Two-dimensional MHD Research Articles

Steady two dimensional MHD stokes flow between two parallel plates under angular velocity with one plate moving uniformly and the other plate at rest and uniform suction at the stationary plate

In this paper, we talk about the investigation of Steady two-dimensional MHD stokes flow between two parallel plates under angular velocity with the assistance of Differential equation and transform methods to discover the arrangement of the Mathematical conditions with one in uniform movement and the other plate at rest with uniform suction at the stationary plate. The velocity profile for the progression of fluid and pressure gradient are talked about for various values, and the impacts of velocity potential are examined and examined graphically.

Generation of bulk vorticity and current density in current-vortex sheet models

Matsuoka et al. [1] present a set of equations governing the evolution of two-dimensional MHD flows using current-vortex sheets. In the resulting model the vorticity ω and current density j are zero except on the current-vortex sheets. We show that this is not true in general and that the term Δ(B × u) in the evolution equation for j does not vanish, leading to the generation of j and ω in the bulk. This means that the evolution of the system is not governed solely by the dynamics of quantities on the current-vortex sheets. A perturbative solution is derived that shows this explicitly.

Numerical Model of Compression Plasma Flows in Channels under a Longitudinal Magnetic Field

We consider a mathematical model of plasma flows in nozzle channels formed by two coaxial electrodes. The acceleration of plasma in an azimuthal magnetic field is accompanied by its compression and heating in the compression zone at the channel axis past the tip of the shorter central electrode. The mathematical apparatus of the model is based on numerically solving two-dimensional MHD problems using the Zalesak flux-corrected transport (Z-FCT) scheme. In the computations, we study the dependence of the compression phenomenon and its quantitative characteristics on the channel geometry, the problem parameters, and the additional longitudinal magnetic field present in the channel.

Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids

We consider a two-dimensional MHD model describing the evolution of viscous, compressible and electrically conducting fluids under the action of vertical magnetic field without resistivity. Existence of global weak solutions is established for any adiabatic exponent γ≥1. Inspired by the approximate scheme proposed in [15], we consider a two-level approximate system with artificial diffusion and pressure term. At the first level, we prove global well-posedness of the regularized system and establish uniform-in-ε estimates to the regular solutions. At the second level, we show global existence of weak solutions to the system with artificial pressure by sending ε to 0 and deriving uniform-in-δ estimates. Then global weak solution to the original system is constructed by vanishing δ. The key issue in the limit passage is the strong convergence of approximate sequence of the density and magnetic field. This is accomplished by following the technique developed in [15,26] and using the new technique of variable reduction developed by Vasseur et al. [33] in the context of compressible two-fluid model so as to handle the cross terms.

A Cancellation Nanoflare Model for Solar Chromospheric and Coronal Heating. II. 2D Theory and Simulations

Abstract Recent observations at high spatial resolution have shown that magnetic flux cancellation occurs on the solar surface much more frequently than previously thought, and so this led Priest et al. (2018) to propose magnetic reconnection driven by photospheric flux cancellation as a mechanism for chromospheric and coronal heating. In particular, they estimated analytically the amount of energy released as heat and the height of the energy release during flux cancellation. In the present work, we take the next step in the theory by setting up a two-dimensional resistive MHD simulation of two canceling polarities in the presence of a horizontal external field and a stratified atmosphere in order to check and improve upon the analytical estimates. Computational evaluation of the energy release during reconnection is found to be in good qualitative agreement with the analytical estimates. In addition, we go further and undertake an initial study of the atmospheric response to reconnection. We find that, during the cancellation, either hot ejections or cool ones or a combination of both hot and cool ejections can be formed, depending on the height of the reconnection location. The hot structures can have the density and temperature of coronal loops, while the cooler structures are suggestive of surges and large spicules.

Multiple slip effects on mixed convection of Oldroyd-B fluid towards a stretchy surface with radiation and chemical reaction using Cattaneo-Christov heat flux

The steady two-dimensional MHD mixed convection flow and heat transfer characteristics of an Oldroyd-B fluid over a linear stretchy sheet is considered. The flow caused by a linear stretchy sheet with effects of velocity, temperature and concentration slips. The thermal radiation, internal heat generation effects are included in the study. The resulting governing PDEs rebuild into ODE model by applying a suitable similarity transformation and then are solved by homotopy analysis method. The effects of the various constants on the velocity, temperature and concentration fields are discussed. It is found that the velocity, temperature and concentration fields decrease when increasing the corresponding slip constants.

