INTRODUCTION:

Double diffusive convective heat and mass transfer with chemical reaction are important in many industrial processes such as energy transfer in wet cooling tower, polymer processing, flow in a desert cooler, distribution of moisture and temperature over agricultural fields, groves of fruit trees, designs of chemical processing equipments and evaporation at the surface of a water body [1-3].

Afify [4] studied the effects of radiation and chemical reactions, in the presence of a transverse magnetic field, on the free convective flow and mass transfer of an optically dense viscous and electrically conducting fluid past a vertical isothermal cone surface. Numerical results for the skin-friction coefficient, the local Nusselt number, the local Sherwood number were given; as well, the velocity, temperature, and concentration profiles. Aboeldahab and Azzam[5] investigated the unsteady three-dimensional combined heat and mass free convective flow over a stretching surface with time-dependent chemical reaction. Abd El-Aziz and Salem[6] investigated the influence of chemical reactions on the coupled heat and mass transfer by natural convection from a vertical stretching surface in the presence of a space or temperature-dependent heat source effect. The sheet was stretched linearly in the presence of a uniform transverse magnetic field. They concluded that the flow field was influenced appreciably by the chemical reaction, heat source, magnetic field, and suction or injection at the sheet. Postelnicu [7] studied numerically the effect of the chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. It was obtained that the thickness of the concentration boundary layer decreased as the Lewis number increased. This phenomenon evidenced also when the chemical reaction was absent. Bég et al. [8] studied the heat and mass transfer characteristics of mixed convection flow of a chemically-reacting Newtonian fluid along vertical and inclined plates in the presence of diffusion-thermo(Dufour) and thermal-diffusion (Soret) effects. Skin friction coefficient was found to be enhanced with a positive increase in concentration-to thermal-buoyancy ratio parameter. Mohamed et al. [9] studied numerically the flow, chemical reaction and mass transfer of a steady laminar boundary layer of an electrically conducting and heat generating fluid driven by a continuously moving porous surface embedded in a non-Darcian porous medium in the presence of a transfer magnetic field. The results obtained were presented graphically for velocity, temperature and concentration profiles, as well as the Sherwood number for various considered parameters.Das[10] considered the effect of chemical reaction and thermal radiation on the heat and mass transfer flow of MHD micropolar fluid in a rotating frame of reference. Mallikarjuna and Bhuvanavijaya[11] investigated numerically the combined effects of non-uniform heat source/sink and higher order chemical reaction in Darcy-Forchheimer porous medium over a vertical plate in a rotating system. Thermal and solutal boundary layer equations took into account the non-uniform heat source/sink and higher order chemical reaction respectively. The effects of Forchhiemer parameter (inertial parameter), non-uniform heat source/sink and higher order chemical reaction were investigated and reported graphically on velocity, temperature and concentration profiles. More references can be found in [12 - 15], motivated by the investigations mentioned above, the purpose of the present work is to consider for the first time, the effect of higher order chemical reaction on double diffusive mixed convective flow over a rotating vertical cone embedded in a porous medium.

FORMULATION OF THE PROBLEM:

Consider steady, laminar, incompressible double diffusive mixed convective heat and mass transfer flow of Newtonian fluid over an impermeable vertical rotating cone embedded in a fluid saturated Darcy porous medium. It is assumed that the vertical cone is rotating in an ambient fluid with an angular velocity (Ω). Figure 1 shows physical configuration and the coordinate system. We consider rectangular curvilinear coordinate system (x, y, z), where x-axis is along a meridional section, y-axis is along a circular section and the z-axis is normal to the cone surface. The wall is maintained at variable temperature and variable concentration which are higher than the ambient fluid temperature and ambient fluid concentration respectively. The fluid saturated porous medium is assumed to be homogeneous and isotropic which is in a local thermodynamic equilibrium with solid matrix. The fluid properties (viscosity, density, thermal conductivity, molecular diffusivity and pressure with negligible body forces) are assumed to be constant except the density in the buoyancy force term of x-momentum equation. In addition, we assume n^th order homogeneous chemical reaction with a constant rate between the fluid and diffusing species. By employing the above assumptions, Oberbeck-Boussinesq approximation and usual buoyancy layer approximations, the governing equations for mass, momentum, energy and concentration are as follows:

(1)

(2)

(3)

(4)

(5)

The corresponding boundary conditions are defined as:-

(6)

where u, v and w are the velocity components along the tangential (x), circumferential or azimuthal (y) and normal (z) directions respectively, ( r ) is the radius of the cone, is the angular velocity of the rotation, is the fluid density, is the dynamic viscosity, () is the specific heat at constant pressure, (g) is the acceleration due to gravity. Also, represents the cone apex half angle, (K) is the permeability of the porous medium, (K_e) is the effective thermal conductivity, (K_r) is the chemical reaction parameter and (D) is the molecular diffusivity.

