Effect of Double Dispersion on Convective Flow over a Cone

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.15(2013) No.4,pp.309-321 Effect of Double Dispersion on Convective Flow over a Cone Ch.RamReddy Department of Mathematics, National Institute of Technology, Rourkela-769008, Odisha, India. (Received 08,July,2012; accepted 24,February,2013) Abstract: The effects of thermal and solutal dispersion on natural convection about an isothermal vertical cone with a fixed apex half angle, pointing downwards in a Newtonian fluid are analyzed in both aiding and opposing buoyancy cases. The non-linear governing equations and their associated boundary conditions are initially cast into dimensionless forms by local similarity variables. The resulting system of equations is then solved numerically using the fourth-order Runge-Kutta integration scheme with the Newton-Raphson shooting technique. The numerical results are compared and found to be in good agreement with previously published results as special cases of the present investigation. The variation of skin friction coefficient, heat and mass transfer rates with the governing parameters such as buoyancy ratio, thermal and solutal dispersion parameters are discussed in a wide range of values of these parameters. Keywords: Natural convection, Thermal dispersion, Solutal dispersion, Cone, Newtonian fluid. 1 Introduction There are several practical applications in which significant temperature and concentration differences between the surface of the body and the free stream exist. These temperature and concentration differences cause density gradients in the fluid medium, and natural convection effects become important in the presence of gravitational body force. Further, laminar natural convection boundary layer flow over a heated vertical surface is encountered in a variety of engineering applications including thermal insulation, cooling of metallic surfaces in a bath and heat dissipation of electronic components. Due to its important applications, one needs to study the convective heat and mass transfer from different geometries. To begin with, axisymmetric bodies such as a cone, horizontal and vertical cylinders, and spheres (which are amicable for the fundamental study using the standard analytical techniques) are used to understand the convective heat transfer mechanism as the heat sources. Depending on the flow and field conditions in the fluid medium, different flow models are being employed. For example, Alamgir [1] investigated the overall heat transfer in laminar natural convection from vertical cone using the integral method. The compressibility effects in laminar free convection from a vertical cone is studied by Pop [2]. Yih [3] examined the effect of radiation on natural convection of Newtonian fluids about a truncated cone. The coupled heat and mass transfer by natural convection of Newtonian fluids about a truncated cone in the presence of a magnetic field and radiation effects is discussed by Chamkha [4]. Eco [5] performed a similarity analysis to investigate the laminar free convection boundary layer flow in the presence of a transverse magnetic field over a vertical down-pointing cone with mixed thermal boundary conditions. Postelnicu [6] studied the free convection about a vertical frustum of a cone in a micropolar fluid with constant wall temperature. Cheng [7] analyzed the heat transfer by natural convection of a micropolar fluid from a vertical truncated cone with power-law variation in temperature. Sohouli [8] discussed the boundary layer approximations of fluid flow and heat transfer of vertical full cone embedded in porous media. Recently, Parand [9] analyzed the problem of natural convection about a cone embedded in a porous medium. They found the local similarity solutions for a full cone with the prescribed wall temperature or surface heat flux being a power function of distance from the vertex of the inverted cone. Corresponding author. E-mail address: chramreddy@nitrkl.ac.in, chittetiram@gmail.com Copyright c World Academic Press, World Academic Union IJNS.2013.06.30/729