Analytical study of the magnetohydrodynamic natural convection of a nanofluid filled horizontal shallow cavity with internal heat generation

An analytical study is presented of the laminar, steady-state two-dimensional MHD natural convection of a nanofluid filled shallow cavity, which is driven by volumetrically internal heat sources and is subject to an external uniform vertical magnetic field. All walls are electrically insulated, with the horizontal ones being adiabatic, while the vertical ones are isothermal. The working nanofluid is water partially filled with copper nanoparticles. The method of the matched asymptotic expansions is used to obtain solutions of the flow and heat transfer problem in the core region of the horizontal cavity taking into account the Brownian motion of the nanoparticles. Overall, the results indicate that convective phenomena are intensified by increasing the internal heating. It is found that in the hydrodynamic regime adding more nanoparticles slightly enhances heat transfer, whereas in the presence of an external magnetic field results in the deterioration of it. In the latter magnetohydrodynamic regime, the nanofluid flow is decelerated, and thus heat conduction dominates over convection as the magnitude of the magnetic field increases or smaller size of nanoparticles are employed. Finally, heat transfer enhancement is observed by ignoring the Brownian motion part of the effective thermal conductivity.

On the effect of parallel shear flow on the plasmoid instability

The effect of symmetric sheared plasma flow parallel to the equilibrium magnetic field on the evolution of plasmoid instability is investigated by using two-dimensional MHD simulations. It is shown that as the velocity amplitude of a shear flow increases from sub-Alfvénic to the level of super-Alfvénic, the current sheet is stabilized against the plasmoid instability. Suppression of plasmoid instability also takes place for a sub-Alfvénic shear flow, provided that the shear length is sufficiently small compared to the magnetic shear. Therefore, an initially elongated thin current sheet with a Lundquist number greater than the critical value does not necessarily guarantee the development of the plasmoid instability. As a result, the Kelvin-Helmholtz (KH) instability is triggered. Although the rate of reconnection is quite fast in the nonlinear regime of the plasmoid instability, the reconnection of twisted field lines during the KH instability is very slow.

Local-in-time well-posedness for compressible MHD boundary layer

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work [27], we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.

Finite Element Method for Two-Dimensional Unsteady MHD Couette Flow of a Generalized Oldroyd-B Fluid

In recent years, non-Newtonian fluids have been widely applied in engineering and industry. Due to the complexity of these fluids, there are several constitutive equations of non- Newtonian fluids. One particular subclass of non-Newtonian fluids is the class of a generalized Oldroyd-B fluids with fractional derivative constitutive equations, which has been found to approximate the response of many dilute polymeric liquids. Although the analytic solution of these equations can be derived, it contains a multi-level sum of an infinite series and Fox H-functions, which is extremely complex and difficult to evaluate. Therefore, seeking the numerical solution of the equation is of great importance. Different from the general time fractional diffusion equation, the constitutive equation is a novel multi-term time fractional diffusion equation and has a special time fractional operator on the spatial derivative, which is challenging to approximate. In this paper, we will consider the finite element method for the two-dimensional multi-term time fractional diffusion equation. Firstly, we utilise the L1 and L2 schemes to approximate the two different time fractional derivatives, respectively. Secondly, we apply the finite element method to the equation and establish the fully variational formulation. Then we adopt linear polynomials on triangular elements to derive the matrix form of the numerical scheme. Furthermore, we present the stability and convergence of the scheme. Finally, two numerical examples are investigated to show the effectiveness of our method, in which a two-dimensional unsteady MHD Couette flow of a generalized Oldroyd-B fluid is considered.

Acceleration of Plasma in Coaxial Channels with Preshaped Electrodes and Longitudinal Magnetic Field

The paper presents a mathematical model and results of numerical simulation of axisymmetric plasma flows in nozzle-type channels formed by two coaxial electrodes. Transonic accelerated flows, which are of interest for the development of plasma accelerators, are considered. The mathematical apparatus of the models are two-dimensional MHD problems solved numerically, the steady-state solutions of which are obtained in the process of relaxation. Some characteristics of the flow in narrow tubes between close trajectories are considered in the quasi-one-dimensional approximation. The main attention is paid to the influence of the longitudinal magnetic field and the curvilinear geometry of the electrodes on the properties of accelerated flows.

