Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field

The two-dimensional, steady, laminar, forced and free convective boundary layer flow of a magnetic fluid over a semi-infinite vertical plate, under the action of a localized magnetic field, is numerically studied. The magnetic fluid is considered to be water-based with temperature dependent viscosity and thermal conductivity. The study of the boundary layer is separated into two cases. In case I the boundary layer is studied near the leading edge, where it is dominated by the large viscous forces, whereas in case II the boundary layer is studied far from the leading edge of the plate where the effects of buoyancy forces increase. The numerical solution, for these two different cases, is obtained by an efficient numerical technique based on the common finite difference method. Numerical calculations are carried out for the value of Prandl number Pr =  49.832 (water-based magnetic fluid) and for different values of the dimensionless parameters entering into the problem and especially for the magnetic parameter Mn, the viscosity/temperature parameter Θ _r and the thermal/conductivity parameter S*. The analysis of the obtained results show that the flow field is influenced by the application of the magnetic field as well as by the variation of the viscosity and the thermal conductivity of the fluid with temperature. It is hoped that they could be interesting for engineering applications.     

Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field

The two-dimensional, steady, laminar, forced and free convective boundary layer flow of a magnetic fluid over a semi-infinite vertical plate, under the action of a localized magnetic field, is numerically studied. The magnetic fluid is considered to be water-based with temperature dependent viscosity and thermal conductivity. The study of the boundary layer is separated into two cases. In case I the boundary layer is studied near the leading edge, where it is dominated by the large viscous forces, whereas in case II the boundary layer is studied far from the leading edge of the plate where the effects of buoyancy forces increase. The numerical solution, for these two different cases, is obtained by an efficient numerical technique based on the common finite difference method. Numerical calculations are carried out for the value of Prandl number Pr =  49.832 (water-based magnetic fluid) and for different values of the dimensionless parameters entering into the problem and especially for the magnetic parameter Mn, the viscosity/temperature parameter Θ _r and the thermal/conductivity parameter S*. The analysis of the obtained results show that the flow field is influenced by the application of the magnetic field as well as by the variation of the viscosity and the thermal conductivity of the fluid with temperature. It is hoped that they could be interesting for engineering applications.     

Spectrum of the Magnetic Schrödinger Operator in a Waveguide with Combined Boundary Conditions

We consider the magnetic Schrödinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann {For the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.submitted 11/05/04, accepted 21/09/04     

On maximum bifurcation delay in real planar singularly perturbed vector fields

In singularly perturbed vector fields, where the unperturbed vector field has a curve of singularities (a “critical curve”), orbits tend to be attracted towards or repelled away from this curve, depending on the sign of the divergence of the vector field at the curve. When at some point, this sign bifurcates from negative to positive, orbits will typically be repelled away immediately after passing the bifurcation point (“turning point”). Atypical behaviour is nevertheless observed as well, when orbits follow the critical curve for some distance after the turning point, before they repel away from it: a delay in the bifurcation is present. Interesting are systems that have a maximum bifurcation delay, i.e. there is a point on the critical curve beyond which orbits cannot stay close to the critical curve. This behaviour is known to appear in some systems in dimension 3 (see [E. Benoît (Ed.), Dynamic Bifurcations, in: Lecture Notes in Mathematics, vol. 1493, Springer-Verlag, Berlin, 1991]), and it is commonly believed that it is not an issue in (real) planar systems. Beside making the observation that it does appear in non-analytic planar systems, it is shown that whenever bifurcation delay appears, it has no non-trivial maximum for analytic planar vector fields. The proof is based on the notion of family blow-up at the turning point, on formal power series in terms of blow-up variables, the study of their Gevrey properties and analytic continuation of their Borel transform. These results complement existing results concerning the equivalence of local and global canard solutions in [A. Fruchard, R. Schäfke, Overstability and resonance, Ann. Inst. Fourier (Grenoble) 53 (1) (2003) 227–264].     

On the Dido Problem and Plane Isoperimetric Problems

This paper is a continuation of a series of papers, dealing with contact sub-Riemannian metrics on R³. We study the special case of contact metrics that correspond to isoperimetric problems on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at least locally. It is equivalent to studying the associated sub-Riemannian spheres of small radius. It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry). Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem of minimizing length, for a prescribed area), these results do not apply. Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities). In an addendum, we say a few words about: (1) the singularities that can appear in general for the Dido problem, and (2) the motion of particles in a nonvanishing constant magnetic field.     

Magnetic bottles for the Neumann problem: The case of dimension 3

The main object of this paper is to analyze the recent results obtained on the Neumann realization of the Schrödinger operator in the case of dimension 3 by Lu and Pan. After presenting a short treatment of their spectral analysis of keymodels, we show briefly how to implement the techniques of Helffer-Morame in order to give some localization of the ground state. This leaves open the question of the localization by curvature effect which was solved in the case of dimension 2 in our previous work and will be analysed in the case of dimension 3 in a future paper.     

