Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 587357 | 19 pages | https://doi.org/10.1155/2012/587357

On the Solution of Double-Diffusive Convective Flow due to a Cone by a Linearization Method

Academic Editor: Renat Zhdanov
Received08 Sep 2011
Accepted14 Oct 2011
Published07 Feb 2012

Abstract

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.

1. Introduction

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 [1]). 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 [4], Baines and Gill [5], Guo et al. [6], Khanafer and Vafai [7], Sunil et al. [8], and Gaikwad et al. [9]. 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 [12], and Ingham and Pop [13]. Chamkha [14] 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 [15] 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 [16]. 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 [17].

In general, the cross-diffusion effects are small compared to the effects described by Fourier and Fick’s laws (Mojtabi and Charrier-Mojtabi [1]) 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 [18], Awad et al. [19], 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. [20] 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. [21] 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 [22] 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 [2325], the Adomian decomposition method [26, 27], homotopy analysis method [28], 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 [31] 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 [32] obtained periodic solutions of nonlinear reaction-diffusion equations arising in mathematical biology using the exp-function method. Yildirim and Sezer [33] 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. [34] 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. [35] solved the nonlinear Jeffery-Hamel flow problem using two semianalytical methods, the variational iteration method (VIM) and the homotopy perturbation method. Ghafoori et al. [36] 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. [37] 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. [38] 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 [41] 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 Ω immersed in a viscous incompressible liquid. The 𝑥-axis is along the surface of the cone, and the 𝑦-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 𝑇𝑤 (>𝑇) where 𝑇 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 𝜕𝜕𝜕𝑥(𝑟𝑢)+𝑢𝜕𝑦(𝑟𝑣)=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝜕𝑦=𝜈2𝑢𝜕𝑦2+g𝛽𝑇𝑇cosΩ+g𝛽𝐶C𝑢cosΩ,𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝜕𝑦=𝛼2𝑇𝜕𝑦2+𝑘1𝜕2𝐶𝜕𝑦2,𝑢𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝜕𝑦=𝐷2𝐶𝜕𝑦2+𝑘2𝜕2𝑇𝜕𝑦2,(2.1) where 𝑢 and 𝑣 are the velocity components in the 𝑥 and 𝑦 directions, respectively, 𝑟=𝑥sinΩ is the local radius of the cone, 𝜈 is the kinematic viscosity, 𝜌 is the density, g is the acceleration due to gravity, 𝛽 is the coefficient of thermal expansion, 𝛽 is the coefficient of solutal expansion, 𝑇 is the temperature, 𝐶 is the concentration, 𝛼 is the thermal diffusivity, 𝐷 is the species diffusivity, and 𝑘1, 𝑘2 are cross-diffusion coefficients.

The boundary conditions for (2.1) have the form 𝑢=𝑣=0,𝑇=𝑇𝑤=𝑇+𝑇𝑟𝑥𝐿,𝐶=𝐶𝑤=𝐶+𝐶𝑟𝑥𝐿at𝑦=0,𝑢0,𝑇𝑇,𝐶𝐶as𝑦.(2.2) Here, the subscripts 𝑤 and refer to the surface and ambient conditions, respectively, 𝑇𝑟 and 𝐶𝑟 are positive constants, and 𝐿 is a characteristic length.

