**Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects**

Faiz GA Awad, Precious Sibanda and Mahesha Narayana *School of Mathematical Sciences, University of KwaZulu-Natal South Africa*

#### **1. Introduction**

80 Mass Transfer - Advanced Aspects

Yonemoto. Y, Kunugi, T. (2010a). Multi-scale modeling of the gas-liquid interface based on

Yonemoto, Y., Kunugi, T. (2010b). Fundamental numerical simulation of microbubble

Yonemoto, Y, Kunugi, T. (2010c). Macroscopic wettability based on an interfacial jump condition, *Phys. Rev. E.*, Vol. 81, May 2010c, pp. 056310-1-056310-8, ISSN 1550-2376

Vol. 22, September 2010b, pp. 397-405, ISSN 1875-0494

69-79

mathematical and thermodynamic approaches, *Open transport phenom.J.*, Vol. 2, pp.

interaction using multi-scale multiphase flow equation, *Microgravity Sci. Technol.*,

The study of double-diffusive convection has received considerable attention during the last several decades since this occurs in a wide range of natural settings. The origins of these studies can be traced to oceanography when hot salty water lies over cold fresh water of a higher density resulting in double-diffusive instabilities known as "salt-fingers," Stern (35; 36). Typical technological motivations for the study of double-diffusive convection range from such diverse fields as the migration of moisture through air contained in fibrous insulations, grain storage systems, the dispersion of contaminants through water-saturated soil, crystal growth and the underground disposal of nuclear wastes. Double-diffusive convection has also been cited as being of particular relevance in the modeling of solar ponds (Akbarzadeh and Manins (1)) and magma chambers (Fernando and Brandt (12)).

Double-diffusive convection problems have been investigated by, among others, Nield (28) Baines and Gill (3), Guo et al. (14), Khanafer and Vafai (17), Sunil et al. (37) and Gaikwad et al. (13). Studies have been carried out on horizontal, inclined and vertical surfaces in a porous medium by, among others, Cheng (9; 10), Nield and Bejan (29) and Ingham and Pop (32). Na and Chiou (24) presented the problem of laminar natural convection in Newtonian fluids over the frustum of a cone while Lai (18) investigated the heat and mass transfer by natural convection from a horizontal line source in saturated porous medium. Natural convection over a vertical wavy cone has been investigated by Pop and Na (33). Nakyam and Hussain (25) studied the combined heat and mass transfer by natural convection in a porous medium by integral methods.

Chamkha and Khaled (4) studied the hydromagnetic heat and mass transfer by mixed convection from a vertical plate embedded in a uniform porous medium. Chamkha (5) investigated the coupled heat and mass transfer by natural convection of Newtonian fluids about a truncated cone in the presence of magnetic field and radiation effects and Yih (38) examined the effect of radiation in convective flow over a cone. Cheng (6) used an integral approach to study the heat and mass transfer by natural convection from truncated cones in porous media with variable wall temperature and concentration. Khanafer and Vafai (17) studied the double-diffusive convection in a lid-driven enclosure filled with a fluid-saturated porous medium. Mortimer and Eyring (22) used an elementary transition state approach to obtain a simple model for Soret and Dufour effects in thermodynamically ideal mixtures of substances with molecules of nearly equal size. In their model the flow of heat in the Dufour effect was identified as the transport of the enthalpy change of activation as molecules diffuse.

a Porous Medium with Cross-Diffusion Effects 3

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 83

and fluid phases are assumed to be in local thermal equilibrium. The governing equations for

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> *<sup>K</sup>*g*<sup>β</sup>* cos <sup>Ω</sup> *ν*

where for a thin boundary layer, *r* = *x* sin Ω, g is the acceleration due to gravity, *c* is an empirical constant, *K* is the permeability, *ν* is kinematic viscosity of the fluid, respectively, *β* and *β*∗ are the thermal expansion and the concentration expansion coefficients, *αy* and *Dy* are the effective thermal and mass diffusivities of the saturated porous medium defined by *α<sup>y</sup>* = *α* + *γdu* and *Dy* = *D* + *ξdu*, respectively, *γ* and *ξ* are coefficients of thermal and solutal dispersions, respectively, *α* and *D* are constant thermal and molecular diffusivities, *kT* is the thermal diffusion ratio, *cs* is concentration susceptibility and *cp* is the specific heat at constant pressure. We assume a nonlinear power-law for temperature and concentration

✕

*x*

✕

*u*

❥

❪

*v*

*T*∞ *C*∞

Porous medium

, *<sup>φ</sup>*(*η*) = *<sup>C</sup>* <sup>−</sup> *<sup>C</sup>*<sup>∞</sup>

*Cw* − *C*<sup>∞</sup>

, (7)

❥

*<sup>v</sup>* <sup>=</sup> 0, *<sup>u</sup>* <sup>=</sup> 0, *<sup>T</sup>* <sup>=</sup> *Tw* <sup>=</sup> *<sup>T</sup>*<sup>∞</sup> <sup>+</sup> *Axn*, *<sup>C</sup>* <sup>=</sup> *Cw* <sup>=</sup> *<sup>C</sup>*<sup>∞</sup> <sup>+</sup> *Bx<sup>n</sup>* on *<sup>y</sup>* <sup>=</sup> 0, *<sup>x</sup>* <sup>≥</sup> 0 (5) *u* = 0, *T* = *T*∞, *C* = *C*<sup>∞</sup> as *y* → ∞, (6)

*Tw* − *T*<sup>∞</sup>

where *A*, *B* > 0 are constants and *n* is the power-law index. The subscripts *w*, ∞ refer to the cone surface and ambient conditions respectively. We introduce the similarity variables

<sup>2</sup> *<sup>f</sup>*(*η*), *<sup>θ</sup>*(*η*) = *<sup>T</sup>* <sup>−</sup> *<sup>T</sup>*<sup>∞</sup>

*y*

Ω

0

✦✦ ✟✟ ✦✦✦✦

✟ ✟✟✟✟✟

*r* ✲

*Tw Cw*

(*rv*) = 0, (1)

, (2)

*<sup>∂</sup>y*<sup>2</sup> , (3)

*<sup>∂</sup>y*<sup>2</sup> , (4)

<sup>+</sup> *<sup>β</sup>*<sup>∗</sup> *β ∂C ∂y* 

*∂*2*C*

*∂*2*T*

*∂T ∂y*

such a flow are (see Yih (38), Cheng (8), Murthy (23), El-Amin (11));

*∂u*<sup>2</sup>

(*ru*) + *<sup>∂</sup> ∂y*

*∂ ∂x*

*∂u ∂y* + *c* √*K ν*

*u ∂T ∂x* + *v ∂T <sup>∂</sup><sup>y</sup>* <sup>=</sup> *<sup>∂</sup> ∂y αy ∂T ∂y* + *DkT cscp*

*u ∂C ∂x* + *v ∂C <sup>∂</sup><sup>y</sup>* <sup>=</sup> *<sup>∂</sup> ∂y Dy ∂C ∂y* + *DkT cscp*

❄

g

Fig. 1. Inverted smooth cone in a porous medium

*<sup>x</sup>* , *ψ* = *αrRax*

*<sup>η</sup>* <sup>=</sup> *<sup>y</sup> x Ra* 1 2

variations within the fluid so that the boundary conditions are

1

The results were found to fit the Onsager reciprocal relationship (Onsager, (30)). Alam et al. (2) investigated the Dufour and Soret effects on steady combined free-forced convective and mass transfer flow past a semi-infinite vertical flat plate of hydrogen-air mixtures. They used the fourth order Runge-Kutta method to solve the governing equations of motion. Their study showed that the Dufour and Soret effects should not be neglected. Mansour et al. (21) studied the effects of a chemical reaction and thermal stratification on MHD free convective heat and mass transfer over a vertical stretching surface embedded in a porous media with Soret and Dufour effects. Narayana and Murthy (26) examined the Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium.

The effects of the Soret and Dufour parameters on free convection along a vertical wavy surface in a Newtonian fluid saturated Darcy porous medium has been investigated by Narayana and Sibanda (27). Their study showed that in both the aiding and opposing buoyancy cases increasing the Soret parameter leads to a reduction in the axial mass transfer coefficient. They further showed that the effect of the Dufour parameter is to increase the heat transfer coefficient at the surface. On the other hand, the mass transfer coefficient increased with the Dufour parameter only up to a certain critical value of the Soret parameter. Beyond this critical value, the mass transfer coefficient decreased with increasing Dufour parameter values.

The thermophoresis effect on a vertical plate embedded in a non-Darcy porous medium with suction and injection and subject to Dufour and Soret effects was investigated by Partha (31). The findings in this study underlined the importance of the Dufour, Soret and dispersion parameters on heat and mass transfer. The results showed that the Soret effect is influential in increasing the concentration distribution in both aiding as well as opposing buoyancy cases.

Cheng (8) studied the Dufour and Soret effects on heat and mass transfer over a downward-pointing vertical cone embedded in a porous medium saturated with a Newtonian fluid and constant wall temperature and concentration.

In this work we investigate heat and mass transfer from an inverted smooth and a wavy cone in porous media. In the case of the smooth cone we extend the work of Murthy and Singh (23) and El-Amin (11) to include cross-diffusion effects.

As with most problems in science and engineering, the equations that describe double-diffusive convection from an inverted cone in a porous medium are highly nonlinear and do not have closed form solutions. For the smooth cone, the equations are solved used the successive linearisation method (see Makukula et al. (19; 20)) which combines a non-perturbation technique with the Chebyshev spectral collocation method to produce an algorithm that is numerically accurate. The accuracy and robustness of the linearisation method is proved by using the Matlab bvp4c numerical routine and a shooting method to solve the equations. For the wavy cone, the governing nonlinear partial differential equations are solved using the well known Keller-box method.

#### **2. Flow over a smooth cone in porous medium**

Consider the problem of double-diffusive convection flow over inverted cone with half-angle Ω, embedded in a saturated non-Darcy porous medium as shown in Figure 1. The origin of the coordinate system is at the vertex of the cone. The *x*-axis measures the distance along the surface of the cone and the *y*-axis measures the distance outward and normal to the surface of the cone. The surface of the cone is subject to a non-uniform temperature *Tw* > *T*<sup>∞</sup> where *T*∞ is the temperature far from the cone surface. The solute concentration varies from *Cw* on the surface of the inverted cone to a lower concentration *C*∞ in the ambient fluid. The solid 2 Will-be-set-by-IN-TECH

The results were found to fit the Onsager reciprocal relationship (Onsager, (30)). Alam et al. (2) investigated the Dufour and Soret effects on steady combined free-forced convective and mass transfer flow past a semi-infinite vertical flat plate of hydrogen-air mixtures. They used the fourth order Runge-Kutta method to solve the governing equations of motion. Their study showed that the Dufour and Soret effects should not be neglected. Mansour et al. (21) studied the effects of a chemical reaction and thermal stratification on MHD free convective heat and mass transfer over a vertical stretching surface embedded in a porous media with Soret and Dufour effects. Narayana and Murthy (26) examined the Soret and Dufour effects on free convection heat and mass transfer from a horizontal flat plate in a Darcy porous medium. The effects of the Soret and Dufour parameters on free convection along a vertical wavy surface in a Newtonian fluid saturated Darcy porous medium has been investigated by Narayana and Sibanda (27). Their study showed that in both the aiding and opposing buoyancy cases increasing the Soret parameter leads to a reduction in the axial mass transfer coefficient. They further showed that the effect of the Dufour parameter is to increase the heat transfer coefficient at the surface. On the other hand, the mass transfer coefficient increased with the Dufour parameter only up to a certain critical value of the Soret parameter. Beyond this critical value, the mass transfer coefficient decreased with increasing Dufour parameter

The thermophoresis effect on a vertical plate embedded in a non-Darcy porous medium with suction and injection and subject to Dufour and Soret effects was investigated by Partha (31). The findings in this study underlined the importance of the Dufour, Soret and dispersion parameters on heat and mass transfer. The results showed that the Soret effect is influential in increasing the concentration distribution in both aiding as well as opposing buoyancy cases. Cheng (8) studied the Dufour and Soret effects on heat and mass transfer over a downward-pointing vertical cone embedded in a porous medium saturated with a Newtonian

In this work we investigate heat and mass transfer from an inverted smooth and a wavy cone in porous media. In the case of the smooth cone we extend the work of Murthy and Singh (23)

As with most problems in science and engineering, the equations that describe double-diffusive convection from an inverted cone in a porous medium are highly nonlinear and do not have closed form solutions. For the smooth cone, the equations are solved used the successive linearisation method (see Makukula et al. (19; 20)) which combines a non-perturbation technique with the Chebyshev spectral collocation method to produce an algorithm that is numerically accurate. The accuracy and robustness of the linearisation method is proved by using the Matlab bvp4c numerical routine and a shooting method to solve the equations. For the wavy cone, the governing nonlinear partial differential equations

Consider the problem of double-diffusive convection flow over inverted cone with half-angle Ω, embedded in a saturated non-Darcy porous medium as shown in Figure 1. The origin of the coordinate system is at the vertex of the cone. The *x*-axis measures the distance along the surface of the cone and the *y*-axis measures the distance outward and normal to the surface of the cone. The surface of the cone is subject to a non-uniform temperature *Tw* > *T*<sup>∞</sup> where *T*∞ is the temperature far from the cone surface. The solute concentration varies from *Cw* on the surface of the inverted cone to a lower concentration *C*∞ in the ambient fluid. The solid

fluid and constant wall temperature and concentration.

and El-Amin (11) to include cross-diffusion effects.

are solved using the well known Keller-box method.

