Incompressible Magnetohydrodynamics Research Articles

Global well-posedness of the 3D incompressible MHD equations with variable density

This paper is dedicated to the global well-posedness of solution to the 3D incompressible magnetohydrodynamic (MHD) equations with variable density in a critical functional framework. It is proved that there exists a global and unique solution to the MHD equations if the initial (ρ0−1,u0,b0) is in M(Ḃp,13p−1(R3))×Ḃp,13p−1(R3)×Ḃp,13p−1(R3) with 1≤p<6, and satisfies some smallness assumption. Here M(Ḃp,13p−1(R3)) is the multiplier space of Besov space Ḃp,13p−1(R3). The main tool we have used is the Lagrangian approach, which enables us to solve the system through the basic contraction mapping theorem.

A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations

We present and analyze a semi-implicit finite element method (FEM) for the dynamical incompressible magnetohydrodynamics (MHD) equations. The finite element approximation is based on mixed conforming elements, where Taylor–Hood type elements are used for the Navier–Stokes equations and Nédélec edge elements are used for the magnetic equation. The divergence free conditions are weakly satisfied at the discrete level. Due to the use of Nédélec edge element, the proposed method is particularly suitable for problems defined on non-smooth and multi-connected domains. For the temporal discretization, we use a semi-implicit scheme which only needs to solve a linear system at each time step. Moreover, the linearized mixed FEM is energy conserving. We establish an optimal error estimate under a very low assumption on the exact solutions and domain geometries. Numerical results are provided to show its effectiveness and verify the theoretical analysis. In particular, a benchmark lid-driven cavity problem is provided to show that the proposed numerical method produces results as good as that of the ones which are divergence free.

An efficient two-level algorithm for the 2D/3D stationary incompressible magnetohydrodynamics based on the finite element method

In the paper, we develop a new two-level finite element algorithm for solving the 2D/3D stationary incompressible magnetohydrodynamics based on the Newton iterative method. This algorithm is consisting of solving one nonlinear system on a coarse mesh with mesh size H and two linearized problems with different loads on a fine mesh with mesh size h. Compared with existing work on the two-level method for the MHD model, our two-level method allows a much high order scaling between the coarse and fine grid sizes. Furthermore, stability and convergence of this present method are analyzed. Finally, the applicability and effectiveness of the present algorithm are illustrated by several numerical experiments.

Optimal convergence analysis of Crank–Nicolson extrapolation scheme for the three-dimensional incompressible magnetohydrodynamics

This paper considers a Crank–Nicolson extrapolation scheme based on mixed finite element method to solve the three-dimensional incompressible magnetohydrodynamics (MHD) equations. We prove that the fully discrete scheme is almost unconditionally stable and convergent, i.e., stable and convergent when the time step is less than or equal to a constant. By a new negative norm technique, the optimal error estimates in L2-norm are derived. Meanwhile, the numerical investigations provide a sufficient support for the theoretical analysis.

Direct numerical simulations of incompressible multiphase magnetohydrodynamics with phase change

A new phase change model has been developed for the simulation of incompressible multiphase magnetohydrodynamics based on the Volume-of-Fluid method. To decrease the pressure oscillations when large density contrasts are present between the liquid phase and the vapor phase, a smooth distribution of sharp mass transfer rate within a narrow region surrounding the interface is adopted, and a ghost-cell approach is used to impose the saturating temperature at the liquid–vapor interface when solving the energy equation. After that, the method has been implemented in an incompressible multiphase magnetohydrodynamics solver developed in our previous work (Zhang and Ni (2014) [3]). Moreover, when computing the electromagnetic fields, a cut-cell approach is implemented to keep the sharpness of the interface, which is treated as an electrically insulating boundary as it translates and deforms with the fluid. The phase change model has been verified for a series of one-dimensional, two-dimensional and three-dimensional problems, while the numerical results agree well with either the theoretical solutions or the experimental data. In particular, by simulating the vapor bubble rising in superheated liquid under nonzero gravity in presence of external magnetic field, the magnetohydrodynamics effect on the vaporization of the rising bubble is investigated and we observe the magnetic fields to suppress the vapor bubble growth during the phase change. At last, both two-dimensional and three-dimensional film boiling simulations are conducted, which show good qualitative agreement with heat transfer correlations, and the vapor bubble is found to elongate along the direction of the magnetic field during its growth, moreover, the time instant for the vapor bubble to detach from the film is also delayed.

