Stabilized finite element formulation applied to the kinematic Ponomarenko dynamo problem

A stabilized finite element (B, q) formulation is developed to solve the kinematic dynamo problem. As a test case, we solve the induction equation for a given solid body helical flow, embedded in a cylindrical conducting shell. This problem corresponds to the well-known Ponomarenko dynamo. It has the interesting property to have an exact dispersion relation giving the magnetic growth rate as a function of the flow properties. Therefore, it is a good benchmark to test our kinematic dynamo code. We calculated the dynamo threshold and plotted the geometry of the generated magnetic field. We also evaluated the residual error due to our stabilized formulation.     

On kinematic method in shakedown theory: I. Duality of extremum problems

A kinematic method for determining the safety factor in shakedown problems is developed. An upper bound kinematic functional is defined on a set of kinematically admissible time-independent velocity fields. Every value of the functional is an upper bound for the safety factor. Using convex analysis methods, conditions are established under which the infimum of the kinematic upper bounds equals the safety factor, in particular, conditions under which it is sufficient to consider only smooth velocity fields for the safety factor calculation. The method generalizes that recently proposed for the case of spherical yield surfaces by Kamenjarzh and Weichert. The extension covers a wide class of yield surfaces and inhomogeneous bodies. A shakedown problem for a beam subjected to a concentrated load is considered as an example.     

Certain problems concerning the origin and maintenance of cosmic magnetic fields

How regular magnetic fields in a plasma originated and are sustained is a question of considerable interest, since various objects in space such as spiral arms of galaxies, many stars, and planets have a regular magnetic field [1–3]. Several theoretical studies have been devoted to this theme [4–9]. However, because of the severe mathematical difficulties which arise in solving this problem, its study in the general form is still not possible, and its kinematic aspects are usually considered. In [5] a study is made of the induction of a toroidal magnetic field from the dipole field of a rotating spherical mass under the influence of radial convection. For the process to be undamped it is necessary to maintain a poloidal field, which according to the Cowling theorem may be induced by asymmetric motions from a toroidal field. Parker [6] suggested a specific model for the field increase by motions of the cyclonic type and indicated the possibility of self-regulation of the field maintenance process. Under certain conditions this process has a wave nature, the so-called dynamo-waves. In [7, 8] Braginskii considered the question of maintaining a quasi-stationary field by motions of the medium which differ little from axisymmetric motions. As a result of the large magnetic Reynolds numbers Re_m1 the field decays slowly, and a weak generation mechanism is adequate for maintaining the field.In [9] Birman and Schlüter suggested a fundamentally new mechanism for the formation of magnetic fields in a plasma as a result of the appearance under certain conditions of a vortical component of the mechanical electromotive force. It appears that this mechanism may be the reason for the appearance of cosmic magnetic fields.In the present paper we make a detailed study of the Birman mechanism for the origin of magnetic fields, and we derive the conditions which are imposed on the motion and the characteristics of the completely ionized plasma which are necessary for the appearance of a vortical emf. It is shown that in this case the axisymmetric motions of a highly conducting plasma may generate toroidal magnetic fields of considerable magnitude. Further, a kinematic solution is given of the problem of the determination of the magnetic field and currents which arise in the spherical plasma volume modeling a star. This volume rotates about some axis with a given angular velocity which depends on the spherical coordinates R and . Characteristics corresponding to the mean characteristics of the sun model are taken for specific calculations.     

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials(FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier's equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement,strains, and stresses are determined by the exact solution to Navier's equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.     

Magnetic field assisted fluidization： A modified Richardson-Zaki equation

Magnetic particles can be uniformly fluidized by coupling the gas flow with an externally imposed magnetic field. Interparticle forces generated by the magnetic field cause aggregation of the particles in chain-like structures preferentially oriented along the magnetic field lines. In the present paper, we study the implications of the formation of these special types of aggregates on the empirical Richardson-Zaki （RZ） equation, originally proposed to describe the expansion of fluidized beds of non-aggregated particles. We have addressed two important issues, namely the flow regime, which is a function of the size of the aggregates, and the effect of shape and orientation of the chain-like aggregates with respect to gas flow on fluid drag. We propose a modified RZ equation （MRZE） in which the velocity scale, given by the terminal settling velocity of the individual aggregates, and the RZ exponent are predetermined as a function of the chain length. The chain length depends on the ratio of the magnetic energy to gravitational energy, and is estimated from the magnetic field intensity, and particle magnetization, size and density. Predictions of the MRZE are successfully compared with published results in the literature on the expansion of magnetic particles in the presence of externally applied magnetic fields.     

Magnetohydrodynamic flow in an infinite channel

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.     

