Thermal relaxation effect on magnetohydrodynamic instability in a rotating micropolar fluid layer heated from below

Summary. The effect of a vertical magnetic field on the onset of convective instability in a conducting micropolar fluid (Oldroyd fluid) layer heated from below confined between two horizontal planes under the simultaneous action of the rotation of the system and a vertical temperature gradient is considered. Linear stability theory and normal mode analysis are used to derive an eigenvalue system of order twelve, and an exact eigenvalue equation for natural instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. The effects of magnetic field, the relaxation time and micropolar parameters on the critical Rayleigh number and critical wave number are discussed and presented graphically. The analysis presented in this paper is more general than any previous investigation.     

Growth rates of Bénard-Marangoni convection in a fluid layer in the presence of a magnetic field

In this paper we analyze the effect of a uniform vertical magnetic field on the linear growth (and decay) rates of the steady Bénard-Marangoni instability in a horizontal layer of quiescent, electrically conducting fluid with a uniform vertical temperature gradient subject to a prescribed heat flux at its lower boundary. Explicit analytical expressions for the linear growth rates of long-waves instability modes are derived for the first time. The numerically-calculated linear growth (or decay) rates showing the stabilizing effect of the magnetic field are also presented.     

The effect of an oblique magnetic field on the surface instability of a finite conducting fluid layer

Summary The effect of an oblique magnetic field on the growth rate of Rayleigh-Taylor instability (RT) at the interface of a finite thickness layer of a viscous electrically conducting fluid in the presence of surface tension has been studied analytically. The effects of aligned and transverse magnetic fields on the coupled differential equations for the velocity and the magnetic field are discussed separately. The numerical results reveal that the nature and strength of the magnetic field and the layer thickness exacerbate or ameliorate the instability characteristic of such a layer.     

The effect of a uniform vertical magnetic field on the onset of oscillatory marangoni convection in a horizontal layer of conducting fluid

Summary In this paper we use classical linear stability theory to undertake a detailed investigation of the effect of a uniform vertical magnetic field on the onset of oscillatory thermocapillary-driven Marangoni convection in a horizontal layer of quiescent, electrically conducting fluid heated from below. For simplicity we restrict our attention to the simplest case of a fluid layer with a non-deformable free surface and perfectly electrically conducting boundaries in which the onset of convection is always steady in the absence of the magnetic field. The present numerical calculations show that the presence of the magnetic field can cause the preferred mode of instability to be oscillatory rather than steady convection. Nevertheless, in all the cases investigated the effect of the magnetic field is always to stabilise the layer relative to the case of no magnetic field.     

Stability of Ferrofluid Flows in a Horizontal Channel Subjected to a Longitudinal Temperature Gradient and an Oblique Magnetic Field

The stability of thermoconvective flows of a ferrofluid in a horizontal channel subjected to a longitudinal temperature gradient and an oblique strong magnetic field is studied. The flows are governed by the mass, momentum, heat balance and Maxwell equations, in the Boussinesq approximation. Strong magnetic fields are characterized by a small parameter measuring the deviation of the magnetization across the layer from the external magnetic field. An approximate solution of the stationary hydrodynamic problem was found in analytical form in Hennenberg et al. (Phys Fluids 18:093602, 2006). The flow depends on the thermal (gravitational) and magnetic Rayleigh numbers. In the present paper the linear stability of that basic flow is studied by the Galerkin method. The analysis shows that for large Prandtl numbers, typical for ferrofluids, and relatively small magnetic Rayleigh numbers, only oscillatory instability can appear. For a given magnetic Rayleigh number, the critical wavenumber does not depend on the inclination of the magnetic field, while the critical thermal Rayleigh number slightly changes. For horizontal and vertical magnetic fields, both critical numbers decrease when increasing the magnetic Rayleigh number.     