**2. Flow over a smooth cone in porous medium**

values.

and fluid phases are assumed to be in local thermal equilibrium. The governing equations for such a flow are (see Yih (38), Cheng (8), Murthy (23), El-Amin (11));

$$
\frac{
\partial
}{
\partial x
}(ru) + \frac{
\partial
}{
\partial y
}(rv) = 0,
\tag{1}
$$

$$\frac{\partial u}{\partial y} + \frac{c\sqrt{K}}{\nu} \frac{\partial u^2}{\partial y} = \frac{K \mathbf{g} \boldsymbol{\beta} \cos \Omega}{\nu} \left( \frac{\partial T}{\partial y} + \frac{\boldsymbol{\beta}^\*}{\boldsymbol{\beta}} \frac{\partial \mathbb{C}}{\partial y} \right), \tag{2}$$

$$
\mu \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{\partial}{\partial y} \left( \alpha\_y \frac{\partial T}{\partial y} \right) + \frac{Dk\_T}{c\_s c\_p} \frac{\partial^2 C}{\partial y^2}, \tag{3}
$$

$$
\mu \frac{\partial \mathbf{C}}{\partial x} + v \frac{\partial \mathbf{C}}{\partial y} = \frac{\partial}{\partial y} \left( D\_y \frac{\partial \mathbf{C}}{\partial y} \right) + \frac{Dk\_T}{c\_s c\_p} \frac{\partial^2 T}{\partial y^2} \tag{4}
$$

where for a thin boundary layer, *r* = *x* sin Ω, g is the acceleration due to gravity, *c* is an empirical constant, *K* is the permeability, *ν* is kinematic viscosity of the fluid, respectively, *β* and *β*∗ are the thermal expansion and the concentration expansion coefficients, *αy* and *Dy* are the effective thermal and mass diffusivities of the saturated porous medium defined by *α<sup>y</sup>* = *α* + *γdu* and *Dy* = *D* + *ξdu*, respectively, *γ* and *ξ* are coefficients of thermal and solutal dispersions, respectively, *α* and *D* are constant thermal and molecular diffusivities, *kT* is the thermal diffusion ratio, *cs* is concentration susceptibility and *cp* is the specific heat at constant pressure. We assume a nonlinear power-law for temperature and concentration *x*

Fig. 1. Inverted smooth cone in a porous medium

variations within the fluid so that the boundary conditions are

$$\text{If } v = 0, \text{ } u = 0, \text{ } T = T\_{\text{ll}} = T\_{\infty} + A \mathbf{x}^{\text{ll}}, \text{ } \mathbf{C} = \mathbf{C}\_{\text{w}} = \mathbf{C}\_{\infty} + B \mathbf{x}^{\text{ll}} \quad \text{on} \quad y = 0, \text{ } \mathbf{x} \ge 0 \tag{5}$$

$$u = 0, \quad T = T\_{\infty}, \quad \mathbb{C} = \mathbb{C}\_{\infty} \quad \text{as} \quad y \to \infty,\tag{6}$$

where *A*, *B* > 0 are constants and *n* is the power-law index. The subscripts *w*, ∞ refer to the cone surface and ambient conditions respectively. We introduce the similarity variables

$$\eta = \frac{\mathcal{Y}}{\mathfrak{x}} \mathrm{Ra}\_{\mathfrak{x}}^{\frac{1}{2}}, \quad \psi = \operatorname{ar} \mathrm{Ra}\_{\mathfrak{x}}^{\frac{1}{2}} f(\eta), \quad \theta(\eta) = \frac{T - T\_{\infty}}{T\_{w} - T\_{\infty}}, \quad \phi(\eta) = \frac{\mathcal{C} - \mathcal{C}\_{\infty}}{\mathcal{C}\_{w} - \mathcal{C}\_{\infty}}, \tag{7}$$

a Porous Medium with Cross-Diffusion Effects 5

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 85

*<sup>i</sup>*→<sup>∞</sup> *<sup>θ</sup><sup>i</sup>* <sup>=</sup> lim

The functions *fm*, *θ<sup>m</sup>* and *φ<sup>m</sup>* (*m* ≥ 1) are approximations that are obtained by recursively solving the linear parts of the equations that result from substituting (16) in equations (9) - (11). Using the above assumptions, nonlinear terms in *fi*, *θi*, *φ<sup>i</sup>* and their corresponding derivatives

which are chosen to satisfy boundary conditions (12), the subsequent solutions for *fi*, *hi*, *θ<sup>i</sup> i* ≥ 1 are obtained by successively solving the linearized form of the governing equations.

The coefficient parameters *ak*,*i*−1, *bk*,*i*−1, *ck*,*i*−<sup>1</sup> (*<sup>k</sup>* = 1, 2, ..., 6), *rj*,*i*−<sup>1</sup> (*<sup>j</sup>* = 1, 2, 3) are given by

*i*−1 ∑ *m*=0 *f* ��

2

2

*i*−1 ∑ *m*=0 *φ*�

*i*−1 ∑ *m*=0

*i*−1 ∑ *m*=0 *θ*�

*i*−1 ∑ *m*=0

*<sup>i</sup>* + *<sup>b</sup>*5,*i*−<sup>1</sup> *<sup>f</sup>* �

*<sup>i</sup>* + *<sup>c</sup>*5,*i*−<sup>1</sup> *<sup>f</sup>* �

<sup>−</sup>*<sup>η</sup>* and *φ*0(*η*) = *e*

*<sup>i</sup>* = *<sup>r</sup>*1,*i*−1, (19)

*<sup>i</sup>* + *<sup>b</sup>*6,*i*−<sup>1</sup> *fi* + *Df <sup>φ</sup>*��

*<sup>i</sup>* + *<sup>c</sup>*6,*i*−<sup>1</sup> *fi* + *Srθ*��

*i*−1 ∑ *m*=0 *f* �� *m*,

> *i*−1 ∑ *m*=0 *θ*�� *<sup>m</sup>* − *n*

*i*−1 ∑ *m*=0 *f* ��

*i*−1 ∑ *m*=0 *φ*�� *<sup>m</sup>* − *n*

*<sup>m</sup>*, (23)

*i*−1 ∑ *m*=0 *θ*� *m*,

*<sup>m</sup>*, (25)

*<sup>m</sup>*, (26)

*i*−1 ∑ *m*=0 *φ*�

*<sup>i</sup>*(∞) = 0, *θi*(0) = *θi*(∞) = *φi*(0) = *φi*(∞) = 0. (22)

*fm* + *Ra<sup>γ</sup>*

*<sup>m</sup>*, (24)

*fm* + *Ra<sup>ξ</sup>*

*<sup>m</sup>*, (27)

*<sup>m</sup>*, *<sup>c</sup>*5,*i*−<sup>1</sup> = *Ra<sup>ξ</sup>*

*<sup>m</sup>*, *<sup>b</sup>*5,*i*−<sup>1</sup> = *Ra<sup>γ</sup>*

*<sup>i</sup>*→<sup>∞</sup> *<sup>φ</sup><sup>i</sup>* <sup>=</sup> 0. (17)

<sup>−</sup>*η*, (18)

*<sup>i</sup>* = *<sup>r</sup>*2,*i*−1, (20)

*<sup>i</sup>* = *<sup>r</sup>*3,*i*−1, (21)

where *fi*, *θi*, *φ<sup>i</sup>* (*i* = 1, 2, 3, . . .) are such that

Starting from the initial guesses

*<sup>a</sup>*1,*i*−<sup>1</sup> *<sup>f</sup>* ��

*<sup>b</sup>*1,*i*−1*θ*��

*<sup>c</sup>*1,*i*−1*φ*��

subject to the boundary conditions

*<sup>a</sup>*1,*i*−<sup>1</sup> = <sup>1</sup> + <sup>2</sup>*<sup>λ</sup>*

*<sup>b</sup>*1,*i*−<sup>1</sup> = <sup>1</sup> + *Ra<sup>γ</sup>*

*<sup>b</sup>*3,*i*−<sup>1</sup> = −*<sup>n</sup>*

*<sup>b</sup>*6,*i*−<sup>1</sup> <sup>=</sup> *<sup>n</sup>* <sup>+</sup> <sup>3</sup>

*<sup>c</sup>*1,*i*−<sup>1</sup> <sup>=</sup> <sup>1</sup>

*<sup>c</sup>*3,*i*−<sup>1</sup> = −*<sup>n</sup>*

*<sup>c</sup>*6,*i*−<sup>1</sup> <sup>=</sup> *<sup>n</sup>* <sup>+</sup> <sup>3</sup>

The linearized equations to be solved are

*<sup>i</sup>* + *<sup>a</sup>*2,*i*−<sup>1</sup> *<sup>f</sup>* �

*<sup>i</sup>* + *<sup>b</sup>*2,*i*−1*θ*�

*<sup>i</sup>* + *<sup>c</sup>*2,*i*−1*φ*�

*fi*(0) = *f* �

*i*−1 ∑ *m*=0 *f* �

> *i*−1 ∑ *m*=0 *θ*�

> > *i*−1 ∑ *m*=0 *f* �

2

*Le* <sup>+</sup> *Ra<sup>ξ</sup>*

*i*−1 ∑ *m*=0 *f* �

> *i*−1 ∑ *m*=0 *φ*�

2

*i*−1 ∑ *m*=0 *f* �

> *i*−1 ∑ *m*=0 *f* �

lim *<sup>i</sup>*→<sup>∞</sup> *fi* <sup>=</sup> lim

*<sup>f</sup>*0(*η*) = <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*η*, *<sup>θ</sup>*0(*η*) = *<sup>e</sup>*

*<sup>i</sup>* − *Nφ*�

*<sup>i</sup>* + *<sup>b</sup>*3,*i*−1*θ<sup>i</sup>* + *<sup>b</sup>*4,*i*−<sup>1</sup> *<sup>f</sup>* ��

*<sup>i</sup>* + *<sup>c</sup>*3,*i*−1*φ<sup>i</sup>* + *<sup>c</sup>*4,*i*−<sup>1</sup> *<sup>f</sup>* ��

*<sup>m</sup>*, *<sup>a</sup>*2,*i*−<sup>1</sup> = <sup>2</sup>*<sup>λ</sup>*

*<sup>m</sup>*, *<sup>b</sup>*4,*i*−<sup>1</sup> = *Ra<sup>γ</sup>*

*<sup>m</sup>*, *<sup>c</sup>*4,*i*−<sup>1</sup> = *Ra<sup>ξ</sup>*

*<sup>m</sup>*, *<sup>b</sup>*2,*i*−<sup>1</sup> <sup>=</sup> *<sup>n</sup>* <sup>+</sup> <sup>3</sup>

*<sup>m</sup>*, *<sup>c</sup>*2,*i*−<sup>1</sup> <sup>=</sup> *<sup>n</sup>* <sup>+</sup> <sup>3</sup>

are considered to be very small and therefore neglected.

*<sup>i</sup>* − *θ*�

where *ψ* is the stream function and *Rax* is the Rayleigh number defined by:

$$u = \frac{1}{r} \frac{\partial \psi}{\partial y'}, \quad v = -\frac{1}{r} \frac{\partial \psi}{\partial x} \quad \text{and} \quad Ra\_X = \frac{\mathbf{g} \beta \mathbf{K} \cos \Omega (T\_w - T\_\infty) x}{\mathbf{a} \nu}. \tag{8}$$

The dimensionless momentum, energy and concentration equations become

$$f'' + 2\lambda f'f'' - \theta' - N\phi' = 0,\tag{9}$$

$$
\theta'' + \frac{n+3}{2} f \theta' - nf' \theta + \text{Ra}\_{\gamma} (f'' \theta' + f' \theta'') + \text{D}\_f \phi'' = 0,\tag{10}
$$

$$\frac{1}{1.e}\phi'' + \frac{n+3}{2}f\phi' - nf'\phi + \text{Ra}\_{\overline{\xi}}(f''\phi' + f'\phi''') + \mathcal{S}\_{\text{I}}\theta'' = 0,\tag{11}$$

subject to the boundary conditions

$$\begin{aligned} f &= 0, \quad \theta = 1 \\ f' &= 0, \quad \theta = 0 \end{aligned} \quad \begin{aligned} \phi &= 1 \quad \text{on} \quad \eta = 0, \\ \phi &= 0 \quad \text{on} \quad \eta \to \infty. \end{aligned} \tag{12}$$

where primes denote differentiation with respect to *η*. The important thermo-physical parameters are the buoyancy ratio *N* (where *N* > 0 represents aiding buoyancy and *N* < 0 represents the opposing buoyancy), the Dufour parameter *Df* , the Soret parameter *Sr*, the pore depended Rayleigh number *Rad* and the Lewis number *Le*. These are defined as

$$N = \frac{\beta^\*}{\beta} \frac{\mathbb{C}\_w - \mathbb{C}\_\infty}{T\_w - T\_\infty},\\ \ D\_f = \frac{Dk\_T}{c\_s c\_p} \frac{\mathbb{C}\_w - \mathbb{C}\_\infty}{a(T\_w - T\_\infty)},\\ \ S\_r = \frac{Dk\_T}{c\_s c\_p} \frac{a(T\_w - T\_\infty)}{\mathbb{C}\_w - \mathbb{C}\_\infty},\tag{13}$$

$$Ra\_d = \frac{\text{g}\beta \text{K} \cos(\Omega)(T\_w - T\_\infty)d}{\text{av}}, \text{ } Le = \frac{a}{D}, \text{ } \hat{\sigma} = \frac{\mathbb{C}\sqrt{\text{K}}a}{\text{v}d},\tag{14}$$

where *Ra<sup>γ</sup>* = *γRad*, *Ra<sup>ξ</sup>* = *ξRad* represent the thermal and solutal dispersions respectively, *λ* = *σ*ˆ *Rad* and *σ*ˆ is an inertial parameter. The parameters of engineering interest in heat and mass problems are the local Nusselt number *Nux* and the local Sherwood number *Shx*. These parameters characterize the surface heat and mass transfer rates respectively. The local heat and mass transfer rates from the surface of the cone are characterized by the Nusselt and Sherwood numbers respectively where

$$\mathrm{Nu}\_{\mathrm{X}} = -\mathrm{Ra}\_{\mathrm{x}}^{\frac{1}{2}} \left[ 1 + \mathrm{Ra}\_{\gamma} f'(0) \right] \theta'(0) \quad \text{and} \quad \mathrm{Sh}\_{\mathrm{X}} = -\mathrm{Ra}\_{\mathrm{x}}^{\frac{1}{2}} \left[ 1 + \mathrm{Ra}\_{\tilde{\zeta}} f'(0) \right] \phi'(0). \tag{15}$$

#### **2.1 Method of solution**

To solve equations (9) - (12), the successive linearisation method (see Makukula et al. (19; 20)) was used. This assumes that the functions *f*(*η*), *θ*(*η*) and *φ*(*η*)) may be expressed as

$$f(\eta) = f\_i(\eta) + \sum\_{m=0}^{i-1} f\_m(\eta)\_{\prime}$$

$$\theta(\eta) = \theta\_i(\eta) + \sum\_{m=0}^{i-1} \theta\_m(\eta)\_{\prime} \tag{16}$$

$$\text{phi}(\eta) = \phi\_i(\eta) + \sum\_{m=0}^{i-1} \phi\_m(\eta)\_{\prime}$$

where *fi*, *θi*, *φ<sup>i</sup>* (*i* = 1, 2, 3, . . .) are such that

$$\lim\_{i \to \infty} f\_i = \lim\_{i \to \infty} \theta\_i = \lim\_{i \to \infty} \phi\_i = 0. \tag{17}$$

The functions *fm*, *θ<sup>m</sup>* and *φ<sup>m</sup>* (*m* ≥ 1) are approximations that are obtained by recursively solving the linear parts of the equations that result from substituting (16) in equations (9) - (11). Using the above assumptions, nonlinear terms in *fi*, *θi*, *φ<sup>i</sup>* and their corresponding derivatives are considered to be very small and therefore neglected.

