Numerical Simulation of Williamson Combined Natural and Forced Convective Fluid Flow between Parallel Vertical Walls with Slip Effects and Radiative Heat Transfer in a Porous Medium

Numerical study of the slip effects and radiative heat transfer on a steady state fully developed Williamson flow of an incompressible Newtonian fluid; between parallel vertical walls of a microchannel with isothermal walls in a porous medium is performed. The slip effects are considered at both boundary conditions. Radiative highly absorbing medium is modeled by the Rosseland approximation. The non-dimensional governing Navier–Stokes and energy coupled partial differential equations formed a boundary problem are solved numerically using the fourth order Runge–Kutta algorithm by means of a shooting method. Numerical outcomes for the skin friction coefficient, the rate of heat transfer represented by the local Nusselt number were presented even as the velocity and temperature profiles illustrated graphically and analyzed. The effects of the temperature number, Grashof number, thermal radiation parameter, Reynolds number, velocity slip length, Darcy number, and temperature jump, on the flow field and temperature field and their effects on the boundaries are presented and discussed.


Introduction
Combined natural (free) and forced convection in channels occurs in many applications [1] and geometries such flow between parallel vertical walls [2], including flow reversal [3], inclined parallel plates [4], and vertical channels [5], etc. Thermal radiative emission from a hot surface to a cold surface plays an important role in many uses, including energy conversion [6], with radiation effects and viscous heating in a channel partially filled by a porous material [7], viscous heating in a porous channel [8], microchannels [9], heat exchangers with vertical hexagonal rod bundle geometries [10], buoyancy-driven vortical flow [11], biofidelity corridors [12], fluid flow control [13,14], in various boundary conditions [15], and pressure dependent viscosity flows [16].Furthermore mixed convection viscoelastic slip flow through a porous medium in a vertical porous channel with thermal radiation flow [17] is found in industrial processes and has acquired substantial importance due to its wideranging applications in fluid flow control [18], propulsion [19,20], and viscous gravity currents inside confining channels and fractures [21,22].
Velocity and thermal slip affects high performance magnetic bearings that are subjected to higher thermal loads and many other research fields [23][24][25][26], EHD (electrohydrodynamics) mixers [27], microfluidics and nanofluidics [28][29][30][31][32], and pseudoplastic material hydrodynamics [33][34][35][36][37][38].The flow slip at boundary conditions is seen in rare gas flow [28] as well as incompressible flow [29].In addition, thermal jump conditions are seen in many experiments [30].Ulmanella and Ho [29] experimentally detected the speed of fluids at several microchannel walls, as a function of shear rate, type of fluid, and wall surface properties.Bocquet and Barrat [30] explained the probability of temperature slip simultaneously with velocity discontinuity at boundary conditions.They presented the amount of velocity slip and the value of temperature jump by bringing together the velocity slip length and temperature slip length, respectively.The slip relations should be applied at the fluid solid boundaries in microchannels in other textbooks [31].
Recently Jha et al. [32] presented an exact solution of steady fully developed natural convection flow of a viscous, incompressible, electrically conducting fluid in a vertical annular microchannel with the effect of porous resistance in the presence of velocity slip and temperature jump at the annular microchannel surfaces [32].They expressed their solution in terms of radius ratio, Darcy number, rarefaction parameter, and fluid-wall interaction parameter effect on the flow.They found that an increase in Darcy number leads to a decrease in the fluid velocity, volume flow rate and skin friction, but they have not studied the effects of parameters on Nusselt number and their geometry is cylindrical.
Of the various kinds of fluids the Williamson fluid is studied in this paper.The Williamson fluid was first introduced by Williamson in 1929 [33].This kind of fluid model was first is used to model pseudoplastics which do not exhibit a real yield value and cannot be modeled as plastics nor Newtonian fluids.Gravity currents such as drainage processes may occur in a variety of natural and industrial activities, including the geological storage of carbon dioxide.This kind of non-Newtonian fluid behavior is observed in gravity currents propagating in confining boundaries [34] in volcanos and geothermal applications [35].King and Woods [36] presented a dipole solution for viscous gravity currents.Longo et al. illustrated the dipole solution for power-law gravity currents in porous formations [37].Buoyancy-driven fluid drainage from a porous medium for V-shaped Hele-Shaw cells where the fluid drains from an edge is discussed in another reference.It flows between walls with a limited gap (with respect to the main length scales) mimicking flows in porous media (Hele-Shaw cell analogy) [38].
Considering all the above, the aim of the current study was the synthesis of the radiative hydrodynamics of a highly absorbing incompressible fluid in a straight up microchannel filled with a saturated porous material.Different wall temperatures are applied on the walls of the channels and the fluid is an optically thick medium.A parameter study on temperature profile, velocity profile, Nusselt number, and friction coefficient is investigated analytically.