Stability of magnetic fluid penetration through a porous medium with uniform magnetic field oblique to the interface

Recent work has shown that the fingering instability, which develops when a more viscous fluid is pushed through the voids of a porous medium or through a Hele-Shaw cell by a less viscous fluid, can be prevented if a magnetic field is applied tangential to a flat fluid interface separating magnetizable and non-magnetizable fluids. This earlier work is extended here by considering equilibrium magnetic field components both perpendicular and parallel to the flat interface. The tangential field component stabilizes those waves traveling along the field lines while the normal field is destabilizing. The analysis is developed through a general set of relations for perturbation field and flow interfacial variables defined for a "prototype" magnetizable fluid layer which can be used to describe the small signal stability characteristics of layered fluid systems. In a uniform tangential magnetic field geometry, experimental results of the most unstable wavelength in a Hele-Shaw cell are shown to agree well with theory.     

The effect of a uniform magnetic field on the onset of steady Bénard-Marangoni convection in a layer of conducting fluid

In this paper we use a combination of analytical and numerical techniques to analyse the effect of a uniform vertical magnetic field on the onset of steady Bénard-Marangoni convection in a horizontal layer of quiescent, electrically conducting fluid subject to a uniform vertical temperature gradient. The critical values of the Rayleigh and Marangoni numbers for the onset of steady convection are calculated and the latter is found to be critically dependent on the non-dimensional Crispation and Bond numbers. The stability of the layer to long wavelength disturbances is analysed and the two different asymptotic limits of strong surface tension (small Crispation number) and strong magnetic field (large Chandrasekhar number) are investigated. In the latter case analytical results for the critical Rayleigh and Marangoni numbers are obtained and are found to be in excellent agreement with the results of numerical calculations. We conclude that the presence of the magnetic field always has a stabilising effect on the layer. Treating the Marangoni number as the critical parameter we show that if the free surface is non-deformable then any particular disturbance can be stabilised with a sufficiently strong magnetic field, but if the free surface is deformable and gravity waves are excluded then the layer is always unstable to infinitely long wavelength disturbances with or without a magnetic field. Including gravity has a stabilising effect on the long wavelength modes, but not all disturbances can be stabilised no matter now strong the magnetic field is.     

Diffusion of carbon and titanium in γ -iron in a magnetic field and a magnetic field gradient

The diffusion coefficients of carbon and titanium in γ -iron were measured in a 6T magnetic field and in magnetic field gradients ranging from 30 to 45 T/m. We have found that the diffusion of carbon in γ -iron is retarded by application of a 6T magnetic field. In contrast with carbon diffusion, no noticeable effect of a magnetic field on the diffusivity of titanium in γ -iron is observed. On the other hand, the diffusion of carbon in γ -iron can be enhanced in a magnetic field gradient when carbon atoms move towards the direction with a higher magnetic field strength. The higher the magnetic field gradient strength becomes, the more the carbon diffusion is enhanced. Nevertheless, a magnetic field gradient causes a decrease in diffusivity of carbon in γ -iron when the opposite magnetic field gradient is applied.     

轴向磁场对硅单晶Czochralski生长过程的影响

利用有限元方法对炉内的传递过程进行了全局数值模拟,假定熔体和气相中的流动都为准稳态轴对称层流,熔体为不可压缩流体,Cz炉外壁温度维持恒定,模拟磁场强度范围为(0～0.3)T,研究了用Czochralski(Cz)法生长单晶硅轴向磁场对熔体流动和氧传输过程的影响.结果表明:轴向磁场可有效地抑制熔体内的流动,但增大加热器功率和结晶界面处晶体内的轴向温度梯度;对于常规Cz炉,轴向磁场可增大结晶界面平均氧浓度,而对于具有气体导板的Cz炉,则会减小结晶界面平均氧浓度.     

磁性液体在均匀磁场中的瞬态双热线导热系数的测试

将封有聚α-烯烃合成油基磁性液体的两玻璃管放置于磁场中,为消除磁场力、重力所引起的磁性液体自然对流的影响,消除端部效应,研制了磁性液体在均匀磁场中瞬态双热线导热系数的实验测量系统,经与蒸馏水、乙醇标准样品的导热系数测量比较,实验装置有较高的测量精度。实验测量了不同方向的均匀磁场对不同体积浓度的磁性液体导热系数的影响。结果显示,当磁场方向与热通量方向一致时,磁场显著强化磁性液体的导热系数,其导热系数随磁场强度的增加而近似线性增加,且体积浓度越大增加量越大;当磁场方向与热通量方向垂直时,磁性液体的导热系数随磁场强度的变化不明显。     