310 International Journal of Nonlinear Science, Vol.15(2013), No.4, pp. 309-321 In the recent past, considerable attention has been paid to the theoretical and numerical study of convective flow, heat and mass transfer in a Newtonian fluid as it plays a crucial role in diverse applications, such as thermal insulation, extraction of crude oil and chemical catalytic reactors etc. Since the hydrodynamic mixing is called dispersion. The transport of energy and concentration due to the hydrodynamic mixing is called thermal and solutal dispersion respectively. These thermal and solutal dispersions cause additional heat and concentration mass transfers, which brings further complications in dealing with transport processes in a fluid medium. The interest in such studies is motivated by numerous engineering applications in several areas such as geothermal engineering, thermal insulation systems, petroleum recovery, packed bed reactors, sensible heat storage beds, ceramic processing and ground water pollution etc. It is also important to note that the natural convection driven by thermal and solutal dispersions play an important role in the overall heat and mass transfer. In the view of the above said possible applications, many authors have reported the importance of thermal and solutal dispersion effects along a vertical plate on fluid flow, heat and mass characteristics in a fluid medium. A detailed analysis regarding the effect of double dispersion on mixed convection heat and mass transfer in non-darcy porous medium, one can refer the works of Murthy [10] (also see the references cited therein). The problem of steady laminar simultaneous heat and mass transfer by natural convection flow over a vertical permeable plate embedded in a uniform porous medium in the presence of inertia and thermal dispersion effects is investigated by Chamkha [11]. El-Amin [12] discussed the double dispersion effect on natural convection heat and mass transfer in non-darcy porous medium. A similarity solution for the flow of a micropolar fluid along an isothermal vertical plate with an exponentially decaying heat generation term and thermal dispersion is presented by El-Hakiem [13]. The effect of double dispersion on natural convection heat and mass transfer from vertical plate in non-darcy porous medium saturated with non-newtonian fluids is investigated by Kairi [14]. Srinivasacharya [15] analyzed hydromagnetic mixed convection heat and mass transfer along a vertical plate embedded in a non-darcy porous medium in the presence of chemical reaction, radiation and thermal dispersion effects. Recently, Srinivasacharya [16] discussed the effects of magnetic field and double dispersion on free convection heat and mass transfer along a vertical plate embedded in a doubly stratified non-darcy porous medium saturated with power-law fluids. But, very little attention has been paid to study the significance of thermal and solutal dispersion effects over a cone on fluid flow, heat and mass characteristics in a fluid medium. The similarity solution for non-darcy mixed convection about an isothermal vertical cone with a fixed apex half angle, pointing downwards in a fluid saturated porous medium with uniform free stream velocity is obtained by Murthy and Singh [17]. From the literature, it seems that the problem of natural convection flow over a cone in a Newtonian fluid with thermal and solutal dispersion effects has not been investigated so far. Motivated by the above referenced work and the vast possible industrial applications, it is of paramount interest in this study to consider the effects of buoyancy parameter, thermal and solutal parameters on natural convective flow over a cone. The presence of thermal and solutal dispersions make the mathematical model of the present physical system a little more complicated leading to the complex interactions of the flow, heat and mass transfer mechanism. Analytical solution is ruled out in the current set up and hence a numerical solution is obtained for the present problem. The effects of buoyancy, thermal and solutal dispersion parameters on the physical quantities of the flow are analyzed. The results are compared with relevant results in the existing literature and are found to be in good agreement. 2 Mathematical Formulation Consider the steady laminar free convection boundary layer flow along a heated vertical down-pointing cone with a halfangle Ω in an otherwise quiescent Newtonian fluid. Choose the coordinate system such that the x-axis is along the surface of the cone, and y-axis coincide with outward normal to the surface of the cone. We also consider that the boundary layer is thin relative to the local cone radius so that the local radius of a point inside the layer, can be replaced with the value at the cone surface. This condition is not satisfied with the neighborhood of the cone tip. On the other hand, the pressure gradient across the boundary layer is considered to be negligible, so that the equations governing the problem are strictly applicable to cones of small apex angle. The geometry and the coordinate system are schematically shown in Fig. (1). The surface of the cone is subject to a linearly varying temperature T w (x) = T + T r ( x L) and concentration C w (x) = C + C r ( x L), where Tr and C r are any reference temperature and concentration, which are chosen unequal to T and C respectively and L is a characteristic length. In addition, thermal and solutal dispersion effects are considered. Using the Boussinesq and boundary layer approximations, the basic equations governing the steady state dynamics of IJNS email for contribution: editor@nonlinearscience.org.uk

