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

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 [23–25], 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 𝑇_𝑤 (>𝑇_∞) where 𝑇_∞ is the ambient temperature.

Figure 1 

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 𝜕𝑢𝜕𝑢𝜕𝑥(𝑟𝑢)+𝜕𝜕𝑦(𝑟𝑣)=0,𝜕𝑥+𝑣𝜕𝜕𝑦𝑢=𝜈𝜕²𝑢𝜕𝑦²𝑢𝜕𝑇+g𝛽𝑇−𝑇_∞cosΩ+g𝛽^∗𝐶−C_∞cosΩ,𝜕𝑥+𝑣𝜕𝜕𝑇𝑦=𝛼𝜕²𝑇𝜕𝑦²+𝑘₁𝜕²𝐶𝜕𝑦²𝑢𝜕𝐶,𝜕𝑥+𝑣𝜕𝜕𝐶𝑦=𝐷𝜕²𝐶𝜕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=𝑇_𝑤=𝑇_∞+𝑇_𝑟𝑥𝐿,𝐶=𝐶^𝑤=𝐶_∞+𝐶_𝑟𝑥𝑢→0,𝑇⟶𝑇_∞𝐿at𝑦=0,,𝐶⟶𝐶_∞as𝑦⟶∞.(2.2) Here, the subscripts 𝑤 and ∞ refer to the surface and ambient conditions, respectively, 𝑇_𝑟 and 𝐶_𝑟 are positive constants, and 𝐿 is a characteristic length.

