused the the local Sherwood Nusselt number number todecreases. decrease Figuredue to6 theb shows steeper the bubble local Sherwood increase. As number the velocities increases of whenfree bubblesNb increases increase, but the has Reynolds a lower valueand Schmidt when Mnumbers increases. decrease. These Based graphs on illustrate the results the from decreasing Figure trend5b and of 6b, the Sh local values Nusselt for Nt number and M againstand local Nb Sherwood are also aligned number to against the Froessling the Brownian curve motionin equation parameter 18. Since as the the magnetic Sherwood parameter number decreasesincreases. because This figure of the also increasing demonstrates Brownian that there motion are nonuniqueparameter, solutionsthe values at for some Nb, level Reynolds of the number,magnetic and parameter, Schmidt where number the presencealso decrease. of those Since solutions the effect depends of the on suction linear parameter heat and velocity and Brownian stability. Thismotion shows parameter that fluid decreases and thermal the velocities motion on of the free wall bubbles, of the the sheet Reynolds decelerated and whenSchmidt we strengthenednumbers are smallerthe effects due of to the a larger magnetic drag parameter. force coefficient.

(a) (b)

Figure 6.6. (a(a)) Reduced Reduced local local Nusselt Nusselt number number with with Nb for Nb di ﬀforerent different M values. M (values.b) Reduced (b) localReduced Sherwood local Sherwoodnumber with number Nb for with diﬀ erentNb for M different values. M values.

TheBased nanoparticle on the results and in temperature Figures2–4, the concentrationSh values for pr Bi,ofiles Nt, for and different Ec against values suction of areEckert aligned Ec are to respectivelythe Froessling shown curve in in Figures Equation 7a (18).and 6b Since for Pra stretching= 1, the Sherwood surface when number all governing is calculated parameters with several are parametersfixed. It is discovered as stated in that Equation there are (18). two The differen effect oft profiles the suction for parameterdifferent layer caused thicknesses the Sherwood with numbera given massto decrease suction due parameter to the steeper S, which bubble supports increase. the As existence the velocities of a dual of free solution, bubbles as increase, illustrated the in Reynolds Figure 5a,b.and SchmidtIt is found numbers that when decrease. Ec increases, Based the on thenanoparticle results from and Figures temperature5b and concentration6b, Sh values profiles for Nt andwill decreaseM against and, Nb arehence, also tend aligned to toincrease the Froessling the nanoparticle curve in Equationand thermal (18). boundary Since the Sherwoodlayer thicknesses. number Figuredecreases 7a becauseillustrates of the increasingpower of the Brownian Eckert motionnumber parameter, Ec on temperature the values in for the Nb, boundary Reynolds layer. number, By studyingand Schmidt the graph, number the also values decrease. of Ec increase Since the when effect the of thewall suction temperature parameter of the and sheet Brownian increases. motion This parameter decreases the velocities of free bubbles, the Reynolds and Schmidt numbers are smaller due to a larger drag force coefficient. The nanoparticle and temperature concentration profiles for different values of Eckert Ec are respectively shown in Figures6b and7a for a stretching surface when all governing parameters are fixed. It is discovered that there are two different profiles for different layer thicknesses with a given mass suction parameter S, which supports the existence of a dual solution, as illustrated in Figure5a,b. It is found that when Ec increases, the nanoparticle and temperature concentration profiles will decrease and, hence, tend to increase the nanoparticle and thermal boundary layer thicknesses. Figure7a illustrates the power of the Eckert number Ec on temperature in the boundary layer. By studying the Symmetry 2019, 11, 1367 11 of 17

