Mixed convection and thermodynamic irreversibilities in MHD nanofluid stagnationpoint flows over a cylinder embedded in porous media
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Abstract
The impingement of CuOwater nanofluid flows upon a cylinder subject to a uniform magnetic field with constant surface temperature and embedded in porous media is investigated for the first time in literature. The surface of the cylinder can feature uniform or nonuniform mass transpiration and is hotter than the incoming nanofluid flow. The gravitational effects are taken into account and the threedimensional governing equations of mixed convection in curved porous media, under magnetohydrodynamic effects, are reduced to those solvable by a finite difference scheme. Through varying a mixed convection parameter, the situations dominated by forced, mixed and free convection are examined systematically. The numerical solutions of these equations reveal the flow velocity and temperature fields as well as the Nusselt number and induced shear stress. These are then used to calculate the rate of entropy generation within the system by viscous and heat transfer irreversibilities. The results show that Nusselt number increases with increasing the concentration of nanoparticles, while it slightly deceases through intensifying the magnetic parameter. Nonuniform transpiration is shown to strongly affect the average rate of heat transfer. Importantly, it is demonstrated that the specific mode of heat convection can majorly influence the intensity of entropy generation and that the irreversibilities are much larger under natural convection compared to those in mixed and forced convection. Calculation of Bejan number shows that this is due to more pronounced relative contribution of viscous irreversibilities when free convection effects dominate the mixed convection process.
Keywords
Nanofluid Stagnationpoint flow Porous media Entropy generation Similarity solution Mixed convectionList of symbols
 A_{1}, A_{2}, A_{3}, A_{4}
Constants
 \(a\)
Cylinder radius
 \(B_{0}\)
Magnetic field strength
 \(B{\text{e}}\)
Bejan number
 \(B{\text{r}}\)
Brinkman number \(B{\text{r}} = \frac{{\mu_{\text{f}} \left( {\bar{k} \cdot a} \right)^{2} }}{{k_{\text{f}} \left( {T_{\rm w}  T_{\infty } } \right)}}\)
 \(C_{\text{p}}\)
Specific heat at constant pressure
 \(f\left( {\begin{array}{*{20}c} {\eta ,} & \varphi \\ \end{array} } \right)\)
Function related to ucomponent of velocity
 \(G\left( {\begin{array}{*{20}c} {\eta ,} & \varphi \\ \end{array} } \right)\)
Function related to vcomponent of velocity
 \(G{\text{r}}\)
Grashof number \(G{\text{r}} = \frac{{g \cdot \beta_{\text{f}} .a^{3} .\left( {T_{\text{w}}  T_{\infty } } \right)}}{{16\upsilon_{\text{f}}^{2} }}\)
 \(g\)
Gravitational acceleration
 \(h\)
Heat transfer coefficient
 \(k\)
Thermal conductivity
 \(\bar{k}\)
Freestream strain rate
 \(k_{1}\)
Permeability of the porous medium
 \(M\)
Magnetic parameter, defined as \(M = \frac{{\bar{\sigma } \cdot B_{0}^{2} }}{{2\rho_{\text{f}} \bar{k}}}\)
 \(N_{G}\)
Entropy generation number \(N_{G} = \frac{{\dot{S}_{\text{gen}}^{{{\prime \prime \prime }}} }}{{S_{\text{0}}^{{{\prime \prime \prime }}} }}\)
 \(Nu\)
Nusselt number
 \(p\)
Fluid pressure
 \(P\)
Nondimensional fluid pressure
 \(P_{0}\)
The initial fluid pressure
 \(P{\text{r}}\)
Prandtl number
 \(q_{\text{w}}\)
Heat flow at the wall
 \(r\)
Radial coordinate
 \(\text{Re}\)
Freestream Reynolds number \(\text{Re} = \frac{{\bar{k}a^{2} }}{{2\upsilon_{\text{f}} }}\)
 \(S\left( \varphi \right)\)
Transpiration rate function \(S\left( \varphi \right) = \frac{{U_{0} \left( \varphi \right)}}{{\bar{k}{{a}} }}\)
 \(\dot{S}_{0}^{{{\prime \prime \prime }}}\)
Characteristic entropy generation rate
 \(\dot{S}_{\text{gen}}^{{{\prime \prime \prime }}}\)
Rate of entropy generation
 T
Temperature
 \(T_{\infty }\)
Freestream temperature
 \(T_{\text{w}}\)
Wall temperature
 \(u,v,w\)
Velocity components along (\(r  \varphi  z\))axis
 U_{0} (φ)
Transpiration
 z
Axial coordinate
Greek symbols
 α
Effective thermal diffusivity of the porous medium
 β
Thermal expansion coefficient
 \(\eta\)
Similarity variable, \(\eta = \left( {\frac{r}{a}} \right)^{2}\)
 \(\theta \left( {\begin{array}{*{20}c} {\eta ,} & {\varphi } \\ \end{array} } \right)\)
Nondimensional temperature
 \(\lambda\)
Permeability parameter, \(\lambda = \frac{{a^{2} }}{{4k_{1} }}\)
 \(\lambda_{1}\)
Dimensionless mixed convection parameter \(\lambda_{1} = \frac{{G{\text{r}} }}{{\text{Re}^{2} }} = \frac{{g \cdot \beta_{\text{f}} \cdot \left( {T_{\text{w}}  T_{\infty } } \right)}}{{4a \cdot \bar{k}^{2} }}\)
 ε
Porosity
 \(\varLambda\)
Dimensionless temperature difference \(\varLambda = \frac{{\left( {T_{\text{w}}  T_{\infty } } \right)}}{{T_{\text{w}} }}\)
 \(\mu\)
Dynamic viscosity
 \(\upsilon\)
Kinematic viscosity
 \(\rho\)
Fluid density
 \(\sigma\)
Shear stress
 \(\bar{\sigma }\)
Electrical conductivity
 \(\phi\)
Nanoparticle volume fraction
 \(\varphi\)
Angular coordinate
Subscripts
 \(w\)
Condition on the surface of the cylinder