Semi analytical analysis for transient Eyring-Powell squeezing flow in a stretching channel due to magnetic field using DTM

This paper deals with heat and mass transfer in unsteady two-dimensional MHD squeezing flow of Eyring-Powell fluid between two parallel infinite plates. Heat generation/absorption, thermal radiation and Joule heating influences are taken into consideration. The governing nonlinear PDEs are converted to nonlinear ODEs using similarity transformations. Analytical solution for the velocity, temperature and concentration distribution has been obtained by means of Differential Transformation Method (DTM).The comparisons between the results of DTM and numerical method (Runge-Kutta Fehlbrg), proved high precision of DTM. The effects of various parameters, including squeeze number, radiation parameter, heat generation/absorption parameter and Hartmann number on the flow field, skin friction coefficient and heat transfer have been studied. The achieved results reveal that Lorentz force generated by magnetic field parameter (Ha), reduces velocity distribution. Also, it is found that the temperature profile reduces with the increase of radiation parameter (R).

Global smooth solutions of the 2D MHD equations for a class of large data without magnetic diffusion

In this paper, we establish the global regularity of classical solutions for the two-dimensional MHD equations with only velocity diffusion for a class of large initial data. Both the initial velocity and magnetic field can be arbitrarily large in H s .

Study of an MHD Flow of the Carreau Fluid Flow Over a Stretching Sheet with a Variable Thickness by Using an Implicit Finite Difference Scheme

The present analysis deals with a two-dimensional MHD flow of the Carreau fluid over a stretching sheet with a variable thickness. The governing partial differential equations are converted into an ordinary differential equation by using the similarity approach. The solution of the differential equation is calculated by using the Keller box method. The solution is studied for different values of the Hartmann number, Weissenberg number, wall thickness parameter, and power-law index. The skin friction coefficient is calculated. The present results are compared with available relevant data.

Quasi-periodic Oscillations in Flares and Coronal Mass Ejections Associated with Magnetic Reconnection

Abstract We propose a mechanism for quasi-periodic oscillations of both coronal mass ejections (CMEs) and flare loops as related to magnetic reconnection in eruptive solar flares. We perform two-dimensional numerical MHD simulations of magnetic flux rope eruption, with three different values of the global Lundquist number. In the low Lundquist number run, no oscillatory behavior is found. In the moderate Lundquist number run, on the other hand, quasi-periodic oscillations are excited both at the bottom of the flux rope and at the flare loop top. In the high Lundquist number run, quasi-periodic oscillations are also excited; in the meanwhile, the dynamics become turbulent owing to the formation of multiple plasmoids in the reconnection current sheet. In high and moderate Lundquist number runs, thin reconnection jets collide with the flux rope bottom or flare loop top and dig them deeply. Steep oblique shocks are formed as termination shocks where reconnection jets are bent (rather than decelerated) in the horizontal direction, resulting in supersonic backflows. The structure becomes unstable, and quasi-periodic oscillations of supersonic backflows appear at locally confined high-beta regions at both the flux rope bottom and flare loop top. We compare the observational characteristics of quasi-periodic oscillations in erupting flux ropes, post-CME current sheets, flare ribbons, and light curves with corresponding dynamical structures found in our simulation.

MHD forced convection ow and heat transfer of ferro fluids over a moving at plate with uniform heat flux and second-order slip effects

In this study, the problem of two-dimensional forced convection MHD flow and heat transfer of ferro fluids over a moving at plate with the influence of uniform heat flux and secondorder slip eff ects is considered. By applying the similarity transformation, the governing equations are reconstructed into the similarity equations and the resulting equations are solved via shooting technique. Then, we implement a stability analysis in order to verify which solutions are stable and physically realizable. The e ects of the magnetic parameter,moving parameter, mass transfer parameter, rst-order surface slip parameter, second-order surface slip parameter and volume fraction of solid ferroparticles on the dimensionless velocity, temperature, skin friction and Nusselt numbers are discussed in the form of tabular and graphical presentation. For this present study, we consider the results based on three preferred ferroparticles, namely magnetite, cobalt ferrite and Mn-Zn ferrite in water- and kerosene-based fluids. The results display the existence of dual solutions for a plate moving towards the origin in which the rst solution is stable and physically realizable, while the second solution is not. Moreover, it is demonstrated that the magnetic, moving, mass transfer and slip eff ects together with the volume fraction of ferro fluids delay the boundary layer separation.