Symmetry 2018, 10, x FOR PEER REVIEW 11 of 17 graph, the values of Ec increase when the wall temperature of the sheet increases. This is due to the isdecrease due to ofthe heat decrease transfer of rateheatat transfer the surface; rate at thus, the thesurface; thickness thus,of the the thickness thermal of boundary the thermal layer boundary increases whenlayer increases the value when of Ec increases.the value of Ec increases. Figure8a indicates the e ffect of the magnetic field on the flow field. With the existence of a transverse magnetic field, the fluid induces the Lorentz force, which opposes the flow. This resistive force favors flow deceleration, which results in the decreasing velocity field seen in Figure6a. We can see that when increasing the value of the magnetic parameter M, the retarding force is also increased along with a reduction in velocity. Figure8a also shows that when the value of magnetic parameter M Symmetryincreases, 2018 the, 10 boundary, x FOR PEER layer REVIEW thickness is reduced. Figure8b,c shows that the transverse magnetic11 of 17 field value causes the thermal boundary layer nanoparticle concentration boundary layer to be thicker. Thisis due can to bethe explained decrease of by heat the temperaturetransfer rate andat the nanoparticle surface; thus, concentration the thickness increment of the thermal in the boundary layer increases when the when magnetic the fieldvalue is of increased. Ec increases.

(a) (b)

Figure 7. (a) Eckert Parameter Ec effect on the temperature profile. (b) Eckert Parameter Ec effect on the nanoparticle concentration profile (Pr = 1, Nt = 0.5, Nb = 0.5, Le = 2, 𝜎= 0.1, n = 2, Du = 0.1, Sr = 0.1, M = 0.1, S = 2.5, and Bi = 0.001 when 𝜀 = 1).

Figure 8a indicates the effect of the magnetic field on the flow field. With the existence of a transverse magnetic field, the fluid induces the Lorentz force, which opposes the flow. This resistive force favors flow deceleration, which results in the decreasing velocity field seen in Figure 6a. We can see that when increasing the value of the magnetic parameter M, the retarding force is also increased along with a reduction in(a) velocity. Figure 8a also shows that when the value(b) of magnetic parameter M increases, the boundary layer thickness is reduced. Figure 8b,c shows that the transverse magnetic Figure 7. (a) Eckert Parameter Ec effect effect on the temperature profile.profile. ( b) Eckert Parameter Ec Ec effect effect on field value causes the thermal boundary layer nanoparticle concentration boundary layer to be the nanoparticlenanoparticle concentrationconcentration profile profile (Pr (Pr= =1, 1, Nt Nt= 0.5,= 0.5, Nb Nb= 0.5,= 0.5, Le Le= 2,= σ2,= 𝜎0.1,= 0.1, n = n2, = Du 2, Du= 0.1, = 0.1, Sr = Sr0.1, = thicker. This can be explained by the temperature and nanoparticle concentration increment in the 0.1,M = M0.1, = 0.1, S = S2.5, = 2.5, and and Bi = Bi0.001 = 0.001 when whenε = 𝜀1). = 1). boundary layer when the magnetic field is increased. Figure 8a indicates the effect of the magnetic field on the flow field. With the existence of a transverse magnetic field, the fluid induces the Lorentz force, which opposes the flow. This resistive force favors flow deceleration, which results in the decreasing velocity field seen in Figure 6a. We can see that when increasing the value of the magnetic parameter M, the retarding force is also increased along with a reduction in velocity. Figure 8a also shows that when the value of magnetic parameter M increases, the boundary layer thickness is reduced. Figure 8b,c shows that the transverse magnetic field value causes the thermal boundary layer nanoparticle concentration boundary layer to be thicker. This can be explained by the temperature and nanoparticle concentration increment in the boundary layer when the magnetic field is increased.

(a)

Figure 8. Cont.

(a)

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(b)

(c)

FigureFigure 8. 8. ((aa)) Magnetic Magnetic parameter parameter M M effect effect on on the the velocity velocity profile. profile. ( (bb)) Magnetic Magnetic parameter parameter M M effect effect on on thethe temperature temperature profile. profile. (c ()c Magnetic) Magnetic parameter parameter M Meffect effect on onthe the nanoparticle nanoparticle concentration concentration profile profile (Pr =(Pr 1, =Nb1, = Nb 0.5,= Nt0.5, = 0.5, Nt =Le0.5, = 2, Le 𝜎= 0.1,2, σ n= =0.1, 2, Du n = = 2,0.1, Du Sr= = 0.1, SrS == 2.5,0.1, Bi S == 0.001,2.5, Bi and= 0.001,Ec = 0.1 and when Ec = 𝜀0.1 = 1).when ε = 1).