In order to get non-dimensional equations we introduce the following non-dimensional transformation:

(7)

where (L) is the cone slant height, is the cone surface temperature and is the concentration at the cone base (x = L). Substituting Eq. (7) in Eqs. (1) – (6), the following non-dimensional equations are obtained:-

(8)

(9)

(10) (11)

where prime denotes ordinary differentiation with respect to , is the inverse of Darcy number, is the Grashof number,is the buoyancy ratio, is the local Reynolds number, is the mixed convection parameter, is the Prandtl number, is the Schmidt number and is the chemical reaction parameter.

The transformed boundary conditions can be written as

(12)

If and N = 0, the problem reduces to the mixed convective heat transfer flow over a vertical rotating cone in a fluid saturated porous medium. The main objective of the present study is to find the parameters of physical interest in fluid flow, heat and mass transport problems which are the local surface skin friction coefficients in x (tangential) and y (azimuthal) directions, local Nusselt number and local Sherwood number. These parameters are given by

, , ,

These parameters can be written in non-dimensional form as follows:-

Skin friction coefficient in x and y-directions, Local Nusselt number and Local Sherwood number are respectively

, ,, (13)

NUMERICAL METHOD OF SOLUTION:

The set of equations (8) – (11) with boundary conditions (12) are solved by using shooting method that uses Runge-kutta method and Newton-Raphson method (Mallikarjuna et. al [16] and [17])

In order to assess the accuracy of the present numerical method, we compared our results with those of Hering and Grosh [18] in the absence of inverse Darcy parameter and concentration equation (N=0). The comparison in this case is found to be in good agreement, as shown in Table-1.

Table-1: The values of -H^ʺ(0), -Gʹ(0) and -θʹ(0) for different values of for Pr = 0.7, Da^-1=0, and N = 0 (in the absence of concentration equation).

	-H^ʺ(0)		-Gʹ(0)		-θʹ(0)
	Hering and Grosh [18]	Present work	Hering and Grosh [18]	Present work	Hering and Grosh [18]	Present work
0	1.0205	1.0203	0.61592	0.61583	0.42852	0.42842
0.1	1.1369	1.1368	0.65489	0.65492	0.46156	0.46141
1.0	2.2078	2.2075	0.85076	0.85080	0.61202	0.61213
10	8.5246	8.5243	1.40370	1.40363	1.01730	1.01748

RESULTS AND DISCUSSION:

Selected computations have been found to study the influence of mixed convection parameter, inverse Darcy number,chemical reaction parameter, buoyancy ratio, order of chemical reaction on velocity, temperature and concentration profiles, skin friction components, Nusselt number and Sherwood numbers. The numerical results are obtained for the mixed convection parameter varying as [0.5 ≤ g_s ≤ 10], inverse Darcy number varying as [0.5 ≤ Da^-1 ≤ 3], order of homogeneous chemical reaction varying as [1 ≤ n ≤ 4], chemical reaction parameter [-1.5 ≤ ≤ 3.5] and the buoyancy ratio [-0.5 ≤ N ≤ 1], while thePrandtl number is considered fixed at [Pr = 0.71].

Figure 2 illustrates the tangential velocity profiles for different values of the mixed convection parameter (g_s) and the inverse Darcy number (Da^-1) when [Pr = 0.71(air), Sc = 0.22 (hydrogen gas), = 3, n = 1 and N = 1]. In fact, the inverse Darcy number represents the influence of the porous medium adjacent to the rotating cone. It can be observed that when the mixed convection parameter (g_s) increases from [g_s= 0.5] to [g_s= 3], an enhancement in the tangential velocity profiles can be observed. The mixed convection parameter (g_s) represents the buoyancy to viscous forces ratio. The increase in the mixed convection parameter leads to reduce the boundary layer thickness and increase the fluid mass flow adjacent the rotating cone. The mixing of these effects cause a strong enhancement in the tangential velocity profiles. From the other hand, the tangential velocity profiles decrease as the inverse Darcy number (Da^-1) increases from [Da^-1 = 0.5] to [Da^-1 = 2.5]. This is due to the reduction in the flow velocity when the inverse Darcy number is high (i.e., Darcy number is low).This means that the effect of the porous medium is high when the inverse Darcy number is high which leads to reduce the flow velocity. Figure 3 shows the circumferential velocity profiles for different values of the mixed convection parameter (g_s) and the inverse Darcy number (Da^-1) when [Pr = 0.71 , Sc = 0.22 , = 3 , n = 1 and N = 1] . It can be seen that when the mixed convection parameter (g_s) increases from [g_s= 0.5] to [g_s= 3], the circumferential velocity profiles begin to decrease.This reduction is due to the increase of the mass flow of the ambient fluid into the boundary layerwhich leads to drop the circumferential velocity profiles. Also, the circumferential velocity profiles decrease as the inverse Darcy number (Da^-1) increases for the same reason explained above. Figure 4 explains the normal velocity profile for different values of the mixed convection parameter (g_s) and the inverse Darcy number (Da^-1) when [Pr = 0.71, Sc = 0.22, = 3, n = 1 and N = 1]. The normal velocity profile begins to decrease as the mixed convection parameter increases for the same reason above, while they increase as the inverse Darcy number increases.Figures 5 and 6 display respectively the temperature and theconcentration profiles for different values of the mixed convection parameter (g_s) and the inverse Darcy number (Da^-1) when [Pr = 0.71 , Sc = 0.22 , = 3 , n = 1 and N = 1]. It can be seen in thesefigures that both the temperature and theconcentration profiles decrease as the mixed convection parameter increases. This behavior is due to the increase in the flow of the ambient fluid adjacent to the rotating cone which leads to reduce both the temperature and the concentration as the mixed convection effect becomes significant.From the other side, the temperature and theconcentration profiles increase as the inverse Darcy number (Da^-1) increases from [Da^-1 = 0.5] to [Da^-1 = 2.5].This is because the porous medium causes a resistance to the flow velocity which leads to slow its motion and increases both its temperature and concentration.