Variable viscosity effect on hydromagnetic flow and heat transfer about a fluid underlying the axisymmetric spreading surface

Summary. The effect of variable viscosity on the flow and heat transfer about a fluid underlying the axisymmetric spreading surface in the presence of an axial magnetic field has been investigated. The viscosity of the fluid is assumed to vary as an inverse linear function of temperature and the magnetic field strength is inversely proportional to the radial coordinate. The partial differential equations, governing the present problem, have been transformed, by suitable similarity variables, into a system of ordinary differential equations. This system is solved numerically by the shooting technique. Numerical results are introduced in graphical form for different values of viscosity parameter, _r, and magnetic field parameter. In the presence of variable viscosity, an increase in Prandtl number leads to a rise in the velocity field. Generally, it leads to a fall in the temperature field. Both magnetic field and variable viscosity raise the heat transfer and suppress the fluid flow.     

On the problem of solar magnetic field generation

Generation of a nonaxisymmetric, predominantly toroidal magnetic field in the convective layer of the Sun is considered assuming that the process is related to a decrease in the energy consumed for the convective heat transfer. It is also assumed that the field is antisymmetric relative to the equatorial plane and the field strength varies in proportion to the sine or cosine of the azimuthal angle. It is shown that the field strength increases when both a radial gradient of the angular rotation velocity and a steady-state axisymmetric meridional substance circulation are present. A change in the direction of the poloidal motion (leading to a change in the overall field sign) is assumed to take place on attaining a limiting field strength. This is probably related to the excitation of the corresponding turbulent medium viscosity. In the proposed model, a restart of the field generation process is induced by a change in the magnetic field sign.     

Effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium

This paper deals with the theoretical investigation of the effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linear stability analysis theory and normal mode analysis method. For the case of stationary convection, the effect of various parameters like medium permeability, rotation, non-buoyancy magnetization, coupling parameter, spin diffusion parameter and micropolar heat conduction has been analyzed. The critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of magnetic parameter M₁ and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.     

Diffusion layer growth at Zn/Cu interface under uniform and gradient high magnetic fields

As a common phenomenon occurring in many material processes, diffusion may induce significant changes in composition and microstructure near the interface. In the present study, liquid/solid (Zn/Cu) interface diffusion experiments in high magnetic fields (up to 12 T) were conducted and the thickness changes of diffusion layer under different magnetic field conditions were examined. It was found that there were no noticeable effects of high magnetic fields on the formation of intermetallic phases at the interface. However, the magnetic flux density exerted a non-linear influence on the diffusion layer thickness. This phenomenon should be attributed to the effect of magnetic fields suppressing natural convection and inducing thermo-electromagnetic convection. In addition, the diffusion of Zn into Cu could be retarded by a magnetic field gradient. These results indicate that both the strength and the gradient of high magnetic fields can be used to control the diffusion behavior.     

Current-induced instabilities of hydromagnetic flow in a small gap between rotating conducting cylinders

Summary.  In considering the stability of an electrically conducting fluid between rotating perfectly conducting cylinders with a current-induced pressure gradient acting in the azimuthal direction and with an applied axial magnetic field, the assumption of small-gap approximation is made and the governing equations with respect to both axisymmetric and non-axisymmetric disturbances are solved by a direct numerical procedure. A parametric study covering wide ranges of Q, the Hartmann number which represents the strength of the axial magnetic field, and β, a parameter characterizing the ratio of current-induced and rotation velocities, is conducted for the situation where the outer cylinder is stationary and the inner cylinder is rotating. It is found that the stability characteristics are thoroughly different from those of the case of weakly conducting walls. The variation of the onset mode is shown in the (β, Q)-plane, and the transition of the corresponding neutral curves is discussed in detail. Results for the critical Taylor number and wavenumber pertaining to the critical disturbances are presented. The critical values of radial current density required for the onset of instability are also determined. Received May 8, 2002; revised November 11, 2002 Published online: May 8, 2003 The financial support for this work from National Science Council of Taiwan, ROC, through Grant No. NSC 89-2212-E-132-006 is gratefully acknowledged.     

