Two-dimensional Magnetohydrodynamics Research Articles

On the existence of simple waves for two-dimensional non-ideal magneto-hydrodynamics

Abstract In this article, a method called characteristic decomposition is used to show the presence of simple waves for the two-dimensional compressible flow in a non-ideal magneto-hydrodynamics system. Here, a steady and pseudo-steady state magneto-hydrodynamics system is considered, and we provide a characteristic decomposition of the flow equations in both systems. This decomposition ensures the presence of a simple wave adjacent to a region of constant state for the system under consideration. Further, this result is extended as an application of the characteristic decomposition in a pseudo-steady state, and we prove the existence of a simple wave in a full magneto-hydrodynamics system by taking the vorticity and the entropy to be constant along the pseudo-flow characteristics. These results extend the fundamental theorem proposed by Courant and Friedrichs for a reducible system (R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, New York, Interscience Publishers, Inc., 1948, p. 464). A motivational work was carried out for an ideal gas by Li et al. (“Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations,” Commun. Math. Phys. Math. Phys., vol. 267, no. 1, pp. 1–12, 2006) and for a non-ideal gas by Zafar and Sharma (“Characteristic decomposition of compressible Euler equations for a non-ideal gas in two-dimensions,” J. Math. Phys., vol. 55, no. 9, pp. 093103–093112, 2014], [M. Zafar, “A note on characteristic decomposition for two-dimensional euler system in van der waals fluids,” Int. J. Non-Linear Mech., vol. 86, pp. 33–36, 2016].

Effect of phase exchange kinetics on Taylor dispersion of chemically reactive solutes in an oscillatory magnetohydrodynamics flow between two parallel plates

The study of kinetic sorptive effects on the transport phenomena of reactive solute has numerous real-world applications, including in the industrial and environmental sectors. These kinds of investigations become more realistic when an oscillatory pressure gradient with both the reversible and irreversible reactions at the channel walls is considered in a magnetohydrodynamics flow. In the past, Ng and Yip [J. Fluid Mech. 446, 321–345 (2001)] studied the effect of sorptive phase exchange at boundaries on the solute transport phenomena in an open-channel flow using Mei's multiple-scale homogenization technique. They considered fluid flows without magnetic field and boundary absorption. This work uses the above-mentioned method to investigate the phase exchange kinetics of Taylor dispersion phenomena in a two-dimensional magnetohydrodynamics fluid flowing through a parallel channel. The paper discusses how various parameters and dimensionless numbers, such as the Hartmann, oscillatory Reynolds, and Damkohler, affect the flow velocity, transport coefficient, multi-dimensional concentration distributions, and transverse variation rate. Due to the strong magnetic field, the flow velocity and Taylor dispersivity are adversely affected and conspicuously reduced. Additionally, for large Damkohler numbers, the total dispersion coefficient and the Taylor dispersion coefficient both decrease. However, the longitudinal concentration distribution rises with the Hartmann number and partition coefficient. It is worth noting that in the presence of unequal boundary absorption, there is no occurrence of transverse symmetry in solute concentration at any given time. Controlling various processes of tracer dispersion in environmental systems, especially water purification and the chemical industry, may benefit from these intriguing findings.

Bio-convection of ternary magnetized nanoparticles thermal conductivity in chemical reaction and activation energy flow with Darcy Forchheimer permeable across a double porous medium

This research focuses on investigating the impact of magnetized nanoparticles’ heat and mass transfer process two-dimensional (2D) porous medium ternary hybrid Magnetohydrodynamics (MHD) ferrofluid flow through a two movable permeable porous surface, with a specific emphasis on the Darcy Forchheimer effect, activation energy, and chemical reaction parameters. Recently, motile microorganisms have become a significant area of interest in the study of heat and mass transfer. This study employs colored plots to analyze the optimum heat transfer over a double porous plate by altering different physical parameters. Using the shooting technique, the study provides numerical solutions to nonlinear systems of equations by transforming the partial differential equations (PDE) into ordinary differential equations (ODEs) via the dimensionless similarity transformation. Tables and graphs demonstrate how the governing parameters impact velocity, temperature, mass concentration profiles, and motile microorganisms, as well as physical quantities like skin friction coefficient, Nusselt number, Sherwood number, and motile number. Ternary hybrid ferrofluid exhibits a significantly improved heat transfer rate compared to both Ferro and hybrid ferrofluid. The convective flux of heat and mass transfer experiences a decrement with an escalation in the values of the activation energy (E) parameter. The presence of positive values for both the similarity variable (n) and Exothermic/Endothermic parameter (λ) leads to an enhancement in the flow of thermal transfer on both porous surfaces. This research will have far-reaching implications in various fields, including coolant technology in the manufacturing industry, heat dissipation in electronic modules, high-performance computing cooling facilities, the automotive sector, and space applications.

