Abstract
The combined effects of a uniform vertical magnetic field and a nonuniform basic temperature profile on the onset of steady Marangoni convection in a horizontal layer of micropolar fluid are studied. The closed-form expression for the Marangoni number for the onset of convection, valid for polynomial-type basic temperature profiles upto a third order, is obtained by the use of the single-term Galerkin technique. The critical conditions for the onset of convection have been presented graphically.
1. Introduction
Convective flow in a thin layer of fluid, free at the upper surface and heated from below, is of fundamental importance and a prototype to a more complex configuration in experiments and industrial processes. The convective flows in a liquid layer can be driven by buoyancy forces due to temperature gradients and/or thermocapillary (Marangoni) forces caused by surface tension gradients. Thermal convective problems have long been studied extensively since the pioneering experimental and theoretical works of Bénard [1], Rayleigh [2], and Pearson [3]. The instability problems have been studied in several other directions (cf. [4–18]).
Most of the previous studies were concerned with convection in Newtonian fluids. However, much less work has been done on convection in non-Newtonian fluids such as the micropolar fluids. The theory of micropolar fluids, as developed by Eringen [19], has been a field of sprightly research for the last few decades especially in many industrially important fluids like paints, polymeric suspensions, colloidal fluids, and also in physiological fluids such as normal human blood and synovial fluids. Rama Rao [20] studied the effect of a magnetic field on convection in a micropolar fluid. The onset of convection as overstable motions in a micropolar fluid was examined in [21]. Sharma and Gupta [22] studied convection in micropolar fluids in a porous medium. Ramdath [23] considered buoyancy-and thermocapillary-driven (Bénard-Marangoni) convection in a layer of micropolar fluid. The effect of throughflow on Marangoni convection in micropolar fluids was analyzed in [24]. Siddheshwar and Sri Krishna [25] presented both linear and nonlinear analyses of convection in a micropolar fluid occupying a porous medium. Sunil et al. [26] studied the effect of rotation on convection in a micropolar ferrofluid.
There has also been much less work focused on the effect of nonuniform temperature gradient on convection. Friedrich and Rudraiah [27] studied the combined effects of nonuniform temperature gradients and rotation on Marangoni convection. The combined effects of nonuniform temperature gradients and a magnetic field on Marangoni convection were investigated by Rudraiah et al. [28]. The work of Friedrich and Rudraiah [27] was further extended to include the effect of buoyancy by Rudraiah and Ramachandramurthy [29]. Dupont et al. [30] studied the effect of a cubic quasisteady temperature profile on Marangoni convection. The effects of nonuniform temperature gradients on the onset of oscillatory Marangoni and Bénard-Marangoni convection in a magnetic field were analyzed in [31, 32], respectively. Chiang [33] investigated the effect of Dupont et al. [30] temperature profile on the onset of stationary and oscillatory Bénard-Marangoni convection.
Thermal convection in micropolar fluids has also been studied. Rudraiah and Siddheshwar [34] analyzed the effects of nonuniform temperature gradients of parabolic- and stepwise-types on the onset of Marangoni convection in a micropolar fluid. This study was later extended by Siddheshwar and Pranesh [35] to include the effect of a magnetic field and buoyancy forces. Very recently, Idris et al. [36] studied the effect of Dupont et al. [30] cubic temperature profile on the onset of Bénard-Marangoni convection in a micropolar fluid.
In this paper, we shall investigate the combined effects of Dupont et al. [30] cubic temperature profile and a magnetic field on the onset of Marangoni convection in a micropolar fluid. The single-term Galerkin technique [37] is employed to obtain a closed-form expression of (Marangoni number) for the onset of convection. Comparisons with the other polynomial-type temperature profiles normally used by previous investigators shall be undertaken.
2. Mathematical Formulation
We wish to examine the stability of a horizontal layer of quiescent micropolar fluid of thickness in the presence of a magnetic field. We assume that the layer is bounded below by a rigid boundary, which is kept at a constant temperature, and above by a perfectly insulated, flat free surface. Moreover, the spin-vanishing boundary condition is assumed at the boundaries.
The governing equations for the problem are the continuity equation, conservation of momentum, conservation of angular momentum, conservation of energy, and magnetic induction, compare [19, 34, 35]: where is the velocity, is the spin, is the temperature, is the magnetic field, is the hydromagnetic pressure, is the coupling viscosity coefficient, is the shear kinematic viscosity coefficient, is the moment of inertia, and are the bulk and shear spin viscosity coefficients, is the micropolar heat conduction coefficient, is the specific heat, is the thermal conductivity, and is the magnetic viscosity (where electrical conductivity and magnetic permeability). All the viscosity coefficients, heat conduction coefficient and thermal conductivity are thermodynamically restricted on the assumption of Clausius-Duhem inequality (see Eringen [19]) and are all positive quantities.
