Dimensional Magnetohydrodynamics Research Articles

Transverse expansion of (1 + 2) dimensional magneto-hydrodynamics flows with longitudinal boost invariance

In the present work, we investigate the effects of the magnetic field on expanding hot and dense nuclear matter as an ideal fluid. We consider QGP, in the particular case of a (1 + 2) dimensional longitudinally boost-invariant fluid expansion, in the background of an inhomogeneous magnetic field that is generated by external sources. We assume the magnetic field points in the direction perpendicular to the reaction plane, follows the power-law decay in proper time, and has two components on the transverse plane. To simplify our calculation, we suppose the investigated fluid has azimuthal symmetry, and magneto-hydrodynamic equations are described in a polar coordinate system on the transverse plane of reaction. Our results depict the space-time evolution of the transverse expansion of the fluid in the presence of an inhomogeneous external magnetic field. Moreover, we show when the magnetic field decays in proper time τ with a power-law τn/2(n<1), two distinct solutions can be found depending on the values of n. We show that n<−2 gives rise to the physical scenario whereas the n>−2 leads to non-physical results.

Asymptotic behavior of the stationary magnetohydrodynamic equations in an exterior domain

In this paper, we study the asymptotic behavior of solutions to the incompressible magnetohydrodynamic (MHD) equations in an exterior domain. We will show that, under some assumption, any nontrivial velocity field u and magnetic field h obey a minimal decaying rate exp(−C|x|2 log |x|) at infinity. Our proof is based on appropriate Carleman estimates. As a consequence, we establish a Liouville-type result for the three dimensional incompressible MHDs in an exterior domain.

Numerical Exploration via Least Squares Estimation on Three Dimensional MHD Yield Exhibiting Nanofluid Model with Porous Stretching Boundaries

The numerical study of a three-dimensional magneto-hydrodynamic (MHD) Casson nano-fluid with porous and stretchy boundaries is the focus of this paper. Radiation impacts are also supposed. A feasible similarity variable may convert a verbalized set of nonlinear “partial” differential equations (PDEs) into a system of nonlinear “ordinary” differential equations (ODEs). To investigate the solutions of the resulting dimensionless model, the least-square method is suggested and extended. Maple code is created for the expanded technique of determining model behaviour. Several simulations were run, and graphs were used to provide a thorough explanation of the important parameters on velocities, skin friction, local Nusselt number, and temperature. The comparison study attests that the suggested method is well-matched, trustworthy, and accurate for investigating the governing model’s answers. This method may be expanded to solve additional physical issues with complicated geometry.

Global solvability of 2D MHD boundary layer equations in analytic function spaces

In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces. When the initial data is a small perturbation around a selected profile, and such a profile is governed by an one dimensional heat equation with a source term, we establish the global in time existence and uniqueness of analytic solutions to the two dimensional (2D) MHD boundary layer equations. It is noted that the far-field state of velocity is not required to be small initially, but decays to zero as time tends to infinity with suitable decay rates. The whole analysis is divided into two parts: When the initial far-field states are not small, we reformulate the original problem into a small perturbation problem by extracting a suitable background profile which is governed by a heat equation, then we prove a long-time existence of solutions, and the lower bound of lifespan of solutions is given in term of the small parameter of initial perturbation; After establishing the long-time existence, combining the decay properties of far-field state and the properties of solutions to the heat equation, it is shown that both the solution obtained in the first part and the background profile indeed satisfy some smallness requirements when time goes to the lower bound of lifespan of solutions. Then we can show the global existence of solutions after this moment.

On the convergence of solutions of globally modified magnetohydrodynamics equations with locally Lipschitz delays terms

Existence and uniqueness of strong solutions for the three dimensional system of globally modified magnetohydrodynamics equations with locally Lipschitz delays terms are established in this article. Galerkin’s method and Aubin Lions compactness theorem are the main mathematical tools we use to prove the existence result. Moreover, we prove that, from a sequence of weak solutions of globally modified magnetohydrodynamics equations with locally Lipschitz delays terms, we can extract a subsequence which converges in an adequate sense to a weak solution of three dimensional magnetohydrodynamics equations with locally Lipschitz delays terms.

