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.     

Propagation of nonlinear capillaary waves in an electrically conducting liquid

The article deals with the propagation of periodic capillary waves with finite amplitude on the surface of an electrically conducting liquid subjected to the effect of a magnetic field. It is shown that the evolution of wave packets is described by perturbed nonlinear Schrödinger equations. Its asymptotic solution is obtained, and it is established that the influence of MHD effects manifests itself in reduced frequency and amplitude of the propagating waves.Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 97–99, 1990.     

Magnetic and drift surfaces in stellarators

In this paper we study the structure of a stellarator field in toroidal geometry. A field line tracing code is developed to explore the structure of magnetic fields on the fine scale of the electron gyroradius p_e, so as to explain anomalous electron transport. The magnetic field is modelled by a simple analytic representation with finite number of parameters, so that we can integrate the field lines to a high accuracy. In a typical Heliac field we find that (i) most of the magnetic surfaces are well behaved on the fine scale, even when there is no two-dimensional symmetry and (ii) the width of an island, formed in the vicinity of a magnetic surface with rational rotational transform i = n/m, decays exponentially with m. Among those numerical studies, we have an example where the island width w is less than the electron gyroradius p_e for m greater than 17. This demonstrates that higher-order islands do not affect the electron transport. Our numerical results indicate that the anomalous electron transport observed in experiments may be due to the presence of an ambipolar electrostatic potential Φ. To reconfirm this proposition we compute the guiding center orbits of electrons and estimate the island widths of the drift surfaces that are swept out. We find that with a small electric potential depending on toroidal and poloidal angles, the drift surface island width w is an order of magnitude larger than that without the electric potential and decays exponentially at a slower rate. Since the drift step size is of the order of the maximum of p_e and w, the electron transport, which scales like the square of the step size, is enhanced when there is an electric potential.     

Propagation of nonlinear magnetoacoustic waves in a compressible medium of finite electrical conductivity

We study the one-dimensional propagation of weakly nonlinear waves in a compressible medium of finite electrical conductivity subjected to the action of a magnetic field. We obtain evolution equations that describe the wave processes under small and finite magnetic Reynolds numbers. It is shown that in a medium of finite conductivity the evolution of perturbations in a fluid is described by the modified Bürger's equation. We find the stationary and automodel solution of this equation and use them as the basis for analyzing the influence of effects of electrical conductivity on the structure of weak shock waves.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 73–76.     

Mixed-mode oscillation genealogy in a compartmental model of bone mineral metabolism

Summary A well-supported self-oscillating eight-compartment model has been proposed by Staub et al. to account for thein vivo rat calcium metabolism (Staub et al.,Am. J. Physiol. 254, R134–139, 1988). The nonlinear nucleus of this model is a three-compartment subunit which represents the dynamic autocatalytic processes of phase transition at the interface between bone and extracellular fluids. The organization of the temporal mixed-mode oscillations which successively appear as the calcium input is varied is analyzed. On one side of the bifurcation diagram, the generation of periodic trajectories with a single large amplitude oscillation is governed by homoclinic tangencies to small amplitude limit cycles and follows the universal sequence (U-sequence) given for the periodic solutions of unimodal transformations of the unit interval into itself. On the other side, the progressive appearance and interweaving of trajectories with multiple large amplitude oscillations per period is linked to homoclinic tangencies to large amplitude unstable cycles. The bifurcation sequence responsible for the temporal pattern generation has been analyzed by modeling the first return map of the differential system associated with the compartmental subunit. We establish that this genealogy does not follow the usual Farey treelike organization and that a comprehensive view of the resulting fractal bifurcation structure can be obtained from the unfolding of singular points of bimodal maps. These theoretical features can be compared with those reported in experiments on dissolution processes, and the extent to which the knowledge of the subunit bifurcation structure provides new conceptual insights in the field of bone and calcium metabolism is discussed.     

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]     

Instability of a planar liquid layer in an alternating longitudinal magnetic field with non-zero mean

The conducting liquid interface is found to undulate in an alternating magnetic field. It was shown earlier that ifM =B ₀ ²/μηω, B₀, ω, μ andη being the amplitude (complex) of the alternating longitudinal magnetic field imposed at the interface, the angular frequency of the field, the magnetic permeability and the viscosity respectively, and ifM _c was the critical value ofM then the planar layer was stable or unstable according asM < M _c orM > M _c. In this paper we have determined the stability criterion when in addition to the alternating longitudinal field there acts a uniform field in the same direction. After comparing our results with those obtained earlier, in the absence of the uniform field, we find that the additional uniform field has a significant destabilizing effect.     

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound‐soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.     

Damping of Periodic Waves in Physically Significant Wave Systems

Damping of periodic waves in the classically important nonlinear wave systems—nonlinear Schrödinger, Korteweg–deVries (KdV), and modified KdV—is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m=m(t) . The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m= 0 to soliton waves when m= 1 .     

Optimal Disturbance Growth in Watertable Flow

The linear stability of a liquid flow down an inclined plane is investigated. The equations governing the evolution of the disturbance are written in vector form where the dependent variables are the normal velocity and the normal vorticity. Similar to other shear flows, it is shown that there can be transient growth in the energy of a disturbance followed by an exponential decay although all eigenvalues predict decay only. Parameter studies reveal that the maximum amplification occur for waves with no streamwise dependence and with a spanwise wavenumber of (1). The mechanism involved in this growth is analyzed. A free surface parameter (S) can be identified that is related to the extent gravity and surface tension influence the free surface. A scaling of the equations is studied which revealed that the maximum transient growth scales with the Reynolds number as Re² if k2 S Re² is kept constant, where k is the absolute value of the wavenumber vector. For small values of S exponential growth of free-surface modes also exists. In general, however, we have found that for moderate times the transient growth dominates over the exponential growth and that its characteristics are similar to the transient growth found in other shear flows, e.g., plane Poiseuille flow.     