¬¬Chemical Reaction and Nonlinear Rosseland Approximation Effects On Double-Diffusive MHD Sisko Nanofluid Flow Over a Nonlinear Stretching Sheet in A Porous Medium with Concentration-Dependent Internal Heat Source

This study investigates the two-dimensional magnetohydrodynamics laminar flow of a Sisko nanofluid over a nonlinear stretching sheet in a porous medium, considering chemical reactions, nonlinear Rosseland approximation, and an internal heat source based on concentration. The governing boundary layer equations and associated boundary conditions are transformed into non-dimensional form using appropriate variables. The resulting non-dimensional equations are solved using the modified Adomian decomposition technique, implemented with the symbolic computing software MATHEMATICA. The influence of various parameters, including Prandtl number, thermophoresis parameter, Brownian motion parameter, chemical reaction parameter, Sisko fluid parameters, heat source parameter, radiation parameter, Darcy number, mass Grashof number, magnetic field, and Grashof number, on the flow fields of velocity, temperature, and nanoparticles volume fractions is depicted through graphical representations. The study reveals that the velocity profile is accelerated by the first value of the Sisko fluid parameter and Darcy number, but it diminishes in the presence of a magnetic field and the second Sisko fluid parameter. The temperature profile decreases with an increment in the Prandtl number, while it improves in proportion to increases in the heat source, thermophoresis, Brownian motion, and radiation parameters respectively. Additionally, an increase in the chemical reaction parameter leads to an increase in the concentration profile of the fluid.

Growth of Massive Molecular Cloud Filament by Accretion Flows. I. Slow-shock Instability versus Ambipolar Diffusion

The Herschel Gould Belt Survey showed that stars form in dense filaments in nearby molecular clouds. Recent studies suggest that massive filaments are bound by the slow shocks caused by accretion flows onto the filaments. The slow shocks are known to be unstable to corrugation deformation of the shock front. Corrugation instability could convert the accretion flow's ram pressure into turbulent pressure that influences the width of the filament, which, according to theory, determines the self-gravitational fragmentation scale and core mass. In spite of its importance, the effect of slow-shock instability on star-forming filaments has not been investigated. In addition, the linear dispersion relation obtained from ideal magnetohydrodynamics (MHD) analysis shows that the most unstable wavelength of shock corrugation is infinitesimally small. In the scale of dense filaments, the effect of ambipolar diffusion can suppress the instability at small scales. This study investigates the influence of ambipolar diffusion on the instability of the slow shock. We perform two-dimensional MHD simulations to examine the linear growth of the slow-shock instability, considering the effect of ambipolar diffusion. The results demonstrate that the most unstable scale of slow-shock instability is approximately 5 times the length scale of ambipolar diffusion ℓ AD calculated using post-shock variables, where ℓ AD corresponds to the scale where the magnetic Reynolds number for ambipolar diffusivity is unity.

Sensitivity analysis for MHD bionanofluid slip flow model in fluid-permeable plate

Diagnostic and therapeutic purpose on human healthcare is moving to a trend of combined engineering principles and design concepts. It is therefore beneficial to refine the study on thermodynamic properties of bionanofluid towards a better state of diagnoses and treatments. This paper aims to determine the characteristics of a magnetohydrodynamics (MHD) bionanofluid slip flow model in a fluid-permeable plate. A Buongiorno mathematical model describing steady two-dimensional MHD bionanofluid slip flow across a permeable plate is constructed. Numerical analysis by shooting technique and sensitivity analysis under the principle of RSM is configured to determine the correlation between input parameters and output responses. The relationship between the four parameters of interest namely skin friction coefficient, local Nusselt number, local Sherwood number and local density of motile microorganism with respective velocity, temperature, nanoparticle concentration and microorganism density distributions to the selected input parameters are investigated. Skin friction coefficient and local density of motile microorganism increase with suction and Schmidt number respectively, signifying positive sensitivity. Local Sherwood number is conditionally more sensitive to Brownian motion parameter or else Lewis number. While, local Nusselt number is always gaining negative sensitivity from Lewis number, thermophoresis and Brownian motion parameters. Extensive numerical studies on boundary layer problem have been done; yet this is the first study invoking sensitivity analysis to discuss the sensitivity of parameters of interest to the input thermophysical parameters.