A Study of Two Dimensional Unsteady MHD Free Convection Flow over a Vertical Plate in the Presence of Radiation

This paper compacts with an exact analysis of radiative effects on the magnetohydrodynamic (MHD) free convection flow of an electrically conducting incompressible viscous fluid over a vertical plate. The non-dimensional continuity, momentum, and energy equations are solved using appropriate transformation. The dimensionless momentum and energy equations are solved numerically through an explicit finite difference method. The stability and convergence analysis also discussed. Finally, outcomes of the parameters on velocity and temperature profiles are displayed graphically and qualitatively.

On magnetohydrodynamics Prandtl fluid flow in the presence of stratification and heat generation

In present article, the mathematical modeling of physical problem focusing on two dimensional magnetohydrodynamics Prandtl fluid flow over a stretching sheet with stratification and heat generation is discussed. The similarity transforms are utilized to non-dimensionalized the modeled​ governing equations. The transformed governing equations are solved by using shooting technique with the aid of fifth order Runge–Kutta algorithm. Finally, the behavior of velocity field, temperature field, coefficient of skin frication and local Nusselt number (for thermally stratified and un-stratified cases) are discussed through graphs as well as tables. The thermal stratification cools down the fluid temperature while heat generation raises it. Also, local Nusselt number is enlarges absolutely versus stratification parameter while it has opposite behavior against heat generation parameter. The comparative study of skin friction coefficient is presented to ensure the validity of adapted methodology.

MHD stagnation point flow towards a quadratically stretching/shrinking surface

A quadratically stretching/shrinking surface of two dimensional magnetohydrodynamics stagnation point flow is investigated numerically. The velocity of the surface is assumed in quadratic form and subject to a linear mass flux. The influences of the governing parameters namely stretching/shrinking parameter ⋋, suction/injection parameter S, fluid temperature index m, and magnetic parameter M on the flow and thermal fields are studied. The model of nonlinear ordinary differential equations is obtained by reducing the partial differential equations of the boundary layer using an suitable similarity transformation. The equations are then solved numerically by boundary value problem solver, bvp4c built in MATLAB software. The numerical results are verified by comparing them with previously reported results. The characteristics of the flow and heat transfer characteristics are shown graphically and analyzed for distinct values of the governing parameters. It is found that both magnetic and temperature index parameters reduce the velocity flow while the magnetic parameter enhances the heat transfer rate. There exist dual solutions for certain range of ⋋. A stability analysis is performed to determine which of these solutions are stable and which are not.

Magnetohydrodynamic waves in braided magnetic fields

Aims. We investigate the propagation of transverse magnetohydrodynamic (MHD) wave fronts through a coronal plasma containing a braided magnetic field. Methods. We performed a series of three dimensional MHD simulations in which a small amplitude, transverse velocity perturbation is introduced into a complex magnetic field. We analysed the deformation of the wave fronts as the perturbation propagates through the braided magnetic structures and explore the nature of Alfvénic wave phase mixing in this regime. We considered the effects of viscous dissipation in a weakly non-ideal plasma and evaluate the effects of field complexity on wave energy dissipation. Results. Spatial gradients in the local Alfvén speed and variations in the length of magnetic field lines ensure that small scales form throughout the propagating wave front due to phase mixing. Additionally, the presence of complex, intricate current sheets associated with the background field locally modifies the polarisation of the wave front. The combination of these two effects enhances the rate of viscous dissipation, particularly in more complex field configurations. Unlike in classical phase mixing configurations, the greater spatial extent of Alfvén speed gradients ensures that wave energy is deposited over a larger cross-section of the magnetic structure. Further, the complexity of the background magnetic field ensures that small gradients in a wave driver can map to large gradients within the coronal plasma. Conclusions. The phase mixing of transverse MHD waves in a complex magnetic field will progress throughout the braided volume. As a result, in a non-ideal regime wave energy will be dissipated over a greater cross-section than in classical phase mixing models. The formation rate of small spatial scales in a propagating wave front is a function of the complexity of the background magnetic field. As such, if the coronal field is sufficiently complex it remains plausible that phase mixing induced wave heating can contribute to maintaining the observed temperatures. Furthermore, the weak compressibility of the transverse wave and the observed phase mixing pattern may provide seismological information about the nature of the background plasma.

