Effect of rotation on a ferromagnetic fluid heated and soluted from below in the presence dust particles

Abstract

This paper deals with the theoretical investigation of the effect of rotation on a layer of a ferromagnetic fluid permeated with dust particles, heated and soluted from below subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linearized stability theory and a normal mode analysis method. For the case of stationary convection, non-buoyancy magnetization and dust particles have always a destabilizing effect, whereas stable solute gradient and rotation have a stabilizing effect on the onset of instability. The critical wave number and the critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetization M₁. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. It is observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles, stable solute gradient and rotation. The oscillatory modes are introduced due to the presence of the dust particles, stable solute gradient and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.

Introduction

In the last millennium, the investigation on ferromagnetic fluid attracted researchers because of the increase of applications of magnetic fluids. The most famous application of magnetic fluids is the sealing of rotating shafts. This advantage is commonly used in various technical applications like the sealing of hard disc drives, rotating X-ray tubes and rotating vacuum feed-throughs where reliable sealing at low friction is required. The use of magnetic fluids as a heat transfer medium that may be magnetically hold in a certain position is nowadays the commercially most important branch of ferrofluid manufacturing. The major application in this field is cooling of loudspeakers, enabling a significant increase of the maximum acoustical power without any geometrical changes of the speaker system. The third important field of application of magnetic fluids is their use in biomedical applications. For examples, their use as a contrast medium in X-ray examinations [1] and for positioning tamponade for retinal detachment repair in eye surgery [2] has been reported.

Rosensweig [3] has given an authoritative introduction to research on magnetic liquids in his monograph and the study of the effect of magnetization yields interesting information. This magnetization, in general, is function of the magnetic field, temperature and density of the fluid. Any variation of these quantities can induce a change of body force distribution in the fluid. This leads to convection in ferrofluids in the presence of magnetic field gradient. This mechanism is known as ferroconvection, which is similar to Bénard convection. A detailed account of the theoretical and experimental results of the onset of thermal instability (Bénard convection) in a fluid (non-magnetic) layer under varying assumptions of hydrodynamics and hydromagnetics has been given in the celebrated monograph by Chandrasekhar [4]. The convective instability of a ferromagnetic fluid for a fluid layer heated from below in the presence of uniform vertical magnetic field has been considered by Finlayson [5].

Thermoconvective stability of ferrofluids without considering buoyancy effects has been investigated by Lalas and Carmi [6], whereas Shliomis [7] analyzed the linearized relation for magnetized perturbed quantities at the limit of instability. The thermal convection in a magnetic fluid has been considered by Zebib [8], whereas the stability of a static magnetic fluid under the action of an external pressure drop has been studied by Polevikov [9]. Schwab et al. [10] investigated experimentally the Finlayson’s problem in the case of a strong magnetic field and detected the onset of convection by plotting the Nusselt number versus the Rayleigh number. Then, the critical Rayleigh number corresponds to a discontinuity in the slope. Later, Stiles and Kagan [11] examined the experimental problem reported by Schwab et al. [10] and generalized the Finlayson’s model assuming that under a strong magnetic field, the rotational viscosity augments the shear viscosity.

The Bénard convection in ferromagnetic fluids has been considered by many authors [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. The ferromagnetic fluid has been considered to be clean in all the above studies. In many situations, the fluid is often not pure but contains suspended/dust particles. Saffman [23] has considered the stability of laminar flow of a dusty gas. Scanlon and Segel [24] have considered the effects of suspended particles on the onset of Bénard convection, whereas Sharma et al. [25] have studied the effect of suspended particles on the onset of Bénard convection in hydromagnetics. The suspended particles were thus found to destabilize the layer. Palaniswamy and Purushotham [26] have studied the stability of shear flow of stratified fluids with fine dust and found the effects of fine dust to increase the region of instability. On the other hand, the multiphase fluid systems are concerned with the motion of a liquid or gas containing immiscible inert identical particles. Of all multiphase fluid systems observed in nature, blood flows in arteries, flow in rocket tubes, dust in gas cooling systems to enhance the heat transfer processes, movement of inert solid particles in atmosphere, sand or other particles in sea or ocean beaches are the most common examples of multiphase fluid systems. Naturally studies of these systems are mathematically interesting and physically useful for various good reasons. The effect of dust particles on non-magnetic fluids has been investigated by many authors [27], [28], [29], [30].

In the standard Bénard problem the instability is driven by a density difference caused by a temperature difference between the upper and lower planes bounding the fluid. If the fluid is additionally has salt dissolved in it then there are potentially two destabilizing sources for the density difference, the temperature field and salt field. When there are two effects such as this, the phenomenon of convection which arises is called double-diffusive convection or thermohaline convection. Vaidyanathan et al. [31] studied ferrothermohaline convection in which a horizontal layer of an incompressible ferromagnetic fluid of thickness d in the presence of transverse magnetic field, heated from below and salted from above is considered. They found that the salinity of a ferromagnetic fluid enables the fluid to get destabilized more when it is salted from above. The really interesting situation from both a physical and a mathematical viewpoint arises when the layer is simultaneously heated from below and salted from below. More recently, Sunil et al. [32] have studied the effect of magnetic field dependent viscosity on thermosolutal convection in a ferromagnetic fluid.

In view of the above investigations, it is attempted to discuss the effect of rotation on a ferromagnetic fluid heated and soluted from below in the presence of dust particles, subjected to a vertical magnetic field. The present study can serve as a theoretical support for experimental investigations e.g. evaluating the influence of impurities in a ferromagnetic fluid on thermal convection phenomena. This problem, to the best of our knowledge, has not been investigated yet.