A novel flux splitting based on wave-particle splitting for ideal magnetohydrodynamics

A novel flux splitting method based on wave-particle splitting is developed for one-dimensional ideal magnetohydrodynamics. While ideal magnetohydrodynamics (MHD) equations are non-convex with non-homogeneous flux as opposed to their hydrodynamic counterparts, the present flux splitting methods cannot develop Riemann solver. The proposed approach based on wave-particle method referred as Advection Magnetic-direction Wave Particle Splitting (AMWPS) scheme splits the flux into four parts: the wave-like total pressure part, the wave-like normal magnetic part, the particle-like advection part and the particle-like tangential magnetic part. The particle-like parts can be solved by the scheme developed by Toro and Vazquez-Cendon (TV scheme) and we propose a novel approach to the solution of Riemann problem formed by the wave-like parts including total pressure sub-flux and normal magnetic sub-flux. As for the ordered wave foliation in MHD and other limitations in wave structure, we also analyze the relation between plasma β and specific heat ratio γ. The advantage of AMWPS is that it can sharply capture isolated, stationary Alfven wave discontinuity. Several one-dimensional and two-dimensional MHD problems have been tested to highlight the accuracy, positivity preservation and robustness of AMWPS scheme and comparative studies show that AMWPS significantly outperforms the Riemann solver named Harten, Lax and Leer for contact wave (HLLC) in most cases.

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.

The significant role of Darcy–Forchheimer and thermal radiation on Casson fluid flow subject to stretching surface: A case study of dusty fluid

This paper addresses the problem of steady two-dimensional magnetohydrodynamics (MHD) migration of dusty fluid across a stretching sheet with the inclusion of Darcy–Forchheimer porosity and Brownian dispersion. The most significant aspect of the ongoing problem is elaborating the entire context with dusty and fluid phases. The governing partial differential equations (PDEs) are transmuted into non-dimensional ordinary differential equations (ODEs) by implementing similarity transformations. The MATLAB script has used the bvp4c strategy to accumulate a visualization of the computational observations. Also, this study illustrates an assessment of the effects of significant parameters on non-Newtonian fluids and fluids with a dusty phase. It is observed that the thermal boundary layer is enhanced with the increasing strength of the mass concentration of dusty granules ([Formula: see text]) and Eckert number (Ec) for both scenarios, while in the case of fluid interaction temperature parameter ([Formula: see text]) the scenario is opposite for dusty and fluid phases. Moreover, the heat transfer rate intensifies with the increasing effect of thermal radiation, and magnetic field. The significant variations in the various physical quantities are represented through tabular and graphs.

A robust and contact resolving Riemann solver for the two-dimensional ideal magnetohydrodynamics equations

This paper presents a new cell-centered numerical method for ideal magnetohydrodynamics (MHD) that can be used in Lagrangian or Eulerian discretization. A two-dimensional Riemann solver based on the HLLC-type MHD method is established, which can be viewed as an extension from the HLLC-2D in hydrodynamics. The main feature of the algorithm is to introduce a nodal contact velocity and ensure the compatibility between edge fluxes and the nodal flow intrinsically. It transforms naturally from Lagrangian setting to the Eulerian setting in terms of grid nodal velocity, and gains benefits of the Lagrangian nature of the scheme. In the Lagrangian approach, the finite volume scheme itself can keep the magnetic field divergence-free strictly, while in the Eulerian case, a special constrained transport (CT) algorithm is constructed from the discontinuous fluxes on cell interfaces to ensure solenoidal nature again. Numerical tests are presented to demonstrate the performance of this new solver and compare the difference between the Lagrangian and Eulerian methods.

The virtual element method for a 2D incompressible MHD system

We present a novel discretization for the two-dimensional incompressible Magnetohydrodynamics (MHD) system coupling an electromagnetic model and a fluid flow model. Our approach follows the framework of the Virtual Element Method and offers two main advantages. The method can be implemented on unstructured meshes making it highly versatile and capable of handling a broad set of problems involving interfaces, free-boundaries, or adaptive refinements of the mesh. The second advantage concerns the divergence of the magnetic flux field and the fluid velocity. Our approach guarantees that the numerical approximation of the magnetic flux field and the fluid velocity are divergence free if their initial states are divergence free. Importantly, the divergence-free condition for the fluid velocity is satisfied in a pointwise sense. We include a theoretical proof of the condition on the magnetic flux field, energy estimates and a well-posedness study. Numerical testing confirms robustness of the method and its convergence properties on a variety of meshes.

