1 Introduction

The structure of the flow near a stagnation-point is a fundamental topic in fluid dynamics,and it has attracted the attention of many researchers during the past several decades becauseof its wide industrial and technical applications,such as heat exchangers placed in a low-velocityenvironment,cooling of nuclear reactors during emergency shutdown,solar central receivers ex-posed to wind currents,cooling of electronic devices by fans,and many hydrodynamic processes.Hiemenz^{[ 1 ]} and Homann^{[ 2 ]} initiated the study of two dimensional and axisymmetric three di-mensional stagnation point flows,respectively. Eckert^{[ 3 ]} extended Hiemenz’s work by includingthe energy equation and obtained an exact similarity solution for the thermal field. Later,theproblem of stagnation point flow was extended numerically by Schlichting and Bussmann^{[ 4 ]} and analytically by Ariel^{[ 5 ]} to include the effect of suction.

In recent years,the dynamics of micropolar fluids has become a popular area of research.The analysis of physical problems in these fluids has revealed several interesting phenomena,which are not found in Newtonian fluids,such as the extrusion of polymer fluids,solidification ofliquid crystals,cooling of a metallic plate in a bath,animal bloods,fluids with additives,exoticlubricants,and colloidal,suspension solutions,and many other situations. Eringen^{[ 6, 7 ]} was thefirst to propose the theory of micropolar fluids,in which the microscopic effects arising fromthe local structure and micromotions of the fluid elements are taken into account. Extensivereviews of this theory and its applications can be found in Refs. ^{[ 8 ]} and ^{[ 9 ]}.

Many attempts were made to find analytical and numerical solutions by applying certainspecial conditions and using different mathematical approaches. Willson^{[ 10 ]} used the Karman-Polhausen approximate integral method to study the micropolar boundary-layer flow near astagnation point. Peddieson and McNitt^{[ 11 ]} numerically studied the boundary-layer flow ata stagnation point under steady-state conditions using a finite difference scheme. A set ofboundary layer equations for two dimensional flow of an incompressible micropolar fluid neara stagnation point was done by Bhargava and Rani^{[ 12 ]}. Ramach and ran and Mathur^{[ 13 ]} studiedthe heat transfer in the stagnation point flow of a micropolar fluid. Heat transfer from non-isothermal surfaces in the stagnation-point flow of a micropolar fluid was studied by Unsworth and Chiam^{[ 14 ]}. Nazar et al.^{[ 15 ]} analyzed the steady stagnation flow towards a permeable verticalsurface immersed in a micropolar fluid.

The porous media heat transfer problems have numerous thermal engineering applicationssuch as geothermal energy recovery,crude oil extraction,thermal insulation,ground waterpollution,thermal energy storage,and flow through filtering devices. Extensive reviews onthis topic were provided in most recent books^{[ 16, 17, 18 ]}. Recently,Gupta and Sharma^{[ 19 ]} studiedthe thermal instability of a micropolar fluid through a porous medium that has a constantthickness. The steady boundary layer flow of a micropolar fluid through a porous medium byusing the generalized Darcys law was examined by Raptis^{[ 20 ]}.

The melting process is encountered in a wide range of technologies,such as metal casting,laser manufacturing(drilling,welding,and selective sintering),seasonal freezing and meltingof soil,lakes and rivers,and thermal energy storage. Epstein and Cho^{[ 21 ]} studied melting heattransfer from a flat plate in a steady laminar case,while Kazmierezack et al.^{[ 22, 23 ]} consideredmelting from a vertical flat plate embedded in a porous medium in both natural and forcedconvection modes.

Cheng and Lin^{[ 24, 25 ]} examined melting effect on mixed convective heat transfer from aporous vertical plate in a liquid-saturated porous medium with aiding and opposing externalflows. Carslaw and Jaeger^{[ 26 ]} discussed melting of a semi-infinite body with constant thermo-physical properties and obtained an analytical solution for Dirichlet boundary conditions. Raisi and Rostaml^{[ 27 ]} investigated numerically the temperature distribution and melt pool size in asemi-infinite body due to a moving Laser heat source. Kearns and Plumb^{[ 28 ]} experimentallystudied direct contact melting of a packed bed. The magnetic and buoyancy effects on meltingprocesses about a vertical wall embedded in a saturated porous medium were investigated byTashtoush^{[ 29 ]}. Very recently,Ishak et al.^{[ 30 ]} studied the steady laminar boundary layer flow and heat transfer from a warm,laminar liquid flow to a melting surface moving parallel to aconstant free stream. Moreover,the melting heat transfer in boundary layer stagnation-pointflow towards a stretching(shrinking)sheet problem was studied by Bachok et al.^{[ 31 ]}.

