Incompressible Magnetohydrodynamics Research Articles

Effects of heat and chemical reactions on peristaltic flow of Newtonian fluid in a diverging tube with inclined MHD

AbstractThe problem of magnetohydrodynamics (MHD) in connection with peristaltic transport has attracted significant attention due to its wide range of engineering applications. In this paper, we have investigated the peristaltic flow of an incompressible MHD Newtonian fluid in an inclined non‐uniform tube in the presence of an inclined magnetic field and heat and mass transfer effects. The flow is investigated in a wave frame of reference moving with the velocity of the wave. The governing equations of a two‐dimensional fluid have been simplified under long wave length and low Reynolds number approximations. Exact solutions have been obtained for the velocity, temperature, and concentration profiles. The expression for pressure gradient is calculated using numerical integration. The graphical results are presented in order to interpret the various physical parameter of interest for five different wave forms. Trapping phenomena have been also discussed. Copyright © 2010 Curtin University of Technology and John Wiley & Sons, Ltd.

The inviscid and non-resistive limit in the cauchy problem for 3-D nonhomogeneous incompressible magneto-hydrodynamics

In this paper, the inviscid and non-resistive limit is justified for the local-in-time solutions to the equations of nonhomogeneous incompressible magneto-hydrodynamics (MHD) in ℝ 3. We prove that as the viscosity and resistivity go to zero, the solution of the Cauchy problem for the nonhomogeneous incompressible MHD system converges to the solution of the ideal MHD system. The convergence rate is also obtained simultaneously.

Well-posedness of the upper convected Maxwell fluid in the limit of infinite Weissenberg number

An iteration scheme is used to show the well-posedness of the initial-boundary value problem for incompressible hypoelastic materials, which arise as a high Weissenberg number limit of viscoelastic fluids. We first assume that the stress is a rank-one matrix T=qqT, q∈ℝn, and develop energy estimates to show that the problem is locally well-posed. This problem is related to incompressible ideal magnetohydrodynamics (MHD). We show that the general case T=CCT, C∈ℝn×n can be handled by a generalization of the method we developed. Copyright © 2010 John Wiley & Sons, Ltd.

On two-dimensional magnetic reconnection with nonuniform resistivity

In this paper, two theoretical approaches for the calculation of the rate of quasi-stationary, two-dimensional magnetic reconnection with nonuniform anomalous resistivity are considered in the framework of incompressible magnetohydrodynamics (MHD). In the first, ‘global’ equations approach, the MHD equations are approximately solved for a whole reconnection layer, including the upstream and downstream regions and the layer center. In the second, ‘local’ equations approach, the equations are solved across the reconnection layer, including only the upstream region and the layer center. Both approaches give the same approximate answer for the reconnection rate. Our theoretical model is in agreement with the results of recent simulations of reconnection with spatially nonuniform resistivity.

Nonlinear evolution of resistive tearing mode with sub‐Alfvenic shear flow

Nonlinear evolution of resistive tearing mode with different sub‐Alfvenic shear flow is investigated using incompressible magnetohydrodynamics simulations. The sub‐Alfvenic shear flow can have either stabilization or destabilization effects on the tearing mode development, which is mostly determined by the shear flow thickness (av). The shear flow tends to boost the tearing mode instability when the thickness of the shear flow is above av ∼ 0.35a (where a is the normalization unit). The boosting effect of the shear flow on the tearing mode becomes strongest at av ∼ 0.6a. But the strength of the boosting and suppressing effects is controlled by the asymptotic velocity (V0) of the shear flow. The suppressing and boosting effects of the shear flow increase with increase of the shear flow velocity. The kinetic energy associated with initial shear flow is reduced when the shear flow has a boosting effect on the tearing mode growth.

