Eur. Phys. J. Plus (2017) 132: 441 DOI 10.1140/epjp/i2017-11660-0 THE EUROPEAN PHYSICAL JOURNAL PLUS Regular Article Numerical analysis of 3D micropolar nanoﬂuid ﬂow induced by an exponentially stretching surface embedded in a porous medium M. Subhania and S. Nadeem Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Received: 3 July 2017 / Revised: 24 August 2017 c Società Italiana di Fisica / Springer-Verlag 2017 Published online: 27 October 2017 – Abstract. The present article is devoted to probe the behavior of a three-dimensional micropolar nanoﬂuid over an exponentially stretching surface in a porous medium. The mathematical model is constructed in the form of partial diﬀerential equations using the boundary layer approach. Then by employing similarity transformations, the modelled partial diﬀerential equations are transformed to ordinary diﬀerential equations. The solution of subsequent ODEs is derived by utilizing the BVP-4C technique alongside the shooting scheme. The graphical illustrations are presented to interpret the salient features of pertinent physical parameters on the concerned proﬁles for diﬀerent kinds of nanoparticles, namely copper, titania and alumina with water as the base ﬂuid. For a better understanding of the ﬂuid ﬂow, the numerical variation in the local skin friction coeﬃcients, Cfx and Cfy , and local Nusselt number is analyzed through tables. We can see, from the present study, that the omission of porous matrix enhances the ﬂow of the ﬂuid. Microrotation has a decreasing impact on the skin friction whereas it increases the rate of the heat transfer of the nanoﬂuid. 1 Introduction In view of the substantial practical applications in industrial and manufacturing ﬁelds, a lot of work has been performed on boundary layer ﬂows for linear and non-linear stretching surfaces. The interminable list of its engineering applications comprises extrusion of polymer sheet, paper production, tinning and annealing of copper wires, manufacturing of metal wires and plastic sheets, cooling of metallic plates inside a cooling reservoir and so forth. Sakiadis [1] propounded the concept of the boundary layer ﬂow past a continuous stretched sheet and devised the boundary layer equations for ﬂow in two dimensions. Tsou et al. [2] inspected the consequences of heat transfer on boundary layer ﬂows past a stretching surface. Gupta et al. [3] incorporated mass transfer analysis over stretching sheet and considered suction or blowing eﬀects. After that, numerous researchers have further delved into boundary layer ﬂows involving important conventional ﬂuids [4,5]. In the contemporary period, ﬂow due to an exponentially stretching surface is gaining much importance due to its vast applications. For instance, in the process of drawing and tinning of copper wires, the rate of heat transfer past a continuously stretching surface which has exponential modiﬁcations in its stretching velocity and temperature, inﬂuences the form of the ﬁnal product. In this regard, Magyari and Keller [6] forwarded the comparison of numerical and analytical solutions. Liu et al. [7] considered the characteristics of ﬂow and heat transfer of 3D ﬂow over a surface which was stretched exponentially. Recently Ahmad et al. [8] explored the inﬂuence of the Cattaneo-Christov heat ﬂux model induced by an exponentially stretching surface. It was the need of the hour to