Global well-posedness and exponential decay rates of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with vacuum in some class of large initial data

In this paper, the initial-boundary value problem for the two-dimensional viscous, compressible, and heat conducting magnetohydrodynamics equations with vacuum is considered. When all the coefficients of viscosity, heat conductivity and magnetic diffusivity are constants, first, we show that for general initial data and any time T>0, the local strong solution in the sense of Sobolev norms will never blow-up provided that the density is bounded from above and the temperature belongs to space L∞(0,T;L2)∩L2(0,T;L∞). Second, based on the above blow-up criterion and the delicate analysis of the nonlinear strong coupling equations, we establish the global existence of strong solutions when the initial mass of the fluid and the initial energy of the magnetic field (i.e., ‖ρ0‖L1 and ‖H0‖L2) are suitably small, where the refined Zlotnik inequality and the decoupling of the full compressible MHD equations play an essential role in proving the uniform upper bound of the density and the higher-order a priori uniform in time estimates, respectively. It is worth mentioning that the initial velocity and temperature can be large. Moreover, we obtain exponential decay under the H1-norm of (u,θ,H) and L2-norm of (ρu˙,ρθ˙,Ht), where the long-time behaviors of the higher-order derivative of velocity are given and it is different from the known results.

Provably Positive Central Discontinuous Galerkin Schemes via Geometric Quasilinearization for Ideal MHD Equations

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such a positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was unknown prior to our analysis and differs from that for the noncentral discontinuous Galerkin (DG) and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124–2147]. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the one-dimensional (1D) case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial, yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov’s modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding 1D, two-dimensional, and three-dimensional numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta.

Oscillations of highly magnetized non-rotating neutron stars

Highly magnetized neutron stars are promising candidates to explain some of the most peculiar astronomical phenomena, for instance, fast radio bursts, gamma-ray bursts, and superluminous supernovae. Pulsations of these highly magnetized neutron stars are also speculated to produce detectable gravitational waves. In addition, pulsations are important probes of the structure and equation of state of the neutron stars. The major challenge in studying the pulsations of highly magnetized neutron stars is the demanding numerical cost of consistently solving the nonlinear Einstein and Maxwell equations under minimum assumptions. With the recent breakthroughs in numerical solvers, we investigate pulsation modes of non-rotating neutron stars which harbour strong purely toroidal magnetic fields of 1015−17 G through two-dimensional axisymmetric general-relativistic magnetohydrodynamics simulations. We show that stellar oscillations are insensitive to magnetization effects until the magnetic to binding energy ratio goes beyond 10%, where the pulsation mode frequencies are strongly suppressed. We further show that this is the direct consequence of the decrease in stellar compactness when the extreme magnetic fields introduce strong deformations of the neutron stars.

Optimal L2 error analysis of first-order Euler linearized finite element scheme for the 2D magnetohydrodynamics system with variable density

In this paper, a first-order Euler semi-implicit finite element scheme is proposed for solving the two-dimensional incompressible magnetohydrodynamics flows with variable density numerically. The proposed FEM algorithm is unconditionally stable at the full discrete level, which is a key issue in designing efficient algorithms for the multi-physical field problems. When the time step size τ and the mesh size h both are sufficiently small, optimal spatial error estimates in L 2 -norm are presented for the density, velocity and magnetic field by using energy techniques based on uniform regularities of solutions to the time-discrete scheme and the method of mathematical induction. Finally, numerical results are given to confirm the theoretical analysis.