Ch.RamReddy: Effect of Double Dispersion on Convective Flow over a Cone 311 Figure 1: Physical model and coordinate system a viscous incompressible liquids are given by (ru) x + (rv) y = 0, (1) u u x + v u y = ν 2 u y 2 + g(β T (T T ) + β C (C C )) cos Ω, (2) u T x + v T y = ( ) T α e, y y (3) u C x + v C y = y ( D e C y ), (4) where u and v are the velocity components in x and y directions respectively, r = x sin Ω is the local radius of the cone, T is the temperature, C is the concentration, g is the acceleration due to gravity, ρ is the density, ν is the kinematic viscosity, β T is the coefficient of thermal expansion, β C is the coefficient of solutal expansions. α e and D e are the effective thermal and solutal diffusivities respectively, and these are written as α e = α + γdu and D e = D + ξdu (see Murthy [10]; Srinivasacharya [16]), d is the pore diameter, α is the thermal diffusivity, D is the molecular diffusivity, γ and ξ are the coefficients of the thermal and solutal dispersions respectively. The value of these quantities lies between 1/7 and 1/3. The boundary conditions are u = 0, v = 0, T = T w (x), C = C w (x) at y = 0 (5a) u 0, T T, C C as y. (5b) Here the subscripts w and indicate the conditions at the surface and at the outer edge of the boundary layer respectively. by The non-dimensional quantities x, y, r, u, v, T and C relate to their dimensional counterparts X, Y, R, U, V, T and C X = x L, Y = y L Gr1/4, R = r L, U = u U 0, V = v U 0 Gr 1/4, T = T T T w T, C = C C C w C, (6a) (6b) (6c) IJNS homepage: http://www.nonlinearscience.org.uk/

312 International Journal of Nonlinear Science, Vol.15(2013), No.4, pp. 309-321 where the reference velocity U 0 and the global Grashof number Gr are given by ( ) 2 U 0 = [gβ T L(T w T ) cos Ω] 1/2 U0 L and Gr =. ν Usage of the variables (6), the boundary layer equations (1)-(4) reduce to (RU) X + (RV ) = 0, (7) Y U U X + V U Y = 2 U Y U T X + V T Y = 1 P r Y U C X + V C Y = 1 Sc Y + T + NC, (8) 2 [ (1 + P rd s U) T ], (9) Y [ (1 + ScD c U) C Y ]. (10) In usual definitions, P r = ν α and Sc = ν D are the Prandtl and Schmidt numbers respectively. Further, N = β C(C w C ) β T (T w T ) is the buoyancy ratio. Thermal buoyancy acts always vertically upwards, the species buoyancy may act in either direction depending on the relative molecular weights. So N > 0 indicates aiding buoyancy where both the thermal buoyancy and solutal buoyancy are in the same direction and N < 0 indicates opposing buoyancy where the solutal buoyancy is in the opposite direction to the thermal buoyancy. When N = 0, the flow is driven by thermal buoyancy alone. Finally, D s = γd L Gr1/2 and D c = ξd L Gr1/2 are the thermal and solutal dispersion parameters respectively. Assuming T w T = T r and C w C = C r, the boundary conditions (5) can be written as U = 0, V = 0, T = X, C = Y at Y = 0 (11a) U 0, T 0, C 0 as Y. (11b) In view of the continuity equation (7), we introduce the stream function ψ by U = 1 ψ R Y, V = 1 ψ R X. (12) Substituting (12) in (8)-(10) and then using the following similarity transformations ψ(x, Y ) = XRf(Y ), T (X, Y ) = Xθ(Y ), C(X, Y ) = Xϕ(Y ), (13) we obtain the following nonlinear system of differential equations f + 2ff (f ) 2 + θ + Nϕ = 0, (14) 1 P r θ + D s X(f θ ) + 2fθ f θ = 0, (15) 1 Sc ϕ + D c X(f ϕ ) + 2fϕ f ϕ = 0. (16) Boundary conditions (11) in terms of f, θ, ϕ become Y = 0 : f = 0, f = 0, θ = 1, ϕ = 1, Y : f 0, θ 0, ϕ 0, where prime in the above equations denote differentiation with respect to Y. A close look at Eqs. (15)-(16) reveals that, in free convection flow due to a cone, the temperature and concentration profiles are not similar because the x-coordinate cannot be eliminated from these equations. Although local non-similarity solutions have been found for some boundary layer flows dealing with convective flow, the technique is difficult to extend to this case. Thus, for ease of analysis, it was decided to proceed with finding local similarity solutions for the governing equations (14)-(16). That is, taking X = x and then varying the X-location, one can still study the effects of various L parameters on different profiles at any given X-location, where L is the characteristic length. If D s = 0, D c = 0, the problem reduces to natural convection flow about the cone. In the limit, as D s = 0, D c = 0 and N = 0, the governing equations (1)-(4) reduce to the corresponding equations for a free convection heat transfer about a cone. Hence, the case of the free convective flow over a cone [5] can be obtained by taking D s = 0, D c = 0 and N = 0. (17a) (17b) IJNS email for contribution: editor@nonlinearscience.org.uk