upgrade the thermal conductivity of some important conventional ﬂuids which are useful in heat transfer, for example water, ethylene glycol and mineral oil which possess poor heat transfer characteristics. A novel solution was found by introducing small metallic solid particles in the ﬂuid; this revolutionized the realm of technology and industry. These nanoparticles elevate the thermal conductivity of ﬂuids improving the heat transfer properties. The ﬂuids so obtained were termed as nanoﬂuids. Therefore, nanoﬂuids are characterized as a solid-ﬂuid mixture with a base ﬂuid of low conductivity and nano-meter sized particles with high thermal conductivity. Due to their higher thermal conductivity, these ﬂuids lower the pumping cost of heat exchangers to a greater extent. This ﬂuid was ﬁrst introduced by Choi [9] in 1995. A thorough investigation of convective transport within nanoﬂuids a e-mail: msubhani@math.qau.edu.pk (corresponding author) Page 2 of 12 Eur. Phys. J. Plus (2017) 132: 441 was done by Buongiorno [10]. Nanoparticles along with their little volume fraction, stability and remarkable useful applications in optical, biomedical and electronic ﬁelds have opened new horizons of research. Recently many scholars have discussed the nanoparticle phenomena in diﬀerent geometries with pertinent physical properties of ﬂuid, see refs. [11–21]. Nadeem et al. [22] discussed the heat transfer phenomenon of a three-dimensional nanoﬂuid over an exponentially stretching surface. Since nanoparticles are minute in size they can haul with them slip velocity through the base ﬂuid molecules (see Buongiorno [10]). Particle rotation is also an essential factor in the heat transfer enhancement which was observed by Ahuja [23]. In view of the fact that nanoparticle can schlep slip velocity with base ﬂuid molecules, the likelihood of translation and microrotation arises. The micropolar theory considers the eﬀects of microrotation hence the application of this hypothesis in case of nanoparticles gives a valuable comprehension to the unusual rise in nanoﬂuid’s thermal conductivity. The theory of micropolar ﬂuids set forth by Eringen [24,25] was a breakthrough in the study of rheologically complex ﬂuids. This class of non-Newtonian ﬂuids comprises randomly oriented particles which are rigid and spherical in shape suspended in a viscous medium. The gist of the theory undertakes the extension of the classical Navier-Stokes equations so that more complex ﬂuids containing certain additives such as liquid crystals, particle suspensions, ﬂuids with materials containing ﬁbrous structures are also considered. The theory of micropolar ﬂuid manifests the intrinsic motion of ﬂuid elements and the microrotational eﬀects. Industrially the examples of micropolar ﬂuid include exotic lubricants, muddy ﬂuids, colloidal solutions, polymer suspensions whereas there are certain biological ﬂuids as well which model micropolar ﬂuid, for instance animal blood. Lukaszewicz [26] and Eringen [27] gave ample details of the micropolar theory in their books. Chamkha et al. [28] analyzed the ﬂow behavior of three-dimensional micropolar ﬂuids. In 2009, Mohamed abd el-aziz [29] deliberated over the eﬀects of viscous dissipation on micropolar ﬂuid past an exponentially extending sheet. Mohanty [30] conducted a numerical research on heat and mass transfer of micropolar ﬂuids past a stretching