FigureFigure 99aa shows shows the the change change in thein the parameter parameter of Brownian of Brownian motion, motion, Nb, on Nb, the on temperature the temperature profile. profile.Based on Based our observation,on our observation, an increase an increase in the Brownian in the Brownian motion parametermotion parameter leads to leads temperature to temperature increase increasein the boundary in the boundary layer. As layer. Brownian As Brownian motion increases,motion increases, nanoparticles nanoparticles emigrate emigrate from the from hot the surface hot surfacetowards towards the cold the ambient cold ambient fluid and, fluid because and, ofbecaus that,e the of temperaturethat, the temperature will increase will in increase the boundary in the boundarylayer. This layer. will helpThis will in thickening help in thickening the boundary the boundary layer. Figure layer.9 bFigure reveals 9b thereveals concentration the concentration profile. profile.The concentration The concentration profile isprofile a decreasing is a decreasing function fu ofnction Nb. The of Nb. reason The for reason this mayfor this be thatmay the be decreasethat the decreasein the Brownian in the Brownian motion parameter motion parameter will reduce will the reduce mass the transfer mass of transfer a nanofluid. of a nanofluid. Figure 10a shows the change in the thermophoresis parameter Nt on the temperature profile. When the thermophoresis parameter increases, the temperature increases in the boundary layer. As the thermophoresis effect increases, nanoparticles emigrate from the hot surface towards the cold ambient fluid and, because of that, the temperature will increase in the boundary layer. This will help in thickening the boundary layer. Figure 10b reveals the variation of the concentration profile. The larger values of thermophoresis parameter Nt in 5a lead to larger temperature differences. The concentration field is driven by the temperature gradient. When the temperature is an increasing function of Nt, a rise in the Nt parameters causes the concentration and the boundary layer thickness to increase.

Symmetry 2018, 10, x FOR PEER REVIEW 13 of 17

Symmetry 2019, 11, 1367 13 of 17 Symmetry 2018, 10, x FOR PEER REVIEW 13 of 17

(a) (b)

Figure 9. (a) Nb effect on the temperature profile. (b) Nb effect on the nanoparticle concentration profile (Pr = 1, Nt = 0.5, Le = 2, 𝜎= 0.1, n = 2, Du = 0.1, Sr = 0.1, M = 0.1, S = 2.5, Bi = 0.001, and Ec = 0.1 when 𝜀 = 1)

Figure 10a shows the change in the thermophoresis parameter Nt on the temperature profile. When the thermophoresis parameter increases, the temperature increases in the boundary layer. As the thermophoresis effect increases, nanoparticles emigrate from the hot surface towards the cold ambient fluid and, because of that, the temperature will increase in the boundary layer. This will help in thickening the boundary(a) layer. Figure 10b reveals the variation of the (concentrationb) profile. The larger values of thermophoresis parameter Nt in 5a lead to larger temperature differences. The Figure 9. ((aa)) Nb Nb effect effect on the temperature profile.profile. ( (bb)) Nb Nb effect effect on the nanoparticle concentration concentration field is driven by the temperature gradient. When the temperature is an increasing profileprofile (Pr == 1, Nt == 0.5, Le = 2, σ𝜎== 0.1, nn == 2, Du == 0.1, Sr = 0.1, M = 0.1, S = 2.5,2.5, Bi = 0.001,0.001, and Ec = 0.10.1 function of Nt, a rise in the Nt parameters causes the concentration and the boundary layer thickness when ε𝜀= =1). 1) to increase. Figure 10a shows the change in the thermophoresis parameter Nt on the temperature profile. When the thermophoresis parameter increases, the temperature increases in the boundary layer. As the thermophoresis effect increases, nanoparticles emigrate from the hot surface towards the cold ambient fluid and, because of that, the temperature will increase in the boundary layer. This will help in thickening the boundary layer. Figure 10b reveals the variation of the concentration profile. The larger values of thermophoresis parameter Nt in 5a lead to larger temperature differences. The concentration field is driven by the temperature gradient. When the temperature is an increasing function of Nt, a rise in the Nt parameters causes the concentration and the boundary layer thickness to increase.

