Effects of Radiation and Magnetic Field on Mixed Convection Stagnation-Point Flow Over a Cylinder in a Porous Medium Under Local Thermal Non-Equilibrium

Journal of Thermal Analysis and Calorimetry (2020) 140:1371–1391

https://doi.org/10.1007/s10973-019-08415-1 (0123456789().,-volV)(0123456789().,- volV)

Effects of radiation and magnetic field on mixed convection stagnation-point flow over a cylinder in a porous medium under local thermal non-equilibrium

1 2 3 Rasool Alizadeh • Nader Karimi • Amireh Nourbakhsh

Received: 3 April 2019 / Accepted: 24 May 2019 / Published online: 3 June 2019 Ó The Author(s) 2019

Abstract Heat transfer enhancement and entropy generation are investigated in a nanofluid, stagnation-point flow over a cylinder embedded in a porous medium. The external surface of cylinder includes non-uniform transpiration. A semi-similarity technique is employed to numerically solve the three-dimensional momentum equations and two-equation model of transport of thermal energy for the flow and heat transfer in porous media. The mathematical model considers nonlinear thermal radiation, magnetohydrodynamics, mixed convection and local thermal non-equilibrium in the porous medium. The nanofluid and porous solid temperature fields as well as those of Bejan number are visualised, and the values of circumferentially averaged Nusselt number are reported. The results show that thermal radiation significantly influences the temperature fields and hence affects Nusselt and Bejan number. In general, more radiative systems feature higher Nusselt numbers and less thermal irreversibilities. It is also shown that changes in the numerical value of Biot number can considerably modify the predicted value of Nusselt number and that the local thermal equilibrium modelling may significantly underpredict the Nusselt number. Magnetic forces, however, are shown to impart modest effects upon heat transfer rates. Yet, they can significantly augment frictional irreversibility and therefore reduce the value of Bejan number. It is noted that the current work is the first systematic analysis of a stagnation-point flow in curved configurations with the inclusion of nonlinear thermal radiation and local thermal non-equilibrium.

Keywords Heat transfer enhancement Á Stagnation-point ﬂow Á Local thermal non-equilibrium Á Convective-radiative heat transfer Á Nonlinear radiation Á MHD

2 List of symbols Br l ðÞkÁa Brinkman number Br ¼ f a Cylinder radius kf ðÞTwÀT1 B0 Magnetic field strength A1; A2; A3; A4 Constants Cp Specific heat at constant pressure asf Interfacial area per unit volume of porous media f ðÞg; u Function related to u-component of Be Bejan number velocity f 0ðÞg; u Function related to w-component of Bem Average Bejan number velocity Bi Biot number Bi ¼ hsf asf Áa 4kf GðÞg; u Function related to v-component of velocity 3 Gr gÁbf Áa ÁT1 Grashof number Gr ¼ 2 16Átf & Nader Karimi h Heat transfer coefficient [email protected] hsf Interstitial heat transfer coefficient 1 Department of Mechanical Engineering, Quchan Branch, k Thermal conductivity Islamic Azad University, Quchan, Iran k Freestream strain rate 2 School of Engineering, University of Glasgow, k1 Permeability of the porous medium Glasgow G12 8QQ, UK kÃ The mean absorption coefficient 3 r B2 Department of Mechanical Engineering, Bu-Ali Sina M Á 0 Magnetic parameter, defined as M ¼ University, Hamedan, Iran 2qf Ák 123 1372 R. Alizadeh et al.

S_000 NG gen Introduction Entropy generation number NG ¼ _000 S0 Nu Nusselt number Convective-radiative heat transfer in porous media is of Nu Circumferentially averaged Nusselt m growing importance in a wide range of technological number applications [1, 2]. The increasing use of porous media in p Fluid Pressure radiative energy systems such as solar collectors, solar P Non-dimensional fluid pressure reactors and porous burners has raised a pressing need for P The initial fluid pressure 0 further understanding and modelling of this combined Pr Prandtl number mode of heat transfer [3–5]. Further, the use of magnetic q Heat flow at the wall w effects in advanced energy technologies (e.g. cooling of q Thermal radiation r nuclear fusion reactors) necessitates inclusion of magne- r Radial coordinate tohydrodynamic effects in heat transfer analyses. The Re kÁa2 Freestream Reynolds number Re ¼ 2t current work aims to respond to these needs through con- Ã 3 Rd 16r T1 Radiation parameter Rd ¼ Ã duction of a numerical analysis on a generic configuration 3k Áks including a vertical cylinder covered by a porous medium SðÞu Transpiration rate function SðÞ¼u U0ðÞu kÁa and subject to an impinging flow. A magnetic field is _000 Characteristic entropy generation rate S0 applied to the system, and nanoparticles are added to fur- _000 Rate of entropy generation Sgen ther enhance electrical and thermal conductivity. The pri- T Temperature mary objective is to understand the influences of pertinent u, v, w Velocity components along (r À u À z)- parameters on this complex multiphysics problem. axis Modelling of thermal radiation in porous media has U0ðÞu Transpiration received a sustained attention over the last few decades z Axial coordinate [6, 7]. Most of the early works was done on porous burners, chiefly because of the importance of thermal radiation in Greek symbols this specific application; see the reviews of the literature in a Thermal diffusivity Refs. [8, 9]. For conciseness, here only the studies pub- b Thermal expansion coefficient lished over the last 10 years are briefly reviewed. Using c Modified conductivity ratio c ¼ kf homotopy technique, Hayat et al. [10] investigated the ÀÁ ks 2 effects of thermal radiation and magnetic fields on a flat g Similarity variable, g ¼ r a plate covered by a porous medium and under an impinging hgðÞ; u Non-dimensional temperature flow. Thermal radiation was modelled through a linear Tw hw Temperature parameter hw ¼ T1 version of Rosseland approximation, and an extensive 2 k Permeability parameter, k ¼ a parametric study was performed. Amongst other findings, 4k1 k1 Dimensionless mixed convection parameter Hayat et al. showed that the influences of radiation and k Gr gÁbf ÁT1 magnetic parameters on the temperature field are quite 1 ¼ Re2 ¼ 16Át2 f similar [10]. Bhattacharyya and Layek [11] analysed e Porosity boundary layer flow over a stretching porous flat surface by l Dynamic viscosity considering the effects of transpiration (fluid suction and t Kinematic viscosity blowing) and thermal radiation. A similarity solution was q Fluid density developed to investigate the effects of transpiration on the r Shear stress velocity and temperature fields [11]. These authors repor- r Electrical conductivity ted that by intensifying fluid suction, the wall temperature Ã r Stefan–Boltzman constant may increase in one class of their proposed solutions. In a / Nanoparticle volume fraction subsequent work, the same group of authors added the u Angular coordinate effects of micropolar fluid flow to their analysis [12]. It was shown that by increasing thermal radiation, the temperature Subscripts and thermal boundary layer thickness decrease and there- 1 Far field fore the heat transfer rate from the sheet enhances [12]. f Base fluid Radiative-conductive thermal boundary condition was nf Nanofluid applied to a cylindrical configuration, and the entropy np Nano-solid-particles generation was analysed analytically by Torabi and Aziz s Solid [13]. In this analysis, the internal heat generation and w Condition on the surface of the cylinder

