Measuring magnetic liquid crests by radioscopy

We present a quantitative method, capable of measuring the three‐dimensional shape of a fluid surface. Utilizing the attenuation of x‐rays in magnetic liquids we are able to map amplitude and relief of large wave crests, a regime which is not accessible by the common optical methods. The technique is applied for measuring the static surface deformations of the Rosensweig instability. It will also be useful for measuring the surface profile of the Faraday instability in magnetic liquids, in standard fluids, in liquid metals, or even in granular material.     

Study of the effect of viscosity and homogeneous horizontal magnetic field on Rayleigh‐Taylor instability

Effect of the viscosity on Rayleigh‐Taylor instability for two contiguous semi‐infinite fluids, in presence of a homogeneous horizontal magnetic field permeating both fluids is investigated. These fluids are incompressible, are arranged in horizontal strata and infinitely conducting. Only the linear terms in the magnetohydrodynamic (MHD) equations are considered. The gravitational acceleration was constant. The dispersion relation that defines the growth rate σ for the system has been defined as a function of the physical parameters of the system and was solved numerically.     

Magnetoviscous potential flow analysis of Kelvin–Helmholtz instability with heat and mass transfer

A linear analysis of the Kelvin–Helmholtz instability of interface between two viscous and magnetic fluids has been carried out where there was heat and mass transfer across the interface while the fluids have been subjected to a constant magnetic field parallel to the streaming direction. The viscous potential flow theory has been used for the investigation. A dispersion relation has been obtained and a stability criterion is given by a critical value of relative velocity as well as the critical value of magnetic field. The resulting plots show the effect of various physical parameters such as wave number, viscosity ratio, ratio of magnetic permeabilities and heat transfer coefficient. It has been observed that heat and mass transfer has a destabilizing effect whereas the horizontal magnetic field stabilizes the system.     

Stability of superposed viscous-viscoelastic (Rivlin-Ericksen) fluids in the presence of suspended particles through a porous medium

The Rayleigh-Taylor instability of a Newtonian viscous fluid overlying a Rivlin-Ericksen viscoelastic fluid containing suspended particles in a porous medium is considered. As in both Newtonian viscous-viscoelastic fluids the system is stable in the potentially stable case and unstable in the potentially unstable case, this holds for the present problem also. The effects of a variable horizontal magnetic field and a uniform rotation are also considered. The presence of magnetic field stabilizes a certain wave-number band, whereas the system is unstable for all wave-numbers in the absence of the magnetic field for the potentially unstable configuration. However, the system is stable in the potentially stable case and unstable in the potentially unstable case for highly viscous fluids in the presence of a uniform rotation.     

Thermal instability in compressible fluids in the presence of rotation and magnetic field

The thermal instability of compressible fluids pervaded by a uniform rotation and a uniform magnetic field, separately, is considered. For $\text{(}\text{Cp}\text{g}\text{)β < 1}$, with Cp, g, and β denoting the specific heat at constant pressure, the acceleration due to gravity, and the uniform temperature gradient, respectively, the system is shown to be stable. The magnetic field as well as rotation introduces oscillatory modes in thermal instability of compressible fluids, which are completely missing for $\text{(}\text{Cp}\text{g}\text{)β > 1}$ in the absence of rotation or magnetic field. The sufficient conditions which do not allow overstable modes are obtained.     

Stability of hydromagnetic laminar flows in an inclined heated layer

Laminar flows of conducting fluids with an imposed magnetic field play an important role in many applications, for instance in geophysics, astrophysics, e.g. when dealing with solar winds, industry, biology, in metallurgy, in biofilms, etc. Also many engineering applications require heating at the boundaries. The inclination has been examined by some authors mainly in theoretical applications, geophysical studies, and materials processing. In Falsaperla et al. (Laminar hydromagnetic flows in an inclined heated layer, 2016) we have investigated analytical solutions of stationary laminar flows of an inclined layer filled with a hydromagnetic fluid heated from below and subject to the gravity field. In this article we study linear instability and nonlinear stability of some of the above solutions and investigate the critical stability/instability thresholds.     

On the Rayleigh-Taylor Instability for Two Uniform Viscous Incompressible Flows

The authors study the Rayleigh-Taylor instability for two incompressible immis- cible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transformation to Lagrangian coordinates, the perturbed equations in Eule- rian coordinates are transformed to an integral form and the two-fluid flow is formulated as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, both fluids being in （unstable） equilibrium is analyzed. By a general method of studying a family of modes, the smooth （when restricted to each fluid domain） solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces are constructed, thus leading to a global instability result for the linearized problem. Then, by using these pathological solutions, the global instability for the corresponding nonlinear problem in an appropriate sense is demonstrated.     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

Thermomagnetic convection in magnetic fluids influenced by a time-modulated magnetic field