Starting from the initial guesses

4 Will-be-set-by-IN-TECH

*θ* + *Raγ*(*f* ��*θ*� + *f* �

*f* = 0, *θ* = 1 , *φ* = 1 on *η* = 0,

where primes denote differentiation with respect to *η*. The important thermo-physical parameters are the buoyancy ratio *N* (where *N* > 0 represents aiding buoyancy and *N* < 0 represents the opposing buoyancy), the Dufour parameter *Df* , the Soret parameter *Sr*, the

> *Cw* − *C*<sup>∞</sup> *α*(*Tw* − *T*∞)

(0) and *Shx* <sup>=</sup> <sup>−</sup>*Ra* <sup>1</sup>

*i*−1 ∑ *m*=0

*i*−1 ∑ *m*=0

*i*−1 ∑ *m*=0 *fm*(*η*),

*φm*(*η*),

pore depended Rayleigh number *Rad* and the Lewis number *Le*. These are defined as

*αν* , *Le* <sup>=</sup> *<sup>α</sup>*

where *Ra<sup>γ</sup>* = *γRad*, *Ra<sup>ξ</sup>* = *ξRad* represent the thermal and solutal dispersions respectively, *λ* = *σ*ˆ *Rad* and *σ*ˆ is an inertial parameter. The parameters of engineering interest in heat and mass problems are the local Nusselt number *Nux* and the local Sherwood number *Shx*. These parameters characterize the surface heat and mass transfer rates respectively. The local heat and mass transfer rates from the surface of the cone are characterized by the Nusselt and

To solve equations (9) - (12), the successive linearisation method (see Makukula et al. (19; 20))

was used. This assumes that the functions *f*(*η*), *θ*(*η*) and *φ*(*η*)) may be expressed as

*f*(*η*) = *fi*(*η*) +

*θ*(*η*) = *θi*(*η*) +

*phi*(*η*) = *φi*(*η*) +

, *Df* <sup>=</sup> *DkT cscp*

(0)]*θ*�

*φ* + *Ra<sup>ξ</sup>* (*f* ��*φ*� + *f* �

and *Rax* <sup>=</sup> <sup>g</sup>*β<sup>K</sup>* cos <sup>Ω</sup>(*Tw* <sup>−</sup> *<sup>T</sup>*∞)*<sup>x</sup>*

*f* � = 0, *θ* = 0 , *φ* = 0 on *η* → ∞. (12)

, *Sr* <sup>=</sup> *DkT cscp*

<sup>√</sup>*K<sup>α</sup>*

2

*<sup>x</sup>* [1 + *Ra<sup>ξ</sup> f* �

*<sup>D</sup>* , *<sup>σ</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>C</sup>*

*f* �� + 2*λ f* � *f* �� − *θ*� − *Nφ*� = 0, (9)

*αν* . (8)

*θ*��) + *Df φ*�� = 0, (10)

*α*(*Tw* − *T*∞) *Cw* − *C*<sup>∞</sup>

*<sup>ν</sup><sup>d</sup>* , (14)

(0)]*φ*�

*θm*(*η*), (16)

, (13)

(0). (15)

*φ*��) + *Srθ*�� = 0, (11)

where *ψ* is the stream function and *Rax* is the Rayleigh number defined by:

The dimensionless momentum, energy and concentration equations become

*r ∂ψ ∂x*

<sup>2</sup> *<sup>f</sup> <sup>θ</sup>*� <sup>−</sup> *n f* �

<sup>2</sup> *<sup>f</sup> <sup>φ</sup>*� <sup>−</sup> *n f* �

*n* + 3

*<sup>∂</sup><sup>y</sup>* , *<sup>v</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup>

*n* + 3

*Cw* − *C*<sup>∞</sup> *Tw* − *T*<sup>∞</sup>

*Rad* <sup>=</sup> <sup>g</sup>*β<sup>K</sup>* cos(Ω)(*Tw* <sup>−</sup> *<sup>T</sup>*∞)*<sup>d</sup>*

*<sup>u</sup>* <sup>=</sup> <sup>1</sup> *r ∂ψ*

*θ*�� +

1 *Le <sup>φ</sup>*�� <sup>+</sup>

subject to the boundary conditions

*<sup>N</sup>* <sup>=</sup> *<sup>β</sup>*<sup>∗</sup> *β*

Sherwood numbers respectively where

2

*<sup>x</sup>* [1 + *Ra<sup>γ</sup> f* �

*Nux* <sup>=</sup> <sup>−</sup>*Ra* <sup>1</sup>

**2.1 Method of solution**

$$f\_0(\eta) = 1 - e^{-\eta}, \quad \theta\_0(\eta) = e^{-\eta} \quad \text{and} \quad \phi\_0(\eta) = e^{-\eta}, \tag{18}$$

which are chosen to satisfy boundary conditions (12), the subsequent solutions for *fi*, *hi*, *θ<sup>i</sup> i* ≥ 1 are obtained by successively solving the linearized form of the governing equations. The linearized equations to be solved are

$$a\_{1,i-1}f\_i^{\prime\prime} + a\_{2,i-1}f\_i^{\prime} - \theta\_i^{\prime} - \mathcal{N}\phi\_i^{\prime} = r\_{1,i-1} \tag{19}$$

$$b\_{1,i-1} \theta\_i^{\prime\prime} + b\_{2,i-1} \theta\_i^{\prime} + b\_{3,i-1} \theta\_i + b\_{4,i-1} f\_i^{\prime\prime} + b\_{5,i-1} f\_i^{\prime} + b\_{6,i-1} f\_i + D\_f \phi\_i^{\prime\prime} = r\_{2,i-1} \tag{20}$$

$$c\_{1,i-1}\phi\_i^{\prime\prime} + c\_{2,i-1}\phi\_i^{\prime} + c\_{3,i-1}\phi\_i + c\_{4,i-1}f\_i^{\prime\prime} + c\_{5,i-1}f\_i^{\prime} + c\_{6,i-1}f\_i + Sr\theta\_i^{\prime\prime} = r\_{3,i-1} \tag{21}$$

subject to the boundary conditions

$$f\_i(0) = f\_i'(\infty) = 0, \ \theta\_i(0) = \theta\_i(\infty) = \phi\_i(0) = \phi\_i(\infty) = 0. \tag{22}$$

The coefficient parameters *ak*,*i*−1, *bk*,*i*−1, *ck*,*i*−<sup>1</sup> (*<sup>k</sup>* = 1, 2, ..., 6), *rj*,*i*−<sup>1</sup> (*<sup>j</sup>* = 1, 2, 3) are given by

$$\begin{aligned} a\_{1,i-1} &= 1 + 2\lambda \sum\_{m=0}^{i-1} f\_{m}' & a\_{2,i-1} &= 2\lambda \sum\_{m=0}^{i-1} f\_{m}'' & \text{(23)}\\ b\_{1,i-1} &= 1 + \text{Ra}\_{\gamma} \sum\_{m=0}^{i-1} f\_{m}' & b\_{2,i-1} &= \frac{n+3}{2} \sum\_{m=0}^{i-1} f\_{m} + \text{Ra}\_{\gamma} \sum\_{m=0}^{i-1} f\_{m\prime}''\\ b\_{3,i-1} &= -n \sum\_{m=0}^{i-1} f\_{m\prime}' & b\_{4,i-1} &= \text{Ra}\_{\gamma} \sum\_{m=0}^{i-1} \theta\_{m\prime}' & b\_{5,i-1} &= \text{Ra}\_{\gamma} \sum\_{m=0}^{i-1} \theta\_{m}' - n \sum\_{m=0}^{i-1} \theta\_{m\prime}'\\ b\_{6,i-1} &= \frac{n+3}{2} \sum\_{m=0}^{i-1} \theta\_{m\prime}' \end{aligned} \tag{24}$$

$$c\_{1,i-1} = \frac{1}{Le} + Ra\_{\tilde{\xi}} \sum\_{m=0}^{i-1} f\_{m\prime}^{\prime} \quad c\_{2,i-1} = \frac{n+3}{2} \sum\_{m=0}^{i-1} f\_m + Ra\_{\tilde{\xi}} \sum\_{m=0}^{i-1} f\_{m\prime}^{\prime\prime} \tag{25}$$

$$\text{Ca}\_{3,i-1} = -n \sum\_{m=0}^{i-1} f'\_{m\prime} \quad \text{c}\_{4,i-1} = \text{Ra}\_{\tilde{\xi}} \sum\_{m=0}^{i-1} \phi'\_{m\prime} \quad \text{c}\_{5,i-1} = \text{Ra}\_{\tilde{\xi}} \sum\_{m=0}^{i-1} \phi''\_{m} - n \sum\_{m=0}^{i-1} \phi'\_{m\prime} \tag{26}$$

$$c\_{6,i-1} = \frac{n+3}{2} \sum\_{m=0}^{i-1} \phi'\_{m\prime} \tag{27}$$

a Porous Medium with Cross-Diffusion Effects 7

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 87

differentiation matrix. Substituting equations (34) - (36) in (19) - (22) leads to the matrix

**<sup>D</sup>***Nk* � *fi*(*ξk*) = 0,

In equation (37), **<sup>A</sup>***i*−<sup>1</sup> is a (3*N*� + <sup>3</sup>) × (3*N*� + <sup>3</sup>) square matrix and **<sup>X</sup>***<sup>i</sup>* and **<sup>R</sup>***<sup>i</sup>* are (3*N*� + <sup>1</sup>) × <sup>1</sup>

In the above definitions, **<sup>a</sup>***k*,*i*−1, **<sup>b</sup>***k*,*i*−1, **<sup>c</sup>***k*,*i*−<sup>1</sup> (*<sup>k</sup>* = 1, 2, .., 6) are diagonal matrices of size (*N*� + 1) × (*N*� + 1) and **I** is an identity matrix of size (*N*� + 1) × (*N*� + 1). After modifying the matrix

Equations (9) - (12) were further solved numerically using the Matlab bvp4c routine and a shooting technique comprising the Runge-Kutta method of four slopes and the Newton-Raphson method. In solving the boundary value problem by the shooting method, the appropriate '∞' was determined through actual computations and differs for each set of

system (37) to incorporate boundary conditions (38) - (37), the solution is obtained as

**X***<sup>i</sup>* = **A**−<sup>1</sup>

⎡ ⎣ **F***i* **Θ***i* **Φ***i*

*N*� ∑ *k*=0

*<sup>θ</sup>i*(*ξN*� ) = *<sup>θ</sup>i*(*ξ*0) = *<sup>φ</sup>i*(*ξN*� ) = *<sup>φ</sup>i*(*ξ*0) = 0. (39)

⎤

**<sup>F</sup>***<sup>i</sup>* = [ *fi*(*ξ*0), *fi*(*ξ*1),..., *fi*(*ξN*�−1), *fi*(*ξN*� )]*T*, (41) **<sup>Θ</sup>***<sup>i</sup>* = [*θi*(*ξ*0), *<sup>θ</sup>i*(*ξ*1),..., *<sup>θ</sup>i*(*ξN*�−1), *<sup>θ</sup>i*(*ξN*� )]*T*, (42) **<sup>Φ</sup>***<sup>i</sup>* = [*φi*(*ξ*0), *<sup>φ</sup>i*(*ξ*1),..., *<sup>φ</sup>i*(*ξN*�−1), *<sup>φ</sup>i*(*ξN*� )]*T*, (43) **<sup>r</sup>**1,*i*−<sup>1</sup> = [*r*1,*i*−1(*ξ*0),*r*1,*i*−1(*ξ*1),...,*r*1,*i*−1(*ξN*�−1),*r*1,*i*−1(*ξN*� )]*T*, (44) **<sup>r</sup>**2,*i*−<sup>1</sup> = [*r*2,*i*−1(*ξ*0),*r*2,*i*−1(*ξ*1),...,*r*2,*i*−1(*ξN*�−1),*r*2,*i*−1(*ξN*� )]*T*, (45) **<sup>r</sup>**3,*i*−<sup>1</sup> = [*r*3,*i*−1(*ξ*0),*r*3,*i*−1(*ξ*1),...,*r*3,*i*−1(*ξN*�−1),*r*3,*i*−1(*ξN*� )]*T*, (46) *<sup>A</sup>*<sup>11</sup> <sup>=</sup> **<sup>a</sup>**1,*i*−1**D**<sup>2</sup> <sup>+</sup> **<sup>a</sup>**2,*i*−1**D**, *<sup>A</sup>*<sup>12</sup> <sup>=</sup> <sup>−</sup>**I**, *<sup>A</sup>*<sup>13</sup> <sup>=</sup> <sup>−</sup>*N***<sup>I</sup>** (47) *<sup>A</sup>*<sup>21</sup> <sup>=</sup> **<sup>b</sup>**4,*i*−1**D**<sup>2</sup> <sup>+</sup> **<sup>b</sup>**5,*i*−1**<sup>D</sup>** <sup>+</sup> **<sup>b</sup>**6,*i*−1**I**, *<sup>A</sup>*<sup>22</sup> <sup>=</sup> **<sup>b</sup>**1,*i*−1**D**<sup>2</sup> <sup>+</sup> **<sup>b</sup>**2,*i*−1**<sup>D</sup>** <sup>+</sup> **<sup>b</sup>**3,*i*−1**I**, (48) *<sup>A</sup>*<sup>23</sup> <sup>=</sup> *Df* **<sup>D</sup>**2, *<sup>A</sup>*<sup>31</sup> <sup>=</sup> **<sup>c</sup>**4,*i*−1**D**<sup>2</sup> <sup>+</sup> **<sup>c</sup>**5,*i*−1**<sup>D</sup>** <sup>+</sup> **<sup>c</sup>**6,*i*−1**I**, (49) *<sup>A</sup>*<sup>32</sup> <sup>=</sup> **<sup>c</sup>**1,*i*−1**D**<sup>2</sup> <sup>+</sup> **<sup>c</sup>**2,*i*−1**<sup>D</sup>** <sup>+</sup> **<sup>c</sup>**3,*i*−1**I**, *<sup>A</sup>*<sup>33</sup> <sup>=</sup> *Sr***D**2. (50)

<sup>⎦</sup> , **<sup>R</sup>***i*−<sup>1</sup> =

*<sup>L</sup>*D with D being the Chebyshev spectral

**D**0*<sup>k</sup> fi*(*ξk*) = 0, (38)

**<sup>A</sup>***i*−1**X***<sup>i</sup>* = **<sup>R</sup>***i*−1, (37)

⎡ ⎣

**<sup>r</sup>**1,*i*−<sup>1</sup> **<sup>r</sup>**2,*i*−<sup>1</sup> **<sup>r</sup>**3,*i*−<sup>1</sup>

*<sup>i</sup>*−1**R***i*−1. (51)

