Locomotion based on the control of the shape of magnetic fluid surfaces and of magnetizable media

The realization of locomotion based on the deformation of a free surface of a magnetic fluid layer in a traveling magnetic field is studied. A plane flow of an incompressible viscous magnetic fluid layer on a horizontal surface in a nonuniform magnetic field and a plane two-layers flow of incompressible viscous magnetic fluids between two parallel solid planes in a magnetic field is considered. Also the flow of an incompressible viscous magnetic fluid layer on a cylinder in a nonuniform magnetic field is an object of investigation. The deformation and the motion of a body made by a magnetizable polymer in an alternating magnetic field are experimentally studied. The cylindrical body (worm) which is located in a cylindrical tube is analyzed. These effects can be used in designing autonomous mobile robots without a hard cover. Such robots can be employed in clinical practice and biological investigations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of the inverse magnetotelluric sounding problem by means of a method using the synthesized magnetic field

A method of solution of the inverse magnetotelluric sounding problem is considered. The method uses the known frequency-dependent magnetic field on the Earth’s surface. The magnetic field on the Earth’s surface is found with the help of a special technique from the known impedance on the Earth’s surface. The method is applied to solve typical inverse problems and provides for efficient determination of final solutions.     

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.     

Free-surface instabilities during electromagnetic shaping of liquid metals

Electromagnetic shaping of free surfaces of liquid metals is a well-known EPM technology used in a couple of metallurgic processes like cold crucible melting, semi-levitation, and electromagnetic slit sealing, among others. However, the stability of such free surfaces is the most important problem and stability control is crucial for success. Within this context we investigate experimentally the stability behavior of liquid metal free surfaces submitted to a high-frequency magnetic field. In this case, the induced Lorentz forces act as an electromagnetic pressure directly on the free surface of the liquid met al. We consider three experimental model configurations: (i) Sessile liquid metal drop (ii) liquid metal ring, and (iii) liquid metal disc. In each model experiment, upon increasing the feeding current beyond a certain threshold value, I_C, we observe that the initial surface contour becomes unstable resulting in (i) drop oscillations (ii) electromagnetic pinching and (iii) static disc deformations. In each configuration the threshold value depends in a similar manner on the frequency of the applied magnetic field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On unsteady MHD flow near an infinite flat wall

The flow of an electrically conducting incompressible viscous fluid near an infinite flat wall in the presence of a magnetic field when the free stream velocity is to be expressed in power series of √ t is considered. The effect of the magnetic field strength on the velocity field induced magnetic field and skin friction is analyzed for two particular cases of the free stream velocity.     

Waves in a Perfectly Conducting Fluid Filling a Half-Space

The constant, maximal, energy preserving boundary conditionsfor the equations of magnetohydrodynamics in a perfectly conductinghalf-space give rise to two essentially different selfadjointoperators in the case when the external magnetic field is orthogonalto the boundary and exactly one such operator when the externalfield is parallel to the boundary. Neither of these problemsadmits surface waves. For a normalized external field, the generalizedeigenfunction expansion is given below. It is shown that, inthe second case, the modes are not coupled by the boundary,while for only one boundary condition for the orthogonal fieldis the wave motion essentially that of free space (in the sensethat solutions are delivered by the group which determines solutionsfor the free space problem for special initial data). The Alvnwave in the parallel field case acts as a grazing wave. Asymptoticwave motion for perturbed problems (inhomogeneous media) isinvestigated as well as local decay of energy (this is not altogethertrivial, since the operators involved are never coercive evenoff their null spaces).     

Inverse problem of determining parameters of inhomogeneity of a body from acoustic field measurements

This work is devoted to development of methods for solving inverse problems in acoustics. Propagation of an acoustic field in a body located in the free space is considered. In the inverse problem, an iterative method for reconstructing the parameters of inhomogeneity of a body from a known acoustic field is applied. The theorem on convergence of the method is proven. Numerical results for inhomogeneous bodies of complex form are presented.     

Dynamics of an ideal liquid with a free surface in conformal variables

Problems of mathematical hydrodynamics with a free surface in conformal variables are studied. Analytical solvability in Hilbert space scale and numerical techniques of finding approximate solutions are considered. The lifetime for solutions, a constructive evaluation, and application of mathematical statistics to the solvability of nonlinear equations are studied. Multiple numerical experiments of the methods considered are shown. A lot of these methods can be applied not only to problems of mathematical hydrodynamics with a free surface but to abstract Cauchy–Kovalevskaya problems in Banach spaces scale as well. Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 28, Hydrodynamics, 2008.     

