Research Article | Open Access
On the Solution of Double-Diffusive Convective Flow due to a Cone by a Linearization Method
The paper details the use of a nonperturbation successive linearization method to solve the coupled nonlinear boundary value problem due to double-diffusive convection from an inverted cone. Diffusion-thermo and thermal-diffusion effects have been taken into account. The governing partial differential equations are transformed into ordinary differential equations using a suitable similarity transformation. The SLM is based on successively linearizing the governing nonlinear boundary layer equations and solving the resulting higher-order deformation equations using spectral methods. The results are compared with the limited cases from previous studies and results obtained using the Matlab inbuilt bvp4c numerical algorithm and a shooting technique that uses Runge-Kutta-Fehlberg (RKF45) and Newton-Raphson schemes. These comparisons reveal the robustness and validate the usage of the linearisation method technique. The results show that the nonperturbation technique in combination with the Chebyshev spectral collocation method is an efficient numerical algorithm with assured convergence that serves as an alternative to numerical methods for solving nonlinear boundary value problems.
The convection driven by two different density gradients with differing rates of diffusion is widely known to as “double-diffusive convection’’ and is an important fluid dynamics phenomenon (see Mojtabi and Charrier-Mojtabi ). The study of double-diffusive convection has attracted attention of many researchers during the recent past due to its occurrence in nature and industry. Oceanography is the root of double-diffusive convection in natural settings. The existence of heat and salt concentrations at different gradients and the fact that they diffuse at different rates lead to spectacular double-diffusive instabilities known as “salt-fingers’’ (see Stern [2, 3]). The formation of salt-fingers can also be observed in laboratory settings. Double-diffusive convection occurs in the sun where temperature and helium diffusions take place at different rates. Convection in magma chambers and sea-wind formations are among other manifestations of double-diffusive convection in nature. Migration of moisture through air contained in fibrous insulations, grain storage systems, the dispersion of contaminants through water-saturated soil, crystal growth, the underground disposal of nuclear wastes, the formation of microstructures during the cooling of molten metals, and fluid flows around shrouded heat-dissipation fins are among other industrial applications of double-diffusive convection.
The inherent instabilities due to double-diffusive convection have been investigated by, among others, Nield , Baines and Gill , Guo et al. , Khanafer and Vafai , Sunil et al. , and Gaikwad et al. . Double-diffusive convection due to horizontal, inclined, and vertical surfaces embedded in a porous medium has been studied by, among others, Cheng [10, 11], Nield and Bejan , and Ingham and Pop . Chamkha  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. Yih  examined the effect of radiation in convective flow over a cone.
Though heat and mass transfer happens simultaneously in a moving fluid, the relations between the fluxes and the driving potentials are generally complicated. It should be noted that the energy flux can be generated by both temperature and composition gradients. The energy flux caused by a composition gradient gives rise to the Dufour or diffusion-thermo effect. Mass fluxes created by temperature gradient lead to the Soret or thermal-diffusion effect. These effects are in collective known as cross-diffusion effects. The cross-diffusion effect has been extensively studied in gases, while the Soret effect has been studied both theoretically and experimentally in liquids, see Mortimer and Eyring . They 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. The results were found to fit the Onsager reciprocal relationship, Onsager .
In general, the cross-diffusion effects are small compared to the effects described by Fourier and Fick’s laws (Mojtabi and Charrier-Mojtabi ) and can therefore be neglected in many heat and mass-transfer processes. However, it has been shown in a number of studies that there are exceptions in areas such as in geosciences where cross-diffusion effects are significant and cannot be ignored, see for instance Kafoussias and Williams , Awad et al. , and the references therein. With this view point, many investigators included cross-diffusion effects in the study of double-diffusive convection in fluid flows involving bodies of various geometries. Alam et al.  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. Shateyi et al.  investigated the effects of diffusion-thermo and thermal-diffusion on MHD fluid flow over a permeable vertical plate in the presence of radiation and hall current. Awad and Sibanda  used the homotopy analysis method to study heat and mass transfer in a micropolar fluid subject to Dufour and Soret effects.