We introduce the dimensionless variables (𝑋,𝑌,𝑅)=𝑥,𝑦Gr^1/4,𝑟𝐿,(𝑈,𝑉)=𝑢,𝑣Gr^1/4𝑈₀,𝑇=𝑇−𝑇^∞𝑇_𝑤−𝑇_∞,𝐶=𝐶−𝐶^∞𝐶_𝑤−𝐶_∞,(2.3) where the reference velocity 𝑈₀ and Grashof number Gr are defined, respectively, as𝑈₀=g𝛽𝐿𝑇_𝑤−𝑇_∞cosΩ^1/2,Gr=𝑈⁰𝐿𝜈².(2.4) On using the variables (2.3), the boundary-layer equations (2.1) reduce to 𝜕𝑈𝜕𝑈𝜕𝑋(𝑅𝑈)+𝜕𝜕𝑌(𝑅𝑉)=0,(2.5)𝜕𝑋+𝑉𝜕𝜕𝑈𝑌=𝜕²𝑈𝜕𝑌²𝑈𝜕+𝑇+𝜆𝐶,(2.6)𝑇𝜕𝑋+𝑉𝜕𝑇𝜕𝑌=P1r𝜕²𝑇𝜕𝑌²+D_f𝜕²𝐶𝑈𝜕𝜕𝑌²,(2.7)𝐶𝜕𝑋+𝑉𝜕𝐶𝜕𝑌=S1c𝜕²𝐶𝜕𝑌²+Sr𝜕²𝑇𝜕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 D_f, and Soret number Sr defined, respectively, as 𝜆=𝛽^∗𝛽𝐶^𝑤−𝐶_∞D_f𝑇_𝑤−𝑇_∞,Pr=𝛼𝜈,Sc=𝐷𝜈,=𝑘¹𝛼𝐶^𝑤−𝐶_∞𝑇_𝑤−𝑇_∞,Sr=𝑘𝐷²𝑇^𝑤−𝑇_∞𝐶_𝑤−𝐶_∞.(2.9) Assuming 𝑇_𝑤−𝑇_∞=𝑇_𝑟 and 𝐶_𝑤−𝐶_∞=𝐶_𝑟, the boundary conditions (2.2) can be written as 𝑈𝑈=→𝑉0=,0,𝑇=𝑋,𝐶=𝑋at𝑌=0,𝑇⟶0,𝐶⟶0as𝑌⟶∞.(2.10) We now introduce the stream function 𝜓(𝑋,𝑌) such that 𝑈=1𝑅𝜕𝜕𝜓𝑌,𝑉=−𝑅1𝜕𝜕𝑋𝜓,(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)𝜕(𝑋,𝑌)+𝑋1𝜕𝜕𝜓𝑌²𝑅𝜕+𝑅²𝑇+𝜆𝐶=0,₂𝑇𝜕𝑌²+Pr𝜕𝜓,𝑇𝜕(𝑋,𝑌)+𝑅D^f𝜕²𝐶𝜕𝑌²𝑅𝜕=0,₂𝐶𝜕𝑌²+Sc𝜕𝜓,𝐶𝜕(𝑋,𝑌)+𝑅Sr𝜕²𝑇𝜕𝑌²=0.(2.12) The boundary conditions (2.10) in terms of the stream function are 𝜕𝜓𝜕𝑋=𝜕𝜕𝜓𝑌=0,𝜕𝜓𝑇=𝑋,𝐶=𝑋at𝑌=0,𝜕𝑌⟶0,𝑇⟶0,𝐶⟶0as𝑌⟶∞.(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 𝜃𝑓^′+2𝑓𝑓^−𝑓′²+𝜃+𝜆𝜙=0,(2.15)_𝜙+Pr2𝑓𝜃^−𝑓′𝜃+D_f𝜙^′=0,(2.16)_𝑓𝑓(0)+=S𝑓c′(02)𝑓=𝜙0^′,−𝑓′𝜙𝜃(0+)=Sr𝜃𝜙^′(0=)0=,1,_(2.17)(∞)⟶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)=𝑓_𝑖(𝑌)+^𝑖−1_𝑚₌₀𝜃(𝑌)=𝜃_𝑖𝐹_𝑚(𝑌),(𝑌)+^𝑖−1_𝑚₌₀𝜙(𝑌)=𝜙_𝑖Θ_𝑚(𝑌),𝑖=1,2,3,…,(𝑌)+^𝑖−1_𝑚₌₀Φ_𝑚(𝑌),(3.1) where 𝑓_𝑖, 𝜃_𝑖, and 𝜙_𝑖 are unknown functions and 𝐹_𝑚, Θ_𝑚, and Φ_𝑚, (𝑚≥1) 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 𝑓^{𝑖}+𝑎_1,𝑖−1𝑓^𝑖+𝑎_2,𝑖−1𝑓^𝑖+𝑎_3,𝑖−1𝑓_𝑖+𝜃_𝑖+𝜆𝜙_𝑖+2𝑓_𝑖𝑓^𝑖−𝑓^2𝑖=𝑟_1,𝑖−1,𝜙𝜃^𝑖_𝑖+𝑏_1,𝑖−1𝜃^𝑖+𝑏_2,𝑖−1𝜃_𝑖+𝑏_3,𝑖−1𝑓^𝑖+𝑏_4,𝑖−1𝑓_𝑖+Pr2𝑓_𝑖𝜃^′_𝑖−𝑓^𝑖𝜃+D_f𝜙^′𝑖=𝑟_2,𝑖−1,+𝑐_1,𝑖−1𝜙^𝑖+𝑐_2,𝑖−1𝜃_𝑖+𝑐_3,𝑖−1𝑓^𝑖+𝑐_4,𝑖−1𝑓_𝑖+Sc2𝑓_𝑖𝜙^′_𝑖−𝑓^𝑖𝜙+Sr𝜃^′𝑖=𝑟_3,𝑖−1,(3.2) where the coefficient parameters 𝑎_{𝑘,𝑖−1} (𝑘=1,2,3), 𝑏_{𝑘,𝑖−1}, 𝑐_{𝑘,𝑖−1} (𝑘=1,…,4), and 𝑟_{𝑘,𝑖−1} (𝑘=1,2,3) are defined as 𝑎_1,𝑖−1=2^𝑖−1_𝑚₌₀𝐹_𝑚,𝑎_2,𝑖−1=−2^𝑖−1_𝑚₌₀𝐹^𝑚,𝑎_3,𝑖−1=2^𝑖−1_𝑚₌₀𝑏_1,𝑖−1𝐹^𝑚,=2Pr^𝑖−1_𝑚₌₀𝐹_𝑚,𝑏_2,𝑖−1=−Pr^𝑖−1_𝑚₌₀𝐹^′_𝑚,𝑏_3,𝑖−1=−Pr^𝑖−1_𝑚₌₀Θ_𝑚,𝑏_4,𝑖−1=2Pr^𝑖−1_𝑚₌₀𝑐_1,𝑖−1Θ^′_𝑚,=2Sc^𝑖−1_𝑚₌₀𝐹_𝑚,𝑐_2,𝑖−1=−Sc^𝑖−1_𝑚₌₀𝐹^′_𝑚,𝑐_3,𝑖−1=−Sc^𝑖−1_𝑚₌₀Φ_𝑚,𝑐_4,𝑖−1=2Sc^𝑖−1_𝑚₌₀𝑟_1,𝑖−1Φ^′_𝑚,=−⎡⎢⎢⎣^𝑖−1_𝑚₌₀𝐹^{𝑚}+2^𝑖−1_𝑚₌₀𝐹_{𝑚𝑖−1𝑚}₌₀𝐹^𝑚−^𝑖−1_𝑚₌₀𝐹^𝑚²+^𝑖−1_𝑚₌₀Θ_𝑚+𝜆^𝑖−1_𝑚₌₀Φ_𝑚𝑟_2,𝑖−1⎤⎥⎥⎦,=−^𝑖−1_𝑚₌₀Θ^𝑚+Pr2^𝑖−1_𝑚₌₀𝐹_{𝑚𝑖−1𝑚}₌₀Θ^′_𝑚−^𝑖−1_𝑚₌₀𝐹^{𝑚𝑖−1}_𝑚₌₀Θ_𝑚+D^f𝑖−1_𝑚=0Φ^′𝑚𝑟_3,𝑖−1,=−^𝑖−1_𝑚₌₀Φ^𝑚+Sc2^𝑖−1_𝑚₌₀𝐹_{𝑚𝑖−1𝑚}₌₀Φ^′_𝑚−^𝑖−1_𝑚₌₀𝐹^{𝑚𝑖−1}_𝑚₌₀Φ_𝑚+Sr^𝑖−1_𝑚=0Θ^′𝑚.(3.3)