(a) (b)

Figure 10. ((a)) Nt effect effect on the temperaturetemperature profile.profile. ( b)) Nt eeffectffect on thethe nanoparticlenanoparticle concentrationconcentration profileprofile (Pr(Pr == 1, NbNb == 0.5, Le == 2, σ𝜎== 0.1,0.1, nn == 2,2, DuDu == 0.1, Sr == 0.1, M = 0.1, S = 2.5, Bi = 0.001,0.001, and Ec = 0.1 when ε𝜀= =1). 1).

Figure 1111aa displaysdisplays thethe variationvariation of of temperature temperature profile profile with with the the Prandtl Prandtl number number Pr. Pr. The The eff effectect of theof the Prandtl Prandtl number, number, Pr, is Pr, that is an that increase an increase in Prandtl in numberPrandtl Prnumber will lead Pr towill a decrease lead to thea decrease temperature the field.temperature Increasing field. the Increasing value of Pr the indicates valuethat of momentumPr indicates di thatffusivity momentum is greater diffusivity compared tois thermalgreater dicomparedffusivity. to As thermal a result diffusivity. of this, the thermalAs a result boundary of th is, layerthe thermal thickness boundary represents layer a decreasing thickness functionrepresents of Pr. Figure 11b shows that(a) the momentum diffusivity dominates as the Prandtl(b) number Pr increases. As a result of this, the nanoparticle concentration and boundary layer thickness represent an increasing functionFigure of Pr.10. (a) Nt effect on the temperature profile. (b) Nt effect on the nanoparticle concentration profile (Pr = 1, Nb = 0.5, Le = 2, 𝜎= 0.1, n = 2, Du = 0.1, Sr = 0.1, M = 0.1, S = 2.5, Bi = 0.001, and Ec = 0.1 when 𝜀 = 1).

Figure 11a displays the variation of temperature profile with the Prandtl number Pr. The effect of the Prandtl number, Pr, is that an increase in Prandtl number Pr will lead to a decrease the temperature field. Increasing the value of Pr indicates that momentum diffusivity is greater compared to thermal diffusivity. As a result of this, the thermal boundary layer thickness represents

Symmetry 2018,, 10,, xx FORFOR PEERPEER REVIEWREVIEW 14 of 17 a decreasing function of Pr. Figure 11b shows that the momentum diffusivity dominates as the Prandtl number Pr increases. As a result of this, the nanoparticle concentration and boundary layer PrandtlSymmetry 2019number, 11, 1367 Pr increases. As a result of this, the nanoparticle concentration and boundary14 layer of 17 thickness represent an increasing function of Pr.

(a) (b)

Figure 11.11. (((aa))) PrPrPr e effectffeffectect on onon the thethe temperature temperaturetemperature profile. profile.profile. (b) Pr((b)) e ffPrPrect effecteffect on the onon nanoparticle thethe nanoparticlenanoparticle concentration concentrationconcentration profile profile(Nb = 0.5,(Nb Nt = 0.5,= 0.5, Nt Le= 0.5,= 2, Leσ = 2,0.1, 𝜎 n= =0.1,2, n Du = 2,= Du0.1, = Sr 0.1,= 0.1,Sr = M 0.1,= M0.1, = S0.1,= 2.5,S = 2.5, Bi = Bi0.001, = 0.001, and and Ec =Ec0.1 = 0.1when whenε = 1).𝜀 == 1).1).