INHOMOGENEOUS NEARLY INCOMPRESSIBLE DESCRIPTION OF MAGNETOHYDRODYNAMIC TURBULENCE

The nearly incompressible theory of magnetohydrodynamics (MHD) is formulated in the presence of a static large-scale inhomogeneous background. The theory is an inhomogeneous generalization of the homogeneous nearly incompressible MHD description of Zank & Matthaeus and a polytropic equation of state is assumed. The theory is primarily developed to describe solar wind turbulence where the assumption of a composition of two-dimensional (2D) and slab turbulence with the dominance of the 2D component has been used for some time. It was however unclear, if in the presence of a large-scale inhomogeneous background, the dominant component will also be mainly 2D and we consider three distinct MHD regimes for the plasma beta ? 1, ? ~ 1, and? 1. For regimes appropriate to the solar wind (? 1, ? ~ 1), compared to the homogeneous description of Zank & Matthaeus, the reduction of dimensionality for the leading-order description from three dimensional (3D) to 2D is only weak, with the parallel component of the velocity field proportional to the large-scale gradients in density and the magnetic field. Close to the Sun, however, where the large-scale magnetic field can be considered as purely radial, the collapse of dimensionality to 2D is complete. Leading-order density fluctuations are shown to be of the order of the sonic Mach number and evolve as a passive scalar mixed by the turbulent velocity field. It is emphasized that the usual pseudosound relation used to relate density and pressure fluctuations through the sound speed as ?? = c 2 s ?p is not valid for the leading-order density fluctuations, and therefore in observational studies, the density fluctuations should not be analyzed through the pressure fluctuations. The pseudosound relation is valid only for higher order density fluctuations, and then only for short-length scales and fast timescales. The spectrum of the leading-order density fluctuations should be modeled as k ?5/3 in the inertial range, followed by a Bessel function solution K ?(k), where for stationary turbulence ? = 1, in the viscous-convective and diffusion range. Other implications for solar wind turbulence with an emphasis on the evolution of density fluctuations are also discussed.

A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics

We introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem, in two and three dimensions. The velocity field is discretized using divergence-conforming Brezzi–Douglas–Marini (BDM) elements and the magnetic field is approximated by curl-conforming Nédélec elements. The H 1-continuity of the velocity field is enforced by a DG approach. A central feature of the method is that it produces exactly divergence-free velocity approximations, and captures the strongest magnetic singularities. We prove that the energy norm error is convergent in the mesh size in general Lipschitz polyhedra under minimal regularity assumptions, and derive nearly optimal a priori error estimates for the two-dimensional case. We present a comprehensive set of numerical experiments, which indicate optimal convergence of the proposed method for two-dimensional as well as three-dimensional problems.

Strong magnetohydrodynamic turbulence with cross helicity

Magnetohydrodynamics (MHD) provides the simplest description of magnetic plasma turbulence in a variety of astrophysical and laboratory systems. MHD turbulence with nonzero cross helicity is often called imbalanced, as it implies that the energies of Alfvén fluctuations propagating parallel and antiparallel the background field are not equal. Recent analytical and numerical studies have revealed that at every scale, MHD turbulence consists of regions of positive and negative cross helicity, indicating that such turbulence is inherently locally imbalanced. In this paper, results from high resolution numerical simulations of steady-state incompressible MHD turbulence, with and without cross helicity are presented. It is argued that the inertial range scaling of the energy spectra (E±) of fluctuations moving in opposite directions is independent of the amount of cross helicity. When cross helicity is nonzero, E+ and E− maintain the same scaling, but have differing amplitudes depending on the amount of cross helicity.

Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations

We propose a convergent implicit stabilized finite element discretization of the nonstationary incompressible magnetohydrodynamics equations with variable density, viscosity, and electric conductivity. The discretization satisfies a discrete energy law, and a discrete maximum principle for the positive density, and iterates converge to weak solutions of the limiting problem for vanishing discretization parameters. A simple fixed point scheme, together with an appropriate stopping criterion is proposed, which decouples the computation of density, velocity, and magnetic field, and inherits the above properties, provided a mild mesh constraint holds. Computational studies are provided.

Conservative regularization of ideal hydrodynamics and magnetohydrodynamics

Inviscid, incompressible hydrodynamics and incompressible ideal magnetohydrodynamics (MHD) share many properties such as time-reversal invariance of equations, conservation laws, and certain topological features. In three dimensions, these systems may lead to singular solutions (involving vortex and current sheets). While dissipative (viscoresistive) effects can regularize the equations leading to bounded solutions to the initial-boundary value (Cauchy) problem which presumably exist uniquely, the time-reversal symmetry and associated conservation properties are certainly destroyed. The present work is analogous to (and suggested by) the Korteweg–de Vries regularization of the one-dimensional, nonlinear kinematic wave equation. Thus the regularizations applied to the original equations of hydrodynamics and ideal MHD retain conservation properties and the symmetries of the original equations. Integral invariants which generalize those known for the original systems are shown to imply bounded enstrophy. The regularization developed can also be applied to the corresponding dissipative models (such as the Navier–Stokes equations and the viscoresistive MHD equations) and may imply interesting regularity properties for the solutions of the latter as well. The models developed thus have intrinsic mathematical interest as well as possible applications to large-scale numerical simulations in systems where dissipative effects are extremely small or even absent.

