Triple- diffusive Convection in a Micropolar Rotating Ferrofluid

S. K. Kango^1*, Sanjay Sharma², Kalpna Chadha³

¹Government College, Barsar (Hamirpur)-174305, Himachal Pradesh, INDIA

²Government College, Darang (Mandi), Himachal Pradesh, INDIA

³Government College, Sujanpur (Hamirpur), Himachal Pradesh, INDIA

*Corresponding Author Email: skkango72@gmail.com

ABSTRACT:

This paper deals with the theoretical investigation of the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stability analysis theory and normal mode analysis method have been carried out to study the onset convection. The influence of various parameters like rotation, solute gradients, and micropolar parameters (i.e. coupling parameter, spin diffusion parameter and micropolar heat conduction parameter) has been analyzed on the onset of stationary convection. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large value of buoyancy magnetization parameter M₁(ratio of the magnetic to gravitational forces). The principle of exchange of stabilities is found to hold true for the micropolar fluid heated from below in the absence of micropolar viscous effect, microinertia, solute gradient and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia, solute gradient and rotation, which were non-existent in their absence.

KEYWORDS: Triple- diffusive convection; Micropolar ferrofluid; Solute gradient; Vertical magnetic field; Rotation; Magnetization.

1. INTRODUCTION:

Micropolar fluids are fluids with internal structures in which coupling between the spin of each particle and the microscopic velocity field is taken into account. They represent fluids consisting of rigid, randomly oriented or spherical particles suspended in viscous medium, where the deformation of fluid particles is ignored (e.g. polymeric suspension, animal blood, liquid crystal). Micropolar fluids have been receiving a great deal of research focus and interest due to their application in a number of processes that occur in industry. Such applications include the extrusion of polymer fluids, solidification of liquid crystal, cooling of metallic plate in a bath, exotic lubricants and colloidal suspension solutions. Micropolar fluid theory was introduced by Eringen [1966] in order to describe some physical systems, which do not satisfy the Navier- Stokes equation. The equations governing the micropolar fluid involve a spin vector and microinertia tensor in addition to the velocity vector. The theory can be used to explain the flow of colloidal fluids, liquid crystals, animal blood etc. The generalization of the theory including thermal effects has been developed by Kazakia and Ariman [1971] and Eringen [1972]. The theory of thermomicropolar convection began with Datta and Sastry [1976] and interestingly continued by Ahmadi [1976], Lebon and Perez- Garcia [1981], Bhattacharya and Jena [1983], Payne and Straughan [1989], Sharma and Kumar [1995, 1997] and Sharma and Gupta [1995]. The above works give a good understanding of thermal convection in micropolar fluids.

In many situations involving suspensions, as in the magnetic fluid case, it might be pertinent to demand an Eringen micropolar description. This was suggested, in fact, by Rosenweig [1995] in his monograph. An interesting possibilities in a planer micro polar ferrofluid flow with an AC magnetic field has been considered by Zahn and Greer [1995].They examined a simpler case where the applied magnetic fields along and transverse to the duct axis are spatially uniform and varying sinusoidally with time. In a uniform magnetic field, the magnetization characteristic depends on particle spin but does not depend on fluid velocity. Micropolar ferrofluid stabilities have become an important field of research these days. A particular stability problem is Rayleigh-Bénard instability in a horizontal thin layer of fluid heated from below. A detailed account of thermal convection in a horizontal thin layer of Newtonian fluid heated from below has been given by Chadrasekhar [1981]. For a ferrofluid, a thermo-mechanical interaction is predicted by Finlayson [1970] in the presence of a uniform vertical magnetic field provided the magnetization is a function of temperature and magnetic field, and a temperature gradient is established across the fluid layer. The thermal convection in Newtonian ferro fluid has been studied by many authors.