Motivated by all these works,we contemplate to study the melting effects on flow and heat transfer of a micropolar fluid near stagnation point embedded in a porous medium in thepresence of internal heat generation(absorption).2 Formulation of problem

Consider stagnation point flow of an incompressible micropolar fluid towards a horizontalplate embedded in a porous medium. It is assumed that the plate constitutes the interfacebetween the liquid phase and the solid phase during melting inside the porous matrix at thesteady state. The coordinate system and flow model are shown in Fig. 1. The x-axis is directedalong the plate and the y-axis is normal to it. It is assumed that the velocity of the external flowis U(x)= ax,where a is a positive constant,and x measures the distance from the leading edgealong the surface of the plate. The plate is at constant temperature Tm at which the materialof the porous matrix melts(the liquid phase temperature is T∞(> Tm) and the temperature ofthe solid far from the interface is T0(< Tm)). The flow is steady,laminar,and two-dimensional.

Under the usual boundary-layer approximations,the basic equations taking into account thepresence of internal heat generation(absorption)in the energy equation for a micropolar fluidcan be written as follows^{[ 32 ]}:

where u and v are the velocity components in the x- and y-directions,respectively. N is thecomponent of the micro-rotation vector normal to the xy-plane,T is the fluid temperature,μ is the dynamic viscosity,k is the gyro-viscosity(or vortex viscosity),ρ is the fluid density,k1 is the permeability of the porous medium,G1 = r*/k is the microrotation constant,α is thethermal diffusivity of the fluid,and cp is the specific heat at constant pressure.Following Epstein and Cho^{[ 21 ]},we assume that the boundary conditions are as follows: and

where m0(0 6 m0 6 1)is the boundary parameter. When the boundary parameter m0 = 0,we obtain N = 0 which is the no-spin condition,i.e.,the microelements in a concentratedparticle flow close to the wall are not able to rotate(as stipulated by Jena and Mathur^{[ 33 ]}).The case m0 = 1/2,represents the weak concentration of microelements. The case correspond-ing to m0 = 1 is used for the modelling of turbulent boundary layer flow(see Peddison and McNitt^{[ 11 ]}). In(6),κ is the thermal conductivity,λ is the latent heat fluid,and cs is the heatcapacity of the solid surface.(6)states that the heat conducted to the melting surface is equalto the heat of melting plus the sensible heat required to raise the solid temperature T0 to itsmelting temperature Tm(see Epstein and Cho^{[ 21 ]} and Bachok et al.^{[ 31 ]}). In order to get a sim-ilarity solution,the dependence of the internal heat generation(absorption)rate of the spacecoordinate can be taken in the form^{[ 34 ]}:

where Q0 is the heat generation or absorption constant.

We introduce the following dimensionless variables:

Through(8),the continuity equation(1)is automatically satisfied. From(2)-(4),we canget

The transformed boundary conditions are then given as follows:

where primes denote differentiation with respect to η,

Here,γ is the heat generation(γ > 0)or absorption(γ < 0)parameter,and M is the dimen-sionless melting parameter which is defined as

The melting parameter is a combination of the two Stefan numbers cf(T∞ − Tm)/λ and cs(Tm − T0)/λ for the liquid and solid phases,respectively.

The physical quantities of interest are the local skin-friction coefficient Cfx,the dimensionlesswall couple stress Mx and the local Nusselt number Nux,which are defined as follows:

where the surface shear stress ιw,the wall couple stress mw,and the heat transfer from theplate qw are defined byUsing the similarity variables(13),we getwhere Rex(= U(x)x/v)is the local Reynolds number.3 Method of solutionThe governing boundary layer equations(9)-(12)have the domain 0 6 η 6 η∞,where η∞is one end of the user specified computational domain. Using the algebraic mapping

χ = 2η/η∞− 1,

the unbounded region [0,1)is mapped into the finite domain [1,−1],and the problem expressedby equations(9)-(12)is transformed to the following system:

The transformed boundary conditions are given as follows:Our technique is accomplished by starting with a Chebyshev approximation for the highestorder derivatives,f′′′,h′′,and θ′′ and generating approximations to the lower order derivativesf′′,f′,f,h′,h,θ′ and θ as follows.Setting

,

then by integration,we obtain

From the boundary condition(20),we obtain

Therefore,we can give approximations to(21)-(27)as follows:

for all i = 0,· · ·,n,where

where χ_i = −cos(iπ/n)are the Chebyshev points.

b²_ij =(χ_i − χ_j)b_ij,

and b_ij are the elements of the matrix B,as given in Ref. ^{[ 35 ]}._ijBy using(28)-(30),one can transform(17)-(19)to the following system of nonlinear equa-tions in the highest derivatives into the following Chebyshev spectral equations:

This system is then solved using Newton’s iteration method with n = 11; the computerprogram is executed in Mathematica 4 running on a PC.4 Results and discussion

In order to assess the accuracy of the present numerical method,we compare our numericalresults obtained for f′′(0)with those reported by Hiemenz^{[ 1 ]} and Bachok et al.^{[ 31 ]} and for θ′(0)with those reported by Yacob et al.^{[ 36 ]} for parameters,K = 0,D_a⁻¹ = 0,and γ = 0 in(9) and (11),and for various values of M. The results show a good agreements,as seen in Table 1 and Table 2.