123 Effects of radiation and magnetic field on mixed convection stagnation-point flow over a… 1373 thermal conductivity were assumed to be linear functions nonlinear thermal radiation model in comparison with its of temperature. The results reflected the strong influence of linear counterpart. Mushtagh et al. [21] considered nonlinear thermal radiation on the thermodynamic irreversibilities of radiation in the problem of solar absorption in a stagnation the system [13]. Heat and mass transfer in a radiative flow nanofluid flow in which Brownian motion of nanoparticles of Maxwell fluid over a titled stretching surface was was considered. Hayat et al. [22] investigated the three-di- studied by Ashraf et al. [14]. This work included exami- mensional viscous flow on a nanofluid over stretching sur- nation of a large number of parameters including Biot face in the presence of nonlinear thermal radiation and number, thermodiffusion parameter, Deborah number, magnetic effects. Their results showed that intensification of inclined stretching angle, radiation parameter, mixed con- thermal radiation leads to increases in the gradient of tem- vection parameter upon the thermal and concentration perature on the surface of the wall [22]. A similar analysis boundary layers. In particular, it was shown that the was reported by Farooq et al. [23]foraviscoelasticnano- thickness of thermal boundary layer decreases at higher fluid flow and Hayat et al. [24]onanunsteadyOldroyd-B thermal radiative powers [14]. Ashraf et al. also argued that fluid. In the latter, it was shown that in the case of convective radiation could have opposite effects upon Nusselt and thermal boundary condition, increases in the radiation Sherwood number [14]. In another theoretical work, parameter result in the thickening of the thermal boundary Zhang et al. [15] considered heat and mass transfer in a layer. Other investigated configurations with nonlinear nanofluid flow in porous media including magnetic and radiation include convectively heated cylinders [25], mixed thermal radiation effects. They also added fluid transpira- convection in stretched flow of an Oldroyd-B fluid with tion to their investigation and found that this effect as well convective condition [26], MHD Carreau fluid over stretched as those of magnetic field and radiation can strongly affect surface [27]. In all these works, it was asserted that imple- the velocity and temperature fields [15]. mentation of nonlinear radiation improves the analysis and Heat transferring, radiative stagnation-point flow of a makes the predictions of heat transfer rates more accurate. nanofluid over a flat plate was modelled by Makinde and The preceding survey of the literature clearly shows that Mishra [16]. These authors included the thermophoresis and a wealth of stagnation-point flow configurations over flat Brownian motion of the nanoparticles in their similarity surfaces with MHD and nonlinear radiation effects has solution [16]. They reported decreasing trends of Nusselt been already investigated. However, the corresponding number and increasing of Sherwood number at higher problem for curved surfaces has received very little atten- thermal radiation. An analogous study was reported by tion. Further, those cases that considered the flow inside a Hayat et al. [17] for non-Newtonian nanofluid flow in the porous medium often used local thermal equilibrium (LTE) presence of a magnetic field. In keeping with the previous assumption and the more accurate local thermal non- investigations, it was shown that application of a magnetic equilibrium (LTNE) approach has been rarely used toge- field results in depletion of momentum and thus weakening ther with radiation and magnetic effects. This is an of the hydrodynamic boundary layer [17]. Stagnation-point important shortcoming particularly in thermochemical flow of unsteady radiative Casson fluid with chemical porous systems for which the existence of local thermal reactions over a stretching surface was analysed by non-equilibrium is well demonstrated [28, 29]. The pri- Abbas et al. [18]. A temperature-dependent chemical reac- mary objective of the current work is to address these tion was considered in this work, and it was shown that issues through using a semi-similarity solution technique. through this effect radiation can significantly influence the mass transfer problem. As an important point, most of the existing studies are concerned with flat surfaces and only Theoretical and numerical methods few investigations have been reported on the boundary layers over curved surface [19]. Carbon nanotubes were Problem configuration, assumptions considered in a boundary layer flow of water over the axis of and governing equations cylinder. It was shown that, for all investigated nanotubes, fluid temperature near the surface of cylinder was higher at Figure 1 shows a schematic view of the problem under larger values of curvature parameter [19]. investigation. A heat transferring, vertical cylinder is A common point in the preceding works is the linear imbedded in porous media and subject to an impinging modelling of thermal radiation. Such simplification has been flow of nanofluid while a magmatic field is acting on the released in a few recent studies. For example, Hussian et al. system. The system is a simplified configuration of solar [20] implemented a nonlinear model of thermal radiation in reactors and solar collectors for energy storage purposes as their analysis of stagnation-point flow over a vertical flat discussed in detail in Ref. [3]. The external surface of the surface. This revealed that heat transfer enhancement could cylinder can include non-uniform transpiration. The fol- be more accurately predicted through incorporation of a lowing assumptions are made throughout this work. 123 1374 R. Alizadeh et al.

Fig. 1 Schematic diagram of a stationary cylinder under radial ψ r stagnation ﬂow and thermal Nanofluid in radiation of nanoﬂuid in porous porous media z media

B 0 B0

(ψ) U0

qr qr Thermal radiation effect

• The nanofluid flow is steady and laminar. q ou v ou v2 ou nf u þ À þ w • The nanofluid is assumed to be Newtonian and single 2 o o o e r r ur z phase. op l o2u 1 ou u 1 o2u o2u ¼À þ nf þ À þ þ • The cylinder is assumed to be infinitely long, and the or e or2 r or r2 r2 ou2 oz2 porous medium is homogenous, isotropic and under l À nf u; ð2Þ local thermal non-equilibrium. k1 • An external axisymmetric radial stagnation-point flow and that in angular direction and by including the magnetic of strain rate of k impinges on the cylinder. However, forces is given by [29] because of the non-uniformity of transpiration, the flow configuration around the cylinder can be non- q ov v ov uv ou nf u þ þ þ w axisymmetric. e2 or r ou r oz • The viscous dissipation of kinetic energy of the flow is 1 op l o2v 1 ov v 1 o2v 2 ou o2v ¼À þ nf þ À þ þ þ ignored. Also, porosity, specific heat, density and r ou e or2 r or r2 r2 ou2 r2 ou oz2 thermal conductivity are assumed to be constant and lnf 2 À v À rB0 Á v: thus the thermal dispersion effects are ignored. k1 • A moderate range of pore-scale Reynolds number is ð3Þ considered in the porous medium, and hence nonlinear effects in momentum transport are negligibly small. The transport of momentum in the axial direction is expressed by the following equation in which buoyancy The details of three-dimensional governing equations force is also included and boundary conditions are provided in the followings. q ow v ow ow The continuity of mass is expressed by nf u þ þ w e2 or r ou oz oðÞru ov ow þ þ r ¼ 0: ð1Þ op l o2w 1 ow 1 o2w o2w or ou oz ¼À þ nf þ þ þ ð4Þ oz e or2 r or r2 ou2 oz2 The momentum equation in the radial direction [30, 31] lnf 2 Æ ðÞqb nfgTðÞÀnf À T1 À rB0 Á w: reads k1