Suppression of Electromagnetically Induced Free-Surface Instabilities in Liquid Metals

The work presented is devoted to studies of foundations of the suppression techniques for electromagnetically induced instabilities of the free liquid metal surfaces. Such instabilities occur, for instance, in a number of technologies for requested refinement of materials like electron beam evaporation or levitation melting, in which the electromagnetic shaping technique is applied. The instability phenomenon in our case has following magnetohydrodynamic background. High-frequency magnetic field causes Lorentz force density generation within so-called skin-layer in the liquid metal, which works as a surface pressure in turn. With increase of the electromagnetic pressure a destabilizing mechanism progresses within the liquid metal domain, which causes a bifurcation. In this paper we discuss application alternating magnetic fields to increase the bifurcation threshold by passive and active redistribution of the Lorentz force density in the near-surface layers of the liquid metal domain. A novel stabilization system on the basis of a local electromagnetic influence on the liquid metal surface in the high frequency magnetic field is presented. The experimental results are discussed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear and weakly nonlinear analysis of the Rosensweig instability

The Rosensweig instability of a layer of magnetic fluid with a free surface manifests itself as a stationary pattern of peaks. The pattern is characterized by a wave vector q whose absolute value gives the wave number q = | q |. Within the frame of a linear stability theory a quantitative analysis of the dependence of the wave number on the strength of the magnetic field is presented. The method of multiple scale analysis is applied for a weakly nonlinear analysis of the Rosensweig instability. For magnetic inductions above the critical value, the stability of different patterns are discussed with respect to the relative permeability of the magnetic fluid and the layer thickness.     

Computational treatment of free convection effects on perfectly conducting viscoelastic fluid

We introduced a magnetohydrodynamic model of boundary-layer equations for a perfectly conducting viscoelastic fluid. This model is applied to study the effects of free convection currents with one relaxation time on the flow of a perfectly conducting viscoelastic fluid through a porous medium, which is bounded by a vertical plane surface. The state space approach is adopted for the solution of one-dimensional problems. The resulting formulation together with the Laplace transform technique is applied to a thermal shock problem and a problem for the flow between two parallel fixed plates, both without heat sources. Also a problem for the semi-infinite space in the presence of heat sources is considered. A discussion of the effects of cooling and heating on a perfectly conducting viscoelastic fluid is given. Numerical results are illustrated graphically for each problem considered.     

Biomagnetic viscoelastic fluid flow over a stretching sheet

Of concern in this paper is an investigation of biomagnetic flow of a non-Newtonian viscoelastic fluid over a stretching sheet under the influence of an applied magnetic field generated owing to the presence of a magnetic dipole. The viscoelasticity of the fluid is characterised by Walter’s B fluid model. The applied magnetic field has been considered to be sufficiently strong to saturate the ferrofluid. The magnetization of the fluid is considered to vary linearly with temperature as well as the magnetic field intensity. The theoretical treatment of the physical problem consists of reducing it to solving a system of non-linear coupled differential equations that involve six parameters, which are solved by developing a finite difference technique. The velocity profile, the skin-friction, the wall pressure and the rate of heat transfer at the sheet are computed for a specific situation. The study shows that the fluid velocity increases as the rate of heat transfer decreases, while the local skin-friction and the wall pressure increase as the magnetic field strength is increased. It is also revealed that fluid viscoelasticity has an enhancing effect on the local skin-friction. The study will have an important bearing on magnetic drug targeting and separation of red cells as well as on the control of blood flow during surgery.     

Numerical modeling of the equilibrium shapes of a ferrofluid drop in an external magnetic field

A numerical solution strategy for calculating equilibrium free surfaces of a magnetic fluid under the action of a magnetic field is proposed and applied to determine shapes of a linear magnetisable ferrofluid drop in a uniform magnetic field. Hysteresis phenomena for the drop deformation and the drop shapes with ends, close to conical, were observed numerically. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical solution of magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface

A new kind of analytic technique, namely the homotopy analysis method (HAM), is employed to give an explicit analytical solution of the steady two-dimensional stagnation-point flow of an electrically conducting power-law fluid over a stretching surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. A uniform transverse magnetic field is applied normal to the surface. An explicit analytical solution is given by recursive formulae for the first-order power-law (Newtonian) fluid when the ratio of free stream velocity and stretching velocity is not equal to unity. For second and real order power-law fluids, an analytical approach is proposed for magnetic field parameter in a quite large range. All of our analytical results agree well with numerical results. The results obtained by HAM suggest that the solution of the problem under consideration converges.     