We introduce the dimensionless variables (𝑋,𝑌,𝑅)=𝑥,𝑦Gr1/4,𝑟𝐿,(𝑈,𝑉)=𝑢,𝑣Gr1/4𝑈0,𝑇=𝑇𝑇𝑇𝑤𝑇,𝐶=𝐶𝐶𝐶𝑤𝐶,(2.3) where the reference velocity 𝑈0 and Grashof number Gr are defined, respectively, as𝑈0=𝑇g𝛽𝐿𝑤𝑇cosΩ1/2𝑈,Gr=0𝐿𝜈2.(2.4) On using the variables (2.3), the boundary-layer equations (2.1) reduce to 𝜕𝜕𝜕𝑋(𝑅𝑈)+𝑈𝜕𝑌(𝑅𝑉)=0,(2.5)𝜕𝑈𝜕𝑋+𝑉𝜕𝑈=𝜕𝜕𝑌2𝑈𝜕𝑌2+𝑇+𝜆𝑈𝜕𝐶,(2.6)𝑇𝜕𝜕𝑋+𝑉𝑇=1𝜕𝑌𝜕Pr2𝑇𝜕𝑌2+Df𝜕2𝐶𝜕𝑌2𝑈𝜕,(2.7)𝐶𝜕𝜕𝑋+𝑉𝐶=1𝜕𝑌𝜕Sc2𝐶𝜕𝑌2𝜕+Sr2𝑇𝜕𝑌2.(2.8) The nondimensional parameters appearing in (2.5)–(2.8) are the buoyancy ratio 𝜆, the Prandtl number Pr, the Schmidt number Sc, Dufour number Df, and Soret number Sr defined, respectively, as 𝛽𝜆=𝛽𝐶𝑤𝐶𝑇𝑤𝑇𝜈,Pr=𝛼𝜈,Sc=𝐷,Df=𝑘1𝛼𝐶𝑤𝐶𝑇𝑤𝑇𝑘,Sr=2𝐷𝑇𝑤𝑇𝐶𝑤𝐶.(2.9) Assuming 𝑇𝑤𝑇=𝑇𝑟 and 𝐶𝑤𝐶=𝐶𝑟, the boundary conditions (2.2) can be written as 𝑈=𝑉=0,𝑇=𝑋,𝐶=𝑋at𝑌=0,𝑈0,𝑇0,𝐶0as𝑌.(2.10) We now introduce the stream function 𝜓(𝑋,𝑌) such that 1𝑈=𝑅𝜕𝜓1𝜕𝑌,𝑉=𝑅𝜕𝜓𝜕𝑋,(2.11) 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 𝑅𝜕3𝜓𝜕𝑌3+𝜕(𝜓,𝜕𝜓/𝜕𝑌)+1𝜕(𝑋,𝑌)𝑋𝜕𝜓𝜕𝑌2+𝑅2𝑇+𝜆𝐶𝑅𝜕=0,2𝑇𝜕𝑌2𝜕+Pr𝜓,𝑇𝜕(𝑋,𝑌)+𝑅Df𝜕2𝐶𝜕𝑌2𝑅𝜕=0,2𝐶𝜕𝑌2𝜕+Sc𝜓,𝐶𝜕𝜕(𝑋,𝑌)+𝑅Sr2𝑇𝜕𝑌2=0.(2.12) The boundary conditions (2.10) in terms of the stream function are 𝜕𝜓=𝜕𝑋𝜕𝜓𝜕𝑌=0,𝑇=𝑋,𝐶=𝑋at𝑌=0,𝜕𝜓𝜕𝑌0,𝑇0,𝐶0as𝑌.(2.13) We further introduce the following similarity variables 𝜓(𝑋,𝑌)=𝑋𝑅𝑓(𝑌),𝑇(𝑋,𝑌)=𝑋𝜃(𝑌),𝐶(𝑋,𝑌)=𝑋𝜙(𝑌).(2.14) Using (2.14), (2.12) along with boundary conditions (2.13) reduces to the following two-point boundary value problem 𝑓+2𝑓𝑓𝑓2𝜃+𝜃+𝜆𝜙=0,(2.15)+Pr2𝑓𝜃𝑓𝜃+Df𝜙𝜙=0,(2.16)+Sc2𝑓𝜙𝑓𝜙+Sr𝜃𝑓=0,(2.17)𝑓(0)=𝑓(0)=0,𝜃(0)=𝜙(0)=1,()0,𝜃()0,𝜙()0.(2.18) The primes in (2.15)–(2.18) denote differentiation with respect to 𝑌.