123 Effects of radiation and magnetic ﬁeld on mixed convection stagnation-point ﬂow over a… 1375

A two-equation model is used here to express the in which / represents the nanoparticle volume fraction. In transport of thermal energy in the porous medium. The Eq. (9) the subscripts ‘‘f’’ and ‘‘np’’ denote base fluid and energy equation for the nanofluid phase is written as [32] solid fraction properties, respectively. oT v oT oT The hydrodynamic boundary conditions are given by the u nf þ nf þ w nf or r ou oz following expressions. k o2T 1 oT 1 o2T o2T r ¼ a: w ¼ 0; v ¼ 0; u ¼ÀU0ðÞu ð10Þ ¼ ÀÁnf nf þ nf þ nf þ nf o 2 o 2 o 2 o 2 ð5Þ 2 qCp r r r r u z a nf r ¼1: w ¼ 2kz ; lim rv ¼ 0; u ¼Àkr À ð11Þ h :a r!1 r þ ÀÁsf sf ðÞT À T ; qC s nf p nf Also, the followings illustrate the boundary conditions while the transport of thermal energy in the solid phase is with respect to u (angular coordinate) expressed by ourðÞ; 0 ourðÞ; 2p urðÞ¼; 0 urðÞ; 2p ; ¼ ; o2 o o2 o2 ou ou Ts 1 Ts 1 Ts Ts ð12Þ ks þ þ þ À hsf Á asfðÞTs À Tnf ovrðÞ; 0 ovrðÞ; 2p or2 r or r2 ou2 oz2 vrðÞ¼; 0 vrðÞ; 2p ; ¼ : ou ou 1 o À ðÞ¼r Á q 0: r or r Equation (10) represents no-slip conditions on the ð6Þ external surface of the cylinder, and Eq. (11) shows that the viscous flow solution approaches, in an analogous way Using Rosseland approximation [33–35] for thermal to the Hiemenz flow, the potential flow solution as r !1 radiation, the radiative heat flux is simplified as follows [36–39]. This can be verified by starting from the conti- Ã o 4 oðÞru o o 4r Ts 1 v w q ¼À : ð7Þ nuity equation in the followings. À r or À ou ¼ oz Con- r 3kÃ or stant ¼ 2kz and integrating in r and z directions with Equation (6) can be now re-written in the form of boundary conditions, w ¼ 0 when z ¼ 0 and u ¼ÀU ðÞu 0 o2T 1 oT 1 o2T o2T when r ¼ a. k s þ s þ s þ s À h Á a ðÞT À T s or2 r or r2 ou2 oz2 sf sf s nf The boundary condition associated with the energy equations in the porous region is given by 1 o 16rÃ oT þ r: T3 s ¼ 0: r or 3kÃ s or Tnf ¼ Tw ¼ Constant, r ¼ a: ð13Þ ð8Þ Ts ¼ Tw ¼ Constant;

It should be noted that in the existing literature, T4 in Tnf ¼ T1; s r ¼1: Eq. (7) is often expanded and linearised about the ambient Ts ¼ T1: temperature T1 [10–12]. However, in the present case this simplification has been avoided and the full nonlinear form and the two boundary conditions with respect to angular coordinate, u are of the expression for thermal radiationÀÁ is implemented. Ã Ã In Eqs. (1–8) p, qnf, lnf , T, qCp nf, knf , b, g, T1, r , k TnfðÞ¼r; 0 TnfðÞr; 2p ; TsðÞ¼r; 0 TsðÞr; 2p ; and qr are the pressure, density, kinematic viscosity of the oT ðÞr; 0 oT ðÞr; 2p oT ðÞr; 0 oT ðÞr; 2p ð14Þ nf ¼ nf ; s ¼ s ; nanofluid, temperature, the heat capacitance of the nanofluid, ou ou ou ou thermal conductivity of the nanofluid, thermal expansion coefficient, gravitational acceleration, prescribed tempera- where Tw is temperature of the cylinder surface and T1 ture at the wall, Stefan–Boltzman constant, constant, the denotes the freestream temperature. mean absorption coefficient and radiative heat flux, respectively. These properties are evaluated inside the boundary Self-similar solutions layer and in the vicinity of the flow impingement point. The nanofluid properties are defined by the followings Reducing the governing Eqs. (1–7) by applying the fol- [33, 34], lowing similarity transformations, ÀÁ ÀÁÀÁ q 1 / q /q ; q C 1 / q C /q C nf ¼ ðÞÀ f þ np Á p nf¼ ðÞÀ ÀÁÁ p fþ Á p np lf knf knp þ 2kf À 2/ÀÁkf À knp lnf ¼ 2:5 ; ¼ ðÞ1 À / kf knp þ 2kf þ 2/ kf À knp ð9Þ

123 1376 R. Alizadeh et al.

o2 o k Á a k Á a 00 1 G 1 f u ¼Àpffiffiffi f ðÞg; u ; v ¼ pffiffiffi GðÞg; u ; w e Á gG þ À þ Re Á A1 g g 4gou2 2g ou koG G oG ¼ 2kf 0ðÞÀg; u z; p ¼ q k2a2P; ð15Þ Á ðÞ1 À / 2:5 f Á G0 À À e2 Á G Á ½k þ M g ou f 2g ou ÀÁ ¼ 0: ð21Þ r 2 in which g ¼ a is the dimensionless radial variable, results in the following dimensionless equations. Relations Considering conditions (10)–(12), the boundary and (15) automatically satisfy the continuity of mass and sub- initial conditions for Eq. (21) can be written as stitution into Eqs. (2), (3) and (4) results in: oGðÞ1; u g ¼ 1 : GðÞ¼1; u 0; ¼ 0 ð22Þ ou 1 o3G 1 oG0 1 oG0 1 oG 1 o2f 0 egf 000 þÀ À þ À þ 2 o 3 o o 2 o o 2 g : G ; u 0 23 8g u 2 u 2g u 2g u 4g u !1 ðÞ¼1 ð Þ f oG0 f oG þ Re Á A Á ðÞ1 À / 2:5 1 þ ff 0 À ðÞf 0 2À þ oGðÞg; 0 oGðÞg; 2p 1 2g ou 2g2 ou GðÞ¼g; 0 GðÞg; 2p ; ¼ ð24Þ # ou ou G of 0 G o2G f 0 oG 1 oG 2 À þ þ À þ e2 Á k½1 À f 0 2g ou 4g2 ou2 2g ou 4g2 ou Equation (5) is non-dimensionalised by using the following transformation 1 oG Æ e2 Á A Á k Á ðÞh À 1 h þ e2 Á Re Á M 1 þ À f 0 ¼ 0; 4 1 w nf o TðÞÀg; u T1 2g u hgðÞ¼; u ; ð25Þ T T ð16Þ w À 1 1 f 2 1 and therefore P À P0 ¼À 2 À 2:5 2e g e Á A Á ðÞ1 À / TðÞ¼g; u T1½1 þ ðÞhw À 1 h ; ð26Þ 1 f 0 1 g 1 o2f By substituting Eqs. (15) and (26) into Eq. (5) and À r dg Re 4Re g2 ou2 neglecting the small dissipation terms, the following 1 1 g 1 oG k g f equation is derived. þ r dg þ r dg 2 o ð17Þ 2 2Re 1 g u Re 1 g 1 o h A G oh "# gh00 þ h0 þ nf þ Re Á Pr Á 2 Á f Á h0 À nf nf nf o 2 nf o 1 1 k z 2 4g u A3 2g u À 2 2 þ 2:5 Bi e A1ðÞ1 À / Re a þ ðÞ¼h À h 0; A s nf 1 1 g 1 of 3 þ r G2 þ G dg; 27 2 2 2 o ð Þ e k 1 g u 2 Tw hsf asf Áa kÁa2 a2 in which hw ¼ is the temperature parameter, Bi ¼ where Re ¼ is the freestream Reynolds number, k ¼ T1 4kf 2tf 4k1 3 16rÃT3 gÁbf Áa ÁT1 1 is the Biot number, Rd ¼ Ã is the radiation parameter, is the reciprocal of Darcy number, Gr ¼ 2 is the 3k Áks 16Átf Gr gÁbf ÁT1 and the thermal boundary conditions for the nanofluid Grashof number, k1 ¼ Re2 ¼ 16 t2 is the dimensionless Á f phase reduce to mixed convection, and prime indicates differentiation with respect to g. g ¼ 1 : hnfðÞ¼1; u 1 ð28aÞ Considering Eqs. (10), (11), and (12), the boundary g !1: hnfðÞ¼1; u 0 ð28bÞ conditions for Eqs. (16) and (17) reduce to: ohnfðÞg; 0 ohnfðÞg; 2p 0 h ðÞ¼g; 0 h ðÞg; 2p ; ¼ ð29a; bÞ g ¼ 1 : f ðÞ¼1; u 0; f ðÞ¼1; u SðÞu ð18Þ nf nf ou ou 0 g !1: f ðÞ¼1; u 0 ð19Þ Substitution of Eqs. (15) and (26) into Eq. (8) results in of ðÞg; 0 of ðÞg; 2p o2 f ðÞ¼g; 0 f ðÞg; 2p ; ¼ ; ð20Þ 00 0 1 hs o o gh þ h þ À Bi Á c Á ðÞhs À hnf u u s s 4g ou2 ð30Þ o hi U0ðÞu 3 where SðÞ¼u is the transpiration rate function. þ R Á g Á ðÞ1 þ ðÞh À 1 h Áh ¼ 0; kÁa d og w s s Combining Eq. (15) with Eqs. (3) and (4) yields a differential equation in terms of GðÞg; u and an expression for where c ¼ kf is the modified conductivity ratio and the ks the pressure thermal boundary conditions for the solid phase of the porous medium are as follows