Ch.RamReddy: Effect of Double Dispersion on Convective Flow over a Cone 313 3 Skin friction, Heat and Mass transfer coefficients The primary objective of this study is to estimate the parameters of engineering interest in fluid flow, heat and mass transport problems are the skin friction coefficient C f, the Nusselt number Nu, and the Sherwood number Sh. These parameters characterize the surface drag, the wall heat and mass transfer rates, respectively. The shearing stress at the surface of the cone τ w is defined as τ w = µ X [ u y ], (18) y=0 and the local heat and mass fluxes from the surface of the cone into the medium can be obtained from q w = k [ ] e T and q m = D [ ] e C, (19a) X y y=0 X y y=0 where µ is the coefficient of viscosity, k e is the effective thermal conductivity of the medium which is the sum of the molecular conductivity k and the dispersion thermal conductivity k d and D e is the effective solutal diffusivity of the medium which is the sum of the molecular diffusivity D and the dispersion solutal diffusivity D d. Sh = The non-dimensional shear stress C f q m L are readily obtained in the form D(C w C ) = 2τ w ρu0 2, the Nusselt number Nu = q w L and the Sherwood number k(t w T ) C f Gr 1/4 = 2f (0), (20a) Nu Gr = [1 + P rxd sf (0)]θ (0), (20b) 1/4 Sh Gr = [1 + ScXD cf (0)]ϕ (0). (20c) 1/4 The effects of the buoyancy parameter, X- location, thermal and solutal dispersion parameters on these coefficients are discussed in the following section. 4 Numerical Procedure The flow Eq. (14) coupled with the energy and concentration Eqs. (15) and (16) constitute a set of nonlinear nonhomogeneous differential equation for which closed-form solution cannot be obtained. Hence the problem has been solved numerically using shooting technique along with fourth order Runge-Kutta integration. The basic idea of shooting method for solving boundary value problem is to find an appropriate initial condition for which the computed solution hit the target so that the boundary conditions at other points are satisfied. Furthermore, the higher order nonlinear differential equations are converted into simultaneous linear differential equations of the first order and they are further transformed into an initial valued problem applying the shooting method incorporating fourth order Runge-Kutta method. The iterative solution procedure was carried out until the error in the solution became less than a predefined tolerance level. The non-linear differential equations (14) - (16) are converted into the following system of linear differential equations of first order by the substitution {f, θ, ϕ, f, θ, ϕ, f } = {z 1, z 2, z 3, z 4, z 5, z 6, z 7 } dz 1 dy = z 4, dz 2 dy = z 5, dz 3 dy = z 6, dz 4 dy = z 7, (21a) (21b) dz 5 dy = [2.z 1.z 5 X.D s.z 5.z 7 z 2.z 4 ], (21c) [1/P r + X.D s.z 4 ] (21d) dz 6 dy = [2.z 1.z 6 X.D c.z 6.z 7 z 3.z 4 ], (21e) [1/Sc + X.D c.z 4 ] (21f) dz 7 dy = z 4.z 4 2.z 1.z 7 z 2 N.z 3. (21g) IJNS homepage: http://www.nonlinearscience.org.uk/