sheet. Alongside a research conducted over micropolar nanoﬂuids gained momentum and [31,32] analyzed the impact of diﬀerent attributes over the micropolar nanoﬂuid. Since the past few decades, many researchers have inquired the characteristics of ﬂow and heat transfer inside a porous medium owing to its incessantly growing industrial and technological applications. Problems associated with porous surfaces incorporate insulation engineering, geo-mechanics such as geo thermal reservoirs and enhanced oil recovery. The main use of a porous medium is to insulate a heated body so that its temperature can be maintained. Porous media are also considered to be useful in vanishing the natural free convection, which would otherwise aﬀect the stretching surface immensely. Kamel et al. [33] and Heruska et al. [34] have examined the micropolar ﬂuid ﬂow across a porous medium. Very recently [35,36] have explored the behavior of micropolar ﬂuid induced by a stretching porous sheet. A comprehensive study of MHD ﬂow of nanoﬂuids through a porous medium was done by Zeeshan et al. [37]. In the vast ﬁeld of literature, there is no research conducted on the behavior of a three-dimensional micropolar ﬂuid with nanoparticles due to an exponentially stretching surface. In the present article, the inﬂuence of a porous medium on the three-dimensional steady ﬂow of a micropolar nanoﬂuid induced by an exponentially stretching surface has been critically examined. Three diﬀerent nanoparticles, namely copper, titania and alumina are compared taking water as the base ﬂuid. Non-dimensional velocities, angular velocities and temperature proﬁles as well as the inﬂuence of various physical parameters on them are exhibited by plotting graphs and are also presented in tabular form. 2 Momentum and temperature description We have considered a steady 3D incompressible boundary layer ﬂow of a micropolar nanoﬂuid embedded in a porous medium over an exponentially stretching surface. It is supposed that the surface is being stretched in two adjacent directions with diﬀerent velocities, Uw which is along the x-axis and Vw which is along the y-axis. Further, Tw is considered to be the temperature near the wall while ambient temperature is assumed to be T∞ . The surface is located at the plane z = 0 and the ﬂow is conﬁned in the region z > 0 (see ﬁg. 1). The transport equations are = 0, ∇·V ρnf ρnf j dV + k(∇ × N ) − νnf V , = −∇p + (μnf + k)∇2 V ∗ dt K dN − 2k N + k(∇ × V ), = χnf ∇2 N dt dT = αnf ∇2 T, dt (1) (2) (3) (4) Eur. Phys. J. Plus (2017) 132: 441 Page 3 of 12 Fig. 1. Physical regime of the problem. where knf μf , μnf = , ρnf = (1 − φ)ρf + φρs , (ρcp )nf (1 − φ)2.5 = (1 − φ)(ρCp )f + φ(ρCp )s , (ks + 2kf ) − 2φ(kf − ks ) μnf , νnf = = , (ks + 2kf ) + φ(kf − ks ) ρnf = (μnf + k/2)j. αnf = (ρCp )nf knf χnf (5) χnf is the spin gradient viscosity, αnf is the nanoﬂuid thermal diﬀusivity, μnf is the dynamic viscosity of the nanoﬂuid, k is the vortex viscosity, ρnf is the density of the nanoﬂuid, knf is the thermal conductivity of the nanoﬂuid, (ρCp )nf is the heat capacity of the nanoﬂuid, φ is the volume fraction of nanoparticles, K ∗ is the permeability of the porous sheet, j = 2νL Uw is the micro-inertia per unit mass. After applying boundary layer approximation, we are left with the following equations: Momentum equations ∂u ∂u ∂u μnf + k ∂ 2 u νnf u ∂N2 k u +v +w = − − , 2 ∂x ∂y ∂z ρnf ∂z ρnf ∂z K∗ ∂v ∂v μnf + k ∂ 2 v νnf v ∂N1 k ∂v +v +w = − + . u 2 ∂x ∂y ∂z ρnf ∂z ρnf ∂z K∗ (6) (7) Angular momentum equations ∂N1 ∂N1 ∂N1 ∂v ∂ 2 N1 +v +w = χnf , − 2kN1 − k ρnf j u 2 ∂x ∂y ∂z ∂z ∂z ∂N2 ∂N2 ∂N2 ∂u ∂ 2 N2 ρnf j u +v +w = χnf . − 2kN2 + k 2 ∂x ∂y ∂z ∂z ∂z Energy equation ∂T ∂T ∂T u +v +w = αnf ∂x ∂y ∂z ∂2T ∂z 2 (8) (9) . (10) Boundary conditions u = Uw , u → 0, v = Vw , v → 0, T = Tw , T → T∞ , ∂v ∂u , N2 = −n at z = 0, ∂z ∂z N1 → 0, N2 → 0, as z → ∞. N1 = n (11) Page 4 of 12 Eur. Phys. J. Plus (2017) 132: 441 The stretching velocities at the wall surface and temperature are described as follows: Uw = U0 e x+y L , Vw = V 0 e x+y L , Tw = T ∞ + T 0 e B(x+y) 2L , (12) where L denotes the reference length, U0 , V0 and T0 are constants, T∞ indicates the ambient temperature, B symbolizes the temperature exponent and n the boundary parameter, n ranges from 0 ≤ n ≤ 1. Here n = 0 symbolizes a strong concentration of microelements at the wall and in this case the microelements at the stretching surface are incapable of rotation. Diminishing of the antisymmetric fragment of the stress tensor is depicted in the case when n = 0.5 and it also exhibits weak concentration of microelements and n = 1 symbolizes the turbulent boundary layer ﬂows. Using appropriate similarity transformations given below: u = U0 e x+y L f (η), v = U0 e x+y L g (η), w=− νU0 2L 12 e x+y 2L {f + ηf + g + ηg }, x+y x+y 1 1 U0 U0 (U0 2νL) 2 e3( 2L ) h1 (η), N2 = (U0 2νL) 2 e3( 2L ) h2 (η), 2νL 2νL 12 B(x+y) x+y U0 T = T∞ + T0 e 2L θ(η), η = e 2L z. 2νL N1 = (13) After applying these transformations to eqs. (6)–(10), the continuity equation is satisﬁed identically while linear momentum, angular momentum and energy equations take the form ρf νnf ρf (A(φ) + R1 )f + f (f + g) − 2f (f + g ) − R 1 h2 − Kp f = 0, ρnf ρnf νf (14) ρf ρf νnf (A(φ) + R1 )g + g (f + g) − 2g (f + g ) + R 1 h1 − Kp g = 0, ρnf ρnf νf (15) ρf R2 h1 − R1 R3 (2h1 + g ) − 3h1 (f + g ) + h1 (f + g) = 0, ρnf (16) ρf R2 h2 − R1 R3 (2h2 − f ) − 3h2 (f + g ) + h2 (f + g) = 0, ρnf (17) 1 (knf /kf )θ − B(f + g )θ + (f + g)θ = 0. Pr (1 − φ) + (φ(ρcp )s /(ρcp )f ) (18) The associated boundary conditions become f (0) = 0, f (0) = 1, g(0) = 0, g (0) = λ, h1 (0) = ng (0), f → 0, h1 → 0, θ(0) = 0, h2 (0) = −nf (0), as η → 0, g → 0, θ → 0, h2 → 0, as η → ∞, (19) where R1 , R2 , R3 , Kp , Pr, λ, φ and n denote vortex viscosity parameter, spin gradient viscosity parameter, microinertia density parameter, porosity parameter, Prandtl number, stretching ratio parameter, solid nanoparticle volume fraction and boundary parameter, respectively. They are deﬁned as R1 = k , μ R2 = χnf , μj R3 = 2νL , jUw Kp = K ∗ Uw , 2νL Pr = (μCp )f , kf λ= V0 U0 and A(φ) = μnf 1 = . μf (1 − φ)2.5 (20) Further, the skin friction coeﬃcients and Nusselt number can be described as Cf x = τwx 1 2 ρ 2 nf Uw , Cf y = τwy 1 2 ρ 2 nf Uw , N ux = xqw , kf (Tw − T∞ ) (21) Eur. Phys. J. Plus (2017) 132: 441 Page 5 of 12 1 0.5 0.9 Cu TiO 0.8 Al2O3 2 Water Water 0.35 0.6 = 0.0, 0.1, 0.2 0.5 g ’() f ’() 2 Al2O3 0.4 0.7 0.4 0.3 0.2 0.15 0.2 0.1 0.1 0.05 0 1 2 (a) 3 4 5 = 0.0, 0.1, 0.2 0.25 0.3 0 Cu TiO 0.45 0 0 1 2 3 4 5 (b) Fig. 2. (a) Impact of φ on velocity proﬁle in x-direction f (η). (b) Impact of φ on g (η). where τwx , τwy and qw are surface shear stress and surface heat ﬂux, which can be deﬁned as ∂u τwx = (μnf + k) + k(N2 )z=0 , ∂z