Kinematic dynamo by large scale tsunami waves in open ocean

Kinematic dynamo problem is studied with tsunami motion in open oceans. Using long wave approximation, a series solution of the dynamo problem is established with fast convergent rate based on a small parameter relating water wave dispersive effects. Taking solitary wave and single wave as typical tsunami wave models, the magnitude of tsunami induced magnetic field is estimated at the order of 10 nano Tesla (nT) just over sea level and 1 nT at altitudes of several hundreds kilometers, respectively, depending on the wave parameters as well as earth magnetic field. The space and time behavior of the magnetic field predicted by present model shows fairly similarity with the field data at Easter Island during 2010 Chile tsunami.     

Spontaneous rotation in the exact solution of magnetohydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0–2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found.     

Magnetohydrodynamic flow on a half-plane

We investigate the magnetohydrodynamic flow (MHD) on the upper, half of a non-conducting plane for the case when the flow is driven by the current produced by an electrode placed in the middle of the plane. The applied magnetic field is perpendicular to the plane, the flow is laminar, uniform, steady and incompressible. An analytical solution has been developed for the velocity field and the induced magnetic field by reducing the problem to the solution of a Fredholm's integral equation of the second kind, which has been solved numerically. Infinite integrals occurring in the kernel of the integral equation and in the velocity and magnetic field were approximated for large Hartmann numbers by using Bessel functions. As the Hartmann number M increases, boundary layers are formed near the non-conducting boundaries and a parabolic boundary layer is developed in the interface region. Some graphs are given to show examples of this behaviour.     

Boundary element method solution of MHD flow in a rectangular duct

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with an external magnetic field applied transverse to the flow has been investigated. The walls parallel to the applied magnetic field are conducting while the other two walls which are perpendicular to the field are insulators. The boundary element method (BEM) with constant elements has been used to cast the problem into the form of an integral equation over the boundary and to obtain a system of algebraic equations for the boundary unknown values only. The solution of this integral equation presents no problem as encountered in the solution of the singular integral equations for interior methods. Computations have been carried out for several values of the Hartmann number (1 ? M ? 10). It is found that as M increases, boundary layers are formed close to the insulated boundaries for both the velocity and the induced magnetic field and in the central part their behaviours are uniform. Selected graphs are given showing the behaviours of the velocity and the induced magnetic field.     

Motion of a cylinder in a viscous electroconductive medium with a magnetic field

The method of force sources is used to consider the planar problem of the motion of a circular cylinder in a viscous electroconductive medium with a magnetic field. The conventional and magnetic Reynolds numbers are assumed to be small. Expressions are obtained for the hydrodynamic reaction forces of the medium, acting on the moving cylinder. It is shown that as a result of the flow anisotropy in the medium, caused by the magnetic field, in addition to the resistance forces on bodies moving at an angle to the field, there are deflecting forces perpendicular to the velocity vector. The velocity field disturbances at great distances from the moving cylinder are determined.The problems of viscous electroconductive flow about solid bodies in the presence of a magnetic field constitute one of the divisions of magnetohydrodynamics. Motion of an electroconductive medium in a magnetic field gives rise to inductive electromagnetic fields and currents which interact with the velocity and pressure hydrodynamic fields in the medium [1, 2]. Under conditions of sufficiently strong interaction, the number of independent flow similarity parameters in MHD is considerably greater than in conventional hydrodynamics. This circumstance complicates the theoretical analysis of MHD flow about bodies, and therefore we must limit ourselves to consideration of individual particular flow cases.Here we consider the linear problem of the motion of an infinite circular cylinder in a viscous incompressible medium with finite electroconductivity located in a uniform magnetic field.There are many studies devoted to the flow of a viscous electroconductive medium with a magnetic field about solid bodies (see, for example, [3–5]). Because of this, some of the results obtained here include previously known results, which will be indicated below. In contrast to the cited studies, the examination is made by the method of force sources, suggested in [6]. This method permits obtaining integral equations for the distribution of the forces acting on the surface of the moving body. Their solution is obtained for small Reynolds and Hartmann numbers. Then the nature of the velocity disturbances at great distances from the body are determined. These results are compared with conventional viscous flow about a cylinder in the Oseen approximation.     

Shear and squeeze rheometry of suspensions of magnetic polymerized chains

We present the first experimental results on the magnetorheology of suspensions of non-Brownian magnetic ellipsoidal particles. These particles are made of spherical iron particles linked by polymers and are called polymerized chains. Steady shear, oscillatory shear, and oscillatory squeeze rheological tests have been performed. The rheological properties of the suspension of polymerized chains have been compared with those of the suspension of spherical iron particles. In shear flow, both suspensions develop nearly the same yield stress, while in squeeze flow, the yield stress is several times higher for the suspension of polymerized chains. We show that the squeezing force of a suspension of spherical particles is an increasing function of the magnetic field intensity at low magnetic fields but decreases dramatically at higher fields. Surprisingly, this phenomenon, attributed to cavitation or air entrainment, does not occur in the suspension of polymerized chains.     