Forward Modeling of SDO/AIA and X-Ray Emission from a Simulated Flux Rope Ejection

Abstract We conduct forward-modeling analysis based on our 2.5 dimensional magnetohydrodynamics (MHD) simulation of magnetic flux rope (MFR) formation and eruption driven by photospheric converging motion. The current sheet (CS) evolution during the MFR formation and eruption process in our MHD simulation can be divided into four stages. The first stage shows the CS forming and gradually lengthening. Resistive instabilities that disrupt the CS mark the beginning of the second stage. Magnetic islands disappear in the third stage and reappear in the fourth stage. Synthetic images and light curves of the seven Solar Dynamics Observatory (SDO)/Atmospheric Imaging Assembly (AIA) channels, i.e., 94 Å, 131 Å, 171 Å, 193 Å, 211 Å, 304 Å, and 335 Å, and the 3–25 keV thermal X-ray are obtained with forward-modeling analysis. The loop-top source and the coronal sources of the soft X-ray are reproduced in forward modeling. The light curves of the seven SDO/AIA channels start to rise once resistive instabilities develop. The light curve of the 3–25 keV thermal X-ray starts to go up when the reconnection rate reaches one of its peaks. Quasiperiodic pulsations (QPPs) appear twice in the SDO/AIA 171 Å, 211 Å, and 304 Å channels, corresponding to the period of chaotic (re)appearance and CS-guided displacements of the magnetic islands. QPPs appear once in the SDO/AIA 94 Å and 335 Å channels after the disruption of the CS by resistive instabilities and in the 193 Å channel when the chaotic motion of the magnetic islands reappears.

Simulation of three dimensional MHD natural convection using double MRT Lattice Boltzmann method

In this study, a three-dimensional mesoscopic simulation of magneto hydrodynamics (MHD) natural convection in a cubic cavity has been studied by new means of the Lattice Boltzmann method with double Multi-Relaxation-Time (MRT) model. In order to solve the momentum and energy equations, two different populations with various lattices have been used. This paper has been conducted for specific values of the Grashof number (Gr=2×103_2×105) and Hartmann number (Ha=0–100), while the Prandtl number is fixed at Pr=0.73. The results are presented in the form of average and local Nusselt number and contours of temperature and velocity at different planes of the cavity. It was found that the double MRT-LBM method is an appropriate approach to solve the studied case. The present results also show that the increase of the Hartmann number causes the heat transfer to drop considerably. Also, the effect of Hartmann number increases by enhancing the Grashof number, as the reduction of average Nusselt number is 12% for Gr=2×103 and 71% for Gr=2×105 when Hartmann number increases from 0 to 100. In contrast with Hartmann number, increasing of Grashof number raises heat transfer rate and the average Nusselt number increases by more than three times by enhancing the Grashof number from 2×103 to 2×105.

Non-Linear Radiation and Chemical Reaction Impacts on Hydromagnetic Particle Suspension Flow in Non-Newtonian Fluids

Abstract A two dimensional (2D) boundary layer two-phase MHD (magneto hydrodynamics) flow of Maxwell and Oldroyd-B fluid over a stretching sheet has been explored. Heat and mass transfer phenomena is inspected through Non-linear radiation, viscous dissipation, joule heating, Soret (thermo-diffusion) and Dufour (diffusion-thermo) Impact. The boundary layer governing differential equations are modelled and transformed to a system of ODE’S with the aid of similarity transformations. The final controlled equations along boundary restrictions are resolved numerically by Runge-Kutta Felhberg method. The graphical analysis has been emphasized for the fluid and dust phase velocity, temperature and concentration fields to the influence of sundry dynamical flow quantities. Furthermore, for authentication of the present computation, the achieved results are distinguished with earlier research works in specific cases and marvellous agreement has been noted. The outcomes conveyed here manifest that velocity and boundary layer thickness escalate with boost up the values of ${K_1}$ . Velocity and boundary layer thickness declines with boost up the values of $M$ . Opposite trend is seen in temperature and concentration profiles. The specific heat ratio parameter $\gamma$ enhances the temperature profile declines. Boost up the values of Soret number $Sr$ temperature profile declines and concentration profile enhances. Skin friction factor declines with increasing values $\beta _v$ verses $M$ .

Study of starspots in fully convective stars using three dimensional MHD simulations

AbstractConcentrated magnetic structures such as sunspots and starspots play a fundamental role in driving solar and stellar activity. However, as opposed to the sun, observations as well as numerical simulations have shown that stellar spots are usually formed as high-latitude patches extended over wide areas. Using a fully spectral magnetohydrodynamic (MHD) code, we simulate polar starspots produced by self-consistent dynamo action in rapidly rotating convective shells. We carry out high resolution simulations and investigate various properties related to stellar dynamics which lead to starspot formation.