 \(\infty\)
Far field
 nf
Nanofluid
 f
Base fluid
 s
Nanosolid particles
Introduction
Convection of nanofluids in porous media is as an attractive area for heat transfer and thermodynamic research communities [1, 2]. Free convection of nanofluids in porous media has already received significant attention [3, 4]. However, the equivalent problem under forced and mixed convection is relatively much less investigated. In particular, evaluation of thermodynamic irreversibilities encountered during forced convection of nanofluids in porous media has been identified as an underdeveloped area demanding more research [2].
The general problem of forced convection of nanofluids through porous media has been, so far, visited by few researchers [5]. The existing studies can be generally categorised into two classes of nanofluid flows in porous conduits [6, 7] and those over rotating porous discs [8, 9]. Only the latter can involve boundary layer flows and thus is further discussed here. In a numerical investigation, Bachok et al. [10] analysed fluid dynamics and heat transfer of nanofluids in a configuration including rotating porous discs. These authors used two different models of the effective thermal conductivity and examined the fluid dynamics and heat transfer behaviours of the system [10]. Hatami et al. [11] studied the nanofluid flow between two counterrotating discs with porous faces. They considered waterbased nanofluids with a number of different metal and metal oxide as nanoparticles [11]. Their investigation included an elaborated study of the effects of nanoparticle size and type on the heat transfer characteristics of the system [11]. The problem of convective heat transfer by nanofluid flows between rotating porous discs was also examined by Hosseini et al. [12]. These authors employed homotopy perturbation method and showed that increasing the concentration of nanoparticles enhances the convection coefficient. This finding was later confirmed by another group of authors in other configurations [6, 7]. According to Hooseini et al. [12], there is a monotonic and nearly linear relationship between the volumetric fraction of nanoparticles and the increase in Nusselt number. A threedimensional investigation was conducted by Saidi and Tamim [8] to predict the heat and mass transfer behaviours of a system involving two rotating porous discs. The Brownian motion of nanoparticles was considered in this study, and the magnetohydrodynamic effects were also investigated [8]. Amongst other findings, it was reported that augmenting the permeability of the porous discs enhances the heat and mass transfer coefficients on the surfaces of the discs [8]. Through considering a moving permeable surface, Khazayinejad et al. [13] solved the governing equations of the transport of momentum in a nanofluid boundary layer. They put forward a similarity solution for the problem and included the influences of nanofluid suction and injection in their analysis [13].
A particular type of boundary layer flows in porous media includes stagnation points [14, 15, 16]. This class of flow finds wide applications in cooling technologies and, therefore, has been investigated by different authors. These have been mostly focused on the stagnation flows of ordinary fluids over flat porous inserts. Here, a concise summary of the literature in this area is presented. A pioneering work on the hydrodynamics of stagnationpoint, isothermal flows on a flat porous insert was conducted by Wu et al. [17]. They assumed a Darcy–Brinkman flow and developed an asymptotic solution for the velocity field in a horizontal porous plate under an impinging jet configuration [17]. In their numerical investigation, Jeng and Tzeng [18] investigated the transport of heat when a slot jet impinges upon the surface of a metallic foam heat sink. These authors reported that the location of the maximum convection coefficient varies with the jet Reynolds number [18]. Jeng and Tzeng, later, set an experimental study of the same problem [19] and demonstrated that by increasing the jet Reynolds number the convection coefficient grows in magnitude. Nevertheless, the flow pressure drop is also intensified [19]. Subsequently, Wong and Saeid [20, 21] conducted a heat transfer optimisation on the problem of jet flow blowing on the surface of a horizontal porous insert heated from below.