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 𝑓(𝑌), 𝜃(𝑌), and 𝜙(𝑌) may be expanded in series form as 𝑓(𝑌)=𝑓𝑖(𝑌)+𝑖1𝑚=0𝐹𝑚(𝑌),𝜃(𝑌)=𝜃𝑖(𝑌)+𝑖1𝑚=0Θ𝑚(𝑌),𝑖=1,2,3,,𝜙(𝑌)=𝜙𝑖(𝑌)+𝑖1𝑚=0Φ𝑚(𝑌),(3.1) where 𝑓𝑖, 𝜃𝑖, and 𝜙𝑖 are unknown functions and 𝐹𝑚, Θ𝑚, and Φ𝑚, (𝑚1) 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 𝑓𝑖+𝑎1,𝑖1𝑓𝑖+𝑎2,𝑖1𝑓𝑖+𝑎3,𝑖1𝑓𝑖+𝜃𝑖+𝜆𝜙𝑖+2𝑓𝑖𝑓𝑖𝑓2𝑖=𝑟1,𝑖1,𝜃𝑖+𝑏1,𝑖1𝜃𝑖+𝑏2,𝑖1𝜃𝑖+𝑏3,𝑖1𝑓𝑖+𝑏4,𝑖1𝑓𝑖+Pr2𝑓𝑖𝜃𝑖𝑓𝑖𝜃+Df𝜙𝑖=𝑟2,𝑖1,𝜙𝑖+𝑐1,𝑖1𝜙𝑖+𝑐2,𝑖1𝜃𝑖+𝑐3,𝑖1𝑓𝑖+𝑐4,𝑖1𝑓𝑖+Sc2𝑓𝑖𝜙𝑖𝑓𝑖𝜙+Sr𝜃𝑖=𝑟3,𝑖1,(3.2) where the coefficient parameters 𝑎𝑘,𝑖1 (𝑘=1,2,3), 𝑏𝑘,𝑖1, 𝑐𝑘,𝑖1 (𝑘=1,,4), and 𝑟𝑘,𝑖1 (𝑘=1,2,3) are defined as 𝑎1,𝑖1=2𝑖1𝑚=0𝐹𝑚,𝑎2,𝑖1=2𝑖1𝑚=0𝐹𝑚,𝑎3,𝑖1=2𝑖1𝑚=0𝐹𝑚,𝑏1,𝑖1=2Pr𝑖1𝑚=0𝐹𝑚,𝑏2,𝑖1=Pr𝑖1𝑚=0𝐹𝑚,𝑏3,𝑖1=Pr𝑖1𝑚=0Θ𝑚,𝑏4,𝑖1=2Pr𝑖1𝑚=0Θ𝑚,𝑐1,𝑖1=2Sc𝑖1𝑚=0𝐹𝑚,𝑐2,𝑖1=Sc𝑖1𝑚=0𝐹𝑚,𝑐3,𝑖1=Sc𝑖1𝑚=0Φ𝑚,𝑐4,𝑖1=2Sc𝑖1𝑚=0Φ𝑚,𝑟1,𝑖1=𝑖1𝑚=0𝐹𝑚+2𝑖1𝑚=0𝐹𝑚𝑖1𝑚=0𝐹𝑚𝑖1𝑚=0𝐹𝑚2+𝑖1𝑚=0Θ𝑚+𝜆𝑖1𝑚=0Φ𝑚,𝑟2,𝑖1=𝑖1𝑚=0Θ𝑚2+Pr𝑖1𝑚=0𝐹𝑚𝑖1𝑚=0Θ𝑚𝑖1𝑚=0𝐹𝑚𝑖1𝑚=0Θ𝑚+Df𝑖1𝑚=0Φ𝑚,𝑟3,𝑖1=𝑖1𝑚=0Φ𝑚2+Sc𝑖1𝑚=0𝐹𝑚𝑖1𝑚=0Φ𝑚𝑖1𝑚=0𝐹𝑚𝑖1𝑚=0Φ𝑚+Sr𝑖1𝑚=0Θ𝑚.(3.3)