Figures (7-11)represent respectively, the variation of chemical reaction parameter () on the tangential, circumferential and normal velocity profiles together with the temperature and concentration profiles for different values of the chemical reaction parameter () when [Pr = 0.71 , Sc = 0.22 , Da^-1= 1, g_s = 10, n = 1 and N = 1]. It can be seen from the results of figures 7 and 11 that the increase in the chemical reaction parameter leads to reduce the tangential velocity and concentration profiles. This behavior is due to the reduction in the fluid motion and its concentration adjacent the cone wall when the chemical reaction parameter increases. From the other side, the results of figures 8, 9 and 10, indicate that there is an increasing in the circumferential and normal velocities and temperature profiles as the chemical reaction parameter increases. This is because; the chemical reaction accelerates the fluid motion in the circumferential and normal directions together with the fluid temperature along the cone wall.

Figure 12 shows the concentration profiles for different values of the order of homogeneous chemical reaction (n) when [Pr = 0.71, Da^-1 = 1, Sc = 0.22, = 3, g_s = 10 and N = 1]. The results show that the concentration profiles increase as the order of homogeneous chemical reaction increases. It is important to note that the magnitude of increase in the concentration profile for the higher order reactions is greater than that of the magnitude of increase for the lower order reactions. Therefore, it can be concluded that the concentration profiles depend on the chemical reaction parameter and the order of chemical reaction rate.

Figures 13 and14 depict respectively , the effect of the mixed convection parameter and the inverse Darcy number on the tangential and azimuthal skin-friction coefficients at different values of the buoyancy ratio when [Pr = 0.71 , Sc = 0.22 , n = 1 and = 3].It can be observed from Fig.13 that the tangential skin-friction coefficient increases as the buoyancy force and mixed convection parameter increase, while it decreases as the inverse Darcy number increases. The increase in the inverse Darcy number means that the effect of the porous medium is high which leads to decrease the motion of the fluid and increases its boundary layer. It should be mentioned that the curve for (N<0) corresponds to the opposing flow, while those for (N>0) corresponds to the aiding flow. With respect to Fig.14,it can be seen that the azimuthal skin-friction coefficient increases as the buoyancy force, mixed convection parameter and inverse Darcy number increase. Therefore, it can be concluded that when the buoyancy ratio increases, the buoyancy induced flow in both tangential and azimuthal directions of the cone increase which leading to increase the skin-friction coefficients in these directions. The reason of this behavior is due to the increase in the velocity gradient in both the tangential and azimuthal directions with increasing values of the buoyancy ratio.

Figures 15 and 16 explain respectively, the effect of the mixed convection parameter and the inverse Darcy number on the local Nusselt and Sherwood numbers for different values of the buoyancy ratio when [Pr = 0.71, g_s = 10, Sc = 0.22, n = 1 and = 3].The results show that the local Nusselt and Sherwood numbers increase as the mixed convection parameter and the buoyancy ratio increase, while they decrease as the inverse Darcy number increases.

Figures 17 and18 show respectively , the effect of the mixed convection and the chemical reaction parameters on the tangential and azimuthal skin-friction coefficients at different values of the buoyancy ratio when [Pr = 0.71 , Da^-1 = 1 , Sc = 0.22 , n = 1]. It can be seen that as the chemical reaction parameter increases, the tangential and azimuthal skin-friction coefficients decrease. This behavior occurs when [N=1]. While, they increase when the chemical reaction parameter increases at [N= - 0.5].From the other hand, the tangential and azimuthal skin-friction coefficients increase when the mixed convection parameter and the buoyancy ratio increase.