Materials and Methods
A two-dimensional, steady state, incompressible and electrically conducting fluid flow with heat transfer by convection between two vertical plates in the presence of radiation in a simple configuration as shown in Figure 1.A stream of cold fluid at temperature T L moving over the left surface of the plate with a slip velocity formed a fully developed laminar flow while the right surface of the plate is heated by convection from a hot fluid at temperature T R .The two parallel planar walls are located at y = ´L and y = +L, with the gap of 2 L. We shall assume that the velocity and the pressure field are of the form: Entropy 2016, 18, 147 3 of 15 The equation which governs the buoyancy-driven flow of an incompressible fluid through a porous medium in the x-direction is: Since in other directions there is no fluid flow, the corresponding equations are not required here.The Navier-Stokes equation for steady state fully developed velocity (Bu/Bx = 0; B 2 u/Bx 2 = 0) for the case for which µ 8 = 0, µ 0 = µ and Γγ' < 1 (the component of extra stress tensor is τ = µγ' (1 ´Γγ') ´1 = µγ' (Γγ' + 1)) [16] can be written as is simplified as: where u is the fluid velocity in vertical direction, ρ 0 is the fluid density at reference temperature (T ref = (T L + T R )/2), µ is the dynamic viscosity, β is the thermal expansion coefficient, and σ c is the electrical conductivity.As well, the temperature equation for optically thick fluid is [18]: where T is the medium temperature, k is the thermal conductivity, σ is the Stefan-Boltzmann constant and χ is the mean absorption coefficient of the medium.The boundary conditions of Equation (1) at the system boundaries are: u py " ´Lq " l v ˆdu dy ˙y"´L where l v is the velocity slip length and for the temperature are: T py " ´Lq " T R `lT where l T is the temperature jump length.The above equations are non-dimensionalized by the following parameters for length: Y " y L (10) for velocity: for temperature: for pressure: and the non-dimensional well-known numbers as Grashof number: Reynolds number: the radiation parameter: temperature parameters: Darcy number: Weissenberg dimensionless number: Which is used for comparison of the evolution of the viscous energy released to the elastic energy stored in the viscoelastic fluid flows, defined as the relation of stress relaxation time of the fluid (Γ) and the fluid flow time (u/L).Physical interpretation of the Weissenberg number is the degree of anisotropy or orientation generated by the deformation, and it is appropriate to describe flows with a constant stretch history, such as simple shear.

Results and Discussion
Research has been carried out into steady state laminar combined free and forced convection fluid flow in a upright channel occupied by a highly absorbing fluid, which are important factors in various industrial applications.For the purpose of obtaining a strong visualization of the physical problem, widespread estimations have been performed to acquire the velocity and thermal field as a function of physical parameters such as combined convection parameter and Darcy number temperature-based parameters such as thermal radiation parameter, thermal parameter, and temperature slip, and a flow parameter, the velocity slip, are exposed comprehensively in Figures 2-7.The non-dimensional governing Equations ( 19)-( 24) are solved numerically using Runge-Kutta-Fehlberg method with shooting technique.The set of simultaneous first order differential equations of equivalent initial-value problem (Z′ = f(Z,Y)) are constructed by the vector Z = [U; U′; θ; θ′] and the first guess of the initial value are assumed as Z(Y = −1) = [0; 0; −1; 0].To benchmark the current numerical method, the comparison of the numerical solution and analytical solution is illustrated in Figure 1.As the analytical solution of the problem for Rd = 0.0, Γ = 0, and λT = 0.0 is:

Results and Discussion
Research has been carried out into steady state laminar combined free and forced convection fluid flow in a upright channel occupied by a highly absorbing fluid, which are important factors in various industrial applications.For the purpose of obtaining a strong visualization of the physical problem, widespread estimations have been performed to acquire the velocity and thermal field as a function of physical parameters such as combined convection parameter and Darcy number temperature-based parameters such as thermal radiation parameter, thermal parameter, and temperature slip, and a flow parameter, the velocity slip, are exposed comprehensively in Figures 2-7.The non-dimensional governing Equations ( 19)-( 24 ?
where pressure drop is calculated from: The numerical results of current work for velocity slip length of 0.01, Da = 0.01, and Gr/Re = 50 are in good agreement with the analytical solution.As shown in Figure 2 the shooting method can capture the precise results with 15 points.Figure 2 presents the consequence of the change of thermal radiation parameter on the numerous feature of temperature and velocity field of the microchannel in the constant value of the Grashof to Reynolds ratio (Gr/Re = 100), and θ R = 10, λ v = 0.01, We = 0.001, λ T = 0.01, Da = 0.1.On behalf of the special case of no absorption of the thermal radiation in the fluid, Entropy 2016, 18, 147 6 of 15 the temperature has a linear profile as shown in Figure 3a.By increasing of thermal radiation number ranging 10 ´4 to 10 ´2 the dimensionless temperature is amplified, particularly close to the left partition.Figure 3b displays a curvy form for the velocity outline.By growth of the amount of thermal radiation, the dimensionless velocity profile alternates from sine shape to parabola shape and its maximum decrease.The maximum value occurs at the middle of parabola profile approaches the Poiseuille profile maximum (i.e., 1.5) by increase of R d .The effect of thermal radiation on the friction factor at walls and pressure gradient is illustrated in Figure 3c.By increasing R d , the friction factor at the walls (the left and right friction coefficients) decrease smoothly while the pressure gradient (the profile with the middle legend), increases dramatically.As well the friction factor for the same Reynolds number is greater at the right wall for low R d and the reverse phenomena is seen for high R d .Both skin-friction values continue to reduce further with the increase of R d although the rate of reduction declines.The Nusselt number profiles are plotted versus R d at the walls in Figure 3d.The augmentation is more sensitive in the right wall.The change in the heat amount at the left and the right surface is the same as the absorbed heat in the system. ( The numerical results of current work for velocity slip length of 0.01, Da = 0.01, and Gr/Re = 50 are in good agreement with the analytical solution.As shown in Figure 2 the shooting method can capture the precise results with 15 points.Figure 2 presents the consequence of the change of thermal radiation parameter on the numerous feature of temperature and velocity field of the microchannel in the constant value of the Grashof to Reynolds ratio (Gr/Re = 100), and θR = 10, λv = 0.01, We = 0.001, λT = 0.01, Da = 0.1.On behalf of the special case of no absorption of the thermal radiation in the fluid, the temperature has a linear profile as shown in Figure 3a.By increasing of thermal radiation number ranging 10 −4 to 10 −2 the dimensionless temperature is amplified, particularly close to the left partition.Figure 3b displays a curvy form for the velocity outline.By growth of the amount of thermal radiation, the dimensionless velocity profile alternates from sine shape to parabola shape and its maximum decrease.The maximum value occurs at the middle of parabola profile approaches the Poiseuille profile maximum (i.e., 1.5) by increase of Rd.The effect of thermal radiation on the friction factor at walls and pressure gradient is illustrated in Figure 3c.By increasing Rd, the friction factor at the walls (the left and right friction coefficients) decrease smoothly while the pressure gradient (the profile with the middle legend), increases dramatically.As well the friction factor for the same Reynolds number is greater at the right wall for low Rd and the reverse phenomena is seen for high Rd.Both skin-friction values continue to reduce further with the increase of Rd although the rate of reduction declines.The Nusselt number profiles are plotted versus Rd at the walls in Figure 3d.The augmentation is more sensitive in the right wall.The change in the heat amount at the left and the right surface is the same as the absorbed heat in the system.The effect of combined heat transfer coefficient (Gr/Re) is revealed in Figure 4.As exposed in Figure 4a with an increase of Gr/Re from 1 to 30 the dimensionless temperature is constant.Figure 4b illustrates the influence of the mixed convection parameter Gr/Re, on the dimensionless velocity profiles for the Rd = 10, and λT = 0.01, λv = 0.01, We = 0.001, θR = 10, Da = 0.1.As seen the parabola shape function changes to a sine-like shape for the velocity profile.It seems that the sine-like shape The effect of combined heat transfer coefficient (Gr/Re) is revealed in Figure 4.As exposed in Figure 4a with an increase of Gr/Re from 1 to 30 the dimensionless temperature is constant.Figure 4b illustrates the influence of the mixed convection parameter Gr/Re, on the dimensionless velocity profiles for the R d = 10, and λ T = 0.01, λ v = 0.01, We = 0.001, θ R = 10, Da = 0.1.As seen the parabola shape function changes to a sine-like shape for the velocity profile.It seems that the sine-like shape for the velocity profile is an odd function but the absolute value of the right side is slightly higher than the left side.In natural convection and heat transfer, the Grashof number (Gr) arises as a dimensionless number which approximates the ratio of the buoyancy to viscous force acting on a fluid.Notwithstanding, in forced convection the Reynolds number governs the fluid flow since in the current mixed convection problem the Gr/Re ratio is an important dimensionless parameter that governs the fluid flow.By increasing Gr/Re, the dimensionless velocity profile maximum increases and the location of the maximum moves from the middle to the right.It is obvious that the velocity is enhanced with increased buoyancy force.An increase of the reversal flow with the surge of Gr/Re and that the reversal flow is absent for small values of Gr/Re.The effect of Gr/Re on the friction factor at the walls and pressure gradient is illustrated in Figure 4c.For high Gr/Re, by increasing Gr/Re, the friction factors at the left and right walls are increased while the friction factor at the left is decreased for low Gr/Re values.Also the friction factor for the same Reynolds number is greater at the right wall rather than the left wall.As obvious from this figure, the natural convection augments the fluid flow near the hot wall and increases the wall skin fraction and diminishes the heat transfer near the cold wall and decreases it.The pressure gradient is constant and independent of Gr/Re.The same trend seen in Nusselt number at walls in Figure 4d, and that augmentation is more sensitive at the right wall.
The influence of Da is revealed in Figure 5.As exposed in Figure 5a by change of Darcy number from 0 to 15 for the λ T = 0.01, We = 0.001, θ R = 10, λ v = 0.01, R d = 10, and Gr/Re = 50, the dimensionless temperature profile is not altered.Figure 5b displays the alteration of the sine-like contour of the velocity profile caused by the existence of the porous media.By increasing the Darcy number, the dimensionless velocity profile conserves its sine shape but its peaks are chamfered and its maximum decreases.As a result the general effect of the Darcy parameter is to decrease the velocity magnitude which is done by the solid matrix.This is as a result of the existence of the porous matrix which generates a resistive force akin to the drag force that acts in the opposite direction of the fluid motion, thus bringing the velocity of the fluid to decrease.
The effect of Darcy number on the friction factor at the walls and pressure gradient is exemplified in Figure 5c.By increasing of Darcy number, the friction factors at both walls are decreased for low Darcy number and increased for high Darcy number.The change of pressure gradients by Darcy number is significant.Physically, it means that with the increasing Ha, the strength of the solid matrix resistance, the resistance force increases which drags the flow backward and resists the effect of natural convection.It is observed that dp/dx increases rapidly with Da.The Darcy number effect on Nusselt number at the walls is shown in Figure 5d.Although the friction factor for the same Darcy number is greater at the right wall rather than the left wall, by increasing the Darcy number the Nusselt number decreases at the right wall and the Nusselt number increases at the left wall where both values approach to the same value.The effect of Darcy number on the friction factor at the walls and pressure gradient is exemplified in Figure 5c.By increasing of Darcy number, the friction factors at both walls are decreased for low Darcy number and increased for high Darcy number.The change of pressure gradients by Darcy number is significant.Physically, it means that with the increasing Ha, the strength of the solid matrix resistance, the resistance force increases which drags the flow backward and resists the effect of natural convection.It is observed that dp/dx increases rapidly with Da.The Darcy number effect on Nusselt number at the walls is shown in Figure 5d.Although the friction factor for the same Darcy number is greater at the right wall rather than the left wall, by increasing the Darcy number the Nusselt number decreases at the right wall and the Nusselt number increases at the left wall where both values approach to the same value.The consequence of changing thermal parameter on the system is recognized in Figure 6.As shown in Figure 6a by increasing θ R from 0.1 to 5 the dimensionless temperature is increased, especially near the left wall.The maximum value occurs at the right wall and the minimum value comes about at the left wall.Figure 6b shows the sine-like shape of the velocity profile for the R d = 0.01, Gr/Re = 100, We = 0.001, λ T = 0.01, λ v = 0.01, and Da = 0.1.As observed, as θ R increases, the dimensionless velocity profile conserves its sine-shape but displays its maximum decrease.As understandable from this figure the natural convection effects are weakened by the increase of θ R .Furthermore, from Figures 3b and 6b it is detected that the increase of both radiation and thermal parameters make the reversal flow smaller, thus one consequence of radiation is the stabilization of the fluid motion as can be understood.The effect of θ R on the pressure gradient and friction factor at walls is illustrated in Figure 6c.By increasing θ R , the friction factor at