Starting from the initial approximations 𝐹0𝑒(𝑌)=1𝑌𝑌𝑒𝑌,Θ0(𝑌)=𝑒𝑌,Φ0(𝑌)=𝑒𝑌,(3.4) which are chosen to satisfy the boundary conditions (2.18), the subsequent solutions 𝐹𝑚, Θ𝑚, Φ𝑚, 𝑚1 are obtained by successively solving the linearized form of (3.2) given below 𝐹𝑖+𝑎1,𝑖1𝐹𝑖+𝑎2,𝑖1𝐹𝑖+𝑎3,𝑖1𝐹𝑖+Θ𝑖+𝜆Φ𝑖=𝑟1,𝑖1,Θ𝑖+𝑏1,𝑖1Θ𝑖+𝑏2,𝑖1Θ𝑖+𝑏3,𝑖1𝐹𝑖+𝑏4,𝑖1𝐹𝑖+DfΦ𝑖=𝑟2,𝑖1,Φ𝑖+𝑐1,𝑖1Φ𝑖+𝑐2,𝑖1Φ𝑖+𝑐3,𝑖1𝐹𝑖+𝑐4,𝑖1𝐹𝑖+SrΘ𝑖=𝑟3,𝑖1,(3.5) subject to the boundary conditions 𝐹𝑖(0)=𝐹𝑖(0)=𝐹𝑖()=0,Θ𝑖(0)=Θ𝑖()=0,Φ𝑖(0)=Φ𝑖()=0.(3.6) Once each solution 𝐹𝑖, Θ𝑖, and Φ𝑖 (𝑖1) has been found from iteratively solving (3.5) for each 𝑖, the functions 𝑓(𝑌), 𝜃(𝑌), and 𝜙(𝑌) are obtained as series 𝑓(𝑌)𝑀𝑖=0𝐹𝑖(𝑌),𝜃(𝑌)𝑀𝑖=0Θ𝑖(𝑌),𝜙(𝑌)𝑀𝑖=0Φ𝑖(𝑌),(3.7) where 𝑀 is the order of SLM approximation. Equations (3.5) are integrated using the Chebyshev spectral collocation method [4244]. The unknown functions are defined by the Chebyshev interpolating polynomials with the Gauss-Lobatto points defined as 𝑌𝑗=cos𝜋𝑗𝑁,𝑗=0,1,,𝑁,(3.8) where 𝑁 is the number of collocation points used. The physical region [0,) is transformed into the domain [1,1] using the domain truncation technique in which the problem is solved on the interval [0,𝑌] instead of [0,). This leads to the mapping 𝑌𝑌=𝜉+12,1𝜉1,(3.9) where 𝑌 is the known number used to invoke the boundary condition at infinity. The unknown functions 𝐹𝑖, Θ𝑖, and Φ𝑖 are approximated at the collocation points by𝐹𝑖(𝜉)𝑁𝑘=0𝐹𝑖𝜉𝑘𝑇𝑘𝜉𝑗,Θ𝑖(𝜉)𝑁𝑘=0Θ𝑖𝜉𝑘𝑇𝑘𝜉𝑗,Φ𝑖(𝜉)𝑁𝑘=0Φ𝑖𝜉𝑘𝑇𝑘𝜉𝑗,𝑗=0,1,,𝑁,(3.10) where 𝑇𝑘 is the 𝑘th Chebyshev polynomial defined as 𝑇𝑘(𝜉)=cos𝑘cos1(𝜉).(3.11) The derivatives of the variables at the collocation points are represented as𝑑𝑛𝐹𝑖𝑑𝑌𝑛=𝑁𝑘=0D𝑛𝑘𝑗𝐹𝑖𝜉𝑘,𝑑𝑛Θ𝑖𝑑𝑌𝑛=𝑁𝑘=0D𝑛𝑘𝑗Θ𝑖𝜉𝑘,𝑑𝑛Φ𝑖𝑑𝑌𝑛=𝑁𝑘=0𝐃𝑛𝑘𝑗Φ𝑖𝜉𝑘,𝑗=0,1,,𝑁,(3.12) where 𝑛 is the order of differentiation and 𝐃=(2/𝑌)𝒟 where 𝒟 is the Chebyshev spectral differentiation matrix (see, [4244]). Substituting (3.8)–(3.12) in (3.5)–(3.6) leads to the matrix equation𝐀𝑖1𝐗𝑖=𝐁𝑖1,(3.13) in which 𝐀𝑖1 is a square matrix of order (3𝑁+3) and 𝐗𝑖, 𝐁𝑖1 are (3𝑁+3)×1 column vectors defined by𝐀𝑖1=𝐴11𝐴12𝐴13𝐴21𝐴22𝐴23𝐴31𝐴32𝐴33,𝐗𝑖=𝐅𝑖𝚯𝑖𝚽𝑖,𝐁𝑖1=𝐫1,𝑖1𝐫2,𝑖1𝐫3,𝑖1,(3.14) with𝐅𝑖=𝐹𝑖𝜉0,𝐹𝑖𝜉1,,𝐹𝑖𝜉𝑁1,𝐹𝑖𝜉𝑁𝑇,𝚯𝑖=Θ𝑖𝜉0,Θ𝑖𝜉1,,Θ𝑖𝜉𝑁1,Θ𝑖𝜉𝑁𝑇,𝚽𝑖=Φ𝑖𝜉0,Φ𝑖𝜉1,,Φ𝑖𝜉𝑁1,Φ𝑖𝜉𝑁𝑇,𝐫1,𝑖1=𝑟1,𝑖1𝜉0,𝑟1,𝑖1𝜉1,,𝑟1,𝑖1𝜉𝑁1,𝑟1,𝑖1𝜉𝑁𝑇,𝐫2,𝑖1=𝑟2,𝑖1𝜉0,𝑟2,𝑖1𝜉1,,𝑟2,𝑖1𝜉𝑁1,𝑟2,𝑖1𝜉𝑁𝑇,𝐫3,𝑖1=𝑟3,𝑖1𝜉0,𝑟3,𝑖1𝜉1,,𝑟3,𝑖1𝜉𝑁1,𝑟3,𝑖1𝜉𝑁𝑇,𝐴11=𝐃3+𝐚1,𝑖1𝐃2+𝐚2,𝑖1𝐃+𝐚3,𝑖1,𝐴12=𝐈,𝐴13𝐴=𝜆𝐈,21=𝐛3,𝑖1𝐃+𝐛4,𝑖1,𝐴22=𝐃2+𝐛1,𝑖1𝐃+𝐛2,𝑖1,𝐴23=Df𝐃2,𝐴31=𝐜3,𝑖1𝐃+𝐜4,𝑖1,𝐴32=Sr𝐃2,𝐴33=𝐃2+𝐜1,𝑖1𝐃+𝐜2,𝑖1.(3.15)