Stability of magnetohydrodynamic stratified shear flows

Summary A study is made of the stability of a stratified shear flow in a perfectly conducting fluid in the presence of an external magnetic field aligned with the flow. A semi-circle theorem for the present hydromagnetic case is proved. The magnetic field is found to have a stabilizing effect on the flow. The Rayleigh-Taylor instability problem in a stratified conducting fluid is discussed. Finally, a study is made of the absorption of wave energy by the mean flow in the hydromagnetic case by considering a shear flow with an anti-symmetric velocity profile given byU=tanhz. Unlike the hydrodynamic case, it is found that, in the critical layer atU=0, the transfer of the wave energy to the mean flow occurs for any value of the Richardson number. This result implies again the stabilizing effect of the magnetic field on the shear flow.     

The onset of transient Marangoni convection in a liquid layer subjected to rotation about a vertical axis

In view of the interesting possibilities of controlling surface tension-driven convection, anticipated in space experiments involving fluid interfaces, the problem of the stability of a thin horizontal fluid layer subjected to rotation about a vertical axis, when the thermal (or concentration) gradient is not uniform is examined by linear stability analysis. Attention is focussed on the situation where the critical Marangoni number is greater than that for the case of uniform thermal gradient and the convection is not, in general, maintained. The case of adiabatic boundary condition is examined because it brings out the effect of surface tension at the free surfaces and allows a simple application of the Galerkin technique, which gives useful results. Numerical results are obtained for special cases and some general conclusions about the destabilizing effects of various basic temperature profiles and the stabilizing effect of coriolis force are presented. The results indicate that the most destabilizing temperature gradient is one for which the temperature gradient is a step function of the depth. Increase in Taylor number and the inverted parabolic basic temperature profile suppress the onset of convection.     

Convective instability in a gravity modulated anisotropic thermally stable porous medium

The effect of gravity modulation on the onset of convection in a horizontal fluid saturated porous layer in which the applied temperature gradient is opposite to that of gravity is investigated. The flow through the porous layer is governed by an extended form of Darcy’s law incorporating Brinkman’s boundary layer correction. The permeability and thermal conductivity of the medium are assumed to be transversely anisotropic. A stability analysis based on the method of small perturbations is performed using normal mode assumption. The study is focussed on low amplitude gravity modulation and the thresholds are found using Mathieu’s functions. The emergence of instability via the synchronous and subharmonic modes and the transition between them are discussed as a function of the physical parameters of interest.     

Combined radiation and mixed convection from a vertical wall with suction/injection in a non–Darcy porous medium

Summary. Mixed convection flow of an absorbing fluid up a uniform non–Darcy porous medium supported by a semi-infinite ideally transparent vertical flat plate due to solar radiation is considered. The external flow field is assumed to be uniform, the effect of the radiation parameter in the boundary layer adjacent to the vertical flat plate with fluid suction/injection through it is analyzed in both aiding and opposing flow situations. It is observed that the similarity solution is possible only when the fluid suction/injection velocity profile varies as x^–1/2. The velocity and temperature profiles in the boundary layer and the heat transfer coefficient are presented for selected values of the parameters. It is observed that the Nusselt number increases with the increase in the radiation parameter and also when the value of the surface mass flux parameter moves from the injection to the suction region.     

Magnetofluiddynamic flow with a pressure gradient and fluid injection

Summary The nonlinear partial differential equation of motion for an incompressible fluid flowing over a flat plate under the influence of a magnetic field and a pressure gradient, and with or without fluid injection (or ejection) through the plate is transformed to a nonlinear, third order ordinary differential equation by using a stream function and a similarity transformation.The necessary boundary conditions are developed for flow with and without fluid injection (or ejection), and an example is presented to illustrate the solution to the flow problem.The controlling equation reduces to the well known Falkner-Skan equation when the magnetic field is zero, and if additionally the pressure gradient is zero, the equation reduces to the Blasius equation.