Numerical study of Hall effects on counter-helicity spheromak merging by two-dimensional Hall-MHD simulations

Hall effects on counter-helicity spheromak merging were investigated by two-dimensional MHD and Hall-MHD simulations of merging two axisymmetric toroidal flux tubes. In Hall-MHD cases, the structure of the reconnection current sheet and reconnection outflow are modified from the MHD case due to the Hall effect. We compared two cases (called “case-O” and “case-I”) of counter-helicity merging, which are distinguished by the polarity of toroidal magnetic fluxes. Radial motion of the reconnection X-point is controlled by poloidal electron flow accompanying the toroidal flux of the merging two spheromaks, and this creates a large difference in the current sheet and flow structure between the two cases of the Hall-MHD regime. The radial shift of the reconnection X-point depending on the polarity of toroidal magnetic flux of the spheromaks breaks the symmetry between the two cases. It was also found that there widely exists separation of ion and electron flow which are affected by the modification of the current sheet structure due to the radial shift of the X-point in the downstream side of the merging, and its spatial scale of the distribution of the Hall electric field is larger than the ion skin depth.

Structure and computation of two-dimensional incompressible extended MHD

A comprehensive study of the extended magnetohydrodynamic model obtained from the two-fluid theory for electrons and ions with the enforcement of quasineutrality is given. Starting from the Hamiltonian structure of the fully three-dimensional theory, a Hamiltonian two-dimensional incompressible four-field model is derived. In this way, the energy conservation along with four families of Casimir invariants is naturally obtained. The construction facilitates various limits leading to the Hamiltonian forms of Hall, inertial, and ideal MHD, with their conserved energies and Casimir invariants. Basic linear theory of the four-field model is treated, and the growth rate for collisionless reconnection is obtained. Results from nonlinear simulations of collisionless tearing are presented and interpreted using, in particular, normal fields, a product of the Hamiltonian theory that gives rise to simplified equations of motion.

The CFD modeling of two-dimensional turbulent MHD channel flow

In this paper, influence of magnetic field on turbulence characteristics of twodimensional flow is investigated. The present study has been undertaken to understand the effects of a magnetic field on mean velocities and turbulence parameters in turbulent 2-D channel flow. Several cases are considered. First laminar flow in a channel and MHD laminar channel flow are analyzed in order to validate model of magnetic field influence on electrically conducting fluid flow. Main part of the paper is focused on MHD turbulence suppression for 2-D turbulent flow in a channel and around the flat plate. The simulations are performed using ANSYS CFX software. Simulations results are obtained with BSL Reynolds stress model for turbulent and MHD turbulent flow around flat plate. The nature of the flow has been examined through distribution of mean velocities, turbulent fluctuations, vorticity, Reynolds stresses and turbulent kinetic energy.

Magnetohydrodynamic dissipative flow across the slendering stretching sheet with temperature dependent variable viscosity

The boundary layer flow across a slendering stretching sheet has gotten awesome consideration due to its inexhaustible pragmatic applications in nuclear reactor technology, acoustical components, chemical and manufacturing procedures, for example, polymer extrusion, and machine design. By keeping this in view, we analyzed the two-dimensional MHD flow across a slendering stretching sheet within the sight of variable viscosity and viscous dissipation. The sheet is thought to be convectively warmed. Convective boundary conditions through heat and mass are employed. Similarity transformations used to change over the administering nonlinear partial differential equations as a group of nonlinear ordinary differential equations. Runge-Kutta based shooting technique is utilized to solve the converted equations. Numerical estimations of the physical parameters involved in the problem are calculated for the friction factor, local Nusselt and Sherwood numbers. Viscosity variation parameter and chemical reaction parameter shows the opposite impact to each other on the concentration profile. Heat and mass transfer Biot numbers are helpful to enhance the temperature and concentration respectively.