the right wall is decreased slightly while the friction factor at the left wall is increased for θ R < 3 and decreased for θ R > 3. Also the friction factor for the θ R < 2.5 is greater at the wall Y = 1 while the friction factor at the wall Y = ´1 is greater for the θ R > 2.5.In addition a slight pressure gradient decrease (the profile with the middle legend) is seen for θ R < 0.5, but a dramatic increase is detected for θ R > 0.5.The increase of θ R has a slight effect on the walls' surface tension but it increases the dp/dx dramatically.Dimensionless temperature profiles are presented in Figures 3a, 4a, 5a and 6a disclose that the first derivative of the temperature profiles increases with the increase of the parameters θ R and R d while it decreases with an increase of Gr/Re.Therefore an increase in Nusselt Entropy 2016, 18, 147 9 of 15 number is seen by the increase in θ R and R d , while the augmentation of the Nusselt number is more at the right wall for θ R < 3.5 (see Figure 5d).
the friction factor for the θR < 2.5 is greater at the wall Y = 1 while the friction factor at the wall Y = −1 is greater for the θR > 2.5.In addition a slight pressure gradient decrease (the profile with the middle legend) is seen for θR < 0.5, but a dramatic increase is detected for θR > 0.5.The increase of θR has a slight effect on the walls' surface tension but it increases the dp/dx dramatically.Dimensionless temperature profiles are presented in Figures 3a-6a disclose that the first derivative of the temperature profiles increases with the increase of the parameters θR and Rd while it decreases with an increase of Gr/Re.Therefore an increase in Nusselt number is seen by the increase in θR and Rd, while the augmentation of the Nusselt number is more at the right wall for θR < 3.5 (see Figure 5d).The outcome of velocity slip in the vertical microchannel velocity and temperature is presented in Figure 7 by changing velocity slip length from 0 to 0.05 for the Rd = 0.01, and λT = 0.01, θR = 0.1, We = 0.001, Da = 0.1, Gr/Re = 50.It can be seen from Figure 7a that for different values of the velocity slip length the dimensionless temperature is not changed.Further, in Figure 7b it is observed that amount of velocity increases at the right wall and decreases at the left wall with an increase of velocity slip, while the sinus shape of the velocity profile is not altered.The maximum value occurs at the right peak and it moves right by increase of the velocity slip.By increase of velocity slip length, the maximum dimensionless velocity profiles to some extent increase.The consequence of velocity slip length on the friction factor at walls and pressure gradient is illustrated  7a that for different values of the velocity slip length the dimensionless temperature is not changed.Further, in Figure 7b it is observed that amount of velocity increases at the right wall and decreases at the left wall with an increase of velocity slip, while the sinus shape of the velocity profile is not altered.The maximum value occurs at the right peak and it moves right by increase of the velocity slip.By increase of velocity slip length, the maximum dimensionless velocity profiles to some extent increase.The consequence of velocity slip length on the friction factor at walls and pressure gradient is illustrated in Figure 7c.It is clearly seen from these figures that dp/dx and the C f at the Y = 1 increases while the C f at the Y = ´1 decrease with increasing velocity slip length.Also the friction factor for the same velocity slip length is greater at the right wall rather than the left wall.In Figure 7d, one can observe that the Nu at the Y = ´1 is not changed considerably with an increase of velocity slip length, but the Nu at the Y = 1 increases rapidly, especially for λ v > 0.03.The above discussion for the gas flow can be regarded by replacement of the velocity slip length by the mean free path of the molecules in a rare gas or velocity slip length by Knudsen number.By an upsurge of the gas viscosity and decrease of the gas density and the sound velocity in it, the velocity slip length increases.
The upshot of temperature slip is verified in Figure 8.As depicted in Figure 8a by increasing λ T from 0 to 0.1 the magnitude of the dimensionless temperature distribution shifted upward with a constant value.This shows that the fluid temperature between the boundaries increases with increasing temperature slip length.Figure 8b illustrates the sine-like outline of the velocity side view with fixed values of other parameters as the R d = 0.01, and Gr/Re = 50, We = 0.001, λ v = 0.01, Da = 0.1, θ R = 0.1.With increasing temperature slip, the dimensionless velocity profile is not changed much.Even though by increasing the temperature, the maximum velocity of the fluid increases, this increase are less than the increase in temperature for common values of the temperature slip.For the rare gas stream between parallel plates the temperature slip is proportional to the velocity slip length ratio or Knudsen number.By growth of the gas Prantdl number and decrease of the specific heat ratio, the temperature's slip length increases.The effect of temperature slip length on the friction factor at boundary conditions and pressure gradient is exemplified in Figure 8c.By increasing the temperature slip, the coefficient of friction at the walls is not changed meaningfully but the pressure gradient decreases slightly.The decrease of the pressure gradient in the gas can be justified by considering the definition of Prandtl number as the ratio of the viscous to the thermal diffusion.As a consequence, the fluid flow is resisted because of this predominant property of the viscous fluid that leads to the decrease in pressure gradient, so the temperature jumps the boundary condition because of the decrease in the required pumping power in comparison with no-jumps boundary condition.Furthermore the friction factor for the same temperature slip length is greater at the right wall than the left wall.The same trend is seen for the Nusselt numbers at walls as shown in Figure 8d.