Material– and flow properties of magnetic fluids can be influenced by applying an external magnetic field. In this work we will particularly consider the onset of convection in magnetic fluids which is influenced by a magnetic force. In a horizontal magnetic fluid layer the force arises if a temperature gradient and an external magnetic field is applied. The behaviour of the onset of convection is investigated for a static and a time–modulated magnetic field. For the case of a static magnetic field the onset of convection depends on the strength of the field and for a time–modulated magnetic field an additional dependence on the frequency of magnetic field variation is found. The experimental results presented here confirm in principle the theoretical predictions about the influence of static and time–modulated magnetic forces on the onset of convection. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of parametric modulation on the onset of thermomagnetic convection

Magnetic fluids – so called ferrofluids – are suspensions of nano-scaled particles in an appropriate carrier liquid. The flow properties of these fluids can be influenced by applying an external magnetic field. It is possible to introduce a magnetic force in a horizontal ferrofluid layer which is able to drive a convective flow. The magnetic force arises in the presence of an external magnetic field if a temperature gradient exists in the fluid gap. The behaviour of the onset of convection depends on the strength of the external magnetic field and on the temperature gradient. In this paper the onset of convection under the influence of time-modulated magnetic field has been investigated. The experimental results presented here show a shift in the onset of convection depending on the frequency of the external magnetic field. This behaviour confirms in principle the theoretical predictions which are also presented here. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability analysis of an interface between immiscible liquids in Hele-Shaw flow in the presence of a magnetic field

Hydrodynamic instabilities may occur when a viscous fluid is driven by a less viscous one through a porous medium. These penetrations are common in enhanced oil recovery, dendrite formation and aquifer flow. Recent studies have shown that the use of magnetic suspensions allow the external control of the instability. The problem is nonlinear and some further improvements of both theory and experimental observations are still needed and continue being a current source of investigation. In this paper we present a generalized Darcy law formulation in order to examine the growth of finger instabilities as a magnetic field is applied to the interface between the fluids in a Hele-Shaw cell. A new linear stability analysis is performed in the presence of magnetic effects and provides a stability criterion in terms of the non-dimensional physical parameters of the examined flow and the wavenumber of the finger disturbances. The interfacial tension inhibits small wavelength instabilities. The magnetic field contributes to the interface stability for moderate wavelength as it is applied parallel to the liquid-interface. In particular, we find an explicit expression, as a function of the susceptibility, for a critical angle between the interface and the magnetic field direction, in which its effect on the interface is neutral. We have developed a new asymptotic solution for the flow problem in a weak nonlinear regime. The first correction captures the second order nonlinear effects of the magnetic field, which tends to align the fingers with the field orientation and have a destabilizing effect. The analysis predicts that the non-linear effects at second order can counterbalance the first order stabilizing effect of a parallel magnetic field which results in a loss of effectiveness for controlling the investigated finger instabilities. The relevant physical parameters for controlling these finger instabilities are clearly identified by our non-dimensional analysis.     

On Stoke's problem in magnetohydrodynamics

Stoke's classic problem involving the impulsive motion of an infinite flat plate in an unbounded viscous incompressible fluid is investigated under the additional specification that the fluid is electrically conducting and the motion is developed in the presence of uniform transverse magnetic field. For the fluids with arbitrary magnetic Prandtl number, the compact expression for the skin friction coefficient at the plate is given in terms of exponential and error functions of complex arguments. For the fluids with unit magnetic Prandtl number, expressions for the induced magnetic field, velocity, current density and induced electric field in the viscous boundary layer region set up near the plate are obtained. The effect of the magnetic field on the skin friction is to make it approach the steady state faster than in nonmagnetic case.     

On Rivlin-Erickson elastico-viscous fluid heated and soluted from below in the presence of compressibility, rotation and hall currents

A layer of compressible, rotating, elastico-viscous fluid heated & soluted from below is considered in the presence of vertical magnetic field to include the effect of Hall currents. Dispersion relation governing the effect of viscoelasticity, salinity gradient, rotation, magnetic field and Hall currents is derived. For the case of stationary convection, the Rivlin-Erickson fluid behaves like an ordinary Newtonian fluid. The compressibility, stable solute gradient, rotation and magnetic field postpone the onset of thermosolutal instability whereas Hall currents are found to hasten the onset of thermosolutal instability in the absence of rotation. In the presence of rotation, Hall currents postpone/hasten the onset of instability depending upon the value of wavenumbers. Again, the dispersion relation is analyzed numerically & the results depicted graphically. The stable solute gradient and magnetic field (and corresponding Hall currents) introduce oscillatory modes in the system which were non-existent in their absence. The case of overstability is discussed & sufficient conditions for non-existence of overstability are derived.     

MHD boundary-layer flow of a micropolar fluid past a wedge with constant wall heat flux

The steady laminar magnetohydrodynamic (MHD) boundary-layer flow past a wedge with constant surface heat flux immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated in this paper. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity variables, and then they are solved numerically by means of an implicit finite-difference scheme known as the Keller-box method. Numerical results show that micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics.     