z=0 ∂v τwy = (μnf + k) + k(N1 )z=0 , ∂z z=0 ∂T qw = −knf . ∂z z=0 After simpliﬁcation, the non-dimensional form of skin friction coeﬃcients and local Nusselt number are −1 1 A + (1 − n)R 1 √ Cf x (Rex ) 2 = f (0), (1 − φ) + φ(ρs /ρf ) 2 −1 1 A + (1 + n)R 1 2 √ Cf y (Rey ) = g (0), (1 − φ) + φ(ρs /ρf ) 2 √ L 1 knf 2 N ux Re− 2 = − θ (0). x kf (22) (23) 3 Numerical scheme The non-linear coupled diﬀerential equations (14)–(18) together with their boundary conditions given in eq. (19) are solved numerically using the BVP-4C technique invoking the shooting methodology. In this process, ﬁrstly the system of eqs. (14)–(18) escorted with boundary conditions are reduced to ﬁrst order equations. Then appropriate initial guesses are adopted which satisfy the boundary conditions. The results obtained depict the impact of various dimensionless parameters such as material parameter R1 , stretching ratio parameter λ, temperature exponent B, porosity parameter Kp and nanoparticle volume fraction radiation φ, on velocity, microrotation and temperature proﬁles. For attaining the convergence criterion of 10−6 the shooting methodology is reiterated. Solutions to the given problem are given in graphical and tabular form. 4 Results and discussion In ﬁgs. 2, 3 and 4 analysis of velocity proﬁles f (η), g (η) in both x- and y-direction, respectively, microrotation proﬁles h1 (η) and h2 (η) and temperature proﬁle θ(η) has been performed for numerous values of the nanoparticle volume fraction φ. It is learnt from ﬁgs. 2(a) and (b) that Cu-water gives a low velocity proﬁle compared with Al2 O3 and TiO2 . Page 6 of 12 Eur. Phys. J. Plus (2017) 132: 441 0.05 0.6 Cu TiO 0 2 0.5 Al2O3 −0.05 Water 0.4 −0.15 = 0.0, 0.1, 0.2 2 = 0.0, 0.1, 0.2 −0.2 0.3 h () h1() −0.1 0.2 −0.25 0.1 −0.3 Cu TiO2 −0.35 Al O 0 2 3 Water −0.4 0 1 2 3 4 −0.1 5 0 1 2 (a) 3 4 5 (b) Fig. 3. (a) Variation of φ on microrotation h1 (η). (b) Eﬀect of φ on microrotation h2 (η). 1 0.9 Cu TiO 0.8 Al2O3 2 Water 0.7 () 0.6 0.5 = 0.0, 0.1, 0.2 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 Fig. 4. Variation of various nanoparticle volume fractions on temperature distribution θ(η). Furthermore, in the absence of nanoparticles, water has a higher velocity proﬁle if compared with all three, i.e. copper, alumina and titania. Therefore, it is discovered that increment in φ will hinder the motion of the ﬂuid so velocity as well as the boundary layer thickness decrease. This is because an increase in nanoparticles makes the thermal conductivity increase resulting in a decrease in the boundary layer thickness. Figures 3(a) and (b) depict the behavior of the microrotation proﬁle with variation of nanoparticle volume fraction for ﬁxed values of vortex viscosity parameter R1 = 3. It is concluded that as we enhance the nanoparticle volume fraction φ, the behavior of h1 (η) is decreasing while h2 (η) is ﬂuctuating within the domain. In case of h2 (η) we learnt that for η < 1, the base ﬂuid has a minimal microrotation proﬁle, whereas in case of η > 1, the base ﬂuid exhibits a boost in the microrotation proﬁle compared with the rest of the mixture (see ﬁg. 3(b)). Furthermore, it is noticed that near the wall, the eﬀects of nanoparticles for the microrotation proﬁle are much sturdier if compared with the case away from the wall. Figure 4 elucidates the impact of φ on temperature distribution θ(η). It is evident through