123 Effects of radiation and magnetic ﬁeld on mixed convection stagnation-point ﬂow over a… 1377

g ¼ 1 : hsðÞ¼1; u 1 ð31aÞ Shear stress and Nusselt number g !1: h ðÞ¼1; u 0 ð31bÞ s The shear stress induced by the nanoﬂuid ﬂow on the ohsðÞg; 0 ohsðÞg; 2p external surface of the cylinder is given by: hsðÞ¼g; 0 hsðÞg; 2p ; o ¼ o ð32; bÞ u u ow r ¼ lnf o ; ð38Þ In Eqs. (16), (17), (21) and (27), A1, A2, A3 and A4 are r r¼a constants in the following forms:

ÀÁ q Á C qnp ÀÁp np A1 ¼ ðÞþ1 À / /; A2 ¼ ðÞþ1 À / / q q Á C f ÀÁ p f ð33Þ q b knf knp þ 2kf À 2/ÀÁkf À knp ðÞÁ np A3 ¼ ¼ ; A4 ¼ ðÞþ1 À / / kf knp þ 2kf þ 2/ kf À knp ðÞq Á b f

It is noted that Eq. (27) is the complete form of where lnf is the nanofluid viscosity. Employing Eq. (15), a Eqs. (14) in Ref. [40]. Equations (16), (21), (27) and (30), semi-similar solution for the shear stress on the surface of and the boundary conditions (18–20), (22–24), (28–29) and the cylinder can be developed. This reads (31–32), were numerically solved by employing an impli- 2 r Á a À2:5 cit, iterative tri-diagonal finite-difference method [41–43]. r ¼ l ½)2kzf 00ðÞ1; u ¼ 1 À / f 0ðÞ1; u : nf a 4lkz GðÞ¼g; u 0 ð39Þ hi 1 o2f 0 egf 000 þ f 00 þ þ Re Á A Á ðÞ1 À / 2:5 1 þ ff 0 À ðÞf 0 2 The local heat transfer coefficient and rate of heat o 2 1 4g u transfer are defined by the following relations 2 0 2 þ e Á k½Æ1 À f e Á A4 Á k1 Á ðÞhw À 1 hnf ÀÁ oTnf qw Àknf o 2knf ohnfðÞ1; u þ e2 Á Re Á M½¼1 À f 0 0; h ¼ ¼ r r¼a ¼À ; ð40Þ Tw À T1 Tw À T1 a og ð34Þ and 1 f 2 1 P P À 0 ¼À 2 À 2:5 2knf ohnfðÞ1; u 2e g e Á A1 Á ðÞ1 À / q ¼À ðT À T Þ: ð41Þ "# ! w a og w 1 f 1 g 1 o2f k g f À r dg þ r dg 35 Re 4Re g2 ou2 Re g ð Þ Therefore, Nusselt number is expressed as "#1 1 h Á a knf 1 1 k z 2 Nunf ¼ ¼À hnfðÞ¼À1; u A3 Á hnfðÞ1; u : ð42Þ 2kf kf À 2 2 þ 2:5 ; e A1ðÞ1 À / Re a 1 o2h A ÀÁ Entropy generation gh00 þ h0 þ nf þ Re Á Pr Á 2 Á f Á h0 nf nf 4g ou2 A nf 3 ð36Þ Bi The volumetric rate of local entropy generation in the þ ðÞ¼hs À hnf 0; A3 porous region of the problem can be expressed by the following relation [44–48]: 1 o2h gh00 þ h0 þ s À Bi Á c Á ðÞh À h s s 4g ou2 s nf ð37Þ o hi þ R Á g Á ðÞ1 þ ðÞh À 1 h 3Áh ¼ 0: d og w s s

123 1378 R. Alizadeh et al.