Entropy 2016, 18, 147 10 of 15 in Figure 7c.It is clearly seen from these figures that dp/dx and the Cf at the Y = 1 increases while the Cf at the Y = −1 decrease with increasing velocity slip length.Also the friction factor for the same velocity slip length is greater at the right wall rather than the left wall.In Figure 7d, one can observe that the Nu at the Y = −1 is not changed considerably with an increase of velocity slip length, but the Nu at the Y = 1 increases rapidly, especially for λv > 0.03.The above discussion for the gas flow can be regarded by replacement of the velocity slip length by the mean free path of the molecules in a rare gas or velocity slip length by Knudsen number.By an upsurge of the gas viscosity and decrease of the gas density and the sound velocity in it, the velocity slip length increases.The upshot of temperature slip is verified in Figure 8.As depicted in Figure 8a by increasing λT from 0 to 0.1 the magnitude of the dimensionless temperature distribution shifted upward with a constant value.This shows that the fluid temperature between the boundaries increases with increasing temperature slip length.Figure 8b illustrates the sine-like outline of the velocity side view with fixed values of other parameters as the Rd = 0.01, and Gr/Re = 50, We = 0.001, λv = 0.01, Da = 0.1, θR = 0.1.With increasing temperature slip, the dimensionless velocity profile is not changed much.Even though by increasing the temperature, the maximum velocity of the fluid increases, this increase are less than the increase in temperature for common values of the temperature slip.For the rare gas stream between parallel plates the temperature slip is proportional to the velocity slip length ratio or Knudsen number.By growth of the gas Prantdl number and decrease of the specific heat ratio, the temperature's slip length increases.The effect of temperature slip length on the friction factor at boundary conditions and pressure gradient is exemplified in Figure 8c.By increasing the temperature slip, the coefficient of friction at the walls is not changed meaningfully but the pressure gradient decreases slightly.The decrease of the pressure gradient in the gas can be justified by considering the definition of Prandtl number as the ratio of the viscous to the thermal diffusion.As a consequence, the fluid flow is resisted because of this predominant property of the viscous fluid that leads to the decrease in pressure gradient, so the temperature jumps the boundary condition because of the decrease in the required pumping power in comparison with no-jumps boundary condition.Furthermore the friction factor for the same temperature slip length is greater at the right wall than the left wall.The same trend is seen for the Nusselt numbers at walls as shown in Figure 8d.The irreversibility in the channel flow of a fluid has two components of energy and momentum.Consequently, entropy production may occur as a result of fluid friction and heat transfer in the direction of finite temperature gradients.Following, Bejan [39,40], the volumetric rate of entropy generation can be expressed as: The irreversibility in the channel flow of a fluid has two components of energy and momentum.Consequently, entropy production may occur as a result of fluid friction and heat transfer in the direction of finite temperature gradients.Following, Bejan [39,40], the volumetric rate of entropy generation can be expressed as: .
The first term in Equation ( 7) describes the heat transfer irreversibility and the second term represents the local entropy generation rate due to fluid friction, respectively.Figure 8 presents the effect of Darcy number on the dimensionless heat transfer component of the entropy, the dimensionless viscous component of the entropy, the total dimensionless heat transfer component of the entropy, and the total dimensionless viscous component of the entropy.The effect of Hartmann number is established in Figure 8.As shown in Figure 9a by the increase of Da from 16 to 18 for the R d = 10, λ T = 0.1, λ v = 0.001, We = 0.001, θ R = 5, and Gr/Re = 1, because the dimensionless temperature shape is not hooked on Darcy number the S θ not altered.This narrow range of Darcy number is to emphasize the maximum exergy of a system.As exposed the S θ (Y) is roughly linear and has a greater value at the left wall and a lesser value at the right wall. Figure 9b displays the distribution of S u by the variation of compactness of the porous medium.The reality of compression of porous media produces a resistive force similar to the drag force that acts in the opposite direction of the fluid motion, thus causing the velocity of the fluid to decrease.By increasing of Darcy number, the peaks of the dimensionless velocity profile are chamfered consequently the velocity gradient inside the channel is decreased which leads to lesser viscous warming, so the overall effect of the Darcy number is to decrease the velocity component of entropy generation and it has an optimum near Da = 17.5.