this graph that the base ﬂuid has a low temperature proﬁle in comparison with Cu-water, TiO2 -water and Al2 O3 -water. Increase in φ ampliﬁes the thermal conductivity of the nanoﬂuid and hence the thermal boundary layer. It is demonstrated through ﬁg. 5 the impact of porosity parameter Kp on velocity distribution in the x- and y-directions, microrotation and temperature proﬁles. Higher values of the porosity parameter correspond to lower velocity in both directions as seen through ﬁgs. 5(a) and (b). The main dependence of the porosity parameter is upon the permeability K ∗ of the medium. An increment in porosity results in lower permeability as they are inversely proportional; this lower permeability instigates a reduction in velocity. Since the existence of a porous medium results Eur. Phys. J. Plus (2017) 132: 441 Page 7 of 12 1 0.5 0.9 K =1 0.45 Kp= 1 K =2 0.4 K =2 Kp= 3 0.35 K =3 p 0.8 p 0.7 g’() f’() 0.6 0.5 0.4 0.25 0.2 0.15 0.2 0.1 0.1 0.05 0 1 2 3 4 p 0.3 0.3 0 p 0 5 0 1 2 (a) 3 4 5 (b) 0.05 1.4 0 K =1 p 1.2 −0.05 Kp= 2 K =3 −0.1 p 1 h2() h1() −0.15 −0.2 −0.25 −0.3 Kp= 1 −0.35 0.6 0.4 K =2 p −0.4 K =3 0.2 p −0.45 −0.5 0.8 0 1 2 3 4 0 5 0 1 2 (c) 3 4 5 (d) 1 0.9 K =1 0.8 K =2 0.7 K =3 ’() p p p 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 (e) Fig. 5. (a) Inﬂuence of Kp on velocity proﬁle in the x-direction. (b) Inﬂuence of Kp on velocity proﬁle in the y-direction. (c) Inﬂuence of Kp on angular velocity h1 (η). (d) Inﬂuence of Kp on angular velocity h2 (η). (e) Inﬂuence of Kp on temperature proﬁle. in a greater hindrance to the ﬂuid ﬂow, velocity decelerates and so the angular velocity as witnessed through ﬁgs. 5(c) and (d). It is observed in ﬁg. 5(e) that an elevation in the porosity parameter rises the temperature noticeably. Thus, it may be concluded that owing to the resistance oﬀered by the porous medium temperature rises. Page 8 of 12 Eur. Phys. J. Plus (2017) 132: 441 1 0.5 0.9 0.45 0.8 0.4 0.7 0.35 n= 0.0, 0.2, 0.4, 0.6, 0.8, 1 0.5 g’() f’() 0.6 0.4 0.3 0.2 0.3 0.15 0.2 0.1 0.1 0.05 0 0 1 2 3 4 n= 0.0, 0.2, 0.4, 0.6, 0.8, 1 0.25 0 5 0 1 2 (a) 3 4 5 (b) 0.1 0.8 0 0.7 −0.2 0.6 −0.3 0.5 h () −0.4 n= 0.0, 0.2, 0.4, 0.6, 0.8, 1 −0.5 2 h1() −0.1 0.4 n= 0.0, 0.2, 0.4, 0.6, 0.8, 1 0.3 −0.6 −0.7 0.2 −0.8 0.1 −0.9 −1 0 1 2 (c) 3 4 5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (d) Fig. 6. (a) Impact of n on velocity proﬁle in the x-direction. (b) Eﬀect of n on velocity proﬁle in the x-direction. (c) Impact of n on angular velocity h1 (η). (d) Inﬂuence of n on angular velocity h2 (η). Now we analyze the impact of boundary parameter n on velocity and angular velocity proﬁles h1 (η) and h2 (η) in x- and y-directions via ﬁg. 6(a)–(d). n indicates the rotation of microelements close to the wall. As mentioned earlier n = 0 signiﬁes concentrated ﬂow of particles i.e. strong concentration wherein the microelements near the surface of the wall are not able to rotate, n = 0.5 is used to specify the diminishing of the antisymmetric fragment of the stress tensor i.e. weak concentration and n = 1 denotes the turbulent ﬂows of the boundary layer. It is illustrated through ﬁgs. 6(a) and (b) that the gradient of velocity at the surface is greater for increasing values of n. And the couple stress h(0) enlarges for higher values of n, as visible in ﬁg. 6(c) and (d). Through ﬁgs. 7(a) and (b) a comparison between 3 diﬀerent types of