In the above definitions, 𝐚𝑘,𝑖1 (𝑘=1,2,3), 𝐛𝑘,𝑖1, 𝐜𝑘,𝑖1 (𝑘=1,,4), Df, and Sr are diagonal matrices of order (𝑁+1) and 𝐈 is the identity matrix of order (𝑁+1). Finally, the solution of the problem is obtained as𝐗𝑖=𝐀1𝑖1𝐁𝑖1.(3.16) Thus, starting with the initial solutions 𝐹0, Θ0, and Φ0, a sequence of approximations 𝑘0𝐹𝑘, 𝑘0Θ𝑘, and 𝑘0Φ𝑘, 𝑘=1,2,,𝑀 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 𝐶𝑓, 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 𝜏𝑤 is defined as 𝜏𝑤=𝜇𝑋𝜕𝑢𝜕𝑦𝑦=0𝜇𝑈0𝐿Gr1/4𝑓(0),(4.1) where 𝜇 is the coefficient of viscosity. The skin friction coefficient at the surface of the cone is defined as Cf=𝜏𝑤(1/2)𝜌𝑈20.(4.2) Using (4.1) in (4.2), we obtain the following relation 𝐶𝑓Gr1/4=2𝑓(0).(4.3) The heat transfer rate at the surface of the cone is defined as 𝑞𝑤=𝑘𝑋𝜕𝑇𝜕𝑦𝑦=0=𝑇𝑘𝑤𝑇𝐿Gr1/4𝜃(0),(4.4) where 𝑘 is the thermal conductivity of the fluid. The Nusselt number is defined as 𝐿Nu=𝑘𝑞𝑤𝑇𝑤𝑇.(4.5) Using (4.4) in (4.5), the dimensionless wall heat transfer rate is obtained as follows: NuGr1/4=𝜃(0).(4.6) The mass flux at the surface of the cone is defined as𝐽𝑤=𝐷𝑋𝜕𝐶𝜕𝑦𝑦=0=𝐶𝐷𝑤𝑇𝐿Gr1/4𝜙(0),(4.7) and the Sherwood is defined as𝐿Sh=𝐷𝐽𝑤𝑇𝑤𝑇.(4.8) Using (4.7) in (4.8), the dimensionless wall mass transfer rate is obtain asShGr1/4=𝜙(0).(4.9)

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 Pr, Schmidt number Sc, buoyancy ratio 𝜆, Dufour number Df, and the Soret number Sr. 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 Pr=Sc=1 and instead focus attention on results pertaining to the other three important parameters. In addition, we restrict ourselves to parameter values in the interval 0Df, Sr1. To highlight the effect of buoyancy, for aiding buoyancy condition, we take 𝜆>0 while, for opposing buoyancy, 𝜆<0.

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 𝑓(0), 𝜃(0), and 𝜙(0) for different parameter values. Table 1 gives the comparison of the SLM results in the absence of cross-diffusion (i.e., Df=Sr=𝜆=0) with those presented by Ece [41]. The SLM solutions are found to be in excellent agreement with those of Ece [41] indicating the accuracy of the linearisation method.