On the shear boundary layer of tension plates

The present paper deals with a generic class of problems for plates subjected to loadings combining a high in-plane tension and a small lateral pressure. It develops the governing differential equations in the singular pertubation form, through the postulation of retaining only one of the Kirchhoff's assumptions, that the plate thickness in the boundary layer region is invariant. The solution by using the standard perturbation method is discussed. The postulation is justified when it is demonstrated that in the shear boundary layer the plate thickness is of higher-order smallness. The general method of solution by the standard perturbation technique is applied to an annular plate problem. Problems of different combinations of supports at the inner and the outer boundaries are solved. The case in which both edges are simply supported is presented as an illustration of the solution technique. In other cases results only are presented. The effect of support on the boundaries is also discussed. The shear effect is found to be most significant at a clamped edge. In the special geometry, it is possible to demonstrate that, when the condition on membrane force is not met as required in the general theory, thagnitude of the boundary layer changes. Specifically, the paper presents a case in which the membrhich the membrane force is zero at the inner edge.     

New examples of Riemannian g.o. manifolds in dimension 7

A Riemannian g.o. manifold is a homogeneous Riemannian manifold (M,g) on which every geodesic is an orbit of a one-parameter group of isometries. It is known that every simply connected Riemannian g.o. manifold of dimension ?5 is naturally reductive. In dimension 6 there are simply connected Riemannian g.o. manifolds which are in no way naturally reductive, and their full classification is known (including compact examples). In dimension 7, just one new example has been known up to now (namely, a Riemannian nilmanifold constructed by C. Gordon). In the present paper we describe compact irreducible 7-dimensional Riemannian g.o. manifolds (together with their “noncompact duals”) which are in no way naturally reductive.     

Numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via Lagrange multipliers

This paper deals with the finite element solution of the eddy current problem in a bounded conducting domain, crossed by an electric current and subject to boundary conditions appropriate from a physical point of view. Two different cases are considered depending on the boundary data: input current density flux or input current intensities. The analysis of the former is an intermediate step for the latter, which is more realistic in actual applications. Weak formulations in terms of the magnetic field are studied, the boundary conditions being imposed by means of appropriate Lagrange multipliers. The resulting mixed formulations are analyzed depending on the regularity of the boundary data. Finite element methods are introduced in each case and error estimates are proved. Finally, some numerical results to assess the effectiveness of the methods are reported.

     

Inverse Problems for the Pauli Hamiltonian in Two Dimensions

We prove in two dimensions that the set of Cauchy data for the Pauli Hamiltonian measured on the boundary of a bounded open subset with smooth enough boundary determines uniquely the magnetic field and the electrical potential provided that the electrical potential is small in an appropriate topology. This result has the immediate consequence, in the case that the magnetic potential and electrical potential have compact support, that we can determine uniquely the magnetic field and the electrical potential by measuring the scattering amplitude at a fixed energy provided that the electrical potential is small in an appropriate topology.     

Energy asymptotics for Type II superconductors

We study the Ginzburg–Landau functional in the parameter regime describing ‘Type II superconductors’. In the exact regime considered minimizers are localized to the boundary — i.e. the sample is only superconducting in the boundary region. Depending on the relative size of different parameters we describe the concentration behavior and give leading order energy asymptotics. This generalizes previous results by Lu–Pan, Helffer–Pan, and Pan.     

三维动边值问题的迎风差分方法

可压缩可混溶油、水三维渗流动边值问题的研究,对重建盆地发育中油气资源运移、聚集的历史和评估油气资源的勘探与开发有重要的价值, 其数学模型是一组非线性耦合偏微分方程组的动边值问题. 该文对有界域的动边值问题提出一类新的二阶修正迎风差分格式, 应用区域变换、 变分形式、能量方法、差分算子乘积交换性理论、高阶差分算子的分解、微分方程先验估计的理论和技巧, 得到了最佳 $l^2$ 误差估计结果. 该方法已成功应用到油资评估的数值模拟中. 它对这一领域的模型分析, 数值方法和软件研制均有重要的价值.     

Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field

Consider a compact manifold M with boundary ∂ M endowed with a Riemannian metric g and a magnetic field Ω. Given a point and direction of entry at the boundary, the scattering relation Σ determines the point and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M,∂ M,g,Ω) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined up to a natural equivalence by Σ. In the case that the magnetic field Ω is taken to be zero, this gives a new rigidity result in Riemannian geometry that is more general than related results in the literature.     