Most boundary value problems in fluid mechanics are solved numerically using either the shooting method or the implicit finite difference scheme in combination with a linearization technique. These methods have their associated difficulties and failures in handling situations where solutions either vary sharply over a domain or problems that exhibit multiple solutions. These limitations necessitate the development of computationally improved semianalytical methods for solving strongly nonlinear problems. There are many different semianalytical methods to solve nonlinear boundary value problems, among them, the variational iteration method, the homotopy perturbation method [23–25], the Adomian decomposition method [26, 27], homotopy analysis method , and the spectral-homotopy analysis methods [29, 30]. These iterative methods may sometimes fail to converge or give slow convergence for strongly nonlinear problems or problems involving large parameters. Yildirim  applied He’s homotopy perturbation method to solve the Cauchy reaction-diffusion problem and compared his results with analytical solutions in certain test cases. Yildirim and Pinar  obtained periodic solutions of nonlinear reaction-diffusion equations arising in mathematical biology using the exp-function method. Yildirim and Sezer  found analytical solutions of linear and nonlinear space-time fractional reaction-diffusion equations (STFRDE) on a finite domain using the homotopy perturbation method (HPM). Yildirim et al.  presented approximate analytical solutions of the biochemical reaction model by the multistep differential transform method (MsDTM) and validated the results by comparing with the fourth-order Runge-Kutta method.
Ganji et al.  solved the nonlinear Jeffery-Hamel flow problem using two semianalytical methods, the variational iteration method (VIM) and the homotopy perturbation method. Ghafoori et al.  solved the equation for a nonlinear oscillator using the differential transform method (DTM). They compared DTM solutions with those obtained using the variational iteration method and the homotopy perturbation method. Joneidi et al.  used three analytical methods, the homotopy analysis method (HAM), homotopy perturbation method, and the differential transform method, to solve the Jeffery-Hamel flow problem. Babaelahi et al.  studied the heat transfer characteristics in an incompressible electrically conducting viscoelastic boundary layer fluid flow over a linear stretching sheet. They solved the flow equations using the optimal homotopy asymptotic method (OHAM) and validated their results by comparing the OHAM solutions with Runge-Kutta solutions.
In this study, we use a nonperturbation, semianalytic successive linearization method (see Makukula et al. [39, 40]) to investigate double-diffusive convection from a cone in a viscous incompressible fluid subject to cross-diffusion effects. The study is an extension of the work by Ece  to include mass transfer and cross-diffusion effects. The linearization method iteratively linearizes the nonlinear equations to give a system of higher-order deformation equations that are then solved using the Chebyshev spectral collocation method.
2. Mathematical Formulation
Consider a vertical down-pointing cone with half-angle immersed in a viscous incompressible liquid. The -axis is along the surface of the cone, and the -axis coincides with the outward normal to the surface of the cone. The origin is at the vertex of the cone, see Figure 1. The surface of the cone is subject to a linearly varying temperature (>) where is the ambient temperature.
Following the usual boundary layer and Boussinesq approximations, the basic equations governing the steady state dynamics of a viscous incompressible liquid are given by where and are the velocity components in the and directions, respectively, is the local radius of the cone, is the kinematic viscosity, is the density, is the acceleration due to gravity, is the coefficient of thermal expansion, is the coefficient of solutal expansion, is the temperature, is the concentration, is the thermal diffusivity, is the species diffusivity, and , are cross-diffusion coefficients.
The boundary conditions for (2.1) have the form Here, the subscripts and refer to the surface and ambient conditions, respectively, and are positive constants, and is a characteristic length.
We introduce the dimensionless variables where the reference velocity and Grashof number are defined, respectively, as On using the variables (2.3), the boundary-layer equations (2.1) reduce to The nondimensional parameters appearing in (2.5)–(2.8) are the buoyancy ratio , the Prandtl number , the Schmidt number , Dufour number Df, and Soret number defined, respectively, as Assuming and , the boundary conditions (2.2) can be written as We now introduce the stream function such that so that the continuity equation (2.5) is satisfied identically. The boundary layer equations (2.6)–(2.8) can be written in terms of the stream function as The boundary conditions (2.10) in terms of the stream function are We further introduce the following similarity variables Using (2.14), (2.12) along with boundary conditions (2.13) reduces to the following two-point boundary value problem The primes in (2.15)–(2.18) denote differentiation with respect to .