A parallel finite element algorithm for nonstationary incompressible magnetohydrodynamics equations

PurposeThe purpose of this paper is to design a parallel finite element (FE) algorithm based on fully overlapping domain decomposition for solving the nonstationary incompressible magnetohydrodynamics (MHD).Design/methodology/approachThe fully discrete Euler implicit/explicit FE subproblems, which are defined in the whole domain with vast majority of the degrees of freedom associated with the particular subdomain, are solved in parallel. In each subproblem, the linear term is treated by implicit scheme and the nonlinear term is solved by explicit one.FindingsFor the algorithm, the almost unconditional convergence with optimal orders is validated by numerical tests. Some interesting phenomena are presented.Originality/valueThe proposed algorithm is effective, easy to realize with low communication costs and preferred for solving the strong nonlinear MHD system.

Note on global regularity of 2D regularized MHD equations with zero magnetic diffusion

In this paper, we establish the global regularity of the Cauchy problem of the two-dimensional regularized incompressible magnetohydrodynamics equations with the Laplacian dissipation in the velocity equation and zero magnetic diffusion.

A Regularity Criterion for the Density-Dependent Magnetohydrodynamics System in Critical Besov Spaces

In this paper, we study the Cauchy problem for incompressible density-dependent magnetohydrodynamics system in $$\mathbb R^N\ (N \ge 2)$$ . We prove the regularity criterion of solutions in the critical Besov spaces. For the proof of the results we use the commutator estimate and product estimate for nonlinear term.

Analyticity of Mild Solution for the 3D Incompressible Magneto-hydrodynamics Equations in Critical Spaces

The Cauchy problem for the 3D incompressible magneto-hydrodynamics equations in critcal spaces is considered. We first prove the global well-posedness of mild solution for the system in some time dependent spaces. Furthermore, we obtain analyticity of the solution.

On the Faedo-Galerkin method for a free boundary problem for incompressible viscous magnetohydrodynamics

The motion of incompressible magnetohydrodynamics (mhd) in a domain bounded by a free surface and coupled through it with external electromagnetic field is considered. Transmission conditions for electric currents and magnetic fields are prescribed on the free surface. Although we show the idea of a proof of local existence by the method of successive approximations, we are not going to prove neither local nor global existence of solutions. The existence of solutions of the linearized problems (the Stokes system for velocity and pressure and the linear transmission problem for the electromagnetic fields) is the main step in the proof of existence to the considered problem. This can be done either by the Faedo-Galerkin method or by the technique of regularizer. We concentrate our considerations to the Faedo-Galerkin method. For this we need an existence of a fundamental basis. We have to find the basis for the Stokes system and mhd system. We concentrate our considerations on the mhd system because this for the Stokes system is well known. We have to emphasize that the considered mhd system is obtained after linearization and transformation to the initial domains by applying the Lagrangian coordinates. This is the main aim of this paper.

Magnetohydrodynamic pressure drop and flow balancing of liquid metal flow in a prototypic fusion blanket manifold

Understanding magnetohydrodynamic (MHD) phenomena associated with the flow of electrically conducting fluids in complex geometry ducts subject to a strong magnetic field is required to effectively design liquid metal (LM) blankets for fusion reactors. Particularly, accurately predicting the 3D MHD pressure drop and flow distribution is important. To investigate these topics, we simulate a LM MHD flow through an electrically non-conducting prototypic manifold for a wide range of flow and geometry parameters using a 3D MHD solver, HyPerComp incompressible MHD solver for arbitrary geometry. The reference manifold geometry consists of a rectangular feeding duct which suddenly expands such that the duct thickness in the magnetic field direction abruptly increases by a factor rexp. Downstream of the sudden expansion, the LM is distributed into several parallel channels. As a first step in qualifying the flow, a magnitude of the curl of the induced Lorentz force was used to distinguish between inviscid, irrotational core flows and boundary and internal shear layers where inertia and/or viscous forces are important. Scaling laws have been obtained which characterize the 3D MHD pressure drop and flow balancing as a function of the flow parameters and the manifold geometry. Associated Hartmann and Reynolds numbers in the computations were ∼103 and ∼101-103, respectively, while rexp was varied from 4 to 12. An accurate model for the pressure drop was developed for the first time for inertial-electromagnetic and viscous-electromagnetic regimes based on 96 computed cases. Analysis shows that flow balance can be improved by lengthening the distance between the manifold inlet and the entrances of the parallel channels by utilizing the effect of flow transitioning to a quasi-two-dimensional state in the expansion region of the manifold.