314 International Journal of Nonlinear Science, Vol.15(2013), No.4, pp. 309-321 Table 1: Comparison of results for a laminar free convection flow over a cone [? ] with N = 0, D s = D c = 0 f (0) θ (0) P r Eco [5] Present results Eco [5] Present results 1.0 0.681482 0.681482 0.638859 0.638859 10.0 0.433269 0.433272 1.275548 1.275554 The boundary conditions in terms of z 1, z 2, z 3, z 4, z 5, z 6 and z 7 are z 1 (0) = 0, z 2 (0) = 1, z 3 (0) = 1, z 4 (0) = 0, z 2 ( ) = 0, z 3 ( ) = 0, z 4 ( ) = 0. (22) Here, Y at is taken as Y max and chosen large enough so that the solution shows little further change for Y larger than Y max. As the initial values for z 5, z 6 and z 7 are not specified in the boundary conditions (22), assume some values for z 5 (0), z 6 (0) and z 7 (0). Then the equations (21) are integrated using the fourth order Runge - Kutta method from Y = 0 to Y = Y max over successive steps Y. The accuracy of the assumed initial values z 5 (0), z 6 (0) and z 7 (0) is then checked by comparing the calculated values of z 5 (0), z 6 (0) and z 7 (0) at Y = Y max with their given value at Y = Y max in (22). If a difference exists, another set of initial values for z 5 (0), z 6 (0) and z 7 (0) must be assumed and the process is repeated. This process is continued until the agreement between the calculated and the given condition at Y = Y max is within the specified degree of accuracy. In the present study, the boundary conditions for Y at are replaced by a sufficiently large value of Y where the velocity, temperature and concentration approach zero. After some trials we imposed a maximal value of Y at of 15. In order to see the effects of step size ( Y ) we ran the code for our model with three different step sizes as Y = 0.001, Y = 0.01 and Y = 0.05. The calculations are repeated until some convergence criterion is satisfied. In each case, we found very good agreement between them on different profiles. A step size of Y = 0.01 was selected to be satisfactory for a convergence criterion of 10 6 in all cases. Extensive calculations have been performed to obtain the wall velocity, temperature and concentration fields for a wide range of parameters. The effects of thermal and solutal dispersion parameters on the convective heat and mass transfer characteristics are studied. 5 Results and Discussion If N = 0 and no thermal and solutal dispersion effects (D s = 0 and D c = 0), the governing equations for the double dispersion on free convective flow over a cone reduce to the simple case of free convective heat transfer over a cone, see Eco [5], who investigated the free convective heat transfer flow over a cone. Our results have been compared with those reported by Eco [5] and Table (1) shows a perfect agreement. To have a better understanding of the flow characteristics, numerical results for the velocity, temperature and concentration are calculated for different values of thermal dispersion parameter D s and solutal dispersion parameter D c in both aiding and opposing buoyancy cases. 5.1 Boundary-layer distributions of velocity, temperature and concentration 5.1.1 With varying thermal dispersion parameter The first set of figures, 2(a) to 2(c), are for D c = 0.3, P r = 1.0, Sc = 1.0 and X = 0.5, and refer to the variation of the non-dimensional velocity f, temperature θ and concentration ϕ across the boundary layer. Figs. 2(a) 2(c) are for both aiding and opposing buoyancy cases. Since D s 0 corresponds to the case of viscous fluid without thermal dispersion. The velocity is less in the absence of thermal dispersion and more that of viscous fluid in the presence of the thermal dispersion. Introducing the effect of thermal dispersion in the energy equation in general favors conduction over convection. In other words, supplementing dispersion effects to the energy equation gives thermal conduction more dominance. One can see from Fig. 2(b) that an increase in the mechanical thermal dispersion coefficient increases the thermal boundary layer thickness i.e., thermal dispersion enhances the transport of heat along the normal direction to the IJNS email for contribution: editor@nonlinearscience.org.uk