The effects of the various parameters such as the inverse Darcy number,the melting param-eter,and the heat generation(absorption)parameter on the velocity,the angular velocity,and the temperature profiles are shown in Figs. 2-10.

Figure 2 presents the effect of the inverse Darcy number D_a⁻¹ on f′. We notice that f′increases with the increase of D_a⁻¹ . This is due to increasing the inverse Darcy number meansan increase of the porosity of the medium. For increasing porosity,the space allowing fluid tomove in a porous medium becomes large. Consequently,the fluid velocity increases. Figure 3displays the influence of D_a⁻¹ on the angular velocity. It is obvious that initially h decreases byincreasing D_a⁻¹ near the surface and the reverse is true away from the surface. It is noticed thatlarge values of Da correspond to high porosity of the porous medium,and the limit Da ! 1corresponds to the case of absence of the porous medium. The presence of porous medium causeshigher retardation to the fluid,which reduces the velocity. Figure 4 illustrates the effects ofD_a⁻¹ on the temperature profiles θ. It is observed that θ increases with the increase of D_a⁻¹ .

Figure 5 shows the variation of f′(η)with η for various values of the melting parameter M. Itcan be seen that f′ decreases with the increase of M. This is in agreement with the fact thatmore intense melting(increasing M)tends to thicken the boundary layer. The angular velocityprofiles for different values of M are illustrated in Fig. 6,which shows that h increases with theincrease of M near the surface and the reverse is true at large distances from the surface. Itcan be noticed that the temperature profiles decrease as M increases as illustrated in Fig. 7.Physically,increasing the melting parameter causes higher acceleration to the fluid flow which,in turn,increases its motion and causes decrease in the temperature profiles. The effects of thethe heat generation parameter(γ > 0) and the absorption parameter(γ < 0)on the velocity,the angular velocity and the temperature are displayed in Figs. 8,9,and 10,respectively. It isseen from Fig. 8 that f′ decreases as the heat generation parameter(γ > 0)increases,but theeffect of the absolute value of the heat absorption parameter(γ < 0)is opposite. In Fig. 9,itis shown that h increases as the heat generation parameter increases,while h decreases as theabsolute value of the heat absorption parameter increases near the surface and the opposite istrue away from it. Figure 10 displays the effect of the heat generation(absorption)parameteron the temperature profiles θ. It is found that as the heat generation parameter increases θincreases near the surface and the reverse is true away from the surface,while the effect of theabsolute value of the absorption parameter is opposite.

Table 3 illustrates the effects of D_a⁻¹,M,and on the local skin-friction coefficient interms of f′′(0),the dimensionless wall couple stress in terms of h′(0),and the local Nusseltnumber in terms of −θ′(0). From this Table,it is observed that the local skin-friction coefficient and the dimensionless wall couple stress increase with increasing D_a⁻¹,but the local Nusseltnumber decreases with increasing D_a⁻¹ . Moreover,it is also found that,increasing M leads toa decrease in the local skin-friction coefficient and the dimensionless wall couple stress,whilethe local Nusselt number increases as M increases. This is because increasing the meltingparameter M increases the thermal boundary layer thickness which results in a reduction intemperature gradient at the surface. Finally,we can see that the local skin-friction coefficient,the dimensionless wall couple stress and the local Nusselt number decrease with increasing theheat generation parameter,but the reverse is true for the absolute values of the heat absorptionparameter.

5 Conclusions

The problem of steady two-dimensional flow of a micropolar fluid at stagnation point em-bedded in a porous medium with melting heat transfer and in the presence of internal heatgeneration(absorption)has been investigated. Using similarity transformations,the governingequations are transformed into a system of coupled non-linear ordinary differential equationswhich is solved numerically by using the Chebyshev spectral method. The results show that thenumerical values of the local skin-friction coefficient and the dimensionless wall couple stressincrease as the inverse Darcy number increases,while it decreases as the melting parameterincreases. The local Nusselt number decreases with increasing the inverse Darcy number,whilethe melting parameter leads to an increase in the local Nusselt number. Also,it can be foundthat the local skin-friction coefficient,the dimensionless wall couple stress and the local Nusseltnumber decrease with increasing of the heat generation parameter,but the opposite is true forthe absolute values of the heat absorption parameter.

Acknowledgements The authors would like to thank the reviewers for their valuable comments,which lead to the improvement of the work.