"# " " "# 2 2 2 2 2 2 knf oTnf 1 oTnf ks oTs 1 oTs Re Á A Á h 1 oh S_000 ¼ þ þ þ Be ¼ 3 w gh02 þ nf gen 2 or r ou T2 or r ou 2 nf 2 o Tnf s ½1 þ ðÞhw À 1 hnf 4g u # " Ã o 2 2 2 16r Ts 1 1 Re h 1 oh þ T3 þ h a ðÞT À T À Á w 02 s Ã s o sf sf s nf þ ghs þ 3k r Tnf Ts 2 4g2 ou "#" c Á ½1 þ ðÞhw À 1 hs # o 2 2 o 2 o 2 2lnf u u w lnf 1 w ÂÃ þ þ þ þ Rd 3 0 2 T or r oz T r ou þ 2 g½1 þ ðÞhw À 1 hs Á ðÞÁhw À 1 hs 1 1 ðÞh À 1 # ,"w (" o 2 o 2 ÂÃ2 w 1 u lnf 2 2 rB0 2 þ þ þ u þ w þ w ; Re Á Br Á hw À2:5 002 02 or r ou k1T1 T1 ðÞ1 À / gf þ 4f ðÞhw À 1 # ð43Þ f 2 ff 0 1 of 0 2 1 of 2 000 þ À2 þ þ S_ 8k T T 2t o 2 o where N gen and S_000 f ÁðÞwÀ 1 f are the character- g g g u 4g u G ¼ S_000 0 ¼ kÁa4ÁT2 0 1 "# #) istic entropy generation rate. Using the similarly variables f 2 þk þ4f 02 þ 2Mf 02 : given in Eqs. (15) and (43), the non-dimensional form of g local entropy generation (NG) is given by "# ð45Þ Re Á A Á h2 1 oh 2 N ¼ 3 w gh02 þ nf G 2 nf 4g2 ou ½1 þ ðÞhw À 1 hnf Grid independency and validation " 2 o 2 Re Á hw 02 1 hs þ 2 ghs þ To verify grid independency of the numerical solution, c Á ½1 þ ðÞh À 1 h 4g2 ou w s # Table 1 reports the average value of Nusselt and Bejan Rd 3 2 number as well as dimensionless shear stress with varying þ 2 g½1 þ ðÞhw À 1 hs Á½ðÞÁhw À 1 h0s ðÞhw À 1 mesh densities of 51 Â 18, 102 Â 36, 204 Â 72, 408 Â 144 2 and 816 Â 288. It is clear from Table 1 that there are no hw þ Bi Á Re ðÞh À h considerable changes of Num and Bem for (g; u) mesh sizes ðÞh À 1 s nf w of (204 Â 72), (408 Â 144) and (816 Â 288). Hence, in this 1 1 Á À work a (408 Â 144) grid in g À u directions was used for 1 þ ðÞhw À 1 hnf 1 þ ðÞhw À 1 hs the computational domain reported. A non-uniform grid (" 2 was implemented in g-direction to resolve the strong gra- Re Á Br Á hw À2:5 002 02 f þ ðÞ1 À / gf þ 4f þ dients around the external surface of the cylinder, and a ðÞhw À 1 g # uniform mesh was applied in u direction. The computa- 2 2 ff 0 1 of 0 1 of tional domain extends over u ¼ 360 and g ¼ 15, À2 þ þ max max o 2 o g g u 4g u where gmax corresponds to g !1. It is noted that, for all "# ) f 2 þk þ4f 02 þ 2Mf 02 ; g Table 1 Grid independence study at Re ¼ 10; k ¼ 10; k1 ¼ 1:0; M ¼ 1:0; Bi ¼ 0:1; Rd ¼ 1:0; hw ¼ 1:2

ð44Þ rmÁa Mesh size Num 4lkz Bem 2 in which Br ¼ lf ðÞkÁa is the Brinkman number. The kf ðÞTwÀT1 51 9 18 8.658791 21.15879 0.54677 Bejan number, deﬁned as the ratio of entropy generation by 102 9 36 8.876546 21.45861 0.58179 heat transfer to the total entropy generation, can be further 204 9 72 9.125783 21.89883 0.60548 expressed as 408 9 144 9.709241 22.52879 0.63035 816 9 88 9.727194 22.55027 0.63005

123 Effects of radiation and magnetic ﬁeld on mixed convection stagnation-point ﬂow over a… 1379

agreement between the two datasets conﬁrms validity of the numerical simulations.

Results and discussion

Two types of fluid including pure water and a CuO–water nanofluid, with varying concentration of nanoparticles, are used in the rest of this study. Table 5 summarises the thermophysical properties of the nanofluid, and Table 6 provides the default values of the pertinent parameters.

Temperature field, Nusselt number and friction coefficient

The hydrodynamics of nanofluids under magnetic effects are already well investigated [32], and therefore they are Fig. 2 The computational grid used in this study not further elaborated in here. Nonetheless, to illustrate the flow field, Fig. 3a depicts the distribution of f (see Eq. 15) for different values of Reynold number. Complexity of the investigated cases, the computational domain fully contains flow field is completely evident in this figure. The radial the entire momentum and thermal boundary layers. Fig- flow field includes a stagnation and a low-velocity region ure 2 shows the computational grid utilised in the current that form as a result of the interactions between the study. A convergence criterion was applied to the numer- impinging uniform flow and the non-uniform transpiration ical simulations, in which when the difference between the (see Fig. 1). This low-velocity region diminishes in size as two consecutive iterations became less than 10À7. The the value of Reynold number increases. Figure 3b illus- solution was considered to have converged and therefore trates the variations in the temperature field of the nano- the iterative process was terminated. The numerical error fluid induced by the changes in Reynold number. As of the implemented numerical scheme is estimated to be of expected, at low Reynolds numbers the temperature field is nearly symmetric, and the thermal boundary layer is uni- OðDgÞ2 [41–43]. The solutions developed in ‘‘Self-similar form and thick. However, increase in Reynolds number solutions’’ and ‘‘Shear stress and Nusselt number’’ sections changes this significantly and results in reducing the were validated by comparing the average Nusselt number, thickness of thermal boundary layer. This is due to the shear stress and velocity parameters with those obtained well-known dependency of the thickness of velocity from the literature for flows over cylinders with no tran- boundary layer upon Reynolds number and the relation spiration and infinitely large permeability. Tables 2, 3 and between the thicknesses of thermal and velocity boundary 4 show the outcomes of this comparison. The excellent layers through Prandtl number [52]. An exception to this

Table 2 Comparison between the current results and those of Gorla [49] on average Nusselt number and average shear stress for different values of magnetic parameter when Re ¼ 100; k ¼ 0; / ¼ 0; k1 ¼ 1:0; Bi ¼ 1000; Rd ¼ 0

rmÁa Pr Num 4lf kz Present work Gorla. [49] Deviation percentage/% Present work Gorla [49] Deviation percentage/%

0.01 18.2115 18.2521 0.2229 1.6521 1.6541 0.1211 0.7 15.3561 15.1712 1.2188 7.1875 7.1941 0.0918 1000 13.1121 13.2514 1.0624 91.1342 91.9241 0.8667

123 1380 R. Alizadeh et al.

Table 3 Comparison between the present work and the results of Gorla [51] in the limit of very large porosity and permeability Re f h Gorla [51] Present work Deviation percentage/% Gorla [51] Present work Deviation percentage/%

0.01 0.12075 0.12051 0.1991 0.84549 0.84557 0.0095 0.1 0.22652 0.22659 0.0309 0.73715 0.73701 0.0190 1.0 0.46647 0.46683 0.0772 0.46070 0.46045 0.0543 10 0.78731 0.78725 0.0076 0.02970 0.02983 0.4377

Table 4 Comparison between the present work and the results of Wang [50] in the limit of very large porosity and permeability g Re = 1.0 Re =10 Wang [50] Present work Wang [50] Present work FF0 ff0 Deviation percentage of ff0 fF0 Deviation percentage of f/% f0/%

1.2 0.02667 0.25302 0.02693 0.25993 0.9749 0.06638 0.58982 0.06631 0.58961 0.0356 1.4 0.09665 0.43724 0.09652 0.43710 0.1347 0.21400 0.84821 0.21393 0.84793 0.0330 1.6 0.19836 0.57315 0.19828 0.57329 0.0403 0.39532 0.94852 0.39541 0.94827 0.0264 1.8 0.32361 0.67444 0.32365 0.67438 0.0123 0.58919 0.98380 0.58914 0.98351 0.0295 2.0 0.46674 0.75054 0.46683 0.75046 0.0193 0.78731 0.99522 0.78735 0.99483 0.0392