Magnetic control of natural convection in the horizontal Bridgman configuration: symmetric and non-symmetric cross sections

This study deals with the electromagnetic damping of free-convective flows in cavities such as those used in the crystal growth horizontal Bridgman configuration. The cavities are filled with a dilute electrically conducting alloy and are subjected to a horizontal temperature gradient. The flow is steady and laminar under an external, vertical, transversal and uniform magnetic field. Several cross sections of the cavities were investigated and can either be centro-symmetric or not. The governing equations for such problems are two coupled partial differential equations, for the velocity and the induced magnetic fields, coupled with a third integral equation for mass conservation. A finite element method has been developed, and the numerical results for the variation of the velocity and the induced magnetic field in terms of the Hartmann number show a considerable decrease in convection intensity as the Hartmann number increases. Results also reveal the presence of the well-known Hartmann and parallel layers. For non-centro-symmetric sections, results show the way the flow reorganises into two cells as the Hartmann number increases.     

Calcul à la rupture en présence d'un écoulement à surface libre : construction de champs de pression approchésYield design for porous media subjected to unconfined flow: construction of approximate pressure fields

We consider the stability of a porous medium submitted to a steady-state flow with free-boundary. Assuming some hypotheses, it is possible to implement the kinematic method by using an approximate pressure field bounding the true pressure field from below. We are interested in finding such approximate pressure fields and in proving that they bound the true pressure field from below without knowing the true pressure field. We use fields which are solutions of a problem with relaxed conditions with regard to the real problem. Under a uniqueness condition of the solution of a weak formulation of the problem, such fields are lower bounds for the true pressure field. Finally, we give the example of a vertical dam. To cite this article: A. Corfdir, C. R. Mecanique 334 (2006).     

Hölder Continuity of Solutions to the Kinematic Dynamo Equations

We study the propagation of regularity of solutions to a three dimensional system of linear parabolic PDE known as the kinematic dynamo equations. The divergence free drift velocity is assumed to be at the critical regularity level with respect to the natural scaling of the equations.     

Numerical simulation and experimental verification of silicone oil flow over magnetic fluid under applied magnetic field

Two-layer flow of magnetic fluid and non-magnetic silicone oil was simulated numerically. The continuity equation, momentum equations, kinematic equation, and magnetic potential equation were solved in two-dimensional Cartesian coordinate. PLIC （piecewise linear integration calculation） VOF （volume of fluid） scheme was employed to track the free interface. Surface tension was treated via a continuous surface force （CSF） model that ensures robustness and accuracy. The influences of applied magnetic field, inlet velocity profile, initial surface disturbance of interface and surface tension were analyzed. The computed interface shapes at different conditions were compared with experimental observation.     

The influence of a magnetic field on the mechanical behavior of a fluid interface

This work focuses on a theoretical investigation of the shape and equilibrium height of a magnetic liquid–liquid interface formed between two vertical flat plates in response to vertical magnetic fields. The formulation is based on an extension of the so called Young–Laplace equation for an incompressible magnetic fluid forming a two-dimensional free interface. A first order dependence of the fluid susceptibility with respect to the magnetic field is considered. The formulation results in a hydrodynamic-magnetic coupled problem governed by a nonlinear second order differential equation that describes the liquid–liquid meniscus shape. According to this formulation, five relevant physical parameters are revealed in this fluid static problem. The standard gravitational Bond number, the contact angle and three new parameters related to magnetic effects in the present study: the magnetic Bond number, the magnetic susceptibility and its derivative with respect to the field. The nonlinear governing equation is integrated numerically using a fourth order Runge-Kutta method with a Newton–Raphson scheme, in order to accelerate the convergence of the solution. The influence of the relevant parameters on the rise and shape of the liquid–liquid interface is examined. The interface shape response in the presence of a magnetic field varying with characteristic wavenumbers is also explored. The numerical results are compared with asymptotic predictions also derived here for small values of the magnetic Bond number and constant susceptibility. A very good agreement is observed. In addition, all the parameters are varied in order to understand how the scales influence the meniscus shape. Finally, we discuss how to control the shape of the meniscus by applying a magnetic field.     

压电、压磁材料球对称问题的通解

研究压电、压磁材料在球坐标系下,不计体力、体电荷和体电流的情况下,由平衡方程、梯度方程、压电和压磁的本构方程导出应力、应变、位移、电位移、电场强度、电位势、磁感强度和磁位势各未知量的通解．考虑不同的边界条件,将其通解应用到应力、电学短路以及位移、电学开路和磁场分布的边界条件中,得到不同边界条件下问题的解。     

On centrifugal separation of particles of two different sizes

We consider flow in a centrifugal force field of a non-dilute suspension with particles or droplets of two sizes. The volume fraction and the velocity fields are determined assuming small convection and shear terms. The resulting flow field is quite different from that in a gravitational settling of a similar mixture. In particular, the volume fraction is a function of time and radius in the sectors separated by kinematic shocks and the settling velocity is a non-monotonic function of the particle size.