⎤

⎦ , (40)

where *s* is the order of differentiation and **D** = <sup>2</sup>

*fi*(*ξN*� ) = 0,

⎡ ⎣

*N*� ∑ *k*=0

*A*<sup>11</sup> *A*<sup>12</sup> *A*<sup>13</sup> *A*<sup>21</sup> *A*<sup>22</sup> *A*<sup>23</sup> *A*<sup>31</sup> *A*<sup>32</sup> *A*<sup>33</sup> ⎤

⎦ , **X***<sup>i</sup>* =

subject to the boundary conditions

column vectors defined by

**<sup>A</sup>***i*−<sup>1</sup> =

equation

where

parameter values.

$$r\_{1,i-1} = -\left[\sum\_{m=0}^{i-1} f\_m^{\prime\prime} + 2\lambda \sum\_{m=0}^{i-1} f\_m^{\prime} \sum\_{m=0}^{i-1} f\_m^{\prime\prime} - \sum\_{m=0}^{i-1} h\_m^{\prime} - N \sum\_{m=0}^{i-1} g\_m^{\prime} \right],\tag{28}$$

$$\begin{split} r\_{2,i-1} &= -\left[\sum\_{m=0}^{i-1} \theta\_m^{\prime\prime} + D\_f \sum\_{m=0}^{i-1} \phi\_m^{\prime\prime} + \frac{n+3}{2} \sum\_{n=0}^{i-1} f\_m \sum\_{n=0}^{i-1} \theta\_n^{\prime} - n \sum\_{m=0}^{i-1} f\_m^{\prime} \sum\_{m=0}^{i-1} \theta\_m \right] \\ &+ \mathcal{R}a\_\gamma \left( f\_m^{\prime\prime} \sum\_{m=0}^{i-1} \theta\_m^{\prime} + f\_m^{\prime} \sum\_{m=0}^{i-1} \theta\_m^{\prime\prime} \right) \,, \end{split}$$

$$r\_{3,i-1} = -\left[\frac{1}{L\varepsilon} \sum\_{m=0}^{i-1} \phi\_m'' + \mathcal{S}\_r \sum\_{m=0}^{i-1} \phi\_m'' + \frac{n+3}{2} \sum\_{m=0}^{i-1} f\_m \sum\_{m=0}^{i-1} \phi\_m' - n \sum\_{m=0}^{i-1} f\_m' \sum\_{m=0}^{i-1} \phi\_m \right] \tag{29}$$

$$+\text{Ra}\_{\gamma} \left( f\_{m}^{\prime\prime} \sum\_{m=0}^{i-1} g\_{m}^{\prime} + f\_{m}^{\prime} \sum\_{m=0}^{i-1} \Phi\_{m}^{\prime\prime} \right) \right].\tag{30}$$

The functions *fi*, *θi*, *φ<sup>i</sup>* (*i* ≥ 1) are obtained by iteratively solving equations (19) - (22). The approximate solutions for *f*(*η*), *θ*(*η*) and *φ*(*η*) are then obtained as

$$f(\eta) \approx \sum\_{m=0}^{\hat{M}} f\_m(\eta), \quad \theta(\eta) \approx \sum\_{m=0}^{\hat{M}} \theta\_m(\eta), \quad \phi(\eta) \approx \sum\_{m=0}^{\hat{M}} \phi\_m(\eta), \tag{31}$$

where *M* is the order of the SLM approximation. Equations (19) - (22) were solved using the Chebyshev spectral collocation method where the unknown functions are approximated using Chebyshev interpolating polynomials at the Gauss-Lobatto points

$$\mathfrak{F}\_{\hat{\jmath}} = \cos \frac{\pi \hat{\jmath}}{\hat{N}}, \quad \hat{\jmath} = 0, 1, \ldots, \hat{N}, \tag{32}$$

where *N* is the number of collocation points. The physical region [0, ∞) is first transformed into the region [−1, 1] using the domain truncation technique in which the problem is solved on the interval [0, *L*] instead of [0, ∞). This is achieved by using the mapping

$$\frac{\eta}{L} = \frac{\xi + 1}{2}, \quad -1 \le \xi \le 1,\tag{33}$$

where *L* is the scaling parameter used to invoke the boundary condition at infinity. The unknown functions *fi*, *θ<sup>i</sup>* and *φ<sup>i</sup>* are approximated at the collocation points by

$$f\_i(\boldsymbol{\xi}) \approx \sum\_{k=0}^{N} f\_i(\boldsymbol{\xi}\_k) T\_k(\boldsymbol{\xi}\_j), \ \theta\_i(\boldsymbol{\xi}) \approx \sum\_{k=0}^{N} \theta\_i(\boldsymbol{\xi}\_k) T\_k(\boldsymbol{\xi}\_j), \ \phi\_i(\boldsymbol{\xi}) \approx \sum\_{k=0}^{N} \phi\_i(\boldsymbol{\xi}\_k) T\_k(\boldsymbol{\xi}\_j), \ j = 0, 1, \dots, \widehat{N}, \tag{34}$$

where *Tk* is the *k*th Chebyshev polynomial defined as

$$T\_k(\xi) = \cos[k \cos^{-1}(\xi)].\tag{35}$$

The derivatives at the collocation points are represented as

$$\frac{d^s f\_{\hat{l}}}{d\eta^s} = \sum\_{k=0}^{\hat{N}} \mathbf{D}\_{\hat{k}j}^s f\_{\hat{l}}(\xi\_k), \quad \frac{d^s \theta\_{\hat{l}}}{d\eta^s} = \sum\_{k=0}^{\hat{N}} \mathbf{D}\_{\hat{k}j}^s \theta\_{\hat{l}}(\xi\_k), \quad \frac{d^s \phi\_{\hat{l}}}{d\eta^s} = \sum\_{k=0}^{\hat{N}} \mathbf{D}\_{\hat{k}j}^s \theta\_{\hat{l}}(\xi\_k), \quad j = 0, 1, \dots, \hat{N}, \tag{36}$$

where *s* is the order of differentiation and **D** = <sup>2</sup> *<sup>L</sup>*D with D being the Chebyshev spectral differentiation matrix. Substituting equations (34) - (36) in (19) - (22) leads to the matrix equation

$$\mathbf{A}\_{i-1}\mathbf{X}\_i = \mathbf{R}\_{i-1} \tag{37}$$

subject to the boundary conditions

$$f\_i(\mathfrak{f}\_{\hat{N}}) = 0, \ \sum\_{k=0}^{\hat{N}} \mathbf{D}\_{\hat{N}k} f\_i(\mathfrak{f}\_k) = 0, \ \sum\_{k=0}^{\hat{N}} \mathbf{D}\_{0k} f\_i(\mathfrak{f}\_k) = 0,\tag{38}$$

$$
\theta\_l(\xi\_{\hat{N}}) = \theta\_l(\xi\_0) = \phi\_l(\xi\_{\hat{N}}) = \phi\_l(\xi\_0) = 0. \tag{39}
$$

In equation (37), **<sup>A</sup>***i*−<sup>1</sup> is a (3*N*� + <sup>3</sup>) × (3*N*� + <sup>3</sup>) square matrix and **<sup>X</sup>***<sup>i</sup>* and **<sup>R</sup>***<sup>i</sup>* are (3*N*� + <sup>1</sup>) × <sup>1</sup> column vectors defined by

$$\mathbf{A}\_{i-1} = \begin{bmatrix} A\_{11} \ A\_{12} \ A\_{13} \\ A\_{21} \ A\_{22} \ A\_{23} \\ A\_{31} \ A\_{32} \ A\_{33} \end{bmatrix}, \quad \mathbf{X}\_{i} = \begin{bmatrix} \mathbf{F}\_{i} \\ \boldsymbol{\Theta}\_{i} \\ \boldsymbol{\Phi}\_{i} \end{bmatrix}, \quad \mathbf{R}\_{i-1} = \begin{bmatrix} \mathbf{r}\_{1,i-1} \\ \mathbf{r}\_{2,i-1} \\ \mathbf{r}\_{3,i-1} \end{bmatrix}, \tag{40}$$

where

6 Will-be-set-by-IN-TECH

*n* + 3 2

*n* + 3 2

The functions *fi*, *θi*, *φ<sup>i</sup>* (*i* ≥ 1) are obtained by iteratively solving equations (19) - (22). The

*M* ∑ *m*=0

where *M* is the order of the SLM approximation. Equations (19) - (22) were solved using the Chebyshev spectral collocation method where the unknown functions are approximated

where *N* is the number of collocation points. The physical region [0, ∞) is first transformed into the region [−1, 1] using the domain truncation technique in which the problem is solved

where *L* is the scaling parameter used to invoke the boundary condition at infinity. The

*kjθi*(*ξk*), *<sup>d</sup>sφ<sup>i</sup>*

*<sup>d</sup>η<sup>s</sup>* <sup>=</sup>

*N* ∑ *k*=0 **D***s*

*θi*(*ξk*)*Tk*(*ξj*), *φi*(*ξ*) ≈

*i*−1 ∑ *m*=0 *h*� *<sup>m</sup>* − *N*

*i*−1 ∑ *n*=0 *fm i*−1 ∑ *n*=0 *θ*� *<sup>n</sup>* − *n*

*i*−1 ∑ *m*=0 *fm i*−1 ∑ *m*=0 *φ*� *<sup>m</sup>* − *n*

*θm*(*η*), *φ*(*η*) ≈

*i*−1 ∑ *m*=0 *g*� *m* 

> *i*−1 ∑ *m*=0 *f* � *m i*−1 ∑ *m*=0 *θm*

> > *i*−1 ∑ *m*=0 *f* � *m i*−1 ∑ *m*=0

. (30)

*M* ∑ *m*=0

*<sup>N</sup>* , *<sup>j</sup>* <sup>=</sup> 0, 1, . . . , *<sup>N</sup>*, (32)

<sup>2</sup> , <sup>−</sup><sup>1</sup> <sup>≤</sup> *<sup>ξ</sup>* <sup>≤</sup> 1, (33)

*N* ∑ *k*=0

*Tk*(*ξ*) = cos[*<sup>k</sup>* cos−1(*ξ*)]. (35)

, (28)

*φm* (29)

*φm*(*η*), (31)

*φi*(*ξk*)*Tk*(*ξj*), *j* = 0, 1, . . . , *N*,

*kjφi*(*ξk*), *j* = 0, 1, . . . , *N*, (36)

(34)

*i*−1 ∑ *m*=0 *f* � *m i*−1 ∑ *m*=0 *f* �� *m* −

*i*−1 ∑ *m*=0 *φ*�� *<sup>m</sup>* +

*i*−1 ∑ *m*=0 *φ*�� *<sup>m</sup>* +

approximate solutions for *f*(*η*), *θ*(*η*) and *φ*(*η*) are then obtained as

*fm*(*η*), *θ*(*η*) ≈

using Chebyshev interpolating polynomials at the Gauss-Lobatto points

*πj*

*ξ<sup>j</sup>* = cos

*η <sup>L</sup>* <sup>=</sup> *<sup>ξ</sup>* <sup>+</sup> <sup>1</sup>

on the interval [0, *L*] instead of [0, ∞). This is achieved by using the mapping

unknown functions *fi*, *θ<sup>i</sup>* and *φ<sup>i</sup>* are approximated at the collocation points by

*N* ∑ *k*=0

*N* ∑ *k*=0 **D***s*

*<sup>r</sup>*1,*i*−<sup>1</sup> = −

*<sup>r</sup>*2,*i*−<sup>1</sup> = −

*<sup>r</sup>*3,*i*−<sup>1</sup> = −

*fi*(*ξ*) ≈

*d<sup>s</sup> fi <sup>d</sup>η<sup>s</sup>* <sup>=</sup>

*N* ∑ *k*=0

> *N* ∑ *k*=0 **D***s*

 *<sup>i</sup>*−<sup>1</sup> ∑ *m*=0 *f* �� *<sup>m</sup>* + 2*λ*

 *<sup>i</sup>*−<sup>1</sup> ∑ *m*=0 *θ*�� *<sup>m</sup>* + *Df*

> *f* �� *m i*−1 ∑ *m*=0 *θ*� *<sup>m</sup>* + *f* � *m i*−1 ∑ *m*=0 *θ*�� *m* ,

*i*−1 ∑ *m*=0 *φ*�� *<sup>m</sup>* + *Sr*

 *f* �� *m i*−1 ∑ *m*=0 *g*� *<sup>m</sup>* + *f* � *m i*−1 ∑ *m*=0 *φ*�� *m* 

*M* ∑ *m*=0

+*Ra<sup>γ</sup>*

 1 *Le*

+*Ra<sup>γ</sup>*

*f*(*η*) ≈

*fi*(*ξk*)*Tk*(*ξj*), *θi*(*ξ*) ≈

*kj fi*(*ξk*), *<sup>d</sup>sθ<sup>i</sup>*

where *Tk* is the *k*th Chebyshev polynomial defined as

The derivatives at the collocation points are represented as

*<sup>d</sup>η<sup>s</sup>* <sup>=</sup>

$$\mathbf{F}\_{\hat{\mathbf{i}}} = \left[ f\_{\hat{\mathbf{i}}}(\xi\_0), f\_{\hat{\mathbf{i}}}(\xi\_1), \dots, f\_{\hat{\mathbf{i}}}(\xi\_{\hat{N}-1}), f\_{\hat{\mathbf{i}}}(\xi\_{\hat{N}}) \right]^T \tag{41}$$

$$\Theta\_{\hat{i}} = \left[ \theta\_{\hat{i}}(\xi\_0), \theta\_{\hat{i}}(\xi\_1), \dots, \theta\_{\hat{i}}(\xi\_{\hat{N}-1}), \theta\_{\hat{i}}(\xi\_{\hat{N}}) \right]^T,\tag{42}$$

$$\Phi\_{\hat{i}} = \left[ \phi\_{\hat{i}}(\xi\_0), \phi\_{\hat{i}}(\xi\_1), \dots, \phi\_{\hat{i}}(\xi\_{\hat{N}-1}), \phi\_{\hat{i}}(\xi\_{\hat{N}}) \right]^T,\tag{43}$$

$$\mathbf{r}\_{1,i-1} = \begin{bmatrix} r\_{1,i-1}(\boldsymbol{\xi}\_0), r\_{1,i-1}(\boldsymbol{\xi}\_1), \dots, r\_{1,i-1}(\boldsymbol{\xi}\_{\widehat{N}-1}), r\_{1,i-1}(\boldsymbol{\xi}\_{\widehat{N}}) \end{bmatrix}^T,\tag{44}$$