Figures 19 and 20 illustrate respectively, the influence of the mixed convection and the chemical reaction parameters on the local Nusselt and Sherwood numbers at different values of the buoyancy ratio when [Pr = 0.71 , Da^-1 = 1 , Sc = 0.22 , n = 1]. It can be noticed in Fig. 19 that as the chemical reaction parameter increases, the local Nusselt number decreases. This behavior occurs when [N=1]. While, they increase when the chemical reaction parameter increases at [N= - 0.5]. This is because, the increase of the chemical reaction parameter leads to accelerate the fluid motion and reduces the tangential and azimuthal skin friction coefficients and the local Nusselt number at N=1, while a reverse behavior can be observed at [N= - 0.5]. With respect to Fig.20, the results indicate that as the chemical reaction parameter increases, the local Sherwood number increases for all values of the buoyancy ratio.This is due to the decreasing in the concentration difference between the cone surface and the fluid when the chemical reaction parameter increases. This leads to the increase in the rate of mass transfer at the cone surface and as a result the local Sherwood number increases. In addition, both the local Nusselt and Sherwood numbers increase when the mixed convection parameter and the buoyancy ratio increase.

Figure 21 displays the effect of the order of chemical reaction and chemical reaction parameteron the local Sherwood number at different values of the buoyancy ratio when [Pr = 0.71, Da^-1 = 1, Sc = 0.22, g_s = 10]. The results illustrate that the local Sherwood number increases as the order of chemical reaction increases for (< 0), while it decreases as the order of chemical reaction increases for (> 0). This notation can be seen for all values of the buoyancy ratio. Therefore, it can be concluded that the local Sherwood number depends on the chemical reaction parameter and order of the chemical reaction rate. It is useful to mention that when (= 0), the chemical reaction effect becomes negligible.

CONCLUSIONS:

The following conclusions can be drawn from the results of the present work.

· When the mixed convection parameter increases, a clear improvement in the tangential flow velocity profiles, tangential and azimuthal skin-friction coefficient, local Nusselt and Sherwood number, while it leads to decrease in the circumferential velocity and the normal velocity profiles.

· When the inverse Darcy number increases, the tangential and circumferential velocity profiles and results of tangential skin-friction coefficient and local Nusselt and Sherwood number are begin to decrease significantly, while it increases normal velocity profile, temperature and concentration and azimuthal skin-friction coefficients.

· The increase in the chemical reaction parameter leads to reduce the tangential velocity and concentration profiles. While, it increases the circumferential and normal velocity and temperature profiles.

· When the chemical reaction parameter increases, the tangential and azimuthal skin-friction coefficients decrease at [N=1]. While, they increase when the chemical reaction parameter increases at [N= - 0.5]. When the chemical reaction parameter increases, the local Nusselt number decreases at [N=1]. While, they increase when the chemical reaction parameter increases at [N= - 0.5]. The local Sherwood number increases for all values of the buoyancy ratio when the chemical reaction parameter increases.

The concentration profiles increase as the order of homogeneous chemical reaction increases. The local Sherwood number increases as the order of chemical reaction increases for (< 0), while it decreases as the order of chemical reaction increases for (> 0).

Fig.1: Physical model and coordinate system.

Fig 2: Tangential velocity profile for g_s and Da^-1.

Fig 3: Circumferential velocity profile for g_s and Da^-1

Fig 4: Normal velocity profile for g_s and Da^-1

Fig 5: Temperature profile forg_s and Da^-1

Fig 6: Concentration profile for g_s and Da^-1

Fig 7: Tangential velocity profile for

Fig 8: Circumferential velocity profile for

Fig 9: Normal velocity profile for

Fig 10: Temperature profile for

Fig 11: Concentration profile for

Fig 12: Concentration profile for n

Fig 13: Effect of g_s and Da^-1 on the tangential skin-friction coefficient

Fig 14: Effect of g_s and Da^-1 on the azimuthal skin-friction coefficient

Fig 15: Effect of g_s and Da^-1 on the local Nusselt number

Fig 16: Effect of g_s and Da^-1 on the local Sherwood number

Fig 17: Effect of g_s and on the tangential skin-friction coefficient

Fig 18: Effect of g_s and on the azimuthal skin-friction coefficient

Fig 19: Effect of g_s and on the local Nusselt number

Fig 20: Effect of g_s and on the local Sherwood number

Fig 21: Effect of n and chemical reaction parameter () on the local Sherwood number

CONFLICT OF INTEREST:

The authors declare that no conflict of interest.

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