Numerical Investigation on Magnetohydrodynamics (MHD) Free Convection Fluid Flow over a Vertical Porous Plate with induced Magnetic Field

In this paper, investigate a two dimensional unsteady Magneto hydro dynamics (MHD) free convection flow of viscous incompressible and electrically conducting fluid flow past an vertical plate in the presence of Grashof Number, Modified Grashof Number, Prandtl Number, Schamidt Number as well as Dufour effects. The governing equations of the problem contain a system of non-linear partial differential equations; have been transformed into a set of coupled non-linear ordinary differential equations which is solved numerically by applying well known explicit finite difference method. The Finite-difference method is an enormously used technique to investigate of the general non linear partial differential equation. Partial differential equations occur in many branches of applied mathematics for example, in hydrodynamics, elasticity, quantum mechanics. Hence, the proposed study is to investigate the numerical results which are performed for various physical parameters such as velocity profiles, temperature distribution and concentration profiles within the boundary layer are separately discussed in graphically.

Travelling Wave Solutions for Three Dimensional Incompressible MHD Equations

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

On Global Dynamics of Three Dimensional Magnetohydrodynamics: Nonlinear Stability of Alfvén Waves

Magnetohydrodynamics (MHD) studies the dynamics of magnetic fields in electrically conducting fluids. In addition to the sound wave and electromagnetic wave behaviors, magneto-fluids also exhibit an interesting phenomenon: They can produce the Alfven waves, which were first described in a physics paper by Hannes Alfven in 1942. Subsequently, Alfven was awarded the Nobel prize for his fundamental work on MHD with fruitful applications in plasma physics, in particular the discovery of Alfven waves. This work studies (and constructs) global solutions for the three dimensional incompressible MHD systems (with or without viscosity) in strong magnetic backgrounds. We present a complete and self-contained mathematical proof of the global nonlinear stability of Alfven waves. Specifically, our results are as follows: We obtain asymptotics for global solutions of the ideal system (i.e.,viscosity \(\mu =0\)) along characteristics; in particular, we have a scattering theory for the system. We construct the global solutions (for small viscosity \(\mu \)) and we show that as \(\mu \rightarrow 0\), the viscous solutions converge in the classical sense to the zero-viscosity solution. Furthermore, we have estimates on the rate of the convergence in terms of \(\mu \). We explain a linear-driving decay mechanism for viscous Alfven waves with arbitrarily small diffusion. More precisely, for a given solution, we exhibit a time \(T_{n_0}\) (depending on the profile of the datum rather than its energy norm) so that at time \(T_{n_0}\) the \(H^2\)-norm of the solution is small compared to \(\mu \) (therefore the standard perturbation approach can be applied to obtain the convergence to the steady state afterwards). The results and proofs have the following main features and innovations: We do not assume any symmetry condition on initial data. The size of initial data (and the a priori estimates) does not depend on viscosity \(\mu \). The entire proof is built upon the basic energy identity. The Alfven waves do not decay in time: the stable mechanism is the separation (geometrically in space) of left- and right-traveling Alfven waves. The analysis of the nonlinear terms are analogous to the null conditions for non-linear wave equations. We use the (hyperbolic) energy method. In particular, in addition to the use of usual energies, the proof relies heavily on the energy flux through characteristic hypersurfaces. The viscous terms are the most difficult terms since they are not compatible with the hyperbolic approach. We obtain a new class of space-time weighted energy estimates for (weighted) viscous terms. The design of weights is one of the main innovations and it unifies the hyperbolic energy method and the parabolic estimates. The approach is ‘quasi-linear’ in nature rather than a linear perturbation approach: the choices of the coordinate systems, characteristic hypersurfaces, weights and multiplier vector fields depend on the solution itself. Our approach is inspired by Christodoulou–Klainerman’s proof of the nonlinear stability of Minkowski space-time in general relativity.