$$\mathbf{r\_{2,i-1}} = \begin{bmatrix} r\_{2,i-1}(\xi\_0), r\_{2,i-1}(\xi\_1), \dots, r\_{2,i-1}(\xi\_{\hat{N}-1}), r\_{2,i-1}(\xi\_{\hat{N}}) \end{bmatrix}^T \tag{45}$$

$$\mathbf{r}\_{3,i-1} = \begin{bmatrix} r\_{3,i-1}(\xi\_0), r\_{3,i-1}(\xi\_1), \dots, r\_{3,i-1}(\xi\_{\widehat{N}-1}), r\_{3,i-1}(\xi\_{\widehat{N}}) \end{bmatrix}^T \tag{46}$$

$$\mathbf{I} = \begin{bmatrix} \mathbf{I} & \mathbf{I} & \mathbf{I} \end{bmatrix} \tag{47}$$

$$A\_{11} = \mathbf{a}\_{1,i-1} \mathbf{D}^2 + \mathbf{a}\_{2,i-1} \mathbf{D}, \quad A\_{12} = -\mathbf{I}, \quad A\_{13} = -\mathbf{N} \tag{47}$$

$$A\_{11} = \mathbf{I}, \quad \mathbf{D}^2 + \mathbf{I}, \quad \mathbf{D} \times \mathbf{I}, \quad \mathbf{I} \quad \mathbf{d} \quad \mathbf{D}^2 + \mathbf{I}, \quad \mathbf{D} \times \mathbf{I}, \quad \mathbf{I} \quad \tag{48}$$

$$A\_{21} = \mathbf{b}\_{4,i-1} \mathbf{D}^2 + \mathbf{b}\_{5,i-1} \mathbf{D} + \mathbf{b}\_{6,i-1} \mathbf{I}\_{\prime} \\ A\_{22} = \mathbf{b}\_{1,i-1} \mathbf{D}^2 + \mathbf{b}\_{2,i-1} \mathbf{D} + \mathbf{b}\_{3,i-1} \mathbf{I}\_{\prime} \\ \tag{48}$$

$$A\_{23} = D\_f \mathbf{D}^2,\ A\_{31} = \mathbf{c}\_{4,i-1} \mathbf{D}^2 + \mathbf{c}\_{5,i-1} \mathbf{D} + \mathbf{c}\_{6,i-1} \mathbf{I},\tag{49}$$

$$A\_{32} = \mathbf{c}\_{1,i-1} \mathbf{D}^2 + \mathbf{c}\_{2,i-1} \mathbf{D} + \mathbf{c}\_{3,i-1} \mathbf{I},\ A\_{33} = \mathbf{S}\_r \mathbf{D}^2. \tag{50}$$

In the above definitions, **<sup>a</sup>***k*,*i*−1, **<sup>b</sup>***k*,*i*−1, **<sup>c</sup>***k*,*i*−<sup>1</sup> (*<sup>k</sup>* = 1, 2, .., 6) are diagonal matrices of size (*N*� + 1) × (*N*� + 1) and **I** is an identity matrix of size (*N*� + 1) × (*N*� + 1). After modifying the matrix system (37) to incorporate boundary conditions (38) - (37), the solution is obtained as

$$\mathbf{X}\_{i} = \mathbf{A}\_{i-1}^{-1} \mathbf{R}\_{i-1}.\tag{51}$$

Equations (9) - (12) were further solved numerically using the Matlab bvp4c routine and a shooting technique comprising the Runge-Kutta method of four slopes and the Newton-Raphson method. In solving the boundary value problem by the shooting method, the appropriate '∞' was determined through actual computations and differs for each set of parameter values.

a Porous Medium with Cross-Diffusion Effects 9

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 89

Table 2 shows the effects of the Dufour and Soret parameters on the heat and mass transfer coefficients when the other parameters are held constant. The accuracy of the method is compared with the Matlab bvp4c solver and a shooting method. Again, the results demonstrate that the SLM is accurate and converges rapidly to the numerical approximations. Furthermore the results show that the heat transfer rate increases with the Soret effect but decreases with the Dufour parameter. On the other hand, mass transfer decreases with increasing Soret numbers while increasing with Dufour numbers. These findings are consistent with those of Narayana and Sibanda (26) where the heat transfer coefficient was observed to increase with increasing values of the Soret parameter while the mass transfer

coefficient decreased with increasing values of the Soret parameter.

λ = 0 λ = 0.5 λ = 1 λ = 2

N = −0.5 N = 0 N = 1 N = 3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 2. Effect of (a) inertia parameter *λ*, (b) power-law index *n*, (c) buoyancy parameter *N*, and (d) the thermal dispersion parameter *Ra<sup>γ</sup>* on the fluid velocity when *Le* = 1, *Sr* = 0.3

Figure 2 shows the effect of (a) the inertia parameter *λ*, (b) the power-law index *n*, (c) the buoyancy parameter *N*, and (d) the modified Rayleigh number *Ra<sup>γ</sup>* on the fluid velocity for the inverted cone in a non-Darcy porous medium. Here *N* < 1 implies that the concentration buoyancy force is less than the thermal buoyancy force, *N* = 1 implies that the buoyancy forces are equal and the case *N* > 1 exists when the concentration buoyancy force exceeds the thermal buoyancy force. It is clear that the boundary layer thickness increases with *λ*, *N* and the Rayleigh number. However, the velocity decreases as the power-law index increases.

f

(η)

f

(η)

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>0</sup>

n = 0 n = 0.25 n = 0.5 n = 1

> Ra<sup>γ</sup> = 0 Ra<sup>γ</sup> = 1 Ra<sup>γ</sup> = 2 Ra<sup>γ</sup> = 3

η

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> <sup>0</sup>

η

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>0</sup>

 N = −0.5 N = 0.5

η

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> <sup>0</sup>

η

0.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

and *Df* = 0.2

f

(η)

f

(η)

1

1.5

#### **2.2 Discussion of smooth cone results**

In the absence of the inertia parameter *λ*, Soret and Dufour effects, the non-Darcy problem reduces to that considered by Yih (38) who solved the governing equations using the Keller-box scheme. The problem would also be a special case of the study by Cheng (8) who used a cubic spline collocation method to solve the governing equations. The results from these previous studies are used as a benchmark to test the accuracy of the linearisation method. The heat and mass transfer coefficients are given in Table 1 for different orders of the linearisation method, buoyancy and Lewis numbers. In general, the linearisation method has fully converged to the numerical results at the seventh order for all parameter values.


Table 1. Benchmark results for *Nux*/*Ra* <sup>1</sup> 2 *<sup>x</sup>* and *Shx*/*Ra* <sup>1</sup> 2 *<sup>x</sup>* when *λ* = 0.0, *n* = 0.0, *Ra<sup>γ</sup>* = 0.0, *Ra<sup>ξ</sup>* = 0.0, *Df* = 0.0 and *Sr* = 0.0


Table 2. Comparison of values of *Nux*/*Ra* <sup>1</sup> 2 *<sup>x</sup>* and *Shx*/*Ra* <sup>1</sup> 2 *<sup>x</sup>* for *λ* = 1.0, *N* = 1.0, *n* = 1.0, *Ra<sup>γ</sup>* = 0.5, *Ra<sup>ξ</sup>* = 0.5 and *Le* = 1.0

8 Will-be-set-by-IN-TECH

In the absence of the inertia parameter *λ*, Soret and Dufour effects, the non-Darcy problem reduces to that considered by Yih (38) who solved the governing equations using the Keller-box scheme. The problem would also be a special case of the study by Cheng (8) who used a cubic spline collocation method to solve the governing equations. The results from these previous studies are used as a benchmark to test the accuracy of the linearisation method. The heat and mass transfer coefficients are given in Table 1 for different orders of the linearisation method, buoyancy and Lewis numbers. In general, the linearisation method has fully converged to the numerical results at the seventh order for all parameter values.

*N Le* SLM Yih (38) Cheng (8)

4 1 1.5990 1.7186 1.7186 1.7186 1.7186 4 10 1.1886 1.1795 1.1795 1.1795 1.1794 1 1 1.0869 1.0870 1.0870 1.0869 1.0870

 10 0.9031 0.9031 0.9031 0.9030 0.9032 100 0.8141 0.8141 0.8141 0.8141 0.8143 1 0.7686 0.7686 0.7686 0.7686 0.7685 10 0.7686 0.7686 0.7686 0.7686 0.7685 1 1.5990 1.7186 1.7186 1.7186 1.7186 10 5.6790 5.6980 5.6980 5.6977 5.6949

 1 1.0869 1.0870 1.0870 1.0869 1.0870 10 3.8141 3.8141 3.8141 3.8139 3.8134 100 12.3653 12.3653 12.3653 12.3645 12.3377 1 0.7686 0.7686 0.7686 0.7686 0.7685 10 0.7686 0.7686 0.7686 0.7686 0.7686

*<sup>x</sup>* and *Shx*/*Ra* <sup>1</sup>

*Sr Df* SLM bvp4c Shooting method

1.5 0.03 1.550183 1.550010 1.550010 1.550010 1.55001

1.0 0.12 1.493268 1.493106 1.493106 1.493106 1.49311 0.5 0.30 1.373266 1.373121 1.373121 1.373121 1.37312 0.1 0.60 1.170132 1.169958 1.169958 1.169958 1.16996 1.5 0.03 0.674035 0.675657 0.675657 0.675657 0.675658

1.0 0.12 0.960995 0.962038 0.962038 0.962038 0.962039 0.5 0.30 1.251253 1.251840 1.251840 1.251840 1.251840 0.1 0.60 1.466009 1.466449 1.466449 1.466449 1.466450

*<sup>x</sup>* and *Shx*/*Ra* <sup>1</sup>

2

2

2

*<sup>x</sup>* when *λ* = 0.0, *n* = 0.0, *Ra<sup>γ</sup>* = 0.0,

*<sup>x</sup>* for *λ* = 1.0, *N* = 1.0, *n* = 1.0,

2

order 3 order 7 order 8

order 3 order 7 order 8

**2.2 Discussion of smooth cone results**

*Nux* √*Rax*

*Shx* √*Rax*

Table 1. Benchmark results for *Nux*/*Ra* <sup>1</sup>

*Ra<sup>ξ</sup>* = 0.0, *Df* = 0.0 and *Sr* = 0.0

*Nux* √*Rax*

*Shx* √*Rax*

Table 2. Comparison of values of *Nux*/*Ra* <sup>1</sup>

*Ra<sup>γ</sup>* = 0.5, *Ra<sup>ξ</sup>* = 0.5 and *Le* = 1.0

Table 2 shows the effects of the Dufour and Soret parameters on the heat and mass transfer coefficients when the other parameters are held constant. The accuracy of the method is compared with the Matlab bvp4c solver and a shooting method. Again, the results demonstrate that the SLM is accurate and converges rapidly to the numerical approximations. Furthermore the results show that the heat transfer rate increases with the Soret effect but decreases with the Dufour parameter. On the other hand, mass transfer decreases with increasing Soret numbers while increasing with Dufour numbers. These findings are consistent with those of Narayana and Sibanda (26) where the heat transfer coefficient was observed to increase with increasing values of the Soret parameter while the mass transfer coefficient decreased with increasing values of the Soret parameter.

Fig. 2. Effect of (a) inertia parameter *λ*, (b) power-law index *n*, (c) buoyancy parameter *N*, and (d) the thermal dispersion parameter *Ra<sup>γ</sup>* on the fluid velocity when *Le* = 1, *Sr* = 0.3 and *Df* = 0.2

Figure 2 shows the effect of (a) the inertia parameter *λ*, (b) the power-law index *n*, (c) the buoyancy parameter *N*, and (d) the modified Rayleigh number *Ra<sup>γ</sup>* on the fluid velocity for the inverted cone in a non-Darcy porous medium. Here *N* < 1 implies that the concentration buoyancy force is less than the thermal buoyancy force, *N* = 1 implies that the buoyancy forces are equal and the case *N* > 1 exists when the concentration buoyancy force exceeds the thermal buoyancy force. It is clear that the boundary layer thickness increases with *λ*, *N* and the Rayleigh number. However, the velocity decreases as the power-law index increases.

a Porous Medium with Cross-Diffusion Effects 11

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 91

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 4. Effect of (a) inertia parameter *λ*, and (b) the thermal dispersion parameter *Ra<sup>γ</sup>* on the

In this section we investigate the case of double-diffusive convection in a fluid around an inverted wavy cone. Figure 6 shows the model of the problem investigated. The wavy surface

where *a*∗ is the amplitude of the wavy surface and 2 is the characteristic length of the wave. The governing momentum, heat and solute concentration equations can be written in the form

> *∂T ∂x*

*∂*2*C <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

> *∂*2*T <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

Here the symbols have their usual meanings. We now use the following non-dimensional

+ *β<sup>t</sup>* sin(Ω)

φ(η)

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> <sup>0</sup>

*y* = *σ*∗(*x*) = *a*∗ sin (*πx*/), (52)

*∂C ∂y*

+ *β<sup>c</sup>* cos(Ω)

, (54)

, (55)

*∂C ∂x* 

, (53)

+ *β<sup>c</sup>* cos(Ω)

*∂*2*C ∂y*<sup>2</sup> 

> *∂*2*T ∂y*<sup>2</sup>

*v* = 0, *T* = *Tw*, *C* = *Cw* on *y* = *σ*∗(*x*) = *a*∗ sin(*πx*/), (56)

*u* = 0, *T* = *T*∞, *C* = *C*<sup>∞</sup> as *y* → ∞. (57)

(*X*, *Y*, *R*, *σ*, *a*) = (*x*, *y*,*r*, *σ*∗, *a*∗) /, (*U*, *V*) = (*u*, *v*) /*α*, (58)

Θ = (*T* − *T*∞)/(*Tw* − *T*∞) and Φ = (*C* − *C*∞)/(*Cw* − *C*∞). (59)

η

Ra<sup>γ</sup> = 0 Ra<sup>γ</sup> = 1 Ra<sup>γ</sup> = 2 Ra<sup>γ</sup> = 3

λ = 0 λ = 0.5 λ = 1 λ = 2

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>0</sup>

 N = 0.5 N = −0.5

η

**3. Flow over a wavy cone in porous media**

of the cone is described by

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>g</sup>*<sup>K</sup> ν* 

subject to boundary conditions

*∂u <sup>∂</sup><sup>y</sup>* <sup>−</sup> *<sup>∂</sup><sup>v</sup>*

*u ∂T ∂x* + *v ∂T <sup>∂</sup><sup>y</sup>* <sup>=</sup> *<sup>α</sup>*

*u ∂C ∂x* + *v ∂C <sup>∂</sup><sup>y</sup>* <sup>=</sup> *<sup>D</sup>*

variables;

concentration profile when *Le* = 1, *Sr* = 0.3 and *Df* = 0.2

*β<sup>t</sup>* cos(Ω)

*∂*2*T <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

 *∂*2*C <sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *∂T ∂y*

*∂*2*T ∂y*<sup>2</sup> + *Dk cscp*

*∂*2*C ∂y*<sup>2</sup> + *Dk cscp*

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

φ(η)

Fig. 3. Effect of (a) inertia parameter *λ*, (b) power-law index *n*, (c) the buoyancy parameter *N*, and (d) the thermal dispersion parameter *Ra<sup>γ</sup>* on the temperature profile when *Le* = 1, *Sr* = 0.3 and *Df* = 0.2

Figures 3 - 4 show the effects of (a) the inertia parameter *λ*, (b) the power-law index *n*, (c) the buoyancy parameter *N*, and (d) the thermal dispersion parameter *Ra<sup>γ</sup>* on the temperature and solute concentration profiles. The temperature profiles decrease with increasing *n*. The concentration profiles increase whereas temperature profile decreases with increasing thermal dispersion parameter.