Figure 12a,b12a,b respectively show the variation in in temperature and and nanoparticle concentration when changing the Lewis number Le. A A greater greater value of Lewis number Le indicates higher thermal diffusivitydiffusivity and and lower lower nanoparticle nanoparticle diffusivity. diffusivity. Le Le > 1> indicates1 indicates the the thermal thermal diffusion diffusion rate rate is above is above the Brownianthe Brownian diffusion diffusion rate. rate.The lower The lowerthe Brownian the Brownian diffusion, diff usion,the lower the the lower mass the transfer mass transferrate. Results rate. fromResults Figure from 12a Figure show 12 athat show thethat thermal the thermal boundary boundary layer thickness layer thickness is an increasing is an increasing function function of Le. However,of Le. However, in Figure in Figure 12b, we12b, can we see can that see that increa increasingsing the the Lewis Lewis number number Le Le results results in in a a thinner nanoparticle concentration boundary layer.

(a) (b)

Figure 12. (((aa):):): LeLe Le effecteffect effect onon thethe temperature temperature profile. profile.profile. (( (bb))) LeLe Le effecteffect effect on on the nanoparticlenanoparticle concentration profileprofile (Pr = 1,1, Nb Nb == 0.5,0.5, Nt Nt = =0.5,0.5, 𝜎σ= =0.1,0.1, n n= 2,= 2,Du Du = 0.1,= 0.1, Sr = Sr 0.1,= 0.1, M = M 0.1,= 0.1,S = 2.5, S = Bi2.5, = Bi0.001,= 0.001, and Ec and = 0.1Ec =when0.1 when 𝜀 == 1).1).ε = 1).

Symmetry 2019, 11, 1367 15 of 17

4. Statistical Properties The influence of magnetic, Eckert, and Biot numbers on the boundary layer flow and heat transfer in a nanofluid over a nonlinearly stretching/shrinking sheet are studied using statistical analysis. To understand what the relationship is between the various physical parameters and also for the local Nusselt number and local Sherwood number, the values of correlation coefficients are calculated. The apparent error of the correlation coefficient can be solved using the formula

P.E (r) = 0.6745 (1 r2)/sqrt(n), (19) − where n is the number of observation and r is the correlation coefficient. r > 0.7 indicates a strong linear relationship between parameter variables. If r is less than P.E, the correlation coefficient is not important; this indicates that there is no relationship between variables. The interaction is said to be remarkable when the value of r is 6 times larger than P.E, and insignificant when r is less than P.E(r). The probable error is used to check the accuracy of the value of the correlation coefficient. The values of r/P.E(r) are shown in Table3. From this table, it is obvious that the M, Ec, and and Bi values for heat and mass transfer fulfil the relationship r/P.E(r) > 6, which indicates that the correlation coefficient is statistically significant.

Table 3. Values of r/P.E.(r).

Local Nusselt Number Local Sherwood Number M Ec Bi M Ec Bi r 0.9947 0.9456 0.9509 0.8126 0.9984 0.9988 P.E(r) 0.0018 0.0178 0.0152 0.05914 0.0005 0.0003 r/P.E(r) 543.9791 53.0041 62.5026 13.7404 1973.8055 2731.6587

5. Conclusions The effects of magnetic, Eckert, and Biot numbers on the heat transfer and boundary layer in a nanofluid over a nonlinearly stretching/shrinking sheet are studied in this work. By using similarity transformation, the governing partial differential equations are transformed into ordinary differential equations. Then, by utilizing the shooting method, the problems are solved numerically. The effect of M, Ec, and Bi numbers on velocity, temperature, and nanoparticle concentration profiles as well as the skin friction coefficient, heat, and mass transfer rates are presented graphically. From the obtained numerical results, similar conclusion is found to be the same as [4]. Results from [4] found that an increase in magnetic parameter would decrease the reduced Nusselt and Sherwood number. The same pattern was also found from [6], where M is a decreasing function, and therefore, it enhances the heat transfer.

Author Contributions: These authors contributed equally to this work Funding: This research was funded by Universiti Kebangsaan Malaysia grant number GGPM-2017-036. Acknowledgments: The authors warmly thank the reviewers for their time spent on reading the manuscript and for their valuable comments and suggestions. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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