Nonlinear Rayleigh-Taylor instability in magnetic fluids with uniform horizontal and vertical magnetic fields

A weakly nonlinear evolution of two dimensional wave packets on the surface of a magnetic fluid in the presence of an uniform magnetic field is presented, taking into account the surface tension. The method used is that of multiple scales to derive two partial differential equations. These differential equations can be combined to yield two alternate nonlinear Schrödinger equations. The first equation is valid near the cutoff wavenumber while the second equation is used to show that stability of uniform wave trains depends on the wavenumber, the density, the surface tension and the magnetic field. At the critical point, a generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. From the latter equation, the various stability crteria are obtained.     

A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship, while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements, and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates. Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999     

A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows

A model equation somewhat more general than Burger's equation has been employed by Herron [1] to gain insight into the stability characteristics of parallel shear flows. This equation, namely, u_t + uu_y = u_xx + u_yy, has an exact solution U(y) = ?2tanh y. It was shown in [1] that this solution is linearly stable, and more recently, Galdi and Herron [3] have proved conditional stability to finite perturbations of sufficiently small initial amplitude using energy methods. The present study utilizes multiple-scaling methods to derive a nonlinear evolution equation for a long-wave perturbation whose amplitude varies slowly in space and time. A transformation to the heat-conduction equation has been found which enables this amplitude equation to be solved exactly. Although all disturbances ultimately decay due to diffusion, it is found that subcritical instability is possible in that realistic disturbances of finite initial amplitude can amplify substantially before finally decaying. This behavior is probably typical of perturbations to shear flows of practical interest, and the results illustrate deficiencies of the energy method.     

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)     

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.      

Well‐posedness of the kinematic dynamo problem

In the framework of magnetohydrodynamics, the generation of magnetic fields by the prescribed motion of a liquid conductor in a bounded region is described by the induction equation, a linear system of parabolic equations for the magnetic field components. Outside G, the solution matches continuously to some harmonic field that vanishes at spatial infinity. The kinematic dynamo problem seeks to identify those motions, which lead to nondecaying (in time) solutions of this evolution problem. In this paper, the existence problem of classical (decaying or not) solutions of the evolution problem is considered for the case that G is a ball and for sufficiently regular data. The existence proof is based on the poloidal/toroidal representation of solenoidal fields in spherical domains and on the construction of appropriate basis functions for a Galerkin procedure. Copyright © 2012 John Wiley & Sons, Ltd.     

A Constitutive Model for Magnetostrictive Materials - Theory and Finite Element Implementation

A 3D macroscopic constitutive law for hysteresis effects in magnetostrictive materials is presented and a finite element implementation is provided. The novel aspect of the thermodynamically consistent model is an additive decomposition of the magnetic and the strain field in a reversible and an irreversible part. Employing the irreversible magnetic field is advantageous for a finite element implementation, where the displacements and magnetic scalar potential are the nodal degrees of freedom. To consider the correlation between the irreversible magnetic field and the irreversible strains a one-to-one relation is assumed. The irreversible magnetic field determines as internal variable the movement of the center of a switching surface. This controls the motion of the domain walls during the magnetization process. The evolution of the internal variables is derived from the magnetic enthalpy function by the postulate of maximum dissipation, where the switching surface serves as constraint. The evolution equations are integrated using the backward Euler implicit integration scheme. The constitutive model is implemented in a 3D hexahedral element which provides an algorithmic consistent tangent stiffness matrix. A numerical example demonstrates the capability of the proposed model to reproduce the ferromagnetic hysteresis loops of a Terfenol-D sample. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polarization of vacuum by an external magnetic field in the (2+1)-dimensional quantum electrodynamics with a nonzero fermion density

The Green's function of the Dirac equation with an external stationary homogeneous magnetic field in the (2+1)-dimensional quantum electrodynamics (QED ₂₊₁) with a nonzero fermion density is constructed. An expression for the polarization operator in an external stationary homogenous magnetic field with a nonzero chemical potential is derived in the one-loopQED ₂₊₁ approximation. The contribution of the induced Chern—Simons term to the polarization operator and the effective Lagrangian for the fermion density corresponding to the occupation of n relativistic Landau levels in an external magnetic field are calculated. An expression of the induced Chern—Simons term in a magnetic field for the case of a finite temperature and a nonzero chemical potential is obtained. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 132–151, October, 2000.     

Influence of variable heat flux on natural convection along a corrugated wall in porous media

In this work the coupled non-linear partial differential equations, governing the free convection from a wavy vertical wall under a power law heat flux condition, are solved numerically. For both Darcy and Forchheimer extended non-Darcy models, a wavy to flat surface transformation is applied and the governing equations are reduced to boundary layer equations. A finite difference scheme based on the Keller Box approach has been used in conjunction with a block tri-diagonal solver for obtaining the solution. Detailed simulations are carried out to investigate the effect of varying parameters such as power law heat flux exponent m, wavelength–amplitude ratio a and the transformed Grashof number Gr′. Both surface undulations and inertial forces increase the temperature of the vertical surface while increasing m reduces it. The wavy pattern observed in surface temperature plots, become more prominent with increasing m or a but reduces as Gr′ increases.     

Gauss‐Green theorem for weakly differentiable vector fields,sets of finite perimeter,and balance laws

We analyze a class of weakly differentiable vector fields F : ?ⁿ → ?ⁿ with the property that F ∈ L^∞ and div F is a (signed) Radon measure. These fields are called bounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure field F over the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ??^{N ? 1} on ?^N, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter E as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem for F holds on E. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N ? 1)‐dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.