A general theoretical model for the magnetohydrodynamic flow of micropolar magnetic fluids. Application to Stokes flow

Many practical applications, which have an inherent interest of physical and mathematical nature, involve the hydrodynamic flow in the presence of a magnetic field. Magnetic fluids comprise a novel class of engineering materials, where the coexistence of liquid and magnetic properties provides us with the opportunity to solve problems with high mathematical and technical complexity. Here, our purpose is to examine the micropolar magnetohydrodynamic flow of magnetic fluids by considering a colloidal suspension of ferromagnetic material (usually non‐conductive) in a carrier magnetic liquid, which is in general electrically conductive. In this case, the ferromagnetic particles behave as rigid magnetic dipoles. Thus, the application of an external magnetic field, apart from the creation of an induced magnetic field of minor significance, will prevent the rotation of each particle, increasing the effective viscosity of the fluid and will cause the appearance of an additional magnetic pressure. Despite the fact that the general consideration consists of rigid particles of arbitrary shape, the assumption of spherical geometry is a very good approximation as a consequence of their small size. Our goal is to develop a general three‐dimensional theoretical model that conforms to physical reality and at the same time permits the analytical investigation of the partial differential equations, which govern the micropolar hydrodynamic flow in such magnetic liquids. Furthermore, in the aim of establishing the consistency of our proposed model with the principles of both ferrohydrodynamics and magnetohydrodynamics, we take into account both magnetization and electrical conductivity of the fluid, respectively. Under this consideration, we perform an analytical treatment of these equations in order to obtain the three‐dimensional effective viscosity and total pressure in terms of the velocity field, the total (applied and induced) magnetic field and the hydrodynamic and magnetic properties of the fluid, independently of the geometry of the flow. Moreover, we demonstrate the usefulness of our analytical approach by assuming a degenerate case of the aforementioned method, which is based on the reduction of the partial differential equations to a simpler shape that is similar to Stokes flow for the creeping motion of magnetic fluids. In view of this aim, we use the potential representation theory to construct a new complete and unique differential representation of magnetic Stokes flow, valid for non‐axisymmetric geometries, which provides the velocity and total pressure fields in terms of easy‐to‐find potentials, via an analytical fashion. Copyright © 2009 John Wiley & Sons, Ltd.     

Gauging magnetorotational instability

Previously (Z. Angew. Math. Phys. 57:615–622, 2006), we examined the axisymmetric stability of viscous resistive magnetized Couette flow with emphasis on flows that would be hydrodynamically stable according to Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum. A uniform axial magnetic field permeates the fluid. In this regime, magnetorotational instability (MRI) may occur. It was proved that MRI is suppressed, in fact no instability at all occurs, with insulating boundary conditions, when a term multipling the magnetic Prandtl number is neglected. Likewise, in the current work, including this term, when the magnetic resistivity is sufficiently large, MRI is suppressed. This shows conclusively that small magnetic dissipation is a feature of this instability for all magnetic Prandtl numbers. A criterion is provided for the onset of MRI.     

Gauging magnetorotational instability

Previously (Z. Angew. Math. Phys. 57:615–622, 2006), we examined the axisymmetric stability of viscous resistive magnetized Couette flow with emphasis on flows that would be hydrodynamically stable according to Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum. A uniform axial magnetic field permeates the fluid. In this regime, magnetorotational instability (MRI) may occur. It was proved that MRI is suppressed, in fact no instability at all occurs, with insulating boundary conditions, when a term multipling the magnetic Prandtl number is neglected. Likewise, in the current work, including this term, when the magnetic resistivity is sufficiently large, MRI is suppressed. This shows conclusively that small magnetic dissipation is a feature of this instability for all magnetic Prandtl numbers. A criterion is provided for the onset of MRI.     

Transport and instability for perfect fluids

Incompressible perfect fluids are described by the Euler equations. We provide a new simple proof for well-posedness for velocities in and linear and nonlinear instability results using transport techniques. The results have an important consequence: the topology of is too fine for interesting questions about large time behavior. Received: 14 September 2001 / Published online: 4 April 2002     

Numerical study of soliton-like surface configurations on a magnetic fluid layer in the Rosensweig instability

In the Rosensweig instability, a perpendicular, uniform magnetic field applied to a pool of magnetic fluid generates an ordered periodic pattern on the liquid–air interface when the field exceeds a critical threshold. Decreasing the field strength, a lower critical threshold is observed at which the developed pattern disappears. In this respect, the deformation of the free surface shows a hysteretic behaviour with changing field strength. Recently, in experiments, a novel soliton-like surface configuration has been generated within the hysteretic regime of the Rosensweig instability. The main objective of this paper is a numerical study of this new configuration by means of two models, an axisymmetric two-dimensional one and a fully three-dimensional one.     

On Nonlinear Rayleigh-Taylor Instabilities

Some of the mathematical properties of the interface between two incompressible inviscid and immiscible fluids with different densities under the influence of a constant gravity field 9 are investigated. The purpose of this paper is to prove that linearly unstable modes for Rayleigh-Taylor instabilities give birth to nonlinear instabilities for the full nonlinear system. The main ingredient is a general instability theorem in an analytic framework which enables us to go from linear to nonlinear instabilities.