On regularity criteria of the 2D generalized MHD equations

In this paper, we consider the two-dimensional incompressible generalized magnetohydrodynamics equations. We obtain some regularity criteria to the corresponding equations in terms of the current density j or the vorticity ω. Our approach is based on a key observation on the nice structure of the incompressible magnetohydrodynamics equations.

Analytic Current–Vortex Sheets in Incompressible Magnetohydrodynamics

In this paper, we address the problem of current-vortex sheets in ideal incompressible magnetohydrodynamics. More precisely, we prove a local-in-time existence and uniqueness result for analytic initial data using a Cauchy-Kowalevskaya theorem.

Global well-posedness of the MHD equations via the comparison principle

In this paper, we prove the global well-posedness of the incompressible magneto-hydrodynamics (MHD) equations near a homogeneous equilibrium in the domain $\mathbb~R^k\times\mathbb{T}^{d-k},~d\geq2,k\geq1$ by using the comparison principle and constructing the comparison function.

Global solutions for the 3-D incompressible nonhomogeneous MHD equations

In this article, we establish the global well-posedness to the 3-D incompressible non-homogeneous magnetohydrodynamics (MHD) equations for a class of large velocity fields and magnetic fields in critical Besov spaces. We improve recent results by Chen et al. (2015) and Zhai et al. (2015) by extending the validity range of the parameters related to the integrability from 1<q≤p<6 to the seemingly optimal range 1<p,q<6. We also relax their assumption on exponential norm of the initial data by replacing it by a more natural polynomial condition.

Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class

We study the zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic (MHD) equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the resistivity coefficient, for the solutions to the viscous incompressible MHD equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible MHD equations converge to that of the ideal incompressible MHD equations as the viscosity and resistivity coefficients go to zero. Moreover, the convergence rate is also given.

Travelling Wave Solutions for Three Dimensional Incompressible MHD Equations

In this paper, the solutions of three dimensional incompressible magnetohydrodynamics (MHD) equations are obtained by using sin method and Riccati auxiliary equation. This paper obtains the soliton solutions by the aid of software Mathematica.

Local free boundary problem for viscous non-homogeneous incompressible magnetohydrodynamics

Local free boundary problem for viscous non-homogeneous incompressible magnetohydrodynamics

Local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow

Based on the partition of unity method (PUM), a local and parallel finite element method is designed and analyzed for solving the stationary incompressible magnetohydrodynamics (MHD). The key idea of the proposed algorithm is to first solve the nonlinear system on a coarse mesh, divide the globally fine grid correction into a series of locally linearized residual problems on some subdomains derived by a class of partition of unity, then compute the local subproblems in parallel, and obtain the globally continuous finite element solution by assembling all local solutions together by the partition of unity functions. The main feature of the new method is that the partition of unity provide a flexible and controllable framework for the domain decomposition. Finally, the efficiency of our theoretical analysis is tested by numerical experiments.

Theory and Transport of Nearly Incompressible Magnetohydrodynamics Turbulence. III. Evolution of Power Anistropy in Magnetic Field Fluctuations throughout the Heliosphere

Abstract A theoretical model that describes the evolution of the power anisotropy in the energy-containing and inertial ranges throughout the heliosphere is developed for three possibilities: (i) no in situ sources of turbulence; (ii) stream-shear sources of 2D and slab turbulence; and (iii) a fully driven turbulence model that includes both stream-shear driving and a pickup ion source of slab turbulence. At the inner boundary (1 au), we assume that the ratios of the 2D to slab fluctuating magnetic energy variances in the energy-containing range are 80:20, 70:30, 60:40, and 55:45. For case (i), in the energy-containing range increases monotonically throughout the heliosphere, whereas the inertial range ratio increases until ∼20 au and then decreases. For case (ii), the energy-containing range ratio increases initially and then remains approximately constant and ordered beyond ∼2 au, according to the inner boundary assumptions. The inertial range ratio for the 80:20 case increases with heliocentric distance, whereas for the 70:30, 60:40, and 55:45 cases, the rations increase between ∼2 to ∼10–20 au, and then generally decrease at larger heliocentric distances. For case (iii), the energy-containing and inertial range ratios increase initially, remain approximately constant and increase slightly, respectively, and then decrease more rapidly between ∼8 and 30 au, and more gradually thereafter, approaching a ratio of ∼1 at 75 au. We present preliminary results that show the power anisotropy in magnetic field fluctuations observed by Ulysses spacecraft increasing with heliocentric distance from ∼1.5 to 4.5 au.