nanoparticles is provided with a variation of n against velocity f (η) and angular velocity h2 (η). It is observed that Cu has a low velocity proﬁle in comparison with Al2 O3 and TiO2 . Figures 8(a) and (b) present the nature of the velocity proﬁle in the x- and y-directions with a variation of a parameter R1 = k/μ, the material parameter which gives the ratio between two viscosities of the ﬂuid under consideration, i.e. dynamic viscosity and vortex viscosity. For R1 these two viscosities possess the same order of magnitude. It was learnt about the velocity that it enhances with an increase in material parameter R1 . This is due to the vortex viscosity that makes ﬂuid particles accelerate whereas the permeability and local inertia coeﬃcient decelerates the ﬂow motion. It is apparent from this ﬁgure that the thickness of the boundary layer grows with R1 . The absolute value of velocity gradient near the surface reduces as we increase R1 . Hence, there is a drag reduction in micropolar ﬂuids in comparison with viscous i.e. Newtonian ﬂuids. The negative velocity gradient close to the surface, f (0) < 0, as displayed in ﬁgs. 8(a) and (b) means that there is resistance to the ﬂuid motion due to the stretching Eur. Phys. J. Plus (2017) 132: 441 Page 9 of 12 1 0.8 Cu TiO 0.9 Cu TiO 0.7 2 Al O 2 Al O 2 3 0.8 2 3 0.6 0.5 0.6 h () n= 0, 1 0.5 2 f ’() 0.7 0.4 0.4 n= 0, 1 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1 2 3 4 5 0 1 2 (a) 3 4 5 (b) Fig. 7. (a) Behavior of diﬀerent nanoparticles w.r.t. n on velocity proﬁle in the x-direction. (b) Behavior of diﬀerent nanoparticles w.r.t. n on angular velocity proﬁle h2 (η). 1 1 =0.5 =1.0 0.9 0.8 0.7 0.7 0.6 0.6 0.5 g’() f’() 0.8 R1= 0, 1, 2 0.4 0.3 0.2 0.1 0.1 1 2 (a) 3 4 5 1 0.4 0.2 0 R = 0, 1, 2 0.5 0.3 0 =0.5 =1.0 0.9 0 0 1 2 3 4 5 (b) Fig. 8. (a) Inﬂuence of R1 on velocity proﬁle in the x-direction. (b) Impact of R1 on velocity proﬁle in the y-direction. surface. We can also notice from the ﬁgures that with a rise in stretching parameter λ there is a decrease in x-direction velocity whereas the velocity proﬁle in the y-direction ampliﬁes. The latter behavior is due to the boundary condition g (0) = λ. Figures 9 and 10 are sketched to portray the behavior of R1 , R2 and R3 on microrotation proﬁles in x- and y-directions. Figures 9(a) and (b) are plotted with a variation of n = 0 and n = 0.5 respectively, with h1 (η). In both cases an increment in the proﬁle behavior is reported for numerous values of micropolar parameters. Similarly, ﬁgs. 10(a) and (b) are sketched to show variation of R1 , R2 and R3 on h2 (η) when n = 0 and n = 0.5. Four diﬀerent variations are evaluated and there is an enhancement witnessed in velocity proﬁles when n = 0 and n = 0.5. This is because as there is an increment in micropolar parameters, rotation of the micropolar components is induced in most of the boundary layer excluding the region near the surface since kinematic viscosity is dominating the ﬂow over there. Table 1 illustrates the thermo-physical properties of the base ﬂuid and nanoparticles. In table 2 the eﬀects of various relevant parameters such as material parameter R1 , boundary parameter (n), porosity parameter (Kp ), volume fraction of nanoparticles (φ) and stretching ratio parameter (λ) on skin frictions Cfx and Cfy and the Nusselt number are displayed. Page 10 of 12 Eur. Phys. J. Plus (2017) 132: 441 