Starting from the initial approximations 𝐹₀(𝑌)=1−𝑒^−𝑌−𝑌𝑒^−𝑌,Θ₀(𝑌)=𝑒^−𝑌,Φ₀(𝑌)=𝑒^−𝑌,(3.4) which are chosen to satisfy the boundary conditions (2.18), the subsequent solutions 𝐹_𝑚, Θ_𝑚, Φ_𝑚, 𝑚≥1 are obtained by successively solving the linearized form of (3.2) given below Θ_𝑖𝐹^{𝑖}+𝑎_1,𝑖−1𝐹^𝑖+𝑎_2,𝑖−1𝐹^𝑖+𝑎_3,𝑖−1𝐹_𝑖+Θ_𝑖+𝜆Φ_𝑖=𝑟_1,𝑖−1,Φ_𝑖+𝑏_1,𝑖−1Θ^𝑖+𝑏_2,𝑖−1Θ_𝑖+𝑏_3,𝑖−1𝐹^𝑖+𝑏_4,𝑖−1𝐹_𝑖+D_fΦ^′𝑖=𝑟_2,𝑖−1,+𝑐_1,𝑖−1Φ^𝑖+𝑐_2,𝑖−1Φ_𝑖+𝑐_3,𝑖−1𝐹^𝑖+𝑐_4,𝑖−1𝐹_𝑖+SrΘ^′𝑖=𝑟_3,𝑖−1,(3.5) subject to the boundary conditions 𝐹_𝑖(0)=𝐹^𝑖(0)=𝐹^𝑖(∞)=0,Θ_𝑖(0)=Θ_𝑖(∞)=0,Φ_𝑖(0)=Φ_𝑖(∞)=0.(3.6) Once each solution 𝐹_𝑖, Θ_𝑖, and Φ_𝑖 (𝑖≥1) 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)≈^𝑀_𝑖=0𝐹_𝑖(𝑌),𝜃(𝑌)≈^𝑀_𝑖=0Θ_𝑖(𝑌),𝜙(𝑌)≈^𝑀_𝑖=0Φ_𝑖(𝑌),(3.7) where 𝑀 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 𝑌_𝑗=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,−1≤𝜉≤1,(3.9) where 𝑌_∞ is the known number used to invoke the boundary condition at infinity. The unknown functions 𝐹_𝑖, Θ_𝑖, and Φ_𝑖 are approximated at the collocation points by𝐹_𝑖(𝜉)≈^𝑁_𝑘=0𝐹_𝑖𝜉_𝑘𝑇_𝑘𝜉_𝑗,Θ_𝑖(𝜉)≈^𝑁_𝑘=0Φ_𝑖Θ_𝑖𝜉_𝑘𝑇_𝑘𝜉_𝑗,(𝜉)≈^𝑁_𝑘=0Φ_𝑖𝜉_𝑘𝑇_𝑘𝜉_𝑗,𝑗=0,1,…,𝑁,(3.10) where 𝑇_𝑘 is the 𝑘th Chebyshev polynomial defined as 𝑇_𝑘(𝜉)=cos𝑘cos⁻¹(𝜉).(3.11) The derivatives of the variables at the collocation points are represented as𝑑^𝑛𝐹_𝑖𝑑𝑌^𝑛=^𝑁_𝑘=0D^𝑛𝑘𝑗𝐹_𝑖𝜉_𝑘,𝑑^𝑛Θ_𝑖𝑑𝑌^𝑛=^𝑁_𝑘=0𝑑_𝑛D^𝑛𝑘𝑗Θ_𝑖𝜉_𝑘,Φ_𝑖𝑑𝑌^𝑛=^𝑁_𝑘=0𝐃^𝑛𝑘𝑗Φ_𝑖𝜉_𝑘,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, [42–44]). Substituting (3.8)–(3.12) in (3.5)–(3.6) leads to the matrix equation𝐀_𝑖−1𝐗_𝑖=𝐁_𝑖−1,(3.13) in which 𝐀_𝑖−1 is a square matrix of order (3𝑁+3) and 𝐗_𝑖, 𝐁_𝑖−1 are (3𝑁+3)×1 column vectors defined by𝐀_𝑖−1=𝐴¹¹𝐴₁₂𝐴₁₃𝐴₂₁𝐴₂₂𝐴₂₃𝐴₃₁𝐴₃₂𝐴₃₃,𝐗^𝑖=𝚯𝐅^𝑖_𝑖𝚽_𝑖,𝐁^𝑖−1=𝐫𝐫^1,𝑖−1_2,𝑖−1𝐫_3,𝑖−1,(3.14) with𝚯𝐅_𝑖𝑖=𝐹_𝑖𝜉₀,𝐹_𝑖𝜉₁,…,𝐹_𝑖𝜉_𝑁−1,𝐹_𝑖𝜉_𝑁^𝑇,𝚽_𝑖=Θ_𝑖𝜉₀,Θ_𝑖𝜉₁,…,Θ_𝑖𝜉_𝑁−1,Θ_𝑖𝜉_𝑁^𝑇,𝐫_1,𝑖−1=Φ_𝑖𝜉₀,Φ_𝑖𝜉₁,…,Φ_𝑖𝜉_𝑁−1,Φ_𝑖𝜉_𝑁^𝑇,𝐫_2,𝑖−1=𝑟_1,𝑖−1𝜉₀,𝑟_1,𝑖−1𝜉₁,…,𝑟_1,𝑖−1𝜉_𝑁−1,𝑟_1,𝑖−1𝜉_𝑁^𝑇,𝐫_3,𝑖−1=𝑟_2,𝑖−1𝜉₀,𝑟_2,𝑖−1𝜉₁,…,𝑟_2,𝑖−1𝜉_𝑁−1,𝑟_2,𝑖−1𝜉_𝑁^𝑇,𝐴₁₁=𝑟_3,𝑖−1𝜉₀,𝑟_3,𝑖−1𝜉₁,…,𝑟_3,𝑖−1𝜉_𝑁−1,𝑟_3,𝑖−1𝜉_𝑁^𝑇,𝐴=𝐃³+𝐚_1,𝑖−1𝐃²+𝐚_2,𝑖−1𝐃+𝐚_3,𝑖−1,𝐴₁₂=𝐈,𝐴₁₃=𝜆𝐈,₂₁𝐴₃₁=𝐛_3,𝑖−1𝐃+𝐛_4,𝑖−1,𝐴₂₂=𝐃²+𝐛_1,𝑖−1𝐃+𝐛_2,𝑖−1,𝐴₂₃=D_f𝐃²,=𝐜_3,𝑖−1𝐃+𝐜_4,𝑖−1,𝐴₃₂=Sr𝐃²,𝐴₃₃=𝐃²+𝐜_1,𝑖−1𝐃+𝐜_2,𝑖−1.(3.15)