Table 5 Thermo-physical Physical properties C /J kg-1 K-1 q/kg m-3 K/W m-1 K-1 b 9 10-5/K-1 properties of the base ﬂuid and p nanoparticle [36] Fluid phase (water) 4179 997.1 0.613 21 CuO 531.8 6320 76.5 1.8

Table 6 Default values of the Simulations parameters gukeRe SðÞu Bi Br h c ; MR k simulation parameters w d 1 1.45 72° 10 0.9 5.0 LnðÞu 0.1 2.0 1.2 1.5 0.1 1.0 1.0 1.0

general behaviour is the low-velocity region in which the thermal diffusivity of the nanofluid is the reason for the thickness of the boundary layer remains large. observed thickening of the boundary layer. Figure 4b Figure 4a shows the influences of volumetric concen- shows that variations in the mixed convection parameter do tration of nanoparticles on the nanofluid temperature field. not have any noticeable influences upon most of the ther- Increasing the volumetric concentration of nanoparticles mal boundary layer. Yet, increases in the value of mixed from 0 to 10% has resulted in a noticeable increase in the convection parameter lead to shrinkage of the high-tem- thickness of the thermal boundary layer. This can be perature spot around the low-velocity region (see Fig. 3a), explained by noting that thermal conductivity of the which could be due to the relative weakening of inertia nanofluid increases by magnifying the volumetric concen- forces. tration of nanoparticles. The resultant increase in the

123 Effects of radiation and magnetic ﬁeld on mixed convection stagnation-point ﬂow over a… 1381

Fig. 3 Effects of Reynolds (a) (b) number on a f ðÞg; u , b hnf ðÞg; u , / ¼ 0:1; Re = 0.1 Re = 0.1 M ¼ 1:0; Rd ¼ 1:0; k 10; k 1:0; Bi 0:1 ¼ 1 ¼ ¼ 1.5 0.8 1.35 0.73 1.2 0.66 1.05 0.59 0.9 0.52 0.75 0.45 0.6 0.38 0.45 0.31 0.24 0.3 0.15 0.17 0.1 0

Re = 1.0 Re = 1.0

1.5 0.8 1.35 0.73 1.2 0.66 1.05 0.59 0.9 0.52 0.75 0.45 0.6 0.38 0.45 0.31 0.3 0.24 0.15 0.17 0 0.1

Re = 10 Re = 10

1.5 0.8 1.35 0.73 1.2 0.66 1.05 0.59 0.9 0.52 0.75 0.45 0.6 0.38 0.45 0.31 0.3 0.24 0.15 0.17 0 0.1

Figure 5 shows the response of the nanofluid and porous the nanofluid phase by means of heat exchanges between solid temperature fields to the increases in radiation the porous solid and nanofluid phase. Figure 6 shows that parameter. By introduction of thermal radiation (increasing Biot number is a key parameter affecting the temperature

Rd from 0 to 1), the thickness of thermal boundary layer in fields of nanofluid and porous solid phase. At low Biot nanofluid and the thickness of the heated porous solid have numbers (Bi = 0.1), the two temperature fields can be increased noticeably. In the current modelling setting, significantly different, in which the temperature of the solid radiation can be viewed as an added thermal diffusivity and phase is much higher than that of nanofluid phase. This is thus introduction of thermal radiation is equivalent to because of the poor heat exchanges between solid and making the medium more diffusive. It is well known that nanofluid phases at low Biot numbers. However, at higher increases in thermal diffusivity lead to deeper thermal Biot numbers (e.g. Bi = 10) the two temperature fields penetration lengths in the solid phase. This increase in feature major similarities, as there is now a strong capa- thermal diffusivity thickens the thermal boundary layer in bility for the exchange of heat between the two phases.

123 1382 R. Alizadeh et al.

Fig. 4 hnf ðÞg; u for varying (a) (b) values of a nanoparticle volume fraction, b mixed convection λ 1 = 0.1 parameter Rd ¼ 2:0; φ = 0 / ¼ 0:05; M ¼ 0; Re ¼ 10; 0.7 Bi ¼ 0:1; k ¼ 10; k1 ¼ 1:0 0.64 0.8 0.58 0.7 0.52 0.6 0.46 0.5 0.4 0.4 0.34 0.3 0.28 0.2 0.22 0.1 0.16 0.1

φ λ = 0.05 1 = 10

0.7 0.64 0.8 0.58 0.7 0.52 0.6 0.46 0.5 0.4 0.4 0.34 0.3 0.28 0.2 0.22 0.1 0.16 0.1

φ = 0.1 λ 1 = 30

0.7 0.64 0.8 0.58 0.7 0.52 0.6 0.46 0.5 0.4 0.4 0.34 0.3 0.28 0.2 0.22 0.1 0.16 0.1

Theoretically, it is expected that in the limit of infinite Biot Nusselt number with these parameters is provided in number the solid and nanofluid temperature fields become Tables 7, 8 and 9. Figure 7 shows that increases in Biot identical, and therefore a local thermal equilibrium (LTE) number lead to a significant drop of Nusselt number for a condition applies. However, as depicted by Fig. 6, the two large fraction of the cylinder circumference. This is an temperature fields at low values of Biot numbers can be important result from the modelling viewpoint, as it indi- quite distinctive. This clearly reflects the importance of cates that LTE models, which effectively assume very employing a local thermal non-equilibrium approach in the large Biot numbers, may highly underpredict the numerical current problem. value of Nusselt number. Hence, the use of LTNE mod- Figures 7 and 8 show the circumferential variation of elling is an important necessity in the current problem. Nusselt number as the pertinent parameters vary. Further According to Fig. 7b, temperature parameter plays a key information on variations of the circumferentially averaged role in determining the value of Nusselt number. Increasing

123 Effects of radiation and magnetic ﬁeld on mixed convection stagnation-point ﬂow over a… 1383

(a) (b)

Rd = 0 Rd = 0

0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

Rd = 1.0 Rd = 1.0

0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

Rd = 3.0 Rd = 3.0

0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

Fig. 5 Effects of radiation parameter on a hnf ðÞg; u , b hsðÞg; u , / ¼ 0:05; M ¼ 0; Re ¼ 10; Bi ¼ 0:1; k ¼ 10; k1 ¼ 1:0 the temperature parameter can lead to large enhancement Figure 7c confirms that increases in the concentration of of heat transfer coefficient when forced and natural con- nanoparticles magnify the value of Nusselt number. This vection is of comparable significance. This is due to the can be attributed to the higher thermal conductivity of direct dependency of energy equation (see Eq. 27) and thus nanofluid with larger concentration of nanofluid. The natural convection upon the wall temperature. At higher observed trend is consistent with that reported in other values of temperature parameters, natural convection is studies of nanofluid flow in porous media, see for example stronger and as long as mixed convection parameter is Refs. [33, 34]. around or below unity, the Nusselt number is enhanced.

123 1384 R. Alizadeh et al.

