Viscous Magnetohydrodynamic Equations Research Articles

Revisiting shear stress tensor evolution: Nonresistive magnetohydrodynamics with momentum-dependent relaxation time

This study aims to develop second-order relativistic viscous magnetohydrodynamics (MHD) derived from kinetic theory within an extended relaxation time approximation (momentum/energy dependent) for the collision kernel. The investigation involves a detailed examination of shear stress tensor evolution equations and associated transport coefficients. The Boltzmann equation is solved using a Chapman-Enskog-like gradient expansion for a charge-conserved conformal system, incorporating a momentum-dependent relaxation time. The derived relativistic nonresistive, viscous second-order MHD equations for the shear stress tensor reveal significant modifications in the coupling with dissipative charge current and magnetic field due to the momentum dependence of the relaxation time. By utilizing a power law parametrization to quantify the momentum dependence of the relaxation time, the anisotropic magnetic field-dependent shear coefficients in the Navier-Stokes limit have been investigated. The resulting viscous coefficients are seen to be sensitive to the momentum dependence of the relaxation time. Published by the American Physical Society 2024

The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain

In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter ε∈(0,∞) , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are μ=O(1) and ν=O(εα) , and the orders of magnetic diffusion coefficients κ and σ are κ=O(1) and σ=O(εα) , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For H1 -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For H1 -initial data with additional regularity (∂zA~0,∂zB~0)∈Lp(Ω)(2<p<∞) , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal. 272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For H2 -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order O(εγ/2) , where γ=min{2,α−2} with α∈(2,∞) . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order O(ε) .

Stability and exponential decay for the compressible viscous non-resistive MHD system

How to construct global solutions of the compressible viscous magnetohydrodynamic (MHD) equations without magnetic diffusion even with small initial data in R3 or T3 is still an extremely challenging open problem. The difficulty comes from the lack of magnetic diffusion and the fact that solutions to inviscid equations generally grow in time. Motivated by this open problem, the present paper focuses on a special case of this MHD system in T3 when the magnetic field is vertical. We establish the global existence and uniqueness of smooth solutions to this system near a steady-state solution. In addition, the solution is shown to be stable and decay exponentially in time. The proof discovers and makes use of the smoothing and stabilizing effect of the steady magnetic field on the perturbations.

Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

Splash singularity for the free boundary incompressible viscous MHD

In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data.

Vanishing dissipation limit of solutions to initial boundary value problem for three dimensional incompressible magneto-hydrodynamic equations with transverse magnetic field

In this paper, we study the vanishing dissipation limit of solutions to the initial boundary value problem of three-dimensional incompressible viscous magnetohydrodynamic equations in the upper half space where the magnetic field is assumed to be transverse to the boundary. When the tangential component of the magnetic field is imposed the zero Neumann boundary condition, we can establish the uniform energy estimates of the solutions to the initial boundary value problem in the conormal Sobolev spaces even if the velocity field satisfies the no-slip boundary condition. Then the vanishing dissipation limit of solutions to the initial boundary value problem can be achieved in L∞ sense based on some compactness arguments. Compared with the previous results, our results show that the strong boundary layer can still be prevented by the transverse magnetic field even with the magnetic diffusion.

Viscous energy dissipation of kink waves due to phase mixing in twisted coronal flux tubes

ABSTRACT We have studied viscous energy damping of kink Magnetohydrodynamic (MHD) waves in weakly twisted magnetic flux tubes. The flux tube has been modelled as a homogeneous internal region with a straight magnetic field, surrounded by a radially non-uniform and magnetically twisted transitional layer embedded in a homogeneous ambient with a straight field. Using a modal expansion technique, we have solved an initial value problem for the incompressible viscous MHD equations and obtained spatio-temporal behaviour of the perturbations of the kink mode in linear regime. In the transitional layer, the perturbations are subject to phase mixing which enhances the viscous effects in the region. We show that magnetic twist can increase or decrease the efficiency of viscous damping of the phase-mixed perturbations in the non-uniform transitional layer. Using the temporal evolution of the total energy, we have obtained the viscous dissipation time as a power function of the Reynolds number. Our results show that magnetic twist could decrease or increase the viscous dissipation time.