In the above definitions, 𝐚_{𝑘,𝑖−1} (𝑘=1,2,3), 𝐛_{𝑘,𝑖−1}, 𝐜_{𝑘,𝑖−1} (𝑘=1,…,4), D_f, 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 as𝐗_𝑖=𝐀^−1𝑖−1𝐁_𝑖−1.(3.16) Thus, starting with the initial solutions 𝐹₀, Θ_0, and Φ₀, a sequence of approximations ∑^𝑘0𝐹_𝑘, ∑^𝑘0Θ_𝑘, and ∑^𝑘0Φ_𝑘, 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 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=𝜇𝑋𝜕𝜕𝑦𝑢_𝑦=0𝜇𝑈₀𝐿Gr^−1/4𝑓^′(0),(4.1) where 𝜇 is the coefficient of viscosity. The skin friction coefficient at the surface of the cone is defined as C_f=𝜏^𝑤(1/2)𝜌𝑈²⁰.(4.2) Using (4.1) in (4.2), we obtain the following relation 𝐶_𝑓Gr^1/4=2𝑓^′(0).(4.3) The heat transfer rate at the surface of the cone is defined as 𝑞_𝑤=−𝑘𝑋𝜕𝜕𝑇𝑦_𝑦=0=−𝑘𝑇^𝑤−𝑇_∞𝐿Gr^−1/4𝜃^(0),(4.4) where 𝑘 is the thermal conductivity of the fluid. The Nusselt number is defined as Nu=𝐿𝑘𝑇_𝑤𝑞−^𝑤𝑇_∞.(4.5) Using (4.4) in (4.5), the dimensionless wall heat transfer rate is obtained as follows: NuGr^−1/4=−𝜃^(0).(4.6) The mass flux at the surface of the cone is defined as𝐽_𝑤=−𝐷𝑋𝜕𝜕𝐶𝑦_𝑦=0=−𝐷𝐶^𝑤−𝑇_∞𝐿Gr^−1/4𝜙^′(0),(4.7) and the Sherwood is defined asSh=𝐿𝐷𝑇_𝑤𝐽−^𝑤𝑇_∞.(4.8) Using (4.7) in (4.8), the dimensionless wall mass transfer rate is obtain asShGr^−1/4=−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 D_f, 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 0≤D_f, Sr≤1. 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., D_f=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.