Figure 5 depicts the variation of the heat transfer rate *NuxRa*−1/2 *<sup>x</sup>* and the mass transfer rate *ShxRa*−1/2 *<sup>x</sup>* with Lewis numbers for different values of the Dufour and Soret parameters. For fixed Soret numbers, it is evident that as *Le* increases, the Nusselt number decreases for any particular value of *Df* . The variation of the Sherwood number with *Le* for different values of *Df* is shown in Figure 5(b). Increasing *Le* enhances the mass transfer rate for any particular value of *Df* . It is also evident that as *Df* increases the Sherwood number increases for all values of *Le*.

The variation of the Nusselt and Sherwood numbers with *Le* and *Sr* when the Dufour number is fixed is shown in Figures 5(c) - 5(d). Increasing *Le* reduces the Nusselt number for all values of *Sr*. Conversely, increasing the Soret parameter enhances the Nusselt number. Also, increasing *Le* contributes to enhancing the mass transfer rate for any particular value of *Sr*. On the other hand, increasing *Sr* reduces the Sherwood number.

Fig. 4. Effect of (a) inertia parameter *λ*, and (b) the thermal dispersion parameter *Ra<sup>γ</sup>* on the concentration profile when *Le* = 1, *Sr* = 0.3 and *Df* = 0.2

#### **3. Flow over a wavy cone in porous media**

10 Will-be-set-by-IN-TECH

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 3. Effect of (a) inertia parameter *λ*, (b) power-law index *n*, (c) the buoyancy parameter *N*, and (d) the thermal dispersion parameter *Ra<sup>γ</sup>* on the temperature profile when *Le* = 1,

Figures 3 - 4 show the effects of (a) the inertia parameter *λ*, (b) the power-law index *n*, (c) the buoyancy parameter *N*, and (d) the thermal dispersion parameter *Ra<sup>γ</sup>* on the temperature and solute concentration profiles. The temperature profiles decrease with increasing *n*. The concentration profiles increase whereas temperature profile decreases with increasing thermal

Figure 5 depicts the variation of the heat transfer rate *NuxRa*−1/2 *<sup>x</sup>* and the mass transfer rate

The variation of the Nusselt and Sherwood numbers with *Le* and *Sr* when the Dufour number is fixed is shown in Figures 5(c) - 5(d). Increasing *Le* reduces the Nusselt number for all values of *Sr*. Conversely, increasing the Soret parameter enhances the Nusselt number. Also, increasing *Le* contributes to enhancing the mass transfer rate for any particular value of *Sr*.

On the other hand, increasing *Sr* reduces the Sherwood number.

*<sup>x</sup>* with Lewis numbers for different values of the Dufour and Soret parameters. For fixed Soret numbers, it is evident that as *Le* increases, the Nusselt number decreases for any particular value of *Df* . The variation of the Sherwood number with *Le* for different values of *Df* is shown in Figure 5(b). Increasing *Le* enhances the mass transfer rate for any particular value of *Df* . It is also evident that as *Df* increases the Sherwood number increases for all

θ(η)

θ(η)

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> <sup>0</sup>

n = 0 n = 0.25 n = 0.5 n = 1

> Ra<sup>γ</sup> = 0 Ra<sup>γ</sup> = 1 Ra<sup>γ</sup> = 2 Ra<sup>γ</sup> = 3

η

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> <sup>0</sup>

η

λ = 0 λ = 0.5 λ = 1 λ = 2

N = −0.5 N = 0 N = 1 N = 3

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>0</sup>

 N = 0.5 N = −0.5

η

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>0</sup>

η

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*Sr* = 0.3 and *Df* = 0.2

dispersion parameter.

*ShxRa*−1/2

values of *Le*.

θ(η), φ(η) θ(η)

In this section we investigate the case of double-diffusive convection in a fluid around an inverted wavy cone. Figure 6 shows the model of the problem investigated. The wavy surface of the cone is described by

$$y = \sigma^\*(\mathbf{x}) = a^\* \sin\left(\pi \mathbf{x}/\ell\right),\tag{52}$$

where *a*∗ is the amplitude of the wavy surface and 2 is the characteristic length of the wave. The governing momentum, heat and solute concentration equations can be written in the form

$$\frac{\partial u}{\partial y} - \frac{\partial v}{\partial \mathbf{x}} = \frac{\mathbf{g}K}{\nu} \left( \beta\_l \cos(\Omega) \frac{\partial T}{\partial y} + \beta\_l \sin(\Omega) \frac{\partial T}{\partial \mathbf{x}} + \beta\_\ell \cos(\Omega) \frac{\partial \mathbf{C}}{\partial y} + \beta\_\ell \cos(\Omega) \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \right), \tag{53}$$

$$u\frac{\partial T}{\partial \mathbf{x}} + v\frac{\partial T}{\partial y} = u\left(\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2}\right) + \frac{Dk}{c\_s c\_p} \left(\frac{\partial^2 C}{\partial \mathbf{x}^2} + \frac{\partial^2 C}{\partial y^2}\right),\tag{54}$$

$$u\frac{\partial \mathbb{C}}{\partial \mathbf{x}} + v\frac{\partial \mathbb{C}}{\partial y} = D\left(\frac{\partial^2 \mathbb{C}}{\partial \mathbf{x}^2} + \frac{\partial^2 \mathbb{C}}{\partial y^2}\right) + \frac{Dk}{c\_s c\_p} \left(\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2}\right),\tag{55}$$

subject to boundary conditions

$$v = 0, \quad T = T\_{\text{w}}, \quad \mathbb{C} = \mathbb{C}\_{\text{w}} \quad \text{on} \quad y = \sigma^\*(\mathbf{x}) = a^\* \sin(\pi \mathbf{x}/\ell), \tag{56}$$

*u* = 0, *T* = *T*∞, *C* = *C*<sup>∞</sup> as *y* → ∞. (57)

Here the symbols have their usual meanings. We now use the following non-dimensional variables;

$$(\mathbf{X}, \mathbf{Y}, \mathbf{R}, \sigma, a) = (\mathbf{x}, \mathbf{y}, \mathbf{r}, \sigma^\*, a^\*) \;/\ell, \; (\mathbf{U}, \mathbf{V}) = (\mathbf{u}, \mathbf{v}) \; \ell/\mathfrak{a},\tag{58}$$

$$
\Theta = (T - T\_{\infty}) / (T\_w - T\_{\infty}) \quad \text{and} \quad \Phi = (\mathbb{C} - \mathbb{C}\_{\infty}) / (\mathbb{C}\_w - \mathbb{C}\_{\infty}).\tag{59}
$$

a Porous Medium with Cross-Diffusion Effects 13

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 93

*σ*∗(*x*) *r*(*x*)

> ✁ ✁ ✁✕

*x*

*Tw*, *Cw*

*R ∂ψ*

*<sup>∂</sup><sup>Y</sup>* <sup>+</sup> tan(Ω)

 *∂*2Φ *<sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

> *∂*2Θ *<sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

*YRa* ¯ <sup>−</sup>1/2 <sup>=</sup> *<sup>Y</sup>* <sup>−</sup> *<sup>σ</sup>*(*X*), (69)

❯

*<sup>U</sup>* <sup>=</sup> <sup>1</sup> *R ∂ψ*

equations (60) - (62) can be written in the following form

*∂*Θ *∂Y* =

*∂*Φ *∂Y* <sup>=</sup> <sup>1</sup>

*∂*2*ψ <sup>∂</sup>Y*<sup>2</sup> <sup>−</sup> *RX R ∂ψ ∂X* 

*∂ψ*

❦ ☛ Ω

The parameters appearing above are given by equations (13) - (14). Introducing the stream

*<sup>∂</sup><sup>Y</sup>* and *<sup>V</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup>

*∂*Θ *<sup>∂</sup><sup>Y</sup>* <sup>+</sup> *<sup>N</sup> <sup>∂</sup>*<sup>Φ</sup>

*∂*2Θ *∂Y*<sup>2</sup>

where *R* is the non-dimensional radius of the cone. The appropriate boundary conditions are

To transform the wavy surface of the cone to a smooth one we introduce the following

*X*¯ = *X*,

*ψ*¯ = *Ra*−1/2*ψ*.

 + *Df*

*∂*2Φ *∂Y*<sup>2</sup>

 + *Sr*

*ψ* = 0, Θ = 1, Φ = 1 on *Y* = *σ*(*X*) = *a* sin(*πX*), (67)

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> 0, <sup>Θ</sup> <sup>=</sup> 0, <sup>Φ</sup> <sup>=</sup> 0 as *<sup>Y</sup>* <sup>→</sup> <sup>∞</sup>. (68)

<sup>=</sup> *Ra*

 *∂*2Θ *<sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

*Le <sup>∂</sup>*2<sup>Φ</sup> *<sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

❑

2

Fig. 6. Schematic sketch of the vertical wavy cone

function *ψ*(*X*,*Y*) defined such that

1 *R*

1 *R*

1 *R*

transformation,

 *∂*2*ψ <sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

 *∂ψ ∂Y ∂*Θ *<sup>∂</sup><sup>X</sup>* <sup>−</sup> *∂ψ ∂X*

 *∂ψ ∂Y ∂*Φ *<sup>∂</sup><sup>X</sup>* <sup>−</sup> *∂ψ ∂X* ✕

✲

 

*u*

*y v* ❄

*<sup>∂</sup><sup>X</sup>* , (63)

*<sup>∂</sup><sup>X</sup>* <sup>+</sup> *<sup>N</sup> <sup>∂</sup>*<sup>Φ</sup> *∂X*

, (64)

, (65)

, (66)

*∂*Θ

*∂*2Φ *∂Y*<sup>2</sup>

*∂*2Θ *∂Y*<sup>2</sup>

g

*T*∞, *C*∞

Fig. 5. The effect of the Dufour and Soret parameters on heat and mass transfers with *λ* = 0.7, *n* = 1, *Ra<sup>γ</sup>* = 0.5, *Ra<sup>ξ</sup>* = 0.5, *Le* = 1 (i) *Sr* = 0.3 and (ii) *Df* = 0.2

The governing equations now become,

$$\frac{\partial \mathcal{U}}{\partial Y} - \frac{\partial V}{\partial X} = \text{Ra} \left[ \frac{\partial \Theta}{\partial Y} + N \frac{\partial \Phi}{\partial Y} + \tan(\Omega) \left( \frac{\partial \Theta}{\partial X} + N \frac{\partial \Phi}{\partial X} \right) \right], \tag{60}$$

$$\mathcal{U}\frac{\partial\Theta}{\partial X} + V\frac{\partial\Theta}{\partial Y} = \left(\frac{\partial^2\Theta}{\partial X^2} + \frac{\partial^2\Theta}{\partial Y^2}\right) + D\_f\left(\frac{\partial^2\Phi}{\partial X^2} + \frac{\partial^2\Phi}{\partial Y^2}\right),\tag{61}$$

$$U\frac{\partial\Phi}{\partial X} + V\frac{\partial\Phi}{\partial Y} = \frac{1}{Le} \left(\frac{\partial^2\Phi}{\partial X^2} + \frac{\partial^2\Phi}{\partial Y^2}\right) + S\_r \left(\frac{\partial^2\Theta}{\partial X^2} + \frac{\partial^2\Theta}{\partial Y^2}\right). \tag{62}$$

Fig. 6. Schematic sketch of the vertical wavy cone

The parameters appearing above are given by equations (13) - (14). Introducing the stream function *ψ*(*X*,*Y*) defined such that

$$
\Delta U = \frac{1}{R} \frac{\partial \psi}{\partial Y} \quad \text{and} \quad V = -\frac{1}{R} \frac{\partial \psi}{\partial X'} \tag{63}
$$

equations (60) - (62) can be written in the following form

Fig. 5. The effect of the Dufour and Soret parameters on heat and mass transfers with

*<sup>∂</sup><sup>Y</sup>* <sup>+</sup> tan(Ω)

 + *Df*

(c) (d)

12 Will-be-set-by-IN-TECH

(a) (b)

*∂*2Φ *∂Y*<sup>2</sup>

 + *Sr*

*∂*2Θ *∂Y*<sup>2</sup>

*∂*Θ

*∂*2Φ *<sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

> *∂*2Θ *<sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

*<sup>∂</sup><sup>X</sup>* <sup>+</sup> *<sup>N</sup> <sup>∂</sup>*<sup>Φ</sup> *∂X*

> *∂*2Φ *∂Y*<sup>2</sup>

*∂*2Θ *∂Y*<sup>2</sup>

, (60)

, (61)

. (62)

*λ* = 0.7, *n* = 1, *Ra<sup>γ</sup>* = 0.5, *Ra<sup>ξ</sup>* = 0.5, *Le* = 1 (i) *Sr* = 0.3 and (ii) *Df* = 0.2

 *∂*Θ *<sup>∂</sup><sup>Y</sup>* <sup>+</sup> *<sup>N</sup> <sup>∂</sup>*<sup>Φ</sup>