Conclusions
The aim of this wprk was to describe the flow of a fully developed non-linear laminar mixed heat transfer of convection and radiation of an incompressible, electrically conducting radiative absorbing fluid in a vertical microchannel in the presence of a porous medium and the effects of thermal radiation heat absorption, and mixed convection.The consequences can be briefly summarized as follows: (1) Temperature increases with increasing Rd and θR.As the radiation parameter increases the capacity of absorption of thermal radiation increases which causes higher temperatures.As well the higher temperature parameter means a higher level of temperature of the system and an increase of radiation heat emitting sources.(2) The heat transfer between the two boundaries of the channel is not simply due to pure conduction and the thermal radiation and mixed convection in a channel filled with a fluid-saturated porous medium has a great impact on heat exchange mechanisms.(3) Dimensionless coefficients suitable for the evaluation of the dimensionless mean velocity, of the

Conclusions
The aim of this wprk was to describe the flow of a fully developed non-linear laminar mixed heat transfer of convection and radiation of an incompressible, electrically conducting radiative absorbing fluid in a vertical microchannel in the presence of a porous medium and the effects of thermal radiation heat absorption, and mixed convection.The consequences can be briefly summarized as follows: (1) Temperature increases with increasing R d and θ R .As the radiation parameter increases the capacity of absorption of thermal radiation increases which causes higher temperatures.As well the higher temperature parameter means a higher level of temperature of the system and an increase of radiation heat emitting sources.(2) The heat transfer between the two boundaries of the channel is not simply due to pure conduction and the thermal radiation and mixed convection in a channel filled with a fluid-saturated porous medium has a great impact on heat exchange mechanisms.(3) Dimensionless coefficients suitable for the evaluation of the dimensionless mean velocity, of the dimensionless bulk temperature and of the Nusselt numbers have been presented.(4) Natural convection helps the fluid flow.As well the increase of temperature through the channel helps natural convection.However the existence of a porous solid matrix increases the pressure loss inside the channel.Since the pressure gradient decreases with increasing Gr/Re; R d and θ R while it increases with an increase of Da. (5) The coefficient of skin friction increases as R d and θ R increase while it decreases with an increase of Gr/Re.The skin friction coefficient and mass transfer rates decrease with an increase in R d whereas heat transfer rate increases with an increase in the parameter R d .(6) Coefficient of skin friction and Nusselt number increase with an increase of R d and θ R while they decrease with the mixed convection parameter.(7) Grashof number, velocity slip, and pressure gradient increase skin friction and the Nusselt number, whereas temperature jump and Reynolds number reduce their values.(8) The shape of velocity profiles is different when Gr/Re changes.By increasing Gr/Re it alternates from a parabola to a sine shape profile.(9) The wall friction and Nusselt numbers may vary monotonically or non-monotonically with R d and θ R , again depending on the values of the other parameters.
Author Contributions: Mohamad Yaghoub Abdollahzadeh Jamalabadi designed the research with theoretical formulations.Payam Hooshmand performed the numerical simulations.Navid Bagheri and HamidReza KhakRah together analyzed the data and wrote the manuscript for initial submission.Majid Dousti had critical contribution in revising the manuscript including the preparation of responses and English correction.All authors have read and approved the final manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.