First-Order System Least Squares for Incompressible Resistive Magnetohydrodynamics

Magnetohydrodynamics (MHD) is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles. Hence, the equations that describe the plasma form a nonlinear system that couples Navier-Stokes equations with Maxwell's equations. This paper shows that the first-order system least squares (FOSLS) finite element method is a viable discretization for these large MHD systems. To solve this system, a nested-iteration-Newton-FOSLS-AMG approach is taken. Most of the work is done on the coarse grid, including most of the linearizations. We show that at most one Newton step and a few V-cycles are all that are needed on the finest grid. Here, we describe how the FOSLS method can be applied to incompressible resistive MHD and how it can be used to solve these MHD problems efficiently. A 3D steady state and a reduced 2D time-dependent test problem are studied. The latter equations can simulate a “large aspect-ratio” tokamak. The goal is to resolve as much physics from the test problems with the least amount of computational work. We show that this is achieved in a few dozen work units or fine grid residual evaluations.

Model of Two-Fluid Reconnection

A theoretical model of quasistationary, two-dimensional magnetic reconnection is presented in the framework of incompressible two-fluid magnetohydrodynamics. The results are compared with recent numerical simulations and experiment.

Investigating the effects of the important parameters on magnetohydrodynamics flow and heat transfer over a stretching sheet

Sheet extrusion is a technique for making flat plastic sheets. Thermoplastic sheet production is a significant sector of plastics processing. In order to improve the physical properties of the sheet, it is consequential to study heat and fluid flow over the sheet as it solidifies. To understand the process, in the present work, the two-dimensional steady-state incompressible viscoelastic boundary layer magnetohydrodynamics flow and heat transfer over a stretching sheet have been studied in the presence of electric and magnetic fields. The highly non-linear momentum and heat transfer equations are solved analytically using the homotopy analysis method. The effect of various physical parameters including the viscoelastic parameter, the Prandtl number, the local Reynolds number, the Eckert number, the Hartmann number, and the local electric parameter on momentum and heat transfer characteristics have been studied. Ultimately some results could be found from this analytical treatment, such as the following: by increasing the values of the local Reynolds number and the electric parameter, the value of the skin friction coefficient is decreased. If the fluid is viscoelastic, the electric field is present, and the Prandtl number is lower, then there would be an obvious decrease of temperature near the boundary sheet. The other result is in the presence of the magnetic field, the effect of the electric field is to decrease the temperature near the stretching boundary.

The Cauchy problem for the 2D magnetohydrodynamic- [formula omitted] equations

In this paper, we consider the Cauchy problem for the 2D incompressible magnetohydrodynamics- α (MHD- α ) equations. We obtain the global solution for the 2D incompressible MHD- α simulation model in the fractional index Sobolev space, and prove that the incompressible MHD- α equations reduce to the incompressible homogeneous MHD equations as α → 0 , and the solution of MHD- α equations will converge to the weak solution of the corresponding MHD equations. Moreover, it is convenient to construct a numerical algorithm based on an iteration scheme in our proof.

EXACT VECTORIAL LAW FOR AXISYMMETRIC MAGNETOHYDRODYNAMICS TURBULENCE

Three-dimensional incompressible magnetohydrodynamics turbulence is investigated under the assumptions of homogeneity and axisymmetry. We demonstrate that previous works of Chandrasekhar may be improved significantly by using a different formalism for the representation of two-point correlation tensors. From this axisymmetric kinematics, the equations ` al a von K´´ an–Howarth are derived from which an exact relation is found in terms of measurable correlations. The relation is then analyzed in the particular case of a medium permeated by an imposed magnetic field B0. We make the ansatz that the development of anisotropy implies an algebraic relation between the axial and the radial components of the separation vector r and we derive an exact vectorial law which is parameterized by the intensity of anisotropy. The critical balance proposed by Goldreich & Sridhar is used to fix this parameter and to obtain a unique exact expression; the particular limits of correlations transverse and parallel to B0 are given for which simple expressions are found. Predictions for the energy spectra are also proposed by a straightforward dimensional analysis of the exact law; it gives a stronger theoretical background to the heuristic spectra previously proposed in the context of the critical balance. We also discuss the wave turbulence limit of an asymptotically large external magnetic field which appears as a natural limit of the vectorial relation. A new interpretation of the anisotropic solar wind observations is eventually discussed.