The surface tension at the free upper surface is
where is the unperturbed value of and . The perturbation (2.1) are nondimensionalised using the following definition:
Following the classical lines of linear stability theory, the linearised and dimensionless governing equations are
where , , , and are, respectively, the amplitudes of the infinitesimal perturbations of velocity, spin, temperature, and magnetic field, is the coupling parameter ( ), is the couple stress parameter (, : finite, real), is the micropolar heat conduction parameter (, : finite, real), is the Chandrasekhar number, is the Prandtl number, is the magnetic Prandtl number, and is a nondimensional basic temperature gradient satisfying the condition .
The infinitesimal perturbations , , , and are assumed to be periodic waves and hence these permit a normal mode solution in the following form:
where and are horizontal components of the wave number .
Substituting (2.5) into (2.4), we get where .
Eliminating between (2.6) and (2.9), we obtain
Equations (2.7), (2.8), and (2.10) are solved subject to the linearized and dimensionless boundary conditions: where is the Marangoni number (where is the temperature difference between the two boundaries).
Following [30], we consider the steady state temperature profile given by
which precisely represents an experimental data, where (-) denotes dimensional quantities, is the temperature at the upper free surface, and , are constants. In nondimensional form, the in this case is given by
The case , , and recovers the classical linear basic state temperature distribution. The different temperature gradients studied in this paper are listed in Table 1.
3. Solution of the Linearized Problem
Equations (2.7), (2.8), and (2.10) subject to the boundary conditions (2.11) constitute an eigenvalue problem. To solve the resulting eigenvalue problem, a single-term Galerkin expansion technique [37] is used to encompass a vast parameter space. Also, the technique employed yields sufficiently accurate and useful results for the purpose in hand with minimum of mathematics [37].
First we multiply (2.7), (2.8) and (2.10) by , , and , respectively. Then we integrate the resulting equations by parts with respect to from 0 to 1. By using the boundary conditions (2.11) and taking , , and , and in which , , and are constants and , , and are trial functions, yields the eigenvalue in the form
where Now with as given in (2.13), we rewrite the expression (3.1) in the closed-form expression for :
where We remark that (3.3) is valid for all polynomial-type basic temperature profiles up to a third order. The critical Marangoni number, , for the onset of convection is the global minimum of over .
4. Discussion
The critical Marangoni number which attains its minimum at is computed from (3.3) for different volumes of , , , and and the results are depicted in Figures 1, 2, and 3. We recover the results of Rudraiah and Siddheshwar [34] for the linear and inverted parabolic temperature gradients when . We observe that as or increases, also increases. Obviously, the onset of convection will be delayed by increasing the concentration of the microelements or heat induced into the fluid by the microelements. But, an increase in leads to a decrease in microrotation, and hence the system becomes more unstable. Also it is observed that Model 4 (Cubic 2), with , , as used by Dupont et al. [30], is less stabilizing than Model 2 (Inverted parabolic), that is, . Based on our results, Model 3 (Cubic 1) with , , is shown to be the most stabilizing of all the considered types of temperature gradients, that is, .
Figures 4–6 illustrate the variations of the critical Marangoni number with the Chandrasekhar number for some assigned values of , , and , respectively. The results indicate that is generally an increasing function of . From Figure 4, we notice that the increase in the concentration of the microelements is to stabilize the system by superposing on the effect of the magnetic field. Figure 5 shows that the effect of on the system is very small compared to the effects of the other microelements. As before, Model 3 (Cubic 1) with , , is shown to be the most stabilizing of all the considered types of temperature gradients, that is, .
5. Conclusion
The problem of Marangoni convection in a micropolar fluid in the presence of a cubic basic state temperature profile and a vertical magnetic field has been studied theoretically. The results indicate that it is possible to delay the onset of convection by the application of a cubic basic state temperature profile. In addition, the presence of a magnetic field is to suppress Magnetomarangoni convection and hence leads to a more stable system. As expected, the presence of the micron-sized suspended particles adds to the stabilizing effect of the magnetic field.
Acknowledgment
The authors acknowledge the financial support received under the Grant UKM-GUP-BTT-07-25-173 and from Universiti Kuala Lumpur (UniKL MICET).