0.08 0.2 Case 1: R =1, R =0.05, R =0.1 1 0.07 2 3 0.1 Case 2: R1=2, R2=0.1, R3=0.2 Case 3: R =2.5, R =0.2, R =0.3 0.06 1 2 3 0 Case 4: R1=4, R2=0.3, R3=0.4 h () 0.04 −0.1 1 h1() 0.05 −0.2 0.03 Case 1: R =1, R =0.05, R =0.1 1 −0.3 0.02 1 0 1 2 3 4 −0.5 5 3 1 2 3 Case 4: R =4, R =0.3, R =0.4 1 0 3 2 Case 3: R =2.5, R =0.2, R =0.3 −0.4 0.01 2 Case 2: R =2, R =0.1, R =0.2 0 1 2 (a) 2 3 3 4 5 (b) Fig. 9. (a) Inﬂuence of R1 , R2 , R3 on angular velocity proﬁle h1 (η). (b) Inﬂuence of R1 , R2 , R3 on angular velocity proﬁle h1 (η). 0.15 0.7 Case 1: R =1, R =0.05, R =0.1 Case 1: R =1, R =0.05, R =0.1 1 2 3 1 2 Case 4: R1=4, R2=0.3, R3=0.4 0.4 2 h () 3 Case 3: R1=2.5, R2=0.2, R3=0.3 0.5 Case 4: R1=4, R2=0.3, R3=0.4 h2() 2 Case 2: R1=2, R2=0.1, R3=0.2 3 Case 3: R1=2.5, R2=0.2, R3=0.3 0.1 1 0.6 Case 2: R =2, R =0.1, R =0.2 0.3 0.05 0.2 0.1 0 0 1 2 3 4 0 5 0 1 (a) 2 3 4 5 (b) Fig. 10. (a) Inﬂuence of microrotation parameters on velocity proﬁle h2 (η). (b) Inﬂuence of microrotation parameters on velocity proﬁle h2 (η). Table 1. Thermophysical properties of Base ﬂuid and Nanoparticles. Thermophysical properties Cu TiO2 Al2 O3 Fluid phase (water) ρ (kg/m ) 8933 4250 3970 997.1 Cp (j/kg)K 385 686.2 765 4179 k (W/mK) 400 8.9538 40 0.613 α × 107 (m /s) 1163.1 30.7 131.7 1.47 3 2 Eur. Phys. J. Plus (2017) 132: 441 Page 11 of 12 Table 2. Eﬀects of vortex viscosity (R1 ), boundary parameter (n), porosity parameter (Kp ), nanoparticle volume fraction (φ) and stretching parameter (λ) on skin friction and Nusselt number. Cfx Cfy θ (0) 1.31474 0.98076 −1.3316 1.34378 1.17343 −1.4267 3 1.39027 1.25559 −1.5172 1 2.00873 1.00437 −1.3157 2.44874 1.22437 −1.4186 2.66306 1.33153 −1.5028 1.39027 1.25559 −1.3316 1.48147 2.59159 −1.3316 1.60774 4.22325 −1.3316 1.39027 1.25559 −1.3316 1.49663 1.29706 −1.3316 1.63583 1.40815 −1.3316 0 1.6855 1.25559 −1.0000 0.1 1.39027 1.17343 −1.3316 0.15 1.18793 1.09006 −1.5266 R1 N λ Kp φ 1 2 2 0.5 0.0 0.5 0.5 1 1 0.1 0.1 3 0.5 3 0.5 1.0 1 0.1 1.5 1 3 0.5 0.5 2 0.1 3 3 0.5 0.5 1 5 Key results A numerical survey is implemented to investigate the three-dimensional ﬂow of a micropolar nanoﬂuid induced by an exponentially stretching surface within a porous medium. The inﬂuence of pertinent physical parameters on velocities in the x-, y-directions, angular velocities as well as on the temperature proﬁle is talked about in detail. The main ﬁndings of the current analysis are itemized as follows. – Increment in nanoparticle volume fraction φ has increasing inﬂuence on velocity proﬁles whereas it decreases the temperature, while microrotation h2 (η) shows a ﬂuctuating behavior at η = 1. – The skin friction coeﬃcient escalates for increasing values of φ, while the local Nusselt number shows a declining behavior. – The velocity proﬁles and momentum boundary layer thickness show an increasing behavior for vortex viscosity parameter R1 . – Microrotation parameter R1 exhibits decreasing eﬀects on the skin friction coeﬃcient while it shows an aggravating impact on the rate of heat transfer of the nanoﬂuid. – The microrotation proﬁles h1 (η) and h2 (η) have a parabolic distribution when n = 0. – Both the velocity and angular velocity proﬁles show a decelerating behavior with increasing values of porosity parameter Kp , while there is a rise in the temperature distribution. Page 12 of 12 Eur. Phys. J. 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