Fig. 6 Effects of Biot number (a) (b) on a hsðÞg; u , b BeðÞg; u , / ¼ 0:05; M ¼ 1:0; Re ¼ 10; Bi = 0.1 Bi = 0.1 Rd ¼ 1:0; Br ¼ 2:0; k ¼ 10; k1 ¼ 1:0; hW ¼ 1:2 0.8 0.8 0.7201 0.73 0.6402 0.66 0.5603 0.59 0.4804 0.52 0.4005 0.45 0.3206 0.38 0.2407 0.31 0.1608 0.24 0.0809 0.17 0.001 0.1

Bi = 1.0 Bi = 1.0

0.8 0.8 0.73 0.7201 0.66 0.6402 0.59 0.5603 0.52 0.4804 0.45 0.4005 0.38 0.3206 0.31 0.2407 0.24 0.1608 0.17 0.0809 0.1 0.001

Bi = 10 Bi = 10

0.8 0.8 0.73 0.7201 0.66 0.6402 0.59 0.5603 0.52 0.4804 0.45 0.4005 0.38 0.3206 0.31 0.2407 0.24 0.1608 0.17 0.0809 0.1 0.001

It is well established that, in general, Reynold number parameter from 0 to 7 increases the value of circumferen- dominates forced convection of heat and thus Nusselt tially average Nusselt number by more than 80%, reflecting number is strongly enhanced by increases in Reynolds the strong influence of this parameter upon the heat transfer number [52]. In keeping with this generic view, Fig. 7d process. As discussed earlier, increases in the radiation shows that increasing Reynolds number can substantially parameter are equivalent to enhancement of the thermal increase the value Nusselt number. Table 9 shows that conductivity of the solid phase which then leads to increasing the value of Reynolds number from 1 to 100 improvement of heat transfer in the nanofluid phase. The leads to an enhancement of Nusselt number by almost 15 values of mixed convection parameter greater than one can times. Figure 8a indicates that increases in radiation slightly enhance Nusselt number and the inverse effect is parameter can result in major enhancement of Nusselt observed for negative values of mixed convection param- number. According to Table 7, increasing radiation eter (see Fig. 7b). Figure 8c and d shows that changes in

123 Effects of radiation and magnetic ﬁeld on mixed convection stagnation-point ﬂow over a… 1385

12 (a) (b) 48 Bi = 0.1 θ Re = 5.0 10 Bi = 1.0 w = 0.4 40 θ λ = 10 Bi = 10 w = 0.8 θ R = 1.0 8 Bi = 100 w = 1.2 d 32 λ1 = 1.0 Bi = 500 θ w = 1.6 θ = 2.0 Bi = 0.1 6 w Nu 24

4 Re = 10 16 λ = 10 2 R = 1.0 d 8 λ1 = 1.0 0 0 20 40 60 80 100 120 140 160 180 0 0 20406080100 120 140 160 180 ϕ ϕ

(c)18 (d) 80 Re = 5.0 φ = 0 Re = 0.1 λ = 10 70 15 φ = 0.05 Re = 1.0 R = 1.0 φ = 0.1 d 60 Re = 10 λ = 1.0 12 φ = 0.125 1 Re = 50 φ = 0.15 Bi = 0.1 50 Re = 100 Nu

9 Nu40 Nu φ = 0.1 30 λ 6 = 10 R = 1.0 20 d λ = 1.0 3 1 10 Bi = 0.1 0 0 0 20406080100 120 140 160 180 0 20406080100 120 140 160 180 ϕ ϕ

Fig. 7 Variation of Nu for different values of a Biot number (Bi), b temperature parameter (hw), c nanoparticle volume fraction (/), d Reynolds number (Re), Re ¼ 5:0; Bi ¼ 0:1; c ¼ 1:5; k ¼ 10

20 9 (a)R = 0 φ = 0.1 (b) d λ λ = 10 1 = 0.1 Rd = 1 λ 16 Re = 5.0 1 = 1.0 R = 2 7 d λ = 1.0 λ1 = 10 R = 3 1 d Bi = 0.1 λ = 30 12 1 Rd = 4 λ1 = – 20 5 Nu Nu 8 φ = 0.1 λ = 10 3 Re = 5.0 4 Rd = 1.0 Bi = 0.1 0 1 02040 60 80 100 120 140 160 180 02040 60 80 100 120 140 160 180 ϕ ϕ

Nu 6 φ Nu M = 4 = 0.1 φ = 0.1 λ = 1.0 λ 1 3.5 1 = 1.0 Re = 5.0 Re = 5.0 3 Rd = 1.0 Rd = 1.0 Bi = 0.1 Bi = 0.1 λ = 10 1.5 0 0 20406080100 120 140 160 180 0 153045607590 ϕ ϕ

Fig. 8 Variation of Nu for different values of a radiation parameter (Rd), b dimensionless mixed convection (k1), c permeability parameter (k), d Magnetic parameter (M), Rd ¼ 1:0; Re ¼ 5:0; Bi ¼ 0:1; c ¼ 1:5; k ¼ 10

123 1386 R. Alizadeh et al.

Table 7 Effects of Biot number, modiﬁed conductivity ratio and on average Nusselt and Bejan numbers number when Re ¼ 1:0; k ¼ 10; k1 ¼ 1:0; M ¼ 1:0; Bi ¼ 0:1; Rd ¼ 1:0; hw ¼ 1:2; / ¼ 0:1

Bi Num Bem c Num Bem Rd Num Bem

0.1 4.380546 0.045425 0.1 2.282902 12.667 0 2.023269 0.94563 1.0 4.361441 0.021875 1.0 2.283273 1.2233 1 2.283467 0.80084 10 4.347604 0.013311 2 2.283652 0.59028 3 2.780153 0.6025 100 4.320692 0.012485 5 2.284617 0.21592 5 3.254344 0.47515 200 4.316352 0.012231 10 2.285825 0.097321 7 3.712680 0.38791

Table 8 Effects of mixed convection, permeability and magnetic parameter on average Nusselt and Bejan number when, Re ¼ 1:0; k ¼ 10; k1 ¼1:0; M ¼ 1:0; Bi ¼ 0:1; Rd ¼ 1:0; hw ¼ 1:2 k1 Num Bem k Num Bem MNum Bem

0.1 4.376399 0.046931 1.0 4.357398 0.02536 0 4.266912 0.047515 1.0 4.380546 0.045425 10 4.380546 0.045425 1 4.380546 0.045425 10 4.420391 0.034127 100 4.417175 0.004072 3 4.383869 0.041768 30 4.500133 0.021613 1000 4.439515 0.000405 5 4.386696 0.038668 - 20 4.274283 0.014048 5000 4.446507 0.000081 7 4.389146 0.036002

Table 9 Effects of Reynolds and Prandtl number and temperature parameter on average Nusselt and Bejan number when, Re ¼ 1:0; k ¼ 10; k1 ¼ 1:0; M ¼ 1:0; Bi ¼ 0:1; Rd ¼ 1:0; hw ¼ 1:2