Harris et al. [22] built a similarity solution for the boundary layer developed near the stagnation point on a porous plate positioned vertically. A numerical work on mixed convection in jet impingement on a flat porous plate revealed that increasing the jet width and the Reynolds number lead to the magnification of the average Nusselt number [23]. It was also shown that decreasing the distance between the jet and the heated section increases the Nusselt number [23]. Kokubun and Fichini [24] presented an analytical solution for the stagnationpoint flow in an infinitely long, horizontal porous insert subject to different thermal boundary conditions. This work showed that a dimensionless parameter, including information on the transport properties of the fluid and solid, dominates the heat transfer process. In an experimental and numerical study, Feng et al. [25] investigated the problem of tube flow impingement on a heated porous insert. They examined metal foam and finned metal foam and demonstrated that by magnifying the thickness of the metal foam heat transfer coefficient decreases. Yet, this was not the true for the metal finned foam [25]. More recently, Buonomo et al. [26] investigated the interactions between a downward vertical, laminar jet and a confined, horizontal porous insert in an axisymmetric configuration. Buonomo et al. illustrated that Peclet number determines the opposing or supporting arrangements of natural and forced convection [26]. Mixed convection of stagnationpoint flows over a vertical plate covered by a porous layer was investigated by Makinde [27] and also by Rosca and Pop [28]. Thermal radiation and magnetic effects have been further considered in the problem of mixed convection on vertical flat, porous walls [29].
All cited literatures, so far, have been entirely focussed on flow configurations over flat porous inserts. A review of literature reveals that the problem of stagnationpoint flow formed upon curved surfaces in porous media has been rarely studied. An exception to this is the most recent work of the authors, in which they developed a semisimilar solution for the stagnation flow upon the surface of cylinder embedded in a homogenous porous medium [5]. This investigation was concerned with the hydrodynamics and heat convection only [5]. Importantly, it was limited to ordinary fluids and did not consider magnetic and gravitational effects nor it involved analysis of entropy generation. Another highly unexplored area includes nanofluid stagnationpoint flow in porous media. The shortage of research in this area extends to both flat and curved configurations. An early investigation of flow over a cylinder embedded in porous media was reported by AbuHijleh [30]. A laminar flow of ordinary fluid through the porous media and over a cylinder was investigated in this work and the rate of entropy generation was calculated numerically [30]. It was demonstrated that increasing the thickness of the porous layer covering an isothermal cylinder reduces the total generation of entropy [30]. Entropy generation in magnetohydrodynamic (MHD) flow of nanofluids in porous media has been analysed in a few recent works. Rashidi and Freidoonmehr [31] considered the MHD and nanofluid equivalent of the classical configuration of Heimenz [32] when the solid plate was replaced by a flat porous insert. Their work was exclusively concerned with the generation of entropy and made the conclusion that the effects of increasing the values of Hartmann, Brinkman and magnetic interaction numbers and reducing Prandtl and Reynolds numbers are similar and lead to an augmentation of the entropy generation. This study was later extended to the configurations including rotating porous discs with ordinary fluids [33] and nanofluid [34]. In addition to these studies, there exists a series of studies on entropy generation by nanofluid flow over permeable surfaces [35, 36, 37]. Although mathematical models similar to those of porous media are used in these works, the physical differences between them and stagnation flows inside porous media are rather significant. Thus, these investigations are not further discussed here.
The preceding review of literature reveals that the general problem of forced and mixed convection of nanofluids in porous media and the particular problem of entropy generation by such flows have been highlighted as largely unexplored fields. Further, there have been already a number of studies on the impingement of external flows in flat porous plates under. However, stagnationpoint flows in curved porous media have so far received very little attention. The existing studies on boundary layer nanofluid flows in porous media are entirely concerned with flat porous inserts or preamble surfaces. Thus, there is currently no study of nanofluid stagnationpoint flow in curved porous inserts. In practice, many curved objects are covered with porous layers and nanofluids are increasingly used as the cooling agents in such configurations [38]. However, there is currently no systematic evaluation of the heat transfer and second law performance of such systems.
The present work, therefore, aims to fill this gap through a study of a cylindrical object embedded in porous media and subject to nonaxisymmetric, nanofluid stagnationpoint flow. The current study builds upon the earlier work of the authors [5] and advances that on four main fronts. These include consideration of a nanofluid flow, addition of gravitational and magnetohydrodynamic effects upon the convection problem and also evaluation of the encountered thermodynamic irreversibilities.
Theoretical and numerical methods
Problem configuration, assumptions and governing equations