Flow of thin liquid film over a rough rotating disk in the presence of a transverse magnetic field

The development of a flow of a viscous conducting fluid over a rough spinning disk in the presence of a transverse magnetic field has been analysed for different patterns of surface roughness of the disk and different initial distributions of the height of the liquid lubricant. The numerical solution of the governing equation of motion subject to initial and boundary conditions has been obtained by a finite-difference method. The temporal evolution of the free surface of the fluid and the rate of retention of the liquid lubricant on the spinning disk have been obtained for different values of the two parameters M , the Hartmann number and N_ratio, the ratio of the surface tension effect to the centrifugation effect. In the absence of the magnetic field, the results have been observed to agree with those of [6]. It has been observed that the effect of surface roughness is to enhance the relative volume of the fluid retained on the spinning disk and this is further enhanced by the presence of the magnetic field.     

Effect of rotation on plane waves of generalized electro-magneto-thermoviscoelasticity with two relaxation times

A new model of the equations of generalized thermovisco-elasticity for a thermally, isotropic and electrically conducting half-space solid whose surface is subjected to a thermal shock is given. The formulation is applied to the generalized thermoelasticity based on the Green and Lindsay (GL) theory under the effect of rotation, where there is an initial magnetic field parallel to the plane boundary of the half-space. The medium is deformed because of thermal shock and due to the application of the magnetic field, it results an induced magnetic and electric fields in the medium. The normal mode analysis is used to obtain the expressions for the variables considered. The distributions of temperature, displacement, stress, induced magnetic and electric fields are represented graphically. Comparisons are made with the results predicted by the coupled theory (CD) in the presence and absence of rotation.     

Exact solutions for equilibrium configurations of the surface of a conducting fluid in a nonuniform magnetic field

We study the two-dimensional magnetic shaping problem for the situation where the free surface of a perfectly conducting fluid is deformed by the magnetic field of a system of linear current-carrying conductors. Equilibrium is achieved due to the balance of capillary and magnetic pressures. We obtain exact solutions of the problem using conformal map techniques. These solutions describe a system of two-dimensional dimples that appear on the initially flat surface of a liquid conductor under the action of a nonuniform magnetic field. We consider the case of two symmetrically located dimples in detail.     

Magnetic field in a turbulent conducting fluid

The problem of magnetic field in conducting turbulent, incompressible fluid is considered. The velocity of the fluid is taken to be independent of the magnetic field and is described by a Gaussian field, ‘white noise’ in time with smooth space correlation. The main result is that no fast dynamo (by which is meant almost sure exponential growth of magnetic field) can exist for an incompressible fluid when the magnetic viscosity is positive. For d = 2, sharper results are obtained; the magnetic field dies out when the magnetic viscosity is strictly positive. Furthermore, when d = 2, existence and characterization of invariant measure are given for d = 2 when the magnetic viscosity is zero. The results are compared to those discussed by Baxendale and Rosovskii in [2]     

Reconstruction of biologic tissue conductivity from noisy magnetic field by integral equation method

Magnetic resonance electrical impedance tomography (MREIT) is a new technique to recover the conductivity of biologic tissue from the induced magnetic flux density. This paper proposes an inversion scheme for recovering the conductivity from one component of the magnetic field based on the nonlinear integral equation method. To apply magnetic fields corresponding to two incoherent injected currents, an alternative iteration scheme is proposed to update the conductivity. For each magnetic field, the regularizing technique on the finite dimensional space is applied to solve an ill-posed linear system. Compared with the well-developed harmonic B_z method, the advantage of this inversion scheme is its stability, since no differential operation is required on the noisy magnetic field. Numerical implementations are given to show the convergence of the iteration and its validity for noisy input data.     

Effect of non-uniform magnetic field on crystal growth by floating-zone method in microgravity

The magnetic damping effect of the non-uniform magnetic field on the floating-zone crystal growth process in microgravity is studied by numerical simulation. The results show that the non-uniform magnetic field with designed configuration can effectively reduce the flow near the free surface and then in the melt zone. At the same time, the designed magnetic field can improve the impurity concentration non-uniformity along the solidification interface. The primary principles of the magnetic field configuration design are also discussed.