Asymptotic behaviors of global solutions to the two-dimensional non-resistive MHD equations with large initial perturbations

This paper is concerned with the asymptotic behaviors of global strong solutions to the incompressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial perturbations in two-dimensional periodic domains in Lagrangian coordinates. First, motivated by the odevity conditions imposed in Pan et al. (2018) [24], we prove the existence and uniqueness of strong solutions under some class of large initial perturbations, where the strength of impressive magnetic fields depends increasingly on the H2-norm of the initial perturbation value of both the velocity and magnetic field. Then, we establish time-decay rates of strong solutions. Moreover, we find that H2-norm of the velocity decays faster than the perturbed magnetic field. Finally, by developing some new analysis techniques, we show that the strong solution converges in a rate of the field strength to a solution of the corresponding linearized problem as the strength of the impressive magnetic field goes to infinity. In addition, an extension of similar results to the corresponding inviscid case with damping is presented.

On the vanishing dissipation limit for the incompressible MHD equations on bounded domains

In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magnetohydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial boundary value problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher-order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD equation as the dissipations tend to zero.

Zero Kinematic Viscosity-Magnetic Diffusion Limit of the Incompressible Viscous Magnetohydrodynamic Equations with Navier Boundary Conditions

We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectly conducting conditions on the magnetic field in a smooth bounded domain Ω ⊂ ℝ3. It is shown that there exists a unique strong solution to the incompressible viscous magnetohydrodynamic equations in a finite time interval which is independent of the viscosity coefficient and the magnetic diffusivity coefficient. The solution is uniformly bounded in a conormal Sobolev space and W1,∞ (Ω) which allows us to take the zero kinematic viscosity-magnetic diffusion limit. Moreover, we also get the rates of convergence in L∞(0, T; L2), L∞(0, T; W1, p) (2 ≤ p < ∞), and L∞((0, T) × Ω) for some T > 0.

Relativistic non-resistive viscous magnetohydrodynamics from the kinetic theory: a relaxation time approach

We derive the relativistic non-resistive, viscous second-order magnetohydrodynamic equations for the dissipative quantities using the relaxation time approximation. The Boltzmann equation is solved for a system of particles and antiparticles using Chapman-Enskog like gradient expansion of the single-particle distribution function truncated at second order. In the first order, the transport coefficients are independent of the magnetic field. In the second-order, new transport coefficients that couple magnetic field and the dissipative quantities appear which are different from those obtained in the 14-moment approximation [1] in the presence of a magnetic field. However, in the limit of the weak magnetic field, the form of these equations are identical to the 14-moment approximation albeit with different values of these coefficients. We also derive the anisotropic transport coefficients in the Navier-Stokes limit.

The Decay Estimates for Magnetohydrodynamic Equations with Coulomb Force

In this article we consider the compressible viscous magnetohydrodynamic equations with Coulomb force. By spectral analysis and energy methods, we obtain the optimal time decay estimate of the solution. We show that the global classical solution converges to its equilibrium state at the same decay rate as the solution of the linearized equations.

Stability of Non-constant Equilibrium Solutions for Compressible Viscous and Diffusive MHD Equations with the Coulomb Force

We consider stability problems for the compressible viscous and diffusive magnetohydrodynamic (MHD) equations arising in the modeling of magnetic field confinement nuclear fusion. In the first part, we investigate the Cauchy problem to the barotropic MHD system. With the help of the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates, we prove that smooth solutions exist globally in time near the non-constant equilibrium solutions. We also obtain the asymptotic behavior of solutions when the time goes to infinity. The result shows that gradients of both the velocity and the magnetic field converge to the equilibrium solutions with the same norm $$ \Vert \cdot \Vert _{H^{s-3}}$$, while the density converge with stronger norm $$ \Vert \cdot \Vert _{H^{s-1}}$$. In the second part, the initial value problem to the full MHD system is studied. By means of the techniques of choosing a non-diagonal symmetrizer and elaborate energy estimates, we prove the existence and uniqueness of global solutions to the system when the initial data are close to the non-constant equilibrium states. We find that both the density and temperature converge to the equilibrium states with the same norm $$ \Vert \cdot \Vert _{H^{s-1}}$$. These phenomena on the charge transport show the essential relationship of the equations between the barotropic and the full MHD systems.