The flow is steady, incompressible and laminar.

The nanofluid is assumed to be Newtonian, electrically conductive and single phase.

The cylinder is assumed to be infinitely long and its axis is parallel to the direction of gravity. Also, the cylinder is subject to a uniform magnetic field.

The porous medium is homogenous, isotropic and under local thermal equilibrium.

The radiation heat transfer and viscous dissipation of kinetic energy of the flow are ignored.

Physical properties such as porosity, specific heat, density and thermal conductivity are assumed to be constant and hence the thermal dispersion effects are negligible.

A moderate range of porescale Reynolds number is considered in the porous medium and hence nonlinear effects in momentum transfer are negligibly small.

The physical mechanisms causing significant deviations from the local thermal equilibrium, such as internal heat generations, are ignored [39, 40].
A threedimensional Darcy–Brinkman model of transport of momentum together with the oneequation model of transport of thermal energy in cylindrical coordinate is used in this work [41, 42, 43]. The governing equations and boundary conditions, in the cylindrical coordinate system shown in Fig. 1, can be summarised as follows.
Thermophysical properties of the base fluid and different nanoparticles [36]
Physical properties  C_{p}/J kg^{−1} K^{−1}  ρ/kg m^{−3}  k/W m^{−1} k^{−1}  β × 10^{−5}/K^{−1} 

Fluid phase (water)  4179  997.1  0.613  21 
CuO  531.8  6320  76.5  1.8 
Equation (7) represents noslip conditions on the external surface of the cylinder. Further, Eq. (8) indicates that the viscous flow solution approaches, in a manner analogous to the Hiemenz flow, the potential flow solution as \(r \to \infty\) [41, 42, 45]. This can be verified by starting from the continuity equation in the following. \( \frac{1}{r}\frac{\partial }{\partial r}\left( {ru} \right)  \frac{\partial v}{\partial \varphi } = \frac{\partial w}{\partial z} =\) Constant \(= 2\bar{k}z\) and integrating in \(r\) and \(z\) directions with boundary conditions,\(w = 0\) when \(z = 0\) and \(u =  U_{0} \left( \varphi \right)\) when \(r = a\).
The boundary condition for the transport of thermal energy is given by
Selfsimilar solutions
A reduction in the governing Eqs. (1–5) is obtained through applying the following similarity transformations.
Shear stress and Nusselt number
Hence, Nusselt number can be written as
Entropy generation
Considering the assumption stated in Sect. 3.1, the volumetric rate of local entropy generation in the problem is given by [49, 50]:
The dimensionless form of volumetric rate of local entropy generation (N_{G}) can be presented as follows.
Grid independency and validation
Comparison between the current results and those of Alizadeh et al. [9] when \(S\left( \varphi \right) = 0\), \(\text{Re} = 1.0\)
\(\text{P}{r}\)  \(N{{{\text{u}}_{\text{m}} }}\)  \(\lambda\)  \(N{{{\text{u}}_{\text{m}} }}\)  

