Effects of nonuniform temperature gradients on the onset of oscillatory Marangoni convection in a magnetic field

Abstract

This article theoretically studies the onset of oscillatory Marangoni convection in a horizontal layer of an electrically conducting fluid, to which a nonuniform thermal gradient and a uniform magnetic field are applied. The top surface of a fluid layer is deformably free and the bottom is rigid. By means of the linear stability theory and a normal mode analysis, the eigenvalue equations of the perturbed state are solved by using the fourth-order Runge-Kutta-Gill's method with the shooting technique. The computational results are compared with those known from the literature, and the agreement is found out to be generally good. The results indicate that the critical Marangoni number -Ma(c) increases with increasing the Chandrasekhar number Q, the Prandtl number Pr, or the Biot number Bi of the upper free surface, but decreases with increasing the Crispation number Cr. As compared with the linear temperature profile, the inverted parabolic temperature profile shows higher -Ma(c) values, while the parabolic temperature profile shows lower -Ma(c) values. In addition, for the piecewise linear temperature profiles, the influences of thermal depth on the critical Marangoni number are also obtained

Numerical Study on High Prandtl Number Liquid Bridge

“…The surface tension on the free interface is a linear function of temperature and it decreases linearly with the temperature increasing. The basic governing equations for the half-floating zone thermocapillary convection include the mass, momentum, and energy conservation equations [6][7] [8]. Basic non-dimensional governing equations are given as following:…”

Combined Effect of Magnetic field and Internal Heat Generation on the Onset of Marangoni Convection

“…Takashima, M [3] presented a detailed numerical study of the linear stability analysis of Benard-Marangoni convection, including stationary and oscillatory modes, and focused the influence of the Crispation number on the conditions for a competition between two of these kinds of modes. Char and Chiang [4] examined the boundary effects on the Benard-Marangoni instability problem in the presence of an electric field, and found that the boundary effects of the solid plate have great influences on the stability of the system. Recently, Hashim and Wilson [5] advanced the analyses of [3,4] to the BenardMarangoni instability of a horizontal liquid layer in the most physically-relevant case when Rayleigh number and Marangoni number are linearly dependent.…”

“…Char and Chiang [4] examined the boundary effects on the Benard-Marangoni instability problem in the presence of an electric field, and found that the boundary effects of the solid plate have great influences on the stability of the system. Recently, Hashim and Wilson [5] advanced the analyses of [3,4] to the BenardMarangoni instability of a horizontal liquid layer in the most physically-relevant case when Rayleigh number and Marangoni number are linearly dependent. All these previous investigators are restricted to the convective system in the absence of a magnetic field.…”

“…Copious literature is available on coupled Benard-Marangoni convection in a horizontal ordinary viscous fluid layer with uniform distribution of internal heat generation (Char and Chiang [4], Wilson [8], Bachok and Arifin [9] and references therein). The problem of penetrative convection in an ordinary viscous fluid-saturated porous layer has also received considerable attention in the recent past because of its applications in many science and engineering problems (with current highly relevant literature including Carr [10], ), Gangadharaiah and Suma [14], ).…”

Effect of magnetic field on thermocapillary convection in a system of two immiscible liquid layers in a rectangular cavity

“…They showed that in addition to an increase in Marangoni number, the unsteadiness also increases with the increase in the concentration of the imposed heat flux. Char and Chen (2003) studied the effect of non‐uniform temperature gradients at the onset of oscillatory Marangoni convection in a perpendicularly applied magnetic field. Since the Lorentz force suppresses the Marangoni convection, they established the result that the critical Marangoni number increases with the increase in the strength of magnetic field, and Prandtl number.…”