Figure 1 .
Figure 1.Schematic illustration of a vertical channel containing fully-developed mixed convection fluid flow.

Figure 1 .
Figure 1.Schematic illustration of a vertical channel containing fully-developed mixed convection fluid flow.
) are solved numerically using Runge-Kutta-Fehlberg method with shooting technique.The set of simultaneous first order differential equations of equivalent initial-value problem (Z 1 = f(Z,Y)) are constructed by the vector Z = [U; U 1 ; θ; θ 1 ] and the first guess of the initial value are assumed as Z(Y = ´1) = [0; 0; ´1; 0].To benchmark the current numerical method, the comparison of the numerical solution and analytical solution is illustrated in Figure 1.As the analytical solution of the problem for R d = 0.0, Γ = 0, and λ T = 0.0 is: U " Gr Re.Da Y ´PX Da

Figure 2 .
Figure 2. Comparison of numerical solution and analytical method.Figure 2. Comparison of numerical solution and analytical method.

Figure 2 .
Figure 2. Comparison of numerical solution and analytical method.Figure 2. Comparison of numerical solution and analytical method.

Figure 3 .
Figure 3. Thermal radiation effect on (a) the dimensionless temperature profile; (b) the dimensionless velocity profile; (c) the friction factor at walls and pressure gradient (middle); (d) Nusselt number at walls.

Figure 3 .
Figure 3. Thermal radiation effect on (a) the dimensionless temperature profile; (b) the dimensionless velocity profile; (c) the friction factor at walls and pressure gradient (middle); (d) Nusselt number at walls.
Entropy 2016, 18, 147 8 of 15 porous matrix which generates a resistive force akin to the drag force that acts in the opposite direction of the fluid motion, thus bringing the velocity of the fluid to decrease.

Figure 4 .
Figure 4. Grashof number to Reynolds number ratio effect on (a) the dimensionless temperature profile; (b) the dimensionless velocity profile; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 5 .
Figure 5. Darcy number effect on (a) the dimensionless temperature profile; (b) the velocity distribution; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.The effect of Darcy number on the friction factor at the walls and pressure gradient is exemplified in Figure5c.By increasing of Darcy number, the friction factors at both walls are decreased for low Darcy number and increased for high Darcy number.The change of pressure gradients by Darcy number is significant.Physically, it means that with the increasing Ha, the strength of the solid matrix resistance, the resistance force increases which drags the flow backward and resists the effect of natural convection.It is observed that dp/dx increases rapidly with Da.The Darcy number effect on Nusselt number at the walls is shown in Figure5d.Although the friction factor for the same Darcy number is greater at the right wall rather than the left wall, by increasing the Darcy number the Nusselt number decreases at the right wall and the Nusselt number increases at the left wall where both values approach to the same value.

Figure 4 .
Figure 4. Grashof number to Reynolds number ratio effect on (a) the dimensionless temperature profile; (b) the dimensionless velocity profile; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 4 .
Figure 4. Grashof number to Reynolds number ratio effect on (a) the dimensionless temperature profile; (b) the dimensionless velocity profile; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 5 .
Figure 5. Darcy number effect on (a) the dimensionless temperature profile; (b) the velocity distribution; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 5 .
Figure 5. Darcy number effect on (a) the dimensionless temperature profile; (b) the velocity distribution; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 6 .
Figure 6.Thermal parameter effect on (a) the dimensionless temperature profile; (b) the dimensionless velocity distribution; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 6 .
Figure 6.Thermal parameter effect on (a) the dimensionless temperature profile; (b) the dimensionless velocity distribution; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 7 .
Figure 7. Velocity slip effect on (a) the dimensionless temperature distribution; (b) the dimensionless velocity profile; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 8 .
Figure 8. Temperature slip effect on (a) the dimensionless temperature distribution; (b) the dimensionless velocity profile; (c) the friction factor at walls and pressure gradient; (d) Nusselt number at walls.

Figure 9 .
Figure 9. Darcy number effect on (a) the dimensionless heat transfer component of the entropy; (b) the dimensionless viscous component of the entropy; (c) the total dimensionless heat transfer component of the entropy; (d) the total dimensionless viscous component of the entropy.

Figure 9 .
Figure 9. Darcy number effect on (a) the dimensionless heat transfer component of the entropy; (b) the dimensionless viscous component of the entropy; (c) the total dimensionless heat transfer component of the entropy; (d) the total dimensionless viscous component of the entropy.