Ch.RamReddy: Effect of Double Dispersion on Convective Flow over a Cone 315 wall as compared with the case where dispersion is neglected (i.e. D s = 0). In general, this work may be useful in showing that, the use of fluid medium with better heat dispersion properties may result in better heat transfer characteristics that may be required in many industrial applications (like those concerned with packed bed reactors, nuclear waste disposal, etc). It is observed that from Fig. 2(c) that the solutal boundary layer thickness is decreased with an increase in the value of thermal dispersion parameter. Further, the temperature in case of viscous fluids without thermal dispersion is less and concentration is more than that of the corresponding Newtonian fluid case with thermal dispersion. Therefore, the effect of varying thermal dispersion is seen to be qualitatively the same for the hydrodynamic, thermal and concentration boundary layers in the comparison between these two categories of aiding and opposing buoyancy. It is remarked that the weaker influence of thermal dispersion on the concentration profiles in the boundary layer. 5.1.2 With varying solutal dispersion parameter The second set of figures, 3(a) to 3(c), are plotted for D s = 0.5, P r = 1.0, Sc = 1.0 and X = 0.5, and refer to the variation of the non-dimensional velocity f, temperature θ and concentration ϕ across the boundary layer. An increase in D c is seen to considerably lower momentum boundary layer thickness closer to the wall and the influence is reversed away from the wall in aiding buoyancy case. But, it can be seen from this figure that increase in D c decreases the momentum boundary layer thickness in opposing buoyancy case. It is seen from Fig. 3(b) that the thermal boundary layer thickness decreases in aiding buoyancy case where as reversed phenomenon is observed for opposing buoyancy case with an increase in D c. A rise in concentration boundary layer thickness is seen from Fig. 3(c) in both aiding and opposing buoyancy cases with increasing values of the coefficient of solutal dispersion parameter D c. Finally, it is noticed that the slightly weaker influence of solutal dispersion on the temperature profiles in the boundary layer. 5.1.3 For various X-locations The third set of figures, 4(a) to 4(c), are plotted for D s = 0.5, D c = 0.3, P r = 1 and Sc = 1.0, and refer to the variation of the non-dimensional velocity f, temperature θ and concentration ϕ across the boundary layer. When X = 0, the flow governing equations are independent of X-location and hence, it shows that the existence of similarity solutions for the present problem. This is an interesting aspect, put into evidence by the present numerical analysis. Fig. 4(a) shows that the hydrodynamic boundary layer thickness increases in the downstream direction, for both aiding and opposing buoyancy cases. Figs. 4(b) and 4(c) demonstrate the similar behavior of the thermal and concentration boundary layers, in comparison with what described above for the hydrodynamic boundary layer. It can be observed from Fig. (4) that a slow stabilization of the velocity, temperature and concentration profiles in the downstream direction. One remark from this figure that it to quickly approach the similarity solutions not far from the leading edge. 5.1.4 Skin friction, Nusselt and Sherwood numbers The streamwise variations of the skin friction, rate of heat and mass transfers at the wall are shown in Figs. 5(a) - 6(c), when P r = Sc = 1 for different values of the other parameters in both aiding and opposing buoyancy cases. All these quantities behave similarly: increase linearly in the streamwise direction with an increase in the value of thermal dispersion D s in the both aiding and opposing cases. Another important fact demonstrated by these three figures is that higher skin friction, heat and mass transfer rates are produced in opposing buoyancy case as compared with aiding buoyancy case. Further, these three quantities increases in the case of aiding buoyancy where as skin friction and heat transfer rate decrease but mass transfer rate increases with the solutal dispersion D c. In all cases represented in Figs. 5(a) to 6(c), it is observed that as the value of thermal dispersion parameter increases, the skin friction, heat and mass transfer coefficients increase where as the value of solutal dispersion parameter decreases the skin friction and heat transfer rate but increases the mass transfer rate. 6 Conclusions In this paper, a boundary layer analysis, for natural convection heat and mass transfer flow over a cone in the presence of thermal and solutal dispersion effects, is presented. Using a set of suitable local similarity variables, the governing equations are transformed into a set of ordinary differential equations depending on several dimensionless parameters. Between these parameters there is also the dimensionless distance along the wall, which is varied between 0 and 1 and this approach is a characteristic feature of the present investigation, where similar solutions cannot be obtained. Consequently, IJNS homepage: http://www.nonlinearscience.org.uk/