P r Ece [41] Present results
𝑓 ( 0 ) 𝜃 ( 0 ) 𝑓 ( 0 ) 𝜃 ( 0 )

1 0.681482 0.638859 0.68148333 0.63885472
10 0.433269 1.275548 0.43327825 1.27552888


Quantiy Df S r SLMbvp4cShooting
𝑀 = 3 𝑀 = 4 𝑀 = 5

𝑓 ( 0 ) 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

𝜃 ( 0 ) 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

𝜙 ( 0 ) 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 2 highlights both the accuracy and the accelerated convergence of the SLM for different values of Df and Sr. The linearisation method converges to the numerical solutions at the fourth-order SLM for all values of Df and Sr. 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 Sr lead to initial decreases in the skin-friction coefficient 𝑓(0) up to Df=Sr=0.5 and then start increasing. The heat transfer coefficient 𝜃(0) shows monotonic decrease, while the mass transfer coefficient exhibits the opposite change when subjected to simultaneous increase in Df and decrease in Sr.

To gain some insight into the dynamics of the problem, the temperature and concentration distributions are shown graphically in Figures 26. The Nusselt number NuGr1/4 and Sherwood number ShGr1/4 which highlight the heat and mass transfer are shown in Figures 7 and 8, as functions of Sr 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 Sr 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 Df=Sr=0 (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 𝜃(𝑌) 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 Sr 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 Sr is observed in both aiding and opposing buoyancy cases.

Figure 7 shows the Nusselt number NuGr1/4 as a function of Sr for different values of Df in aiding and opposing buoyancy conditions. In the opposing buoyancy situation, NuGr1/4 decreases with Sr for the case of pure thermophoresis (Df=0) and increases in the cross-diffusion case (Df0). In the aiding buoyancy situation, NuGr1/4 increases monotonically with Sr for both Df=0 and Df0. The Dufour number reduces the heat transfer coefficient NuGr1/4 in both aiding and opposing flow situations. Further, we observe enhanced heat transfer in the case of aiding buoyancy (𝜆>0) as compared to the opposing buoyancy (𝜆<0) case.

Figure 8 shows the mass transfer coefficient ShGr1/4 as a function of Sr for different values of Df in aiding and opposing buoyancy conditions. In both aiding and opposing buoyancy situations, ShGr1/4 is a decreasing function of Sr and an increasing function of Df. There is also an increased mass transfer in the case of aiding buoyancy (𝜆>0) as compared to the opposing buoyancy (𝜆<0) case.

6. Conclusions

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.

Nomenclature

𝐶:Concentration
𝐶:Dimensionless concentration
𝐶𝑓:Local skin friction coefficient
𝐶𝑟:Concentration difference, 𝐶𝑤𝐶
𝑓:Boundary layer stream function
𝐷:Solutal diffusivity
Df:Dufour number
g:Acceleration due to gravity
Gr:Grashof number
𝐽: Mass flux
𝑘:Thermal conductivity
𝑘1, 𝑘2:Cross-diffusion coefficients
𝐿:Characteristic length
𝑀:Order of successive linearization method
𝑁:Number of collocation points
Nu:Local Nusselt number
Pr:Prandtl number
𝑞:Heat flux
𝑟: Local radius of the cone, 𝑥sinΩ
𝑅:Dimensionless local radius of the cone, 𝑋sinΩ
Sc:Schmidt number
Sh:Local Sherwood number
Sr:Soret number
𝑇:Temperature
𝑇:Dimensionless temperature
𝑇𝑟:Temperature difference, 𝑇𝑤𝑇
𝑈0:Reference velocity
𝑢, 𝑣:Velocity component in the 𝑥,𝑦 directions
𝑈, 𝑉:Dimensionless velocity component in the 𝑋,𝑌 directions
𝑥, 𝑦:Coordinate measured along the surface and normal to it
𝑋, 𝑌:Dimensionless coordinates.
Greek Symbols
𝛼:Thermal diffusivity of the fluid
𝛽: Coefficient of thermal expansion of the fluid
𝛽: Coefficient of solutal expansion
Ω:Vertex half angle of the cone
𝜆: Buoyancy ratio
𝜇:Coefficient of viscosity
𝜈:Coefficient of kinematic viscosity, 𝜈=𝜇/𝜌
𝜃:Boundary layer temperature
𝜌:Density of the fluid
𝜓:Dimensionless stream function
𝜙:Boundary layer concentration
𝜉:Collocation point
𝜏:Shearing stress.
Subscripts
𝑤:Quantities at the surface of the cone
:Quantities far away from the surface of the cone.

Acknowledgments

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.


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