Effect of helical force on the stationary convection in a rotating ferrofluid
Ferrofluids, also called magnetic fluids, are stable colloidal suspensions of ferromagnetic particles in a carrier liquid.The discipline that deals with these fluids is ferrohydrodynamics (FHD); which differs from magnetohydrodynamics (MHD) and electrohydrodynamics (EHD) in that there is usually no electric current or free electric charge in the fluid. Magnetic fluids enjoy an important property. In the absence of a magnetic field, the magnetic dipoles of the particles in the fluid are randomly oriented and the fluid does not exhibit macroscopic magnetization. In the presence of an external magnetic field, the magnetic dipoles are aligned and a magnetization of the fluid is now observable. Ferrofluids are not found in their natural state; we will have to synthesize them. Ferrofluids typically contain 1023 particles per cubic meter and their industrial applications are numerous. They are used in vibration dampers, hard disk seals or electrodynamic loudspeakers to dampen vibrations; they are also used in magnetic inks for bill printing, in truck brakes, energy conversion devices . Heat transfer by ferrofluids has been one of the areas of scientific study that has attracted the attention of many researchers because of its technological applications , . In the area of medicine, ferrofluids are used to transport drugs to specific places in the human body; other additional applications of these fluids where the carrier liquid is blood with particular rheological properties can be found in Mahmoudi et al. , Kim et al. , Alexiou et al. , 7], 8], Popel and Johnson , Baskurt and Meiselman , Bishop et al. . The characterization and synthesis of hard ferromagnetic (M-type strontium hexaferrite) co-doped Zn and Zr for magnetic hyperthermia applications were studied by Majid et al . The exceptional results of their work are numerous and they have shown that a sharp drop in the anisotropy and nanoparticles of SrFe12O19 single domain could be used in magnetic hyperthermia applications. The first ferrofluid studies were carried out by Neuringer and Rosensweig . These authors have established the different equations of ferrohydrodynamics starting from the different principles of physics. Finlayson  was the first to study the classical Rayleigh–Bénard convection problem of a layer of magnetic fluids in the presence of a uniform vertical magnetic field. Through a linear stability study, he highlighted the coupling of buoyancy and magnetic mechanisms and then proved that the magnetic mechanism predominates over that of flotability in very thin layers. The effect of rotation on the thermo - convective instabilities of a horizontal layer of magnetic fluids heated from below and in the presence of a uniform vertical magnetic field was considered by Gupta et al. , and Venkatasubramanian and Kaloni . They noted that the rotation stabilizes the system but at a slower rate than that of ordinary fluids. Auernhammer and Brand  also examined the influence of the rotation of a magnetic fluid layer by a linear and weakly nonlinear analysis and showed that the instability of Küppers–Lortz also appears in the ferrofluids. The complete equations describing a ferrofluid as a binary mixture in an external magnetic field were developed by Ryskin and Pleiner . These authors took into account the magnetophoretic effect, as well as the magnetic stresses in the static and dynamic parts of the magnetic fluid equations. Neglecting magnetophoretic effect Laroze et al.  used the Ryskin approach to study Rayleigh–Bénard convection of the binary liquid mixture of rotating magnetic fluids. They determined in the stationary case for the idealized boundary conditions, the analytic expression of the Rayleigh number as a function of the parameters of the system. Other interesting studies done by Laroze and his collaborators on magnetic fluids are found in Martinez-Mardones et al. , Mardones et al. , Laroze et al. , 23], Pérez et al. , Laroze et al. . Tlili et al modeled the free convection of hybrid nanofluids in a permeable medium via the control of the finite element method in volume, including the magnetic effect. They described the impacts of dimensionless parameters on the migration of nanomaterials.
On the other hand, the classical problem of Rayleigh–Bénard convection of viscous fluids in a rotating layer is studied by Chandrasekhar ; it has been shown that rotation has a stabilizing effect on the system. However, in the planetary atmosphere, the motion of turbulent convection is destined to become helical. Properties generating small-scale helical turbulence were first discovered in magnetohydrodynamics by Steenbeck et al. . They called this phenomenon the alpha effect. This discovery favored the development of MHD theory. The publications on the hydrodynamic alpha effect were made by Levina and her collaborators , , , . In a convective system, it has been shown that the force responsible for helical turbulence is pseudo - vector in nature, called helical force. This force appears as a result of averaging of turbulence in the mean-field equations for large-scale motions. So, the mathematical origin of this helical force is similar to the origin of the electromotive force in the alpha-effect mean-field equations in magnetohydrodynamics; with a principal difference that the electromotive force is a real force with a clear physical interpretation, what has been confirmed by many experiments. The action of this force in the Rayleigh–Bénard convection could induce a new type of instability, called helico - vortex instability. In , for three different types of boundary conditions, linear stability analysis showed that the helical force decreases the instability threshold. Essoun and Chabi Orou  examined the influence of the helical force on a layer of viscous fluid in rotation and found that the helical force has no monotonous effect on the beginning of convection for all Taylor number values, but decreases the corresponding critical wave number for any Taylor number value.
The aim of this work is to analyze the linear stability of a horizontal layer in a rotating ferrofluid under the effect of the helical force. We focus on stationary convection with free boundary conditions. For this, we determined the explicit expression of the Rayleigh number and examined the effect of each dimensionless parameter on the onset of convection.
The paper is structured as follows: in Section 2, the hydrodynamic equations for the convection of a ferrofluid are presented. The linear stability analysis is performed in Section 3. Section 4 is devoted to the main results obtained in the paper followed by the discussions. Finally, conclusion is presented in Section 5.
Mathematical formulation of the problem
Consider a horizontal layer of infinite extension, of thickness d, of an incompressible magnetic fluid (described as a binary mixture of a host fluid and the embedded magnetic particles), in rotation at constant angular velocity ϖ around a vertical axis in the gravitational field g and under the influence of a magnetic field H parallel to ẑ axis. The coordinates at the interfaces of the layer are for the lower plate and for the upper one. The plates are subjected to the
Linear stability analysis
In order to proceed to the analysis of the linear stability, we only need the linear parts of the Eqs. (13)–(17). In addition, the pressure and two components of the velocity will be eliminated by applying the curl operator () and double curl () to the Navier - Stokes equation; then we consider only the component z of the resulting equations, the component w of the velocity and the component ζ of vorticity. After calculations, the linear equations are:
Main results and discussion
Our analysis consists of studying theoretically the criterion of appearance of the stationary convection in a rotating horizontal layer of a magnetic fluid heated from the bottom, in the presence of the helical force.
The Figs. 2 and 3 respectively represent the neutral stability curves in the plane (k, Ra) for different values of the helicoidal force intensity and the Taylor number. The coordinates of the lowest point of each curve correspond to the values of the critical wave number and the
In this work, we performed the analysis of the linear stability of a horizontal rotating layer of a magnetic fluid in the presence of the helical force and subjected to a vertical temperature gradient at free boundary conditions. The analytical expression of the Rayleigh number is determined and the effect of each dimensionless parameter is studied. The helical force, the binary parameter ψ and the magnetic parameters M1, M3 and ψm make the system unstable by accelerating the start of
Declaration of Competing Interest
The authors declare that they have no conflict of interest.
The authors thank IMSP-UAC for financial support. We also thank Dr. L.A. Hinvi and Mr. Segning Hilaire for their collaborations. We would also like to thank very much the anonymous referees whose useful criticisms, comments and suggestions have helped strengthen the content and the quality of the paper.
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