316 International Journal of Nonlinear Science, Vol.15(2013), No.4, pp. 309-321 numerical solutions have been presented for a wide range of (D s, D c ) for aiding buoyancy (N = 0.5) and opposing buoyancy (N = 0.5), at different X-locations along the wall. The major conclusion is that the skin friction, heat and mass transfer rates increase with the increasing value of the thermal dispersion parameter in both aiding and opposing buoyancy cases. On the other hand, an increasing value of the solutal dispersion parameter, decrease the skin friction and heat transfer rate while increase the mass transfer rate in the case of opposing buoyancy. But the skin friction, heat and mass transfer rates increase with an increase in the value of salutal dispersion parameter in the case of aiding buoyancy. Acknowledgments My sincere thanks to the reviewers for their encouraging comments and constructive suggestions to improve the manuscript. References [1] M Alamgir: Overall heat transfer from vertical cones in laminar free convection: an approximate method. ASME Journal of Heat Transfer, 101(1989):174-176. [2] I Pop, HS Takhar: Compressibility effects in laminar free convection from a vertical cone. Appl. Sci. Res, 48(1991):71-82. [3] KA Yih: Effect of radiation on natural convection about a truncated cone. Int. J. Heat Mass Transf, 42(1999):4299-4305. [4] AJ Chamkha: Coupled heat and mass transfer by natural convection about a truncated cone in the presence of magnetic field and radiation effects. Numer. Heat Transf. Part: A Appl, 39(2001):511-530. [5] MC Ece: Free convection flow about a cone under mixed thermal boundary conditions and a magnetic field. Applied Mathematical Modelling, 29(2005):1121-1134. [6] A Postelnicu: Free convection about a vertical frustum of a cone in a micropolar fluid. Int. J. Eng. Sci, 44(2006):672-682. [7] CY Cheng: Natural convection of a micropolar fluid from a vertical truncated cone with power-law variation in temperature. Int. Commun. Heat Mass Transf, 35(2008):39-46. [8] AR Sohouli, M Famouri, A Kimiaeifar, G Domairry: Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux. Commun Nonlinear Sci Numer Simulat, 15(2010):1691-1699. [9] K Parand, Z Delafkar, JA Rad, S Kazem: Numerical study on wall temperature and surface heat flux natural convection equations arising in porous media by rational Legendre Collocation Approach. International Journal of Nonlinear Science, 13(2012):39-50. [10] PVSN Murthy: Effect of double dispersion on mixed convection heat and mass transfer in non-darcy porous medium. ASME Journal of Heat Transfer, 122(2000):476-484. [11] AJ Chamkha, M Mujtaba, A Quadri: Simultaneous heat and mass transfer by natural convection from a plate embedded in a porous medium with thermal dispersion effects. Heat and Mass Transfer, 39(2003):561-569. [12] MF El-Amin: Double dispersion effects on natural convection heat and mass transfer in non-darcy porous medium. Applied Mathematics and Computation, 156(2004):1-17. [13] MA El-Hakiem: Natural convection in a micropolar fluid with thermal dispersion and internal heat generation. Int. Comm.Heat Mass Transfer, 31(2004):1177-1186. [14] RR Kairi, PAL Narayana, PVSN Murthy: The effect of double dispersion on natural convection heat and mass transfer in a non-newtonian fluid saturated non-darcy porous medium. Transport in Porous Media, 76(2009):377-390. [15] D Srinivasacharya, J Pranitha, Ch RamReddy: Chemical reaction and radiation effects on MHD mixed convection heat and mass transfer in a non-darcy porous medium. Int. J. of Fluid Mechanics, 2(2010):1-8. [16] D Srinivasacharya, J Pranitha, Ch RamReddy: Magnetic and double dispersion effects on free convection in a non- Darcy porous medium saturated with power-law fluid. Int. J. for Computational Methods in Engineering Science and Mechanics, 13(2012):210-218. [17] PVSN Murthy, P Singh: Thermal dispersion effects on non-darcy convection over a cone. Computers and Mathematics with Applications, 40(2000):1433-1444. IJNS email for contribution: editor@nonlinearscience.org.uk

Ch.RamReddy: Effect of Double Dispersion on Convective Flow over a Cone 317 (a) (b) (c) Figure 2: (a) Velocity, (b) Temperature and (c) Concentration profiles for various values of N and D s IJNS homepage: http://www.nonlinearscience.org.uk/

318 International Journal of Nonlinear Science, Vol.15(2013), No.4, pp. 309-321 (a) (b) (c) Figure 3: (a) Velocity, (b) Temperature and (c) Concentration profiles for various values of N and D c IJNS email for contribution: editor@nonlinearscience.org.uk

Ch.RamReddy: Effect of Double Dispersion on Convective Flow over a Cone 319 (a) (b) (c) Figure 4: (a) Velocity, (b) Temperature and (c) Concentration profiles for various values of N and X IJNS homepage: http://www.nonlinearscience.org.uk/

320 International Journal of Nonlinear Science, Vol.15(2013), No.4, pp. 309-321 (a) (b) (c) Figure 5: Variation of (a) Skin friction coefficient, (b) Heat transfer and (c) Mass transfer rates at the wall versus X for different values of N and D s IJNS email for contribution: editor@nonlinearscience.org.uk

Ch.RamReddy: Effect of Double Dispersion on Convective Flow over a Cone 321 (a) (b) (c) Figure 6: Variation of (a) Skin friction coefficient, (b) Heat transfer and (c) Mass transfer rates at the wall versus X for different values of N and D c IJNS homepage: http://www.nonlinearscience.org.uk/