The Richtmyer–Meshkov instability in magnetohydrodynamics

In ideal magnetohydrodynamics (MHD), the Richtmyer–Meshkov instability can be suppressed by the presence of a magnetic field. The interface still undergoes some growth, but this is bounded for a finite magnetic field. A model for this flow has been developed by considering the stability of an impulsively accelerated, sinusoidally perturbed density interface in the presence of a magnetic field that is parallel to the acceleration. This was accomplished by analytically solving the linearized initial value problem in the framework of ideal incompressible MHD. To assess the performance of the model, its predictions are compared to results obtained from numerical simulation of impulse driven linearized, shock driven linearized, and nonlinear compressible MHD for a variety of cases. It is shown that the analytical linear model collapses the data from the simulations well. The predicted interface behavior well approximates that seen in compressible linearized simulations when the shock strength, magnetic field strength, and perturbation amplitude are small. For such cases, the agreement with interface behavior that occurs in nonlinear simulations is also reasonable. The effects of increasing shock strength, magnetic field strength, and perturbation amplitude on both the flow and the performance of the model are investigated. This results in a detailed exposition of the features and behavior of the MHD Richtmyer–Meshkov flow. For strong shocks, large initial perturbation amplitudes, and strong magnetic fields, the linear model may give a rough estimate of the interface behavior, but it is not quantitatively accurate. In all cases examined the accuracy of the model is quantified and the flow physics underlying any discrepancies is examined.

Wavelet-based coherent vorticity sheet and current sheet extraction from three-dimensional homogeneous magnetohydrodynamic turbulence

A method for extracting coherent vorticity sheets and current sheets out of three-dimensional homogeneous magnetohydrodynamic (MHD) turbulence is proposed, which is based on the orthogonal wavelet decomposition of the vorticity and current density fields. Thresholding the wavelet coefficients allows both fields to be split into coherent and incoherent parts. The fields to be analyzed are obtained by direct numerical simulation (DNS) of forced incompressible MHD turbulence without mean magnetic field, using a classical Fourier spectral method at a resolution of 5123. Coherent vorticity sheets and current sheets are extracted from the DNS data at a given time instant. It is found that the coherent vorticity and current density preserve both the vorticity sheets and the current sheets present in the total fields while retaining only a few percent of the degrees of freedom. The incoherent vorticity and current density are shown to be structureless and of mainly dissipative nature. The spectral distributions of kinetic and magnetic energies of the coherent fields only differ in the dissipative range, while the corresponding incoherent fields exhibit near-equipartition of energy. The probability distribution functions of total and coherent fields for both vorticity and current density coincide almost perfectly, while the incoherent fields have strongly reduced variances. Studying the energy flux confirms that the nonlinear dynamics is fully captured by the coherent fields only.

DYNAMIC ALIGNMENT AND EXACT SCALING LAWS IN MAGNETOHYDRODYNAMIC TURBULENCE

Magnetohydrodynamic (MHD) turbulence is pervasive in astrophysical systems. Recent high-resolution numerical simulations suggest that the energy spectrum of strong incompressible MHD turbulence is E(k ⊥) ∝ k –3/2 ⊥. So far, there has been no phenomenological theory that simultaneously explains this spectrum and satisfies the exact analytic relations for MHD turbulence due to Politano & Pouquet. Indeed, the Politano-Pouquet relations are often invoked to suggest that the spectrum of MHD turbulence instead has the Kolmogorov scaling –5/3. Using geometrical arguments and numerical tests, here we analyze this seeming contradiction and demonstrate that the –3/2 scaling and the Politano-Pouquet relations are reconciled by the phenomenon of scale-dependent dynamic alignment that was recently discovered in MHD turbulence.

Two‐level finite element method with a stabilizing subgrid for the incompressible MHD equations

AbstractWe consider the Galerkin finite element method (FEM) for the incompressible magnetohydrodynamic (MHD) equations in two dimension. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level FEM with a stabilizing subgrid of a single node is described and its application to the MHD equations is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems including the MHD cavity flow and the MHD flow over a step. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Furthermore, the approximate solutions obtained show the well‐known characteristics of the MHD flow. Copyright © 2009 John Wiley & Sons, Ltd.

Theory and simulations of whistler wave propagation

AbstractA linear theory of whistler waves is developed within the paradigm of a two-dimensional incompressible electron magnetohydrodynamics model. Exact analytic wave solutions are obtained for small-amplitude whistler waves that exhibit magnetic field topological structures consistent with the observations and our simulations in a linear regime. In agreement with experiment, we find that the parallel group velocity of the wave is large compared to its perpendicular counterpart. Numerical simulations of collisional interactions demonstrate that the wave magnetic field either coalesces or repels depending upon the polarity of the associated current. In the nonlinear regime, our simulations demonstrate that the evolution of the wave magnetic field is governed essentially by the nonlinear Hall force.