Detection of magnetohydrodynamic waves by using convolutional neural networks

Nonlinear wave interactions in magnetohydrodynamics (MHD), such as shock refraction at an inclined density interface, lead to a plethora of wave patterns with numerous wave types. Identification of different types of MHD waves is an important and challenging task in such complex wave patterns. Moreover, owing to the multiplicity of solutions and their admissibility for different systems, especially for intermediate-type MHD shock waves, the identification of MHD wave types is complicated if one relies on the Rankine–Hugoniot jump conditions. MHD wave detection is further exacerbated by nonphysical smearing of discontinuous shock waves in numerical simulations. This paper proposes two MHD wave detection methods based on convolutional neural network to enable wave classification and identify their locations. The first method separates the output into regression (location prediction) and classification problems, assuming the number of waves for each training data is fixed. In contrast, the second method does not specify the number of waves a priori, and the algorithm predicts wave locations and classifies types using only regression. We use one-dimensional input data (density, velocity, and magnetic fields) to train the two models that successfully reproduce a complex two-dimensional MHD shock refraction structure. The first fixed output model efficiently provides high precision and recall, achieving total neural network accuracy up to 99%, and the classification accuracy of some waves approaches unity. The second detection model has relatively low performance, with more sensitivity to the setting of parameters, such as the number of grid cells Ngrid and the thresholds of confidence score and class probability, etc. The detection model achieves better than 90% accuracy with F1 score >0.95. The proposed two methods demonstrate very strong potential for MHD wave detection in complex wave structures and interactions.

Effect of Soret on MHD rotating fluid flow through a porous medium with heat and mass transfer

PurposeThe purpose of this paper to analyze the effect of Soret with heat and mass transfer on an unsteady two-dimensional Magnetohydrodynamics flow through a porous medium under the influence of the uniform transverse magnetic field in a rotating parallel plate is considered.Design/methodology/approachA mathematical model was developed using the slip conditions under unsteady state situations. Analytical expressions for the velocity, temperature and concentration profiles, wall shear stress, rates of heat and mass transfer and volumetric flow rate were obtained and computationally discussed with respect to the non-dimensional parameters. Further, the velocity reduces with increasing Hartmann number M and increases with Grashof number Gr and permeability parameter K.FindingsIt is observed that temperature reduces with an increase in Prandtl number Pr and ω. It is noted that the thermal radiation increases with increase in Soret number Sr, Schmidt number Sc, Prandtl number pr and ω.Originality/valueConcentration decreases with an increase in radiation parameter R and chemical reaction parameter Kc.

Global well-posedness of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with large viscosity

Global well-posedness of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with large viscosity

Two-dimensional MHD equilibria of diamagnetic bubble in gas-dynamic trap

This article presents a magnetohydrodynamic (MHD) two-dimensional numerical model of diamagnetic bubble equilibria in an axisymmetric open trap. The theoretical model consists of the Grad–Shafranov equilibrium equation and the transport equation obtained within the resistive single-fluid MHDs with isotropic pressure. Found are the numerical solutions corresponding to the diamagnetic confinement mode. In particular, the equilibria of the diamagnetic bubble in the gas-dynamic multimirror trap are calculated. We investigate the effect of magnetic field corrugation on the equilibrium; the corrugation of the vacuum field is shown to lead to a rather moderate corrugation of the bubble boundary if the period of corrugation is sufficiently small. A valuable numerical result is the distribution of the diamagnetic field, which would be useful for optimizing the position of the wall-stabilization plates.

Component of Energy Flow from Supercritical Accretion Disks Around Rotating Stellar Mass Black Holes

By performing two-dimensional axisymmetric general relativistic radiation magnetohydrodynamics simulations with spin parameter a* varying from −0.9 to 0.9, we investigate the dependence on the black hole spin of the energy flow from a supercritical accretion disk around a stellar mass black hole. It is found that optically and geometrically thick disks form near the equatorial plane, and a part of the disk matter is launched from the disk surface in all models. The gas ejection is mainly driven by the radiative force, but magnetic force cannot be neglected when ∣a*∣ is large. The energy outflow efficiency (total luminosity normalized by and c are the mass-accretion rate at the event horizon and the light speed) is higher for rotating black holes than for nonrotating black holes. This is 0.7% for a* = −0.7, 0.3% for a* = 0, and 5% for a* = 0.7 for (L Edd is the Eddington luminosity). Furthermore, although the energy is mainly released by radiation when a* ∼ 0, the Poynting power increases with ∣a*∣ and exceeds the radiative luminosity for models with a* ≥ 0.5 and a* ≤ −0.7. The faster the black hole rotates, the higher the power ratio of the kinetic luminosity to the isotropic luminosity tends to be. This implies that objects with a high (low) power ratio may have rapidly (slowly) rotating black holes. Among ultraluminous X-ray sources, IC342 X-1, is a candidate with a rapidly rotating black hole.