Table 1 

Comparison of SLM results for single component convection (𝜆=0 and D_f=Sr=0) with Ece [41].

Table 2 

Comparison of 𝑓^(0), −𝜃^(0), and −𝜙^(0) obtained by SLM with bvp4c and shooting methods for different values of D_f and Sr with 𝜆=Pr=Sc=1.

Table 2 highlights both the accuracy and the accelerated convergence of the SLM for different values of D_f and Sr. The linearisation method converges to the numerical solutions at the fourth-order SLM for all values of D_f 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 D_f and decreasing Sr lead to initial decreases in the skin-friction coefficient 𝑓^′(0) up to D_f=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 D_f and decrease in Sr.

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 NuGr^−1/4 and Sherwood number ShGr^−1/4 which highlight the heat and mass transfer are shown in Figures 7 and 8, as functions of Sr for different values of D_f in the aiding and opposing buoyancy cases.

(a)

(b)

Figure 2 

Cross-diffusion effect on (a) temperature and (b) concentration profiles with 𝜆=−0.5 (dashed lines) and 𝜆=0.5 (solid lines).

Figure 3 

Effect of Dufour parameter Df on 𝜃(𝑌) with Sr=0.2, 𝜆=−0.5 (dashed lines), and 𝜆=0.5 (solid lines).

Figure 4 

Effect of Dufour parameter Df on 𝜙(𝑌) with Sr=0.2, 𝜆=−0.5 (dashed lines), and 𝜆=0.5 (solid lines).

Figure 5 

Effect of Sr on 𝜃(𝑌) with Df=0.3, 𝜆=−0.5 (dashed lines), and 𝜆=0.5 (solid lines).

Figure 6 

Effect of Sr on 𝜙(𝑌) with Df=0.3, 𝜆=−0.5 (dashed lines), and 𝜆=0.5 (solid lines).

Figure 7 

Variation of NuGr−1/4 with Sr for different values of Df.

Figure 8 

Variation of ShGr−1/4 with Sr for different values of Df.

The variation of temperature and concentration profiles subject to a simultaneous increase in the cross-diffusion parameters D_f 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 D_f=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 D_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 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 NuGr^−1/4 as a function of Sr for different values of D_f in aiding and opposing buoyancy conditions. In the opposing buoyancy situation, NuGr^−1/4 decreases with Sr for the case of pure thermophoresis (D_f=0) and increases in the cross-diffusion case (D_f≠0). In the aiding buoyancy situation, NuGr^−1/4 increases monotonically with Sr for both D_f=0 and D_f≠0. The Dufour number reduces the heat transfer coefficient NuGr^−1/4 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 ShGr^−1/4 as a function of Sr for different values of D_f in aiding and opposing buoyancy conditions. In both aiding and opposing buoyancy situations, ShGr^−1/4 is a decreasing function of Sr and an increasing function of D_f. 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, 𝐶𝑤−𝐶∞
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, 𝑇𝑤−𝑇∞
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.