Present work  Alizadeh et al. [9]  Present work  Alizadeh et al. [9]  
0.1  3.59781  3.59774  0.1  3.84221  3.84219 
0.4  3.74124  3.74114  1.0  3.84888  3.84888 
0.7  3.84888  3.84888  10  3.88855  3.88851 
1.0  3.93790  3.93788  50  3.95264  3.95263 
10  5.07669  5.07670  100  3.98295  3.98287 
Results and discussion
Default values of the simulation parameters
Simulations parameters  \(\eta\)  \(\phi\)  \(z\)  \(\varphi\)  \(\lambda\)  \(M\)  \(\lambda_{1}\)  Re  \(\varepsilon\)  \(S\left( \varphi \right)\) 

1.45  0.1  a  72°  10  1.0  1.0  10  0.9  \(Ln\left( \varphi \right)\) 
Flow velocity, temperature fields and heat convection coefficient
Figure 9 shows the angular variation of Nusselt number with respect to permeability parameter indicating that by increasing the permeability parameter Nusselt number grows to a small extent. Increase in Nusselt number with decreasing the permeability (or increasing the permeability parameter) has been already reported in studies of convection of nanofluid in straight porous flow conduits [6, 7]. The present study extends this to curved surfaces embedded in porous media. Figure 9b shows that, as expected, by increasing the permeability parameter the dimensionless stress increases. The amount of this increase is particularly large at small values of \(\varphi\) and appears to feature a jump at \(\lambda = 100.\)
Effects of the nanoparticle volume fraction on average Nusselt number and average shear stress (\({\raise0.7ex\hbox{${\sigma_{\rm m} .a}$} \!\mathord{\left/ {\vphantom {{\sigma_{\rm m} .a} {4\mu \bar{k}z}}}\right.\kern0pt} \!\lower0.7ex\hbox{${4\mu \bar{k}z}$}}\)) for \(\text{Re} = 1\), \(\phi = 0.1\), \(\lambda_{1} = 1\), \(\lambda = 100\)
\(\phi\)  \({\raise0.7ex\hbox{${\sigma_{\text{m}} .a}$} \!\mathord{\left/ {\vphantom {{\sigma_{\text{m}} .a} {4\mu_{\text{f}} \bar{k}z}}}\right.\kern0pt} \!\lower0.7ex\hbox{${4\mu_{\text{f}} \bar{k}z}$}}\)  \(N{{{\text{u}}_{\text{m}} }}\)  

\(S\left( \varphi \right) = Ln\left( \varphi \right)\)  \(S\left( \varphi \right) = 1\)  \(S\left( \varphi \right) = 0\)  \(S\left( \varphi \right) = Ln\left( \varphi \right)\)  \(S\left( \varphi \right) = 1\)  \(S\left( \varphi \right) = 0\)  
0.0  5.43016  5.30734  5.34345  1.43516  1.57439  1.67149 
0.05  6.18125  6.03449  6.07846  1.79811  1.95149  2.04587 
0.1  7.07958  6.90923  6.96085  2.25555  2.42114  2.51241 
0.15  8.16591  7.97235  8.03133  2.84526  3.02139  3.10921 
Effects of the dimensionless mixed convection in nanofluid waterCuo on average Nusselt number and average shear stress (\({\raise0.7ex\hbox{${\sigma_{\text{m}} \cdot a}$} \!\mathord{\left/ {\vphantom {{\sigma_{\text{m}} \cdot a} {4\mu \bar{k}z}}}\right.\kern0pt} \!\lower0.7ex\hbox{${4\mu \bar{k}z}$}}\)) when \(\phi = 0.1\), \(\lambda = 100\), \(\text{Re} = 1\)
\(\lambda_{1}\)  \({\raise0.7ex\hbox{${\sigma_{\text{m}} \cdot a}$} \!\mathord{\left/ {\vphantom {{\sigma_{\text{m}} \cdot a} {4\mu_{\text{f}} \bar{k}z}}}\right.\kern0pt} \!\lower0.7ex\hbox{${4\mu_{\text{f}} \bar{k}z}$}}\)  \(N{{{\text{u}}_{\text{m}} }}\)  

