Research Article Effects of Controller and Nonuniform Temperature Profile on the Onset of Rayleigh-Bénard-Marangoni Electroconvection in a Micropolar Fluid H. M. Azmi and R. Idris School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia Correspondence should be addressed to R. Idris; ruwaidiah@umt.edu.my Received 5 February 2014; Accepted 11 May 2014; Published 16 June 2014 Academic Editor: Li Weili Copyright © 2014 H. M. Azmi and R. Idris. Tis 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. Linear stability analysis is performed to study the efects of nonuniform basic temperature gradients on the onset of Rayleigh- B´ enard-Marangoni electroconvection in a dielectric Eringen’s micropolar fuid by using the Galerkin technique. In the case of Rayleigh-B´ enard-Marangoni convection, the eigenvalues are obtained for an upper free/adiabatic and lower rigid/isothermal boundaries. Te infuence of various parameters has been analysed. Tree nonuniform basic temperature profles are considered and their comparative infuence on onset of convection is discussed. Diferent values of feedback control and electric number are added up to examine whether their presence will enhance or delay the onset of electroconvection. 1. Introduction Electrohydrodynamics is a branch of fuid mechanics which involved the efects of electrical forces. It is the study of the dynamics of electrically charged fuids involving the movements of ionized particles or molecules and their interactions with electric felds and the surrounding fuids. In dielectric media, hydrostatic pressure (or motion) is created by electrostatic felds. When media are fuids, a fow is produced. Generally, this phenomenon is related to the direct conversion of electrical energy into kinetic energy and vice versa. An electric feld which is generated by electrically charged particles as well as time-varying magnetic feld is the onset of natural convection. Magnetic efects will be dominant in highly conductive fuids. Ironically, electric efects will control the motion in dielectric fuids with such low values of the conductivity. Te efect of electric felds on the liquid fow is directly converted into the kinetic energy [1]. Evolution of convection in a fuid has been greatly studied by many authors; see [2–8]. Convection is the transfer of thermal energy from one particle to another by the motion of fuids such as liquids or gases in a media. In general, convection can be divided into two which are natural convec- tion and force convection. An example of natural convection is electroconvection. Electroconvection is the movements of fuid in an electric feld. Many authors have studied the onset of convection in a dielectric feld layer such as Roberts [2], Char and Chiang [3], Douiebe et al. [9], and El-Sayed [10]. In 1900, B´ enard has observed a hexagonal pattern of convection cells below afer heating some wax in a hot metal dish above some critical temperature. It shows the process of thermal convection [11]. Te mechanism of thermograv- itational (buoyancy-driven) convection and thermocapillary (surface tension driven) convection with temperature has been the subject of a great deal of theoretical and experi- mental investigations since the pioneering pieces of work of Rayleigh in 1916 [12] and Pearson in 1958 [13]. Te analy- sis of thermogravitational and thermocapillary convection, respectively, has many important applications in the feld of engineering such as fying of airplanes and submarine in physical engineering and production of colloids, paints, and polymeric suspensions in chemical engineering [14]. Rayleigh-B´ enard instability is a mechanism associated with buoyancy since there appears a nondimensional Rayleigh Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 571437, 8 pages http://dx.doi.org/10.1155/2014/571437