3. Successive Linearization Method
The successive linearization method (see Makukula et al. [39, 40]) is used to solve the boundary value problem (2.15)–(2.18). We assume that the functions , , and may be expanded in series form as where , and are unknown functions and , and , are approximations that are obtained by recursively solving the linear part of the equation that results from substituting (3.1) in (2.15)–(2.17). Substituting (3.1) in the governing (2.15)–(2.17), we obtain where the coefficient parameters (), , (), and () are defined as
Starting from the initial approximations which are chosen to satisfy the boundary conditions (2.18), the subsequent solutions , , , are obtained by successively solving the linearized form of (3.2) given below subject to the boundary conditions Once each solution , , and () has been found from iteratively solving (3.5) for each , the functions , , and are obtained as series where is the order of SLM approximation. Equations (3.5) are integrated using the Chebyshev spectral collocation method [42–44]. The unknown functions are defined by the Chebyshev interpolating polynomials with the Gauss-Lobatto points defined as where is the number of collocation points used. The physical region is transformed into the domain using the domain truncation technique in which the problem is solved on the interval instead of . This leads to the mapping where is the known number used to invoke the boundary condition at infinity. The unknown functions , , and are approximated at the collocation points by where is the th Chebyshev polynomial defined as The derivatives of the variables at the collocation points are represented as where is the order of differentiation and where is the Chebyshev spectral differentiation matrix (see, [42–44]). Substituting (3.8)–(3.12) in (3.5)–(3.6) leads to the matrix equation in which is a square matrix of order and , are column vectors defined by with
In the above definitions, (), , (), , and are diagonal matrices of order and is the identity matrix of order . Finally, the solution of the problem is obtained as Thus, starting with the initial solutions , and , a sequence of approximations , , and , are obtained until (3.7) holds. The convergence of this iteration process depends on the parameter values, that is, for small parameter values, the iterates converge faster as compared to large parameter values.
4. Skin Friction, Heat and Mass Transfer Coefficients
The parameters of engineering interest in heat and mass transport problems are the skin friction coefficient , the Nusselt number and the Sherwood number . These parameters characterize the surface drag, the wall heat and mass transfer rates, respectively.
The shearing stress at the surface of the cone is defined as where is the coefficient of viscosity. The skin friction coefficient at the surface of the cone is defined as Using (4.1) in (4.2), we obtain the following relation The heat transfer rate at the surface of the cone is defined as where is the thermal conductivity of the fluid. The Nusselt number is defined as Using (4.4) in (4.5), the dimensionless wall heat transfer rate is obtained as follows: The mass flux at the surface of the cone is defined as and the Sherwood is defined as Using (4.7) in (4.8), the dimensionless wall mass transfer rate is obtain as
5. Results and Discussion
The successive linearization method (SLM) has been applied to solve the nonlinear coupled boundary value problem arising due to double-diffusive convection from a vertical cone immersed in a viscous liquid. Cross-diffusion effects are taken into consideration. The parameters controlling the flow dynamics are the Prandtl number , Schmidt number , buoyancy ratio , Dufour number Df, and the Soret number . We, however, do not discuss the effects of parameters such as the Prandtl and Schmidt numbers whose significance has been widely studied in the literature on double-diffusive convection in viscous liquids. We have thus fixed and instead focus attention on results pertaining to the other three important parameters. In addition, we restrict ourselves to parameter values in the interval , . To highlight the effect of buoyancy, for aiding buoyancy condition, we take while, for opposing buoyancy, .
We first establish the robustness and accuracy of the successive linearization method (SLM) by comparing the SLM results with those obtained numerically and previous related studies in the literature. The Matlab inbuilt bvp4c routine and the shooting technique with Runge-Kutta-Fehlberg (RKF45) and Newton-Raphson schemes are used to obtain the numerical solutions.
Tables 1 and 2 show the results of , , and for different parameter values. Table 1 gives the comparison of the SLM results in the absence of cross-diffusion (i.e., ) with those presented by Ece . The SLM solutions are found to be in excellent agreement with those of Ece  indicating the accuracy of the linearisation method.
Table 2 highlights both the accuracy and the accelerated convergence of the SLM for different values of Df and . The linearisation method converges to the numerical solutions at the fourth-order SLM for all values of Df and . However, for larger values, convergence may require extra terms in the SLM solution series. It is evident that the SLM results are highly accurate as they match with those obtained by the bvp4c and the shooting technique up to the sixth significant digit.
It is to be noted from Table 2 that simultaneously increasing Df and decreasing lead to initial decreases in the skin-friction coefficient up to and then start increasing. The heat transfer coefficient shows monotonic decrease, while the mass transfer coefficient exhibits the opposite change when subjected to simultaneous increase in Df and decrease in .
To gain some insight into the dynamics of the problem, the temperature and concentration distributions are shown graphically in Figures 2–6. The Nusselt number and Sherwood number which highlight the heat and mass transfer are shown in Figures 7 and 8, as functions of for different values of Df in the aiding and opposing buoyancy cases.