 *∂*2Θ *<sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

> *∂*2Φ *<sup>∂</sup>X*<sup>2</sup> <sup>+</sup>

*<sup>∂</sup><sup>X</sup>* <sup>=</sup> *Ra*

*<sup>∂</sup><sup>Y</sup>* <sup>=</sup>

*<sup>∂</sup><sup>Y</sup>* <sup>=</sup> <sup>1</sup> *Le*

*<sup>∂</sup><sup>X</sup>* <sup>+</sup> *<sup>V</sup> <sup>∂</sup>*<sup>Θ</sup>

*<sup>∂</sup><sup>X</sup>* <sup>+</sup> *<sup>V</sup> <sup>∂</sup>*<sup>Φ</sup>

The governing equations now become,

*∂U <sup>∂</sup><sup>Y</sup>* <sup>−</sup> *<sup>∂</sup><sup>V</sup>*

*<sup>U</sup> <sup>∂</sup>*<sup>Θ</sup>

*<sup>U</sup> <sup>∂</sup>*<sup>Φ</sup>

$$\frac{1}{R} \left( \frac{\partial^2 \psi}{\partial X^2} + \frac{\partial^2 \psi}{\partial Y^2} - \frac{R\_X}{R} \frac{\partial \psi}{\partial X} \right) = Ra \left[ \frac{\partial \Theta}{\partial Y} + N \frac{\partial \Phi}{\partial Y} + \tan(\Omega) \left( \frac{\partial \Theta}{\partial X} + N \frac{\partial \Phi}{\partial X} \right) \right],\tag{64}$$

$$\frac{1}{R} \left( \frac{\partial \psi}{\partial Y} \frac{\partial \Theta}{\partial X} - \frac{\partial \psi}{\partial X} \frac{\partial \Theta}{\partial Y} \right) = \left( \frac{\partial^2 \Theta}{\partial X^2} + \frac{\partial^2 \Theta}{\partial Y^2} \right) + D\_f \left( \frac{\partial^2 \Phi}{\partial X^2} + \frac{\partial^2 \Phi}{\partial Y^2} \right), \tag{65}$$

$$\frac{1}{R} \left( \frac{\partial \psi}{\partial Y} \frac{\partial \Phi}{\partial X} - \frac{\partial \psi}{\partial X} \frac{\partial \Phi}{\partial Y} \right) = \frac{1}{Le} \left( \frac{\partial^2 \Phi}{\partial X^2} + \frac{\partial^2 \Phi}{\partial Y^2} \right) + S\_r \left( \frac{\partial^2 \Theta}{\partial X^2} + \frac{\partial^2 \Theta}{\partial Y^2} \right), \tag{66}$$

where *R* is the non-dimensional radius of the cone. The appropriate boundary conditions are

$$
\psi = 0, \ \Theta = 1, \ \Phi = 1 \quad \text{on} \quad Y = \sigma(X) = a \sin(\pi X), \tag{67}
$$

$$\frac{\partial \Psi}{\partial y} = 0, \ \Theta = 0, \ \Phi = 0 \quad \text{as} \quad Y \to \infty. \tag{68}$$

To transform the wavy surface of the cone to a smooth one we introduce the following transformation,

$$\begin{aligned} \bar{X} &= X, \\ \bar{Y}Ra^{-1/2} &= Y - \sigma(X), \\ \bar{\psi} &= Ra^{-1/2}\psi. \end{aligned} \tag{69}$$

a Porous Medium with Cross-Diffusion Effects 15

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 95

Figure 7 shows the effect of the Dufour number *Df* on heat and mass transfer for two different values of the amplitude *a*. The effect of increasing the amplitude, on average, is to reduce the heat and mass transfer rates as compared with the limiting case of a smooth cone. Figures 7(c) and 7(d) highlight the same. Figures 7(a) and 7(b) show that for *a* = 0 (smooth cone) both *NuxRa*−1/2 and *ShxRa*−1/2 increase steadily with *<sup>ξ</sup>* whereas for the wavy cone (i.e., *<sup>a</sup>* �<sup>=</sup> 0) we observe oscillations in *NuxRa*−1/2 and *ShxRa*−1/2 over the three complete cycles of undulations from *ξ* = 0 to *ξ* = 6 having length two. These results represent the nonlinear coupling of the change in fluid velocity and orientation of the gravitation. The results are in agreement with those reported by Cheng (6) and Pop and Na (34). The Dufour number *Df* reduces *NuxRa*−1/2 and *NumRa*−1/2. The opposite is true in the case of *ShxRa*−1/2 and

The effect of *Df* on heat and mass transfer is depicted in Figure 8 for two different values of the cone half angle Ω. From 8(c) and 8(d) it is clear that increasing the half angle Ω, on average, reduces the heat and mass transfer rates. Figures 8(a) and 8(b) show that there is an increase in oscillations of *NuxRa*−1/2 and *ShxRa*−1/2 for higher values of Ω. In this case the

Figure 9 demonstrates the effect of *Df* on heat and mass transfer for two different values of buoyancy ratio *N*. It is evident that the buoyancy ratio amplifies heat and mass transfer from the cone. Again, the Dufour number contributes to lowering heat transfer while enhancing

The effect of *Df* on the heat and mass transfer is highlighted for two different values of Lewis numbers in Figure 10. We observe that *Le* reduces heat transfer whereas the opposite is true in the case of mass transfer. For large values of *Le*, higher values of *Df* (≥ 0.5) produce negative heat transfer rates indicating that heat diffuses from fluid to the cone in such cases. Figures 10(a) and 10(c) confirm and reinforce the same fact. The effect of Soret number *Sr* on heat and mass transfer for two different values of amplitude *a* is projected in Figure 11. The decreasing effect of the amplitude *a* on heat and mass transfer rates observed in this situation also. The Soret number *Sr* contributes to increasing *NuxRa*−1/2 and *NumRa*−1/2

The effect of *Sr* on heat and mass transfer is shown in Figure 12 for two different values of cone half angle Ω. The fact that Ω reduces the heat and mass transfer rates is observed in plots 12(a) and 12(d). The Soret number *Sr* has the effect of increasing the heat transfer and

Figure 13 shows the effect of *Sr* on heat and mass transfer rates for two different values of the buoyancy ratio *N*. From 13(a) - 13(d) we observe that the buoyancy ratio enhances both heat and mass transfer rates. For selected values of *N*, *Sr* contributes towards enhancing the heat

The effect of *Sr* on the heat and mass transfer rates is shown in Figure 14 for selected values of the Lewis number *Le*. It is evident that *Le* reduces the heat transfer whereas the opposite is true in case of mass transfer. At large values of *Le* there is a critical value of *Sr* up to which *NuxRa*−1/2 and *NumRa*−1/2 increases and beyond this critical value, both *NuxRa*−1/2 and *NumRa*−1/2 start to fall as can be more clearly seen in Figures 14(a) and 14(c). From Figures 14(b) - 14(d) we observe that the effect of *Sr* is to reduce the rate of mass transfer from the

Dufour number also reduces the heat transfer while enhancing mass transfer.

while reducing *ShxRa*−1/2 *ShmRa*−1/2 as can be seen in Figures 11(a) - 11(d).

reducing the mass transfer for all values of Ω.

surface of the wavy cone.

transfer rate while reducing the mass transfer rate.

*ShmRa*−1/2.

mass transfer rates.

Substituting the transformations (70) into equations (64) - (66) and letting *Ra* → ∞, we obtain the following equations

$$\frac{1+\sigma\_X^2}{R}\frac{\partial^2 \vec{\Psi}}{\partial \vec{Y}^2} = \left[1 - \sigma\_{\vec{X}} \tan(\Omega)\right] \left(\frac{\partial \Theta}{\partial \vec{Y}} + N \frac{\partial \Phi}{\partial \vec{Y}}\right),\tag{70}$$

$$(1+\sigma\_X^2)\left(\frac{\partial^2\Theta}{\partial Y^2} + D\_f\frac{\partial^2\Phi}{\partial Y^2}\right) = \frac{1}{R}\left(\frac{\partial\bar{\Psi}}{\partial\bar{Y}}\frac{\partial\Theta}{\partial X} - \frac{\partial\bar{\Psi}}{\partial\bar{X}}\frac{\partial\Theta}{\partial\bar{Y}}\right),\tag{71}$$

$$(1+\sigma\_{\mathcal{X}}^2)\left(\frac{1}{Le}\frac{\partial^2\Phi}{\partial\bar{Y}^2} + S\_r\frac{\partial^2\Theta}{\partial\bar{Y}^2}\right) = \frac{1}{R}\left(\frac{\partial\bar{\psi}}{\partial\bar{Y}}\frac{\partial\Phi}{\partial\bar{X}} - \frac{\partial\bar{\psi}}{\partial\bar{X}}\frac{\partial\Phi}{\partial\bar{Y}}\right).\tag{72}$$

We may further simplify equations (70) - (72) by introducing the following transformation

$$\tilde{\xi} = \bar{X}, \ \eta = \bar{Y}/[(1+\sigma\_{\tilde{\xi}}^2)\tilde{\xi}^{1/2}], \ \bar{\Psi} = R\tilde{\xi}^{1/2}f(\tilde{\xi}, \eta), \ \Theta = \theta(\tilde{\xi}, \eta), \ \Phi = \phi(\tilde{\xi}, \eta). \tag{73}$$

Substituting equation (73) into equations (70) - (72), gives the nonlinear system of differential equations;

$$f^{\prime\prime} = [1 - \sigma\_{\tilde{\xi}} \tan(\Omega)](\theta^{\prime} + N\phi^{\prime}),\tag{74}$$

$$
\xi \theta'' + \frac{3}{2} f \theta' + D\_f \phi'' = \xi (f' \theta\_{\tilde{\xi}} - \theta' f\_{\tilde{\xi}}) \, \tag{75}
$$

$$\frac{1}{1\mathcal{L}}\boldsymbol{\phi}^{\prime\prime} + \frac{3}{2}f\boldsymbol{\phi}^{\prime} + \mathcal{S}\_r\boldsymbol{\theta}^{\prime\prime} = \boldsymbol{\xi}(f^{\prime}\phi\_{\tilde{\boldsymbol{\xi}}} - \boldsymbol{\phi}^{\prime}f\_{\tilde{\boldsymbol{\xi}}}),\tag{76}$$

with boundary conditions

$$\begin{array}{ll} f(\not\subset, 0) = 0, & \theta(\not\subset, 0) = 1, & \phi(\not\subset, 0) = 1, \\ f'(\not\subset, \infty) = 0, & \theta(\not\subset, \infty) = 0, & \phi(\not\subset, \infty) = 0. \end{array} \tag{77}$$

The associated local Nusselt and Sherwood numbers are given by

$$Nu\_{\mathfrak{X}} = -Ra^{1/2} \frac{\mathfrak{T}^{1/2} \theta'(\mathfrak{F}, 0)}{(1 + \sigma\_{\mathfrak{F}}^2)^{\frac{1}{2}}} \quad \text{and} \quad Sh\_{\mathfrak{X}} = -Ra^{1/2} \frac{\mathfrak{T}^{1/2} \phi'(\mathfrak{F}, 0)}{(1 + \sigma\_{\mathfrak{F}}^2)^{\frac{1}{2}}}.\tag{78}$$

The mean Nusselt and Sherwood numbers from the leading edge to streamwise position *x* are given by

$$\frac{Nu\_m}{Ra^{1/2}} = -\frac{\text{x}}{\ell} \frac{\int\_0^{\frac{\pi}{\ell}} \xi^{-1/2} \theta'(\xi, 0) d\xi}{\int\_0^{\frac{\pi}{\ell}} (1 + \sigma\_{\xi}^2)^{\frac{1}{2}} d\xi}, \quad \frac{Sh\_m}{Ra^{1/2}} = -\frac{\text{x}}{\ell} \frac{\int\_0^{\frac{\pi}{\ell}} \xi^{-1/2} \phi'(\xi, 0) d\xi}{\int\_0^{\frac{\pi}{\ell}} (1 + \sigma\_{\xi}^2)^{\frac{1}{2}} d\xi}. \tag{79}$$

#### **3.1 Discussion of wavy cone results**

The governing equations (74) - (76) along with the boundary conditions (77), were solved numerically using the Keller-box method (see Keller (16)) for various parameter combinations. Two hundred uniform grid points of step size 0.05 were used in the *η*- direction. A uniform grid with 120 nodes was used in the *ξ* direction. At every *ξ* grid line, the iteration process is carried out until an accuracy of 10−<sup>6</sup> is achieved for all the variables. The computations carried out are given in Figures 7 to 14.

14 Will-be-set-by-IN-TECH

Substituting the transformations (70) into equations (64) - (66) and letting *Ra* → ∞, we obtain

*∂*Θ

 *∂ψ*¯ *∂Y*¯ *∂*Θ *<sup>∂</sup>X*¯ <sup>−</sup> *∂ψ*¯ *∂X*¯ *∂*Θ *∂Y*¯ 

*<sup>∂</sup>Y*¯ <sup>+</sup> *<sup>N</sup> <sup>∂</sup>*<sup>Φ</sup> *∂Y*¯ 

> *∂ψ*¯ *∂Y*¯ *∂*Φ *<sup>∂</sup>X*¯ <sup>−</sup> *∂ψ*¯ *∂X*¯ *∂*Φ *∂Y*¯

*<sup>ξ</sup>* )*ξ*1/2], *<sup>ψ</sup>*¯ <sup>=</sup> *<sup>R</sup>ξ*1/2 *<sup>f</sup>*(*ξ*, *<sup>η</sup>*), <sup>Θ</sup> <sup>=</sup> *<sup>θ</sup>*(*ξ*, *<sup>η</sup>*), <sup>Φ</sup> <sup>=</sup> *<sup>φ</sup>*(*ξ*, *<sup>η</sup>*). (73)

(*ξ*, ∞) = 0, *θ*(*ξ*, ∞) = 0, *φ*(*ξ*, ∞) = 0. (77)

and *Shx* <sup>=</sup> <sup>−</sup>*Ra*1/2 *<sup>ξ</sup>*1/2*φ*�

, (70)

), (74)

*θξ* − *θ*� *f<sup>ξ</sup>* ), (75)

*φξ* − *φ*� *f<sup>ξ</sup>* ), (76)

(*ξ*, 0)

(*ξ*, 0)*dξ*

. (78)

. (79)

(1 + *σ*<sup>2</sup> *ξ* ) 1 2

, (71)

. (72)

the following equations

1 + *σ*<sup>2</sup> *X*¯ *R*

(1 + *σ*<sup>2</sup> *X*¯ )

(1 + *σ*<sup>2</sup> *X*¯ ) 1 *Le*

*ξ* = *X*¯ , *η* = *Y*¯/[(1 + *σ*<sup>2</sup>

with boundary conditions

*Num Ra*1/2 <sup>=</sup> <sup>−</sup> *<sup>x</sup>*

**3.1 Discussion of wavy cone results**

carried out are given in Figures 7 to 14.