Investigation on effects of stochastic loading at the boundary under thermoelasticity with two relaxation parameters

The main objective of the present work is to study the responses of stochastic type mechanical distribution at the boundary of an elastic half space in the context of generalized thermoelasticity. In order to compare the results under the stochastic mechanical distribution, we have also considered the case of deterministic mechanical distribution prescribed at the boundary. The stochastic mechanical distribution is considered in such a way that it reduces to a deterministic type distribution as a special case. Laplace transform technique is used to solve the problem and its inversion is carried out by using asymptotic expansions valid for short times to obtain the solution of all the physical field variables like, stress and temperature distributions in the physical domain. Numerical results are found out to compare the effects of stochastic and deterministic load prescribed at the boundary of the elastic half space.     

Numerical simulations of thermal convection under the influence of an inclined magnetic field by using solenoidal bases

The effect of an inclined homogeneous magnetic field on thermal convection between rigid plates heated from below under the influence of gravity is numerically simulated in a computational domain with periodic horizontal extent. The numerical technique is based on solenoidal (divergence‐free) basis functions satisfying the boundary conditions for both the velocity and the induced magnetic field. Thus, the divergence‐free conditions for both velocity and magnetic field are satisfied exactly. The expansion bases for the thermal field are also constructed to satisfy the boundary conditions. The governing partial differential equations are reduced to a system of ordinary differential equations under Galerkin projection and subsequently integrated in time numerically. The projection is performed by using a dual solenoidal bases set such that the pressure term is eliminated in the process. The quasi‐steady relationship between the velocity and the induced magnetic field corresponding to the liquid metals or melts is used to generate the solenoidal bases for the magnetic field from those for the velocity field. The technique is validated in the linear case for both oblique and vertical case by reproducing the marginal stability curves for varying Chandrasekhar number. Some numerical simulations are performed for either case in the nonlinear regime for Prandtl numbers Pr = 0.05 and Pr = 0.1. Copyright © 2014 John Wiley & Sons, Ltd.     

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     

Effect of variable density on hydromagnetic mixed convection flow of a non-Newtonian fluid past a moving vertical plate

The effect of temperature-dependent density on MHD mixed convection flow of power-law fluid past a moving semi-infinite vertical plate for high temperature differences between the plate and the ambient fluid is studied. The fluid density is assumed to decrease exponentially with temperature. The usual Boussinesq approximations are not considered due to the large temperature differences. The surface temperature of the moving plate was assumed to vary according to a power-law form, that is, T_w(x) = T_∞ + Ax^γ. The fluid is permeated by a uniform magnetic field imposed perpendicularly to the plate on the assumption of small magnetic Reynolds number. A numerical shooting algorithm for two unknown initial conditions with fourth-order Runge–Kutta integration scheme has been used to solve the coupled non-linear boundary value problem. The effects of various parameters on the velocity and temperature profiles as well as the local skin-friction coefficient and the local Nusselt number are presented graphically and in the tabular form. The results show that application of Boussinesq approximations in a non-Newtonian fluid subjected to high temperature differences gives a significant error in the values of the skin-friction coefficient and the application of an external magnetic field reduces this error markedly in the case of shear-thickening fluid.     

Global well-posedness and existence of uniform attractor for magnetohydrodynamic equations

We study the global well-posedness and existence of uniform attractor for magnetohydrodynamic (MHD) equations. The hydrodynamic system consists of the Navier–Stokes equations for the fluid velocity and pressure coupled with a reduced from of the Maxwell equations for the magnetic field. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the magnetic field is subject to a time-dependent Dirichlet boundary condition. We first establish the global existence of weak and strong solutions to Equations (1.1)-(1.4). And at this stage, we further derive the existence of a uniform attractor for Equations (1.1)-(1.4).     

On the Ginzburg‐Landau critical field in three dimensions

We study the three‐dimensional Ginzburg‐Landau model of superconductivity. Several “natural” definitions of the (third) critical field, H, governing the transition from the superconducting state to the normal state, are considered. We analyze the relation between these fields and give conditions as to when they coincide. An interesting part of the analysis is the study of the monotonicity of the ground state energy of the Laplacian with constant magnetic field and with Neumann (magnetic) boundary condition in a domain Ω. It is proved that the ground state energy is a strictly increasing function of the field strength for sufficiently large fields. As a consequence of our analysis, we give an affirmative answer to a conjecture by Pan. © 2008 Wiley Periodicals, Inc.     

Effects of slip on steady Bödewadt flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady flow and heat transfer arising due to the rotation of a non-Newtonian fluid at a larger distance from a stationary disk is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The momentum equation gives rise to a highly nonlinear boundary value problem. Numerical solution of the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip, non-Newtonian fluid characteristics and the magnetic interaction parameter on the momentum boundary layer and thermal boundary layer are discussed in detail and shown graphically. It is observed that slip has prominent effects on the velocity and temperature fields.