\(S\left( \varphi \right) = Ln\left( \varphi \right)\)  \(S\left( \varphi \right) = 0\)  \(S\left( \varphi \right) = Ln\left( \varphi \right)\)  \(S\left( \varphi \right) = 0\)  
0.01  7.04650  6.94845  2.25569  2.51271 
0.1  7.04950  6.94958  2.25568  2.51269 
1  7.07958  6.96085  2.25555  2.51241 
10  7.37972  7.07375  2.25426  2.50967 
100  10.29258  8.22825  2.24233  2.48544 
− 1  7.01273  6.93581  2.25584  2.51303 
− 10  6.71127  6.82336  2.25716  2.51585 
Effects of the magnetic parameter in nanofluid waterCuo on average Nusselt number and average shear stress (\({\raise0.7ex\hbox{${\sigma_{\text{m}} \cdot a}$} \!\mathord{\left/ {\vphantom {{\sigma_{\text{m}} \cdot a} {4\mu \bar{k}z}}}\right.\kern0pt} \!\lower0.7ex\hbox{${4\mu \bar{k}z}$}}\)) when \(\phi = 0.1\), \(\text{Re} = 1\), \(\lambda = 10\), \(\lambda_{1} = 10\)
\(M\)  \({\raise0.7ex\hbox{${\sigma_{\text{m}} \cdot a}$} \!\mathord{\left/ {\vphantom {{\sigma_{\text{m}} \cdot a} {4\mu_{\text{f}} \bar{k}z}}}\right.\kern0pt} \!\lower0.7ex\hbox{${4\mu_{\text{f}} \bar{k}z}$}}\)  \(N{{{\text{u}}_{\text{m}} }}\)  

\(S\left( \varphi \right) = Ln\left( \varphi \right)\)  \(S\left( \varphi \right) = 0\)  \(S\left( \varphi \right) = Ln\left( \varphi \right)\)  \(S\left( \varphi \right) = 0\)  
0  3.64667  3.88497  2.50179  2.57364 
0.1  3.65876  3.89552  2.50143  2.57123 
1  3.76536  3.98846  2.50111  2.57051 
10  4.64622  4.75670  2.50062  2.56991 
100  7.64299  7.45287  2.49973  2.56711 
Thermodynamic irreversibilities
Conclusions

In keeping with that reported for other porous configurations, the Nusselt number was observed to increase in magnitude by increasing the concentration of nanoparticles.

Intensifying the magnetic field was shown to result in reducing the flow temperature slightly and also causing a small decrease in the averaged Nusselt number.

The functional form of mass transpiration was shown to have important effects upon the average Nusselt number.

By increasing the numerical value of mixed convection parameter, \(\lambda_{1} ,\) the numerical value of the dimensionless temperature and that of the average Nusselt number decreases. That indicates that, as expected, under free convection the flow is colder and the rate of heat transfer is smaller than that under mixed and forced convection.

The entropy generation was found to substantially increase at high values of mixed convection parameter, which means free convection in the investigated configuration involves much more irreversibility compared to mixed and forced convection.

It was argued that the share of viscous irreversibility in entropy generation under free convection is significantly higher than that of thermal irreversibility.

Strong magnetic effects were shown to generate large irreversibilities, while they reduce Bejan number and magnify the relative importance of viscous irreversibilities.
Notes
Acknowledgements
N. Karimi acknowledges the partial financial support by EPSRC through Grant Number EP/N020472/1 (Thermapump).
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