Mass concentration phenomenon to the two-dimensional Cauchy problem of the compressible Magnetohydrodynamic equations

This concerns the global strong solutions to the Cauchy problem of the compressible Magnetohydrodynamic (MHD) equations in two spatial dimensions with vacuum as far field density. We establish a blow-up criterion in terms of the integrability of the density for strong solutions to the compressible MHD equations. Furthermore, our results indicate that if the strong solutions of the two-dimensional (2D) viscous compressible MHD equations blowup, then the mass of the MHD equations will concentrate on some points in finite time, and it is independent of the velocity and magnetic field. In particular, this extends the corresponding Du's et al. results (Nonlinearity, 28, 2959-2976, 2015, [4]) to bounded domain in $ \mathbb{R}^2 $ when the initial density and the initial magnetic field are decay not too show at infinity, and Ji's et al. results (Discrete Contin. Dyn. Syst., 39, 1117-1133, 2019, [10]) to the 2D Cauchy problem of the compressible Navier-Stokes equations without magnetic field.

Up-down asymmetric exact solutions to the Navier-Stokes equations

Exact up-down asymmetric solutions to the Navier-Stokes equations for a viscous and incompressible fluid with time-dependent viscosity ν(t) are derived. Transformations of the exact solutions are defined that produce an infinite sequence of new solutions from each known one. The solutions are presented in terms of elementary functions and have no singularities. Three infinite-dimensional families of new exact axisymmetric unsteady solutions to the viscous magnetohydrodynamics equations are derived. Dynamics of vortex rings and vortex blobs is studied for some exact up-down asymmetric incompressible viscous fluid flows and viscous plasma flows.

Conditional regularity for the 3D MHD equations in the critical Besov space

In this paper, we establish regularity conditions of weak solutions for the 3D incompressible viscous magnetohydrodynamics (MHD) equations, and show that weak solutions to this system are actually regular if the pressure or the gradient of velocity and magnetic fields satisfies some integrable conditions with the logarithmic form in the critical Besov space Ḃ∞,∞−1.

High-speed shear-driven dynamos. Part 2. Numerical analysis

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (J. Fluid Mech., vol. 868, 2019, pp. 176–211). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous resistive MHD equations in plane Couette flow have been utilised. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll–streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfvén wave interaction theory proposed in the companion paper. When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of ‘self-sustained shear-driven dynamos’ at the zero external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in the companion paper. Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions.

Asymptotic limits of the isentropic compressible viscous magnetohydrodynamic equations with Navier-slip boundary conditions

In this paper, two kinds of asymptotic limits to the isentropic compressible viscous magnetohydrodynamic equations in a three-dimensional bounded domain Ω with Navier-slip boundary conditions are discussed. One is the incompressible limit with ill-prepared initial data, and the other is the combined inviscid and incompressible limit with well-prepared initial data. In the first case, we show that the weak solutions of the compressible viscous magnetohydrodynamic equations converge weakly to the weak solutions of the incompressible viscous magnetohydrodynamic equations provided the index of the fluid friction coefficient α1≥1 as the Mach number goes to 0. Moreover, the convergence of the velocity in L2(0,∞;L2(Ω)) is indeed strong under some geometrical assumptions on the domain Ω and α1≥12. In the second case, we show that the weak solution of the compressible viscous magnetohydrodynamic equations converges to the local strong solution of the ideal incompressible magnetohydrodynamic equations. Furthermore, the convergence rates are also obtained.

Global Existence and Optimal Decay Rate of the Compressible Magnetohydrodynamic Equation with Potential Force

In this paper, we consider the three dimensional compressible viscous magnetohydrodynamic equations (MHD) with the external potential force. We first derive the corresponding non-constant stationary solutions. Next, we show global wellposedness of the initial value problem for the three dimensional compressible viscous magnetohydrodynamic equations, provided that the initial data is close to the stationary solution. Finally, based on the elaborate energy estimates for the nonlinear system and Lp - Lq decay estimates of the linearized equation, we show the optimal convergence rates of the solution in Lq-norm with 2⩽q⩽6 and its first derivative in L2-norm when the initial perturbation is bounded in Lp-norm with 1⩽ p<6 5.

Global well-posedness for the 3D viscous nonhomogeneous incompressible magnetohydrodynamic equations

In this paper, we investigate the Cauchy problem for the 3D viscous nonhomogeneous incompressible magnetohydrodynamic equations in critical Besov spaces. We aim at proving the local and global well-posedness for respectively large and small initial data having critical Besov regularity, without assumptions of small density variation.