The Onset of Buoyancy and Surface Tension Driven Convection in a Ferrofluid Layer by Influence of General Boundary Conditions

This paper investigated the buoyancy and surface tension-driven ferro-thermal-convection (FTC) in a ferrofluid (FF) layer due to influence of general boundary conditions. The lower surface is rigid with insulating to temperature perturbations, while the upper surface is stress-free and subjected to general thermal boundary condition. The numerically Galerkin technique (GT) and analytically regular perturbation technique (RPT) are applied for solving the problem of eigenvalue. It is analyzed that increasing Biot number, decreases the magnetic and Marangoni number is to postponement the onset. Additionally, magnetization nonlinearity parameter has no effect on FTC in the non-existence of Biot number. The results under the limiting cases are found to be in good agreement with those available in the literature.


Introduction
Until recently, there were liquids which could be magnetized to be comparable with the magnetization of magnetic nanoparticles. They have developed colloidal suspensions containing magnetic nanoparticles with a carrier liquid like water, hydrocarbon such as mineral oil or kerosene, or fluorocarbon referred as ferrofluids (FFs). Hence, FFs subjects have obtained much attention among the scientific communities [1] [2] [3] [4]. The magnetization of FFs depends on its magnetic field, temperature and density. Whereas when a horizontal FF layer is pre-sent with a magnetic field, it is heated from below and convective motions might take place which is called as FTC [5].
Thereby, FTC can also be induced by providing surface-tension and later with the function of temperature. Qin and Kaloni [6] have investigated both linear and non-linear stability of combined effects of buoyancy and surface tension forces in a FF layer. Hennenberg et al. [7] have examined the coupling effects on Marangoni and Rosensweig instabilities by considering two semi-infinite immiscible and incompressible viscous fluids. The results of different basic temperature gradients on FTC which is driven by buoyancy and surface tension forces discussed by Shivakumara et al. [8] with an plan following indulgent control of FTC concept. Shivakumara and Nanjundappa [9] have also examined the initiation of Marangoni FTC with differing initial temperature gradients. A very less number of researches address the effects of Bouyancy and surface tension forces on FTC (see [10]  The intent of the present work is to investigate Bénard-Marangoni FTC in a FF layer due to influence of general boundary conditions. The numerically Galerkin technique (GT) and analytically regular perturbation technique (RPT) are applied for solving the problem of eigenvalue when both the surfaces insulated to temperature perturbations.

Formulation of the Problem
Consider an incompressible FF horizontal layer of thickness d with temperatures . Where T is the temperature, 0 q is the conductive thermal flux, 1 k the overall thermal conductivity, t h the heat transfer coefficient and T ∞ the temperature in the bulk of the environment.

Cartesian coordinates ( )
, , x y z system are chosen (see Figure 1). Gravity acts vertically downwards and is given by where t α is the thermal expansion coefficient and 0 ρ is the density at where p is the pressure, t is the time and 0 µ the magnetic permeability of vacuum.
where C is the specific heat, is the specific heat at constant volume and magnetic field, and The magnetic field ( H  ) in magnetic fluid obeys the Maxwell equations in the absence electric field and current are is the magnetic induction and ϕ is the magnetic potential.
Since the magnetization ( M  ) depends on the magnitude of magnetic field and temperature, we have The linearized equation of magnetic state about 0 H and 0 T is . It is clear that there exists the following solution for the basic state: where T d β = ∆ is the temperature gradient and the subscript b denotes the basic state.
To study the stability of the system, we perturb all the variables in the form are perturbed variables and are assumed to be small.
Substituting Equation (11) into Equations (8) and (9), and using Equation (7), we obtain (after dropping the primes) 1 . x Again substituting Equation (11) into momentum Equation (4), linearizing, eliminating the pressure term by operating curl twice and using Equation (12) the z-component of the resulting equation can be obtained as (after dropping the primes): is the horizontal Laplacian operator. The temperature Equation (5), after using Equation (11) and linearizing, takes the form (after dropping the primes): where 0 0 . Equations 6(a, b), after substituting Equation (11) and using Equation (12), may be written as (after dropping the primes) ( ) The normal mode expansion of the dependent variables is assumed in the form where  and m are wave numbers in the x and y directions, respectively, and ω is the growth rate with is complex. On substituting Equation (16) where 0 v µ ρ = is the kinematic viscosity and is the effective thermal diffusivity, we obtain (after dropping the asterisks for simplicity) Here, , , W Θ Φ are respectively the z-component perturbed amplitudes of velocity, temperature and magnetization term. In addition d d D z ≡ differential operator, where,

Numerical Solution
The Galerkin method is applied to solve the problem of eigenvalue constituted by Equations (18)

Numerical Results and Discussion
Equation (29)  (33) Here we note that the minimum of Ra corresponding to c a is to be found that for various physical parameters Ra a obtained here are compared with Sparrow et al. [27].
The results established are in admirable agreement and thus validate the exactness of the numerical technique utilized (see Table 1)  Figure 2(b) illustrates that increasing in c a as Bi increases, hence its effect is to diminish the size of convection cells.
In Figure 3

Conclusions
The influence of general boundary conditions on buoyancy and surface tensiondriven FTC in a FF layer is investigated numerically Galrkin technique based on weighted residual technique. The following conclusions were resulting: • The initiation of FTC is inhibited with increasing Biot number Bi .