The variation of temperature and concentration profiles subject to a simultaneous increase in the cross-diffusion parameters Df and is shown in Figure 2. We observe enhanced heat and mass transfer in the presence of the cross-diffusion effect as compared to the case (no cross-diffusion). Increasing the cross-diffusion parameters increases both the thermal and species boundary layer thickness in both the aiding and opposing buoyancy situations. Hence, the cross-diffusion effect plays an important role in enhancing heat and mass transfer in double-diffusion convection processes.
Figure 3 shows the effect of the Dufour number on the temperature distributions. The energy flux created by the concentration gradient gives rise to the Dufour effect or diffusion-thermo effect and due to the increase in the energy flux created by concentration gradients, the temperature in the boundary layer increases significantly. The Dufour effect thus serves to thicken the thermal boundary layer. This trend is true for both aiding and opposing buoyancy scenarios.
Due to the coupling between the momentum, energy, and species balance equations, the Dufour parameter has an effect on the concentration boundary layer as well. This is shown in Figure 4 where it is evident that Df reduces the concentration in the boundary layer in both the cases of aiding and opposing buoyancy.
The effect of the Soret number on the temperature distribution is shown in Figure 5. The Soret parameter has a mixed effect on profiles. In the case of opposing buoyancy, increasing Soret parameter results in the thickening of the thermal boundary layer, while, in the aiding buoyancy case, the effect of is exactly the opposite.
Figure 6 shows the effect of the Soret number on the species distribution in aiding and opposing buoyancy cases. The mass flux created by the temperature gradient gives rise to Soret or thermal-diffusion or thermophoresis effect. The thermophoretic force developed due to temperature gradients drives solute particles into the boundary layer region thereby increasing the concentration boundary layer as can be seen from Figure 6. The increase in concentration boundary with is observed in both aiding and opposing buoyancy cases.
Figure 7 shows the Nusselt number as a function of for different values of Df in aiding and opposing buoyancy conditions. In the opposing buoyancy situation, decreases with for the case of pure thermophoresis () and increases in the cross-diffusion case (). In the aiding buoyancy situation, increases monotonically with for both and . The Dufour number reduces the heat transfer coefficient in both aiding and opposing flow situations. Further, we observe enhanced heat transfer in the case of aiding buoyancy () as compared to the opposing buoyancy () case.
Figure 8 shows the mass transfer coefficient as a function of for different values of Df in aiding and opposing buoyancy conditions. In both aiding and opposing buoyancy situations, is a decreasing function of and an increasing function of Df. There is also an increased mass transfer in the case of aiding buoyancy () as compared to the opposing buoyancy () case.
The problem of double-diffusive convection from a vertical cone was solved using a successive linearization algorithm in combination with a Chebyshev spectral collocation method. A comparison with results in the literature and numerical approximations showed that the SLM is highly accurate with assured and accelerated convergence rate thus confirming the SLM as an alternative semianalytic technique for solving nonlinear boundary value problems with a strong coupling. We found that the Dufour parameter reduces the heat transfer coefficient while increasing the mass transfer rate. In general, the effect of the Soret parameter is to increase the heat transfer coefficient and to reduce the mass transfer coefficient. Aiding buoyancy enhances heat and mass transfer compared to the opposing buoyancy condition.
|:||Local skin friction coefficient|
|:||Boundary layer stream function|
|:||Acceleration due to gravity|
|, :||Cross-diffusion coefficients|
|:||Order of successive linearization method|
|:||Number of collocation points|
|Local Nusselt number|
|:||Local radius of the cone,|
|:||Dimensionless local radius of the cone,|
|Local Sherwood number|
|, :||Velocity component in the directions|
|, :||Dimensionless velocity component in the directions|
|, :||Coordinate measured along the surface and normal to it|
|, :||Dimensionless coordinates.|
|:||Thermal diffusivity of the fluid|
|:||Coefficient of thermal expansion of the fluid|
|:||Coefficient of solutal expansion|
|:||Vertex half angle of the cone|
|:||Coefficient of viscosity|
|:||Coefficient of kinematic viscosity,|
|:||Boundary layer temperature|
|:||Density of the fluid|
|:||Dimensionless stream function|
|:||Boundary layer concentration|
|:||Quantities at the surface of the cone|
|:||Quantities far away from the surface of the cone.|
The authors wish to thank University of KwaZulu-Natal and the National Research Foundation (NRF) for financial support.
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Copyright © 2012 Mahesha Narayana and Precious Sibanda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.