Rayleigh-Bénard convection in a micropolar ferrofluid layer permeated by a uniform, vertical magnetic field with free-free, isothermal, spin-vanishing, magnetic boundaries has been considered by Abraham [2002]. She observed that the micropolar ferro fluid layer heated from below is more stable as compared with the classical Newtonian ferrofluid. The effect of rotation on thermal convection in micropolar fluids is important in certain chemical engineering and biochemical situations. Qin and Kaloni [1992] have considered a thermal instability problem in a rotating micropolar fluid. They found that, depending upon the values of various micropolar parameters and the low values of the Taylor number, the rotation has a stabilizing effect. The effect of rotation on thermal convection in micropolar fluids has also been studied by Sharma and Kumar [1994], whereas the numerical solution of thermal instability of rotating micropolar fluid has been discussed by Sastry and Rao [1983] without taking into account the rotation effect in angular momentum equation. But we also appreciate the work of Bhattacharyya and Abbas [1985] and Qin and Kaloni, they have considered the effect of rotation in angular momentum equation. More recently, Sunil et al., [2006, 2008 & 2009] have studied the effect of rotation on the thermal convection problems in ferrofluid.

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, additionally has salt dissolved in it, then there are potentially two destabilizing sources for the density difference, the temperature field and salt field. The solution behavior in the double-diffusive convection problem is more interesting than that of the single component situation in so much as new instability phenomena may occur which is not present in the classical Bénard problem. When temperature and two or more component agencies, or three different salts, are present then the physical and mathematical situation becomes increasingly richer. Very interesting results in triply diffusive convection have been obtained by Pearlstein et al., [1989]. The results of Pearlstein et al., are remarkable. They demonstrate that for triple diffusive convection linear instability can occur in discrete sections of the Rayleigh number domain with the fluid being linearly stable in a region in between the linear instability ones. This is because for certain parameters the neutral curve has a finite isolated oscillatory instability curve lying below the usual unbounded stationary convection one. Straughan and Walker [1997] derive the equations for non-Boussinesq convection in a multi- component fluid and investigate the situation analogous to that of Pearlstein et al., but allowing for a density non linear in the temperature field. Lopez et al., [1990] derive the equivalent problem with fixed boundary conditions and show that the effect of the boundary conditions breaks the perfect symmetry. In reality the density of a fluid is never a linear function of temperature, and so the work of Straughan and Walker applies to the general situation where the equation of state is one of the density quadratic in temperature. This is important, since they find that departure from the linear Boussinesq equation of state changes the perfect symmetry of the heart shaped neutral curve of Pearlstein et al.

In view of the recent increase in the number of non iso-thermal situations wherein magnetic fluid are put to use in place of classical fluids, we intend to extend our work to the problem of thermal convection in Eringen^,s micropolar fluid to the triple-diffusive convection in a mocropolar ferrofluid in the presence of rotation. In the present analysis, for mathematically simplicity, we have not considered the effect of rotation in angular momentum equation.

6. CONCLUSIONS:

In this paper, the effect of rotation on triple –diffusive convection in a micropolar ferrofluid layer heated and soluted from below subjected to a transverse uniform magnetic field has been investigated. The behavior of various parameters like rotation parameter, solute gradients, non-buoyancy magnetization, coupling parameter, spin diffusive parameter and micropolar heat conduction on the onset of convection has been analyzed analytically and numerically. The results show that for the state of stationary convection, the non-buoyancy magnetization, spin diffusive parameter have destabilizing effect under certain condition(s), whereas the rotation, coupling parameter and solute gradients have a stabilizing effect under certain condition(s). However, the micropolar heat conduction always has a stabilizing effect. The principle of exchange of stabilities is found to hold true for the micropolar ferrofluid heated from below in the absence of micropolar viscous effect, microinertia, rotation and solute gradient. Thus oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia, rotation and solute gradients, which were non-existent in their absence. Finally, we conclude that the rotation and micropolar parameters have a profound influence on triple- diffusive convection in a micropolar ferrofluid layer heated and soluted from below. The micropolar rotating ferrofluid stabilities do deserve a fresh look as related microgravity environment applications.

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Received on 21.08.2016 Accepted on 06.09.2016

Int. J. Tech. 2016; 6(2): 123-132.

DOI: 10.5958/2231-3915.2016.00020.1