equations;

given by

*∂*2*ψ*¯

 *∂*2Θ *<sup>∂</sup>Y*¯ <sup>2</sup> <sup>+</sup> *Df*

*θ*�� + 3

1 *Le <sup>φ</sup>*�� <sup>+</sup>

*f* �

*Nux* <sup>=</sup> <sup>−</sup>*Ra*1/2 *<sup>ξ</sup>*1/2*θ*�

 *x* <sup>0</sup> *<sup>ξ</sup>*−1/2*θ*�

 *x* <sup>0</sup> (<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*<sup>2</sup> *ξ* ) 1 <sup>2</sup> *dξ*

The associated local Nusselt and Sherwood numbers are given by

(1 + *σ*<sup>2</sup> *ξ* ) 1 2

*∂*2Φ *<sup>∂</sup>Y*¯ <sup>2</sup> <sup>+</sup> *Sr*

*<sup>∂</sup>Y*¯ <sup>2</sup> = [<sup>1</sup> <sup>−</sup> *<sup>σ</sup>X*¯ tan(Ω)]

*∂*2Φ *∂Y*¯ <sup>2</sup> <sup>=</sup> <sup>1</sup> *R*

> *∂*2Θ *∂Y*¯ <sup>2</sup> <sup>=</sup> <sup>1</sup> *R*

We may further simplify equations (70) - (72) by introducing the following transformation

Substituting equation (73) into equations (70) - (72), gives the nonlinear system of differential

*f* �� = [1 − *σξ* tan(Ω)](*θ*� + *Nφ*�

<sup>2</sup> *<sup>f</sup> <sup>θ</sup>*� <sup>+</sup> *Df <sup>φ</sup>*�� <sup>=</sup> *<sup>ξ</sup>*(*<sup>f</sup>* �

<sup>2</sup> *<sup>f</sup> <sup>φ</sup>*� <sup>+</sup> *Srθ*�� <sup>=</sup> *<sup>ξ</sup>*(*<sup>f</sup>* �

*f*(*ξ*, 0) = 0, *θ*(*ξ*, 0) = 1, *φ*(*ξ*, 0) = 1,

The mean Nusselt and Sherwood numbers from the leading edge to streamwise position *x* are

The governing equations (74) - (76) along with the boundary conditions (77), were solved numerically using the Keller-box method (see Keller (16)) for various parameter combinations. Two hundred uniform grid points of step size 0.05 were used in the *η*- direction. A uniform grid with 120 nodes was used in the *ξ* direction. At every *ξ* grid line, the iteration process is carried out until an accuracy of 10−<sup>6</sup> is achieved for all the variables. The computations

, *Shm*

*Ra*1/2 <sup>=</sup> <sup>−</sup> *<sup>x</sup>*

 *x* <sup>0</sup> *<sup>ξ</sup>*−1/2*φ*�

 *x* <sup>0</sup> (<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*<sup>2</sup> *ξ* ) 1 <sup>2</sup> *dξ*

3

(*ξ*, 0)

(*ξ*, 0)*dξ*

Figure 7 shows the effect of the Dufour number *Df* on heat and mass transfer for two different values of the amplitude *a*. The effect of increasing the amplitude, on average, is to reduce the heat and mass transfer rates as compared with the limiting case of a smooth cone. Figures 7(c) and 7(d) highlight the same. Figures 7(a) and 7(b) show that for *a* = 0 (smooth cone) both *NuxRa*−1/2 and *ShxRa*−1/2 increase steadily with *<sup>ξ</sup>* whereas for the wavy cone (i.e., *<sup>a</sup>* �<sup>=</sup> 0) we observe oscillations in *NuxRa*−1/2 and *ShxRa*−1/2 over the three complete cycles of undulations from *ξ* = 0 to *ξ* = 6 having length two. These results represent the nonlinear coupling of the change in fluid velocity and orientation of the gravitation. The results are in agreement with those reported by Cheng (6) and Pop and Na (34). The Dufour number *Df* reduces *NuxRa*−1/2 and *NumRa*−1/2. The opposite is true in the case of *ShxRa*−1/2 and *ShmRa*−1/2.

The effect of *Df* on heat and mass transfer is depicted in Figure 8 for two different values of the cone half angle Ω. From 8(c) and 8(d) it is clear that increasing the half angle Ω, on average, reduces the heat and mass transfer rates. Figures 8(a) and 8(b) show that there is an increase in oscillations of *NuxRa*−1/2 and *ShxRa*−1/2 for higher values of Ω. In this case the Dufour number also reduces the heat transfer while enhancing mass transfer.

Figure 9 demonstrates the effect of *Df* on heat and mass transfer for two different values of buoyancy ratio *N*. It is evident that the buoyancy ratio amplifies heat and mass transfer from the cone. Again, the Dufour number contributes to lowering heat transfer while enhancing mass transfer rates.

The effect of *Df* on the heat and mass transfer is highlighted for two different values of Lewis numbers in Figure 10. We observe that *Le* reduces heat transfer whereas the opposite is true in the case of mass transfer. For large values of *Le*, higher values of *Df* (≥ 0.5) produce negative heat transfer rates indicating that heat diffuses from fluid to the cone in such cases. Figures 10(a) and 10(c) confirm and reinforce the same fact. The effect of Soret number *Sr* on heat and mass transfer for two different values of amplitude *a* is projected in Figure 11. The decreasing effect of the amplitude *a* on heat and mass transfer rates observed in this situation also. The Soret number *Sr* contributes to increasing *NuxRa*−1/2 and *NumRa*−1/2 while reducing *ShxRa*−1/2 *ShmRa*−1/2 as can be seen in Figures 11(a) - 11(d).

The effect of *Sr* on heat and mass transfer is shown in Figure 12 for two different values of cone half angle Ω. The fact that Ω reduces the heat and mass transfer rates is observed in plots 12(a) and 12(d). The Soret number *Sr* has the effect of increasing the heat transfer and reducing the mass transfer for all values of Ω.

Figure 13 shows the effect of *Sr* on heat and mass transfer rates for two different values of the buoyancy ratio *N*. From 13(a) - 13(d) we observe that the buoyancy ratio enhances both heat and mass transfer rates. For selected values of *N*, *Sr* contributes towards enhancing the heat transfer rate while reducing the mass transfer rate.

The effect of *Sr* on the heat and mass transfer rates is shown in Figure 14 for selected values of the Lewis number *Le*. It is evident that *Le* reduces the heat transfer whereas the opposite is true in case of mass transfer. At large values of *Le* there is a critical value of *Sr* up to which *NuxRa*−1/2 and *NumRa*−1/2 increases and beyond this critical value, both *NuxRa*−1/2 and *NumRa*−1/2 start to fall as can be more clearly seen in Figures 14(a) and 14(c). From Figures 14(b) - 14(d) we observe that the effect of *Sr* is to reduce the rate of mass transfer from the surface of the wavy cone.

Fig. 8. Effect of *Df* on heat and mass transfer with *a* = 0.2, *N* = 1, *Le* = 2and *Sr* = 0.2

(c) (d)

(a) (b)

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 97

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects

Fig. 7. Effect of *Df* on heat and mass transfer with Ω = *π*/9, *N* = 1, *Le* = 2 and *Sr* = 0.2

Fig. 8. Effect of *Df* on heat and mass transfer with *a* = 0.2, *N* = 1, *Le* = 2and *Sr* = 0.2

Fig. 7. Effect of *Df* on heat and mass transfer with Ω = *π*/9, *N* = 1, *Le* = 2 and *Sr* = 0.2

(c) (d)

16 Will-be-set-by-IN-T

96 Mass Transfer - Advanced Aspects

Heat and Mass Transfer from an Inverted Cone in

Fig. 10. Effect of *Df* on heat and mass transfer with *a* = 0.2, Ω = *π*/9, *N* = 1 and *Sr* = 0.2

(c) (d)

a Porous Medium with Cross-Diffusion Effects 1

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 99

Fig. 9. Effect of *Df* on heat and mass transfer with *a* = 0.2, Ω = *π*/9, *Le* = 2 and *Sr* = 0.2

Fig. 10. Effect of *Df* on heat and mass transfer with *a* = 0.2, Ω = *π*/9, *N* = 1 and *Sr* = 0.2

Fig. 9. Effect of *Df* on heat and mass transfer with *a* = 0.2, Ω = *π*/9, *Le* = 2 and *Sr* = 0.2

(c) (d)

18 Will-be-set-by-IN-T

98 Mass Transfer - Advanced Aspects

Fig. 12. Effect of *Sr* on heat and mass transfer with *a* = 0.2, *N* = 1, *Le* = 2and *Df* = 0.3

(c) (d)

a Porous Medium with Cross-Diffusion Effects 2

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 101

(a) (b)

Heat and Mass Transfer from an Inverted Cone in

Fig. 11. Effect of *Sr* on heat and mass transfer with Ω = *π*/9, *N* = 1, *Le* = 2 and *Df* = 0.3

Fig. 12. Effect of *Sr* on heat and mass transfer with *a* = 0.2, *N* = 1, *Le* = 2and *Df* = 0.3

Fig. 11. Effect of *Sr* on heat and mass transfer with Ω = *π*/9, *N* = 1, *Le* = 2 and *Df* = 0.3

(c) (d)

(a) (b)

20 Will-be-set-by-IN-TEC

100 Mass Transfer - Advanced Aspects

Fig. 14. Effect of *Sr* on heat and mass transfer with *a* = 0.2, Ω = *π*/9, *N* = 1 and *Df* = 0.3

(c) (d)

(a) (b)

a Porous Medium with Cross-Diffusion Effects 23

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 103

Heat and Mass Transfer from an Inverted Cone in

Fig. 13. Effect of *Sr* on heat and mass transfer with *a* = 0.2, Ω = *π*/9, *Le* = 2 and *Df* = 0.3

Fig. 14. Effect of *Sr* on heat and mass transfer with *a* = 0.2, Ω = *π*/9, *N* = 1 and *Df* = 0.3

Fig. 13. Effect of *Sr* on heat and mass transfer with *a* = 0.2, Ω = *π*/9, *Le* = 2 and *Df* = 0.3

(c) (d)

22 Will-be-set-by-IN-TEC

102 Mass Transfer - Advanced Aspects

a Porous Medium with Cross-Diffusion Effects 25

Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects 105

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#### **4. Conclusions**

Double-diffusive convection from inverted smooth and wavy cones in Darcy porous media has been investigated. A similarity analysis is performed to reduce the governing equations to coupled nonlinear differential equations that are solved by using the successive linearisation method (SLM), the Matlab bvp4c, a shooting technique and the Keller-box method.

For the smooth cone the effects of the governing parameters on the velocity, temperature and concentration profiles have been studied. The effects of Dufour and Soret effect on the rate of heat and mass transfer were determined. Comparison between our results and earlier results has been made. The findings suggest that the successive linearisation method is a reliable method for solving nonlinear ordinary differential equations.

In the case of the wavy cone we have studied the effects of cross-diffusion on the heat and the mass transfer rates. From the present study we can see that *Df* reduces heat transfer and increases mass transfer. The effect of *Sr* is exactly the opposite except at high Lewis numbers when the heat transfer rate increases up to a critical value of *Sr* and then starts decreasing beyond that value.

#### **5. References**


24 Will-be-set-by-IN-TECH

Double-diffusive convection from inverted smooth and wavy cones in Darcy porous media has been investigated. A similarity analysis is performed to reduce the governing equations to coupled nonlinear differential equations that are solved by using the successive linearisation

For the smooth cone the effects of the governing parameters on the velocity, temperature and concentration profiles have been studied. The effects of Dufour and Soret effect on the rate of heat and mass transfer were determined. Comparison between our results and earlier results has been made. The findings suggest that the successive linearisation method is a reliable

In the case of the wavy cone we have studied the effects of cross-diffusion on the heat and the mass transfer rates. From the present study we can see that *Df* reduces heat transfer and increases mass transfer. The effect of *Sr* is exactly the opposite except at high Lewis numbers when the heat transfer rate increases up to a critical value of *Sr* and then starts decreasing

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**4. Conclusions**

beyond that value.

pp. 1 - 12.

893 - 898.

**5. References**


**6** 

*Ukraine* 

**Mass Transfer Between** 

Roman Vengrenovich, Bohdan Ivanskii,

β

. Thus, in his paper

**Clusters Under Ostwald's Ripening** 

Anatolii Moskalyuk, Sergey Yarema and Miroslav Stasyk *Chernivtsi National University after Yu. Fed'kovich, Chernivtsi,* 

Decay of oversaturated solid solutions with forming a new phase includes three stages, *viz.* nucleation of centers (clusters, nucleation centers, extractions), independent growth of them and, at last, development of these centers interconnecting to each other. This last stage, socalled late stage of decay of oversaturated solid solution has been firstly revealed by Ostwald (Ostwald, 1900). Its peculiarity consists in the following. Diffusion mass transfer of a matter from clusters with larger magnitudes of surface curvature to ones with smaller magnitudes of surface curvature (owing to the Gibbs-Thomson effect) results in dissolving and disappearing small clusters that causes permanent growth of the mean size of extractions. In accordance with papers (Sagalovich, Slyozov, 1987; Kukushkin, Osipov, 1998), interaction between clusters is realized through the 'generalized self-consistent diffusion field'. This process, when large clusters grow for account of small ones is referred to as the Ostwald's ripening. Investigation of the Ostwald's ripening resulted in determination of the form of the size distribution function in respect of the mass transfer mechanisms. The first detailed theory of the Ostwald's ripening for the diffusion mass transfer mechanism has been developed by Lifshitz and Slyozov (Lifshitz and Slyozov, 1958, 1961). Under diffusion mass transfer mechanism, atoms of a solved matter reaching clusters by diffusion are then entirely absorbed by them, so that cluster growth is controlled by matrix diffusion and, in part, by the volume diffusion coefficient, *Dv* . In paper (Wagner, 1961), Wagner has firstly showed that it is possible, if the atoms crossing the interface 'cluster-matrix' and falling at a cluster surface in unit of time have a time to form chemical connections necessary for reproduction of cluster matter structure. If it is not so, solved atoms are accumulated near the interface 'cluster-matrix' with concentration *C* that is equal to the mean concentration of a solution, *C* . For that, growing process is not controlled by

the volume diffusion coefficient, *Dv* , but rather by kinetic coefficient,

published three years later than the papers by Lifshitz and Slyozov, Wagner considered other mechanism of cluster growth controlled by the rate of formation of chemical connection at cluster surface. The quoted papers (Lifshitz, Slyozov, 1958, 1961; Wagner, 1961) form the base of the theory of the Ostwald's ripening that is conventionally referred to as the Lifshitz-Slyozov-Wagner (LSW) theory. Within the framework of this theory, several

**1. Introduction** 