Re Num Bem Pr Num Bem hw Num Bem

0.1 3.904873 0.013189 0.1 3.906162 0.011565 1 4.380546 0.045425 1.0 4.380546 0.045425 0.4 4.063833 0.018016 2 22.02809 0.170111 10 9.731774 1.05851 0.7 4.221933 0.029383 3 93.86373 0.345881 50 34.83388 2.19961 1.0 4.380546 0.045425 4 263.9523 0.504231 100 64.18778 2.32591 10 9.460663 1.34883 5 583.6297 0.590221

the permeability of the porous medium and intensity of the Nusselt number. The resultant increase in heat transfer rate magnetic field have modest effects upon Nusselt number. relaxes the temperature gradients in the system and thus reduces the thermal irreversibilities. This is depicted by Thermodynamic irreversibilities Fig. 9a and in the reported values of average Bejan number in Table 7. The complex nature of entropy generation in Figure 9 shows the distribution of Bejan number and the current problem is well reflected by Fig. 9b. Switching entropy generation number in the domain with varying from pure water to nanofluid appears to have a strong effect values of radiation parameter, while Table 7 provides the upon the distribution of Bejan number (see Fig. 10a). In the average value of Bejan number. As discussed in the pre- absence of nanoparticles (/ ¼ 0Þ large values of Bejan vious section, increases in radiation parameter boosts number can be found in most of the domain. Addition of

123 Effects of radiation and magnetic ﬁeld on mixed convection stagnation-point ﬂow over a… 1387

Fig. 9 Effects of radiation (a) (b) parameter on a BeðÞg; u , b NGðÞg; u , Br ¼ 2:0; / ¼ Rd = 0 Rd = 0 0:05; M ¼ 0; Re ¼ 10; Bi ¼ 0:1; k ¼ 10; k1 ¼ 1:0 0.1 180 0.089 160 0.078 140 0.067 120 0.056 100 0.045 80 0.034 60 0.023 40 0.012 20 0.001 0

Rd = 1.0 Rd = 1.0

0.1 180 0.089 160 0.078 140 0.067 120 0.056 100 0.045 80 0.034 60 0.023 40 0.012 20 0.001 0

Rd = 3.0 Rd = 3.0

0.1 180 0.089 160 0.078 140 0.067 120 0.056 100 0.045 80 0.034 60 0.023 40 0.012 20 0.001 0

nanoparticles with / ¼ 0:05, results in a considerable Consequently, the temperature gradients are partially reduction of Bejan number, while increasing the volumetric relaxed and thus less thermal irreversibility is encountered. concentration to 0.1 eliminates all large values of Bejan Figure 11 depicts the effects of intensity of the magnetic number. This can be readily explained by noting the field on the distribution of Bejan number and temperature influences of nanoparticles on heat transfer coefficient. It of nanofluid. According to this figure, increasing the was shown in ‘‘Temperature field, Nusselt number and magnetic parameter by an order of magnitude does not friction coefficient’’ section that increases in the concen- result in any noticeable modification in Bejan number and tration of nanoparticles results in augmentation of Nusselt nanofluid temperature. Nonetheless, intensification of the number and therefore enhances the heat transfer process. magnetic field by another two orders of magnitude causes a

123 1388 R. Alizadeh et al.

Fig. 10 Effects of nanoparticle (a) (b) volume fraction on a BeðÞg; u , b NGðÞg; u Rd ¼ 2:0; Br ¼ 2:0; M ¼ 1:0; Re ¼ 10; φ = 0 φ = 0 Bi ¼ 0:1; k ¼ 10; hW ¼ 1:2 1.5 5000 1.35 4500 1.2 4000 1.05 3500 0.9 3000 0.75 2500 0.6 2000 0.45 1500 0.3 1000 0.15 500 0 0

φ = 0.05 φ = 0.05

1.5 5000 1.35 4500 1.2 4000 1.05 3500 0.9 3000 0.75 2500 0.6 2000 0.45 1500 0.3 1000 0.15 500 0 0

φ = 0.1 φ = 0.1

1.5 5000 1.35 4500 1.2 4000 1.05 3500 0.9 3000 0.75 2500 0.6 2000 0.45 1500 0.3 1000 0.15 500 0 0

considerable reduction of Bejan number (as also confirmed point flows, magnetohydrodynamic effects on the investi- by Table 8). Figure 8d shows that enhancement of Nusselt gated heat transfer processes are not significant. Hence, the number with increases in magnetic field is rather small. observed changes in Bejan number are induced by fric- Further, it can be seen from Fig. 11b that nanofluid tem- tional irreversibility. It has been already demonstrated that perature field hardly changes with variation of magnetic intensifying the magnetic field increases frictional forces parameter for a few orders of magnitude. Magnetic field and hinders the nanofluid flow [31]. As a result, hydrody- does not have any direct effect on the transport of thermal namic entropy generation increases significantly, while energy in the current problem and its indirect influences are thermal irreversibly has remained nearly unchanged. The manifested through the velocity field. Nonetheless, since net effect is reduction of Bejan number as shown in low velocities are an inherent characteristic of stagnation- Fig. 11.

123 Effects of radiation and magnetic ﬁeld on mixed convection stagnation-point ﬂow over a… 1389

Fig. 11 Effects of magnetic (a) (b) parameter on a BeðÞg; u , b hnf ðÞg; u , / ¼ 0:05; M ¼ M = 0.1 M = 0.1 1:0; Re ¼ 100; Rd ¼ 1:0; Br ¼ 2:0; k ¼ 10; k1 ¼ 1:0; 0.5 0.8 Bi ¼ 0:1 0.45 0.73 0.4 0.66 0.35 0.59 0.3 0.52 0.25 0.45 0.2 0.38 0.15 0.31 0.1 0.24 0.05 0.17 0 0.1

M = 1.0 M = 1.0

0.5 0.8 0.45 0.73 0.4 0.66 0.35 0.59 0.3 0.52 0.25 0.45 0.2 0.38 0.15 0.31 0.1 0.24 0.05 0.17 0 0.1

M = 100 M = 100

0.5 0.8 0.45 0.73 0.4 0.66 0.35 0.59 0.3 0.52 0.25 0.45 0.2 0.38 0.15 0.31 0.1 0.24 0.05 0.17 0 0.1

Conclusions circumferentially averaged values of Nusselt number were reported. The key findings of this study can be summarised Transfer of heat and generation of entropy by the combined as follows. mode of radiation-convection was investigated numeri- • Increases in the volumetric concentration of nanopar- cally. The investigated problem included a stagnation-point ticles result in the thickening of thermal boundary layer, flow over a cylinder embedded in a porous medium and enhancement of Nusselt number and reduction of Bejan subject to non-uniform transpiration. The three-dimen- number. sional Darcy–Brinkman model of momentum transport in • Increases in radiation parameter strongly increase the porous media along with two-equation (LTNE) model of value of Nusselt number, while it also reduces the transport of thermal energy were solved in cylindrical average Bejan number. coordinate through using a semi-similarity technique. The • Variation in Biot number can considerably affect the mathematical model considered non-thermal linear radia- value of Nusselt and Bejan number. Hence, the use of tion, magnetohydrodynamics and mixed convection. local thermal non-equilibrium (LTNE) approach is Temperature fields of nanofluid and porous solid as well as essential for ensuring accurate prediction of heat Bejan number field were presented under varying param- transfer rates. eters. Further, the angular distribution and the

123 1390 R. Alizadeh et al.

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