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

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 inherent instabilities due to double-diffusive convection have been investigated by, among others, Nield [

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 [

In general, the cross-diffusion effects are small compared to the effects described by Fourier and Fick’s laws (Mojtabi and Charrier-Mojtabi [

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 [

Ganji et al. [

In this study, we use a nonperturbation, semianalytic successive linearization method (see Makukula et al. [

Consider a vertical down-pointing cone with half-angle

Schematic of the problem.

Following the usual boundary layer and Boussinesq approximations, the basic equations governing the steady state dynamics of a viscous incompressible liquid are given by

The boundary conditions for (

We introduce the dimensionless variables
_{f}, and Soret number

The successive linearization method (see Makukula et al. [

Starting from the initial approximations

In the above definitions,

The parameters of engineering interest in heat and mass transport problems are the skin friction coefficient

The shearing stress at the surface of the cone

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 _{f,} and the Soret number

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

Comparison of SLM results for single component convection (

Ece [ | Present results | |||

1 | 0.681482 | 0.638859 | 0.68148333 | 0.63885472 |

10 | 0.433269 | 1.275548 | 0.43327825 | 1.27552888 |

Comparison of _{f} and

Quantiy |
D_{f} | SLM | bvp4c | Shooting | |||

0.00 | 1.00 | 1.244372648 | 1.244372629 | 1.244372629 | 1.244372633 | 1.244373 | |

0.25 | 0.75 | 1.233210690 | 1.233210687 | 1.233210687 | 1.233210690 | 1.233211 | |

0.50 | 0.50 | 1.229002907 | 1.229002906 | 1.229002906 | 1.229002911 | 1.229003 | |

0.75 | 0.25 | 1.233210690 | 1.233210687 | 1.233210687 | 1.233210690 | 1.233211 | |

1.00 | 0.00 | 1.244372648 | 1.244372629 | 1.244372629 | 1.244372633 | 1.244373 | |

0.00 | 1.00 | 0.803753575 | 0.803753516 | 0.803753516 | 0.803753488 | 0.803754 | |

0.25 | 0.75 | 0.750979670 | 0.750979649 | 0.750979649 | 0.750979625 | 0.750980 | |

0.50 | 0.50 | 0.663129905 | 0.663129902 | 0.663129902 | 0.663129885 | 0.663130 | |

0.75 | 0.25 | 0.553122477 | 0.553122494 | 0.553122494 | 0.553122487 | 0.553122 | |

1.00 | 0.00 | 0.444121263 | 0.444121327 | 0.444121327 | 0.444121329 | 0.444121 | |

0.00 | 1.00 | 0.444121263 | 0.444121327 | 0.444121327 | 0.444121329 | 0.444121 | |

0.25 | 0.75 | 0.553122477 | 0.553122494 | 0.553122494 | 0.553122487 | 0.553122 | |

0.50 | 0.50 | 0.663129905 | 0.663129902 | 0.663129902 | 0.663129885 | 0.663130 | |

0.75 | 0.25 | 0.750979670 | 0.750979649 | 0.750979649 | 0.750979625 | 0.750980 | |

1.00 | 0.00 | 0.803753575 | 0.803753516 | 0.803753516 | 0.803753488 | 0.803754 |

Table _{f} and _{f} and

It is to be noted from Table _{f} and decreasing _{f} and decrease in

To gain some insight into the dynamics of the problem, the temperature and concentration distributions are shown graphically in Figures _{f} in the aiding and opposing buoyancy cases.

Cross-diffusion effect on (a) temperature and (b) concentration profiles with

Effect of Dufour parameter D_{f} on

Effect of Dufour parameter D_{f} on

Effect of

Effect of

Variation of _{f}.

Variation of _{f}.

The variation of temperature and concentration profiles subject to a simultaneous increase in the cross-diffusion parameters D_{f} and

Figure

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 _{f} 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

Figure

Figure _{f} in aiding and opposing buoyancy conditions. In the opposing buoyancy situation,

Figure _{f} in aiding and opposing buoyancy conditions. In both aiding and opposing buoyancy situations, _{f}. There is also an increased mass transfer in the case of aiding buoyancy (

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.

Concentration

Dimensionless concentration

Local skin friction coefficient

Concentration difference,

Boundary layer stream function

Solutal diffusivity

_{f}:

Dufour number

Acceleration due to gravity

Grashof number

Mass flux

Thermal conductivity

Cross-diffusion coefficients

Characteristic length

Order of successive linearization method

Number of collocation points

Local Nusselt number

Prandtl number

Heat flux

Local radius of the cone,

Dimensionless local radius of the cone,

Schmidt number

Local Sherwood number

Soret number

Temperature

Dimensionless temperature

Temperature difference,

Reference velocity

Velocity component in the

Dimensionless velocity component in the

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

Buoyancy ratio

Coefficient of viscosity

Coefficient of kinematic viscosity,

Boundary layer temperature

Density of the fluid

Dimensionless stream function

Boundary layer concentration

Collocation point

Shearing stress.

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.