Three dimensional MHD upper-convected Maxwell nanofluid flow with nonlinear radiative heat flux

In the present article, three dimensional upper-convected Maxwell (UCM) nanofluid flow over a stretching surface has been considered to examine the effects of nanoparticles and magnetohydrodynamics (MHD) on heat and mass transfer. A nonlinear radiative heat flux is incorporated in the formulation of energy equation. Similarity transformation reduces the nonlinear partial differential equations of the problem to the ordinary differential equations, which are then solved by the well known shooting technique through Runge-Kutta integration procedure of order four. To strengthen the reliability of our results, the MATLAB built-in function bvp4c is also used. Effects of some prominent parameters such as Brownian motion parameter, Prandtl number, thermophoresis parameter, magnetic parameter on velocity, temperature and concentration are discussed graphically and numerically. It is witnessed that flow velocity is diminishing function of linear and nonlinear thermal radiation parameter. Moreover mounting values of Brownian motion parameter lower the nanoparticle concentration profile.

Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations

Consider the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations \begin{eqnarray*} \frac{\partial{\bf u}}{\partial t}-\alpha\triangle{\bf u}+({\bf u}\cdot\nabla){\bf u}+\nabla p={\bf f}({\bf x},t),\qquad {\bf u}({\bf x},0)={\bf u}_0({\bf x}). \end{eqnarray*} In this system, the dimension $n\geq 3$, ${\bf u}({\bf x},t)=(u_1({\bf x},t),u_2({\bf x},t),\cdots,u_n({\bf x},t))$ and ${\bf f}({\bf x},t)=(f_1({\bf x},t),f_2({\bf x},t),\cdots,f_n({\bf x},t))$ are real vector valued functions of ${\bf x}=(x_1,x_2,\cdots,x_n)$ and $t$. Additionally, $\alpha>0$ is a positive constant. Suppose that the initial function and the external force satisfy appropriate conditions. &nbsp The main purpose of this paper is to make complete use of the uniform energy estimates of the global smooth solutions and couple together a well known Gronwall's inequality to improve the Fourier splitting method to accomplish the decay estimates with sharp rates. The decay estimates with sharp rates of the global smooth solutions of the Cauchy problems for the $n$-dimensional magnetohydrodynamics equations may be established very similarly.

A Study of Mixing in a Magnetohydrodynamic Microfluidic Cell By Numerical Simulation

A comprehensive study of a two dimensional MHD (magnetohydrodynamics) micro stirrer with different electrode configurations covering a wide parameter space has been conducted by using computational fluid dynamics simulations. The wall of the circular cavity stirrer acts as the counter electrode. Cylindrical rods ranging in number from one to four, placed inside the cavity act as working electrodes. When two fluids are placed initially in the two halves of the cavity separated by an axial plane, the periodic flow reversal causes stretching, folding and breaking of the interface between the two fluids; an increase in the interfacial area; and enhanced mixing. Parametric studies conducted by varying the time period of the potential difference and the magnitude of the magnetic field intensity show that both significantly influence mixing quality. By having more complex geometries by increasing the number of working electrodes and choosing their locations inside the cavity, and by implementing specific switching schemes for the potential difference on the electrode pairs, secondary flows could be generated which introduced complex chaotic advection and enhanced mixing compared to the configuration with a single working electrode. This work adds to the relatively small body of work that relies on numerical simulations to study MHD microfluidic mixing problems. It also establishes numerical simulations as a tool for optimal design of MHD-based lab-on-a-chip applications. The core microfluidic functions in lab-on-a-chip applications include pumping, mixing, and controlling fluid flow [1]. Among these, efficient and rapid mixing is an especially important task since it has an effect on chemical reaction rates, when multiple species are present. However, due to the small size of these devices (usually millimeters or even micrometers), the Reynolds number is low, and therefore turbulent mixing techniques cannot be implemented. Techniques for speeding up mixing by molecular diffusion, an inherently slow process, are needed to improve the mixing and chemical reaction functionalities of the LOAC devices. By applying time periodic potential differences to the electrode pairs, complex flow features and chaotic advection are introduced. The post-processed results from one simulation follow. The time evolution of species mass fraction with T=4s is shown in Figure 1. Though during the first 2s the flow generates only a few striations, the fluids are well mixed subsequently. This can be attributed to the fact that after the first 2s the fluid moves in the next 2s in a pattern that is not just a reversal of the pattern in the previous two seconds. This mode is repeated four times during the 16s observation time resulting in a more chaotic flow. The results from a series of simulations show that a certain level of mixing can be accomplished by the simple use of the sinusoidal potential boundary conditions on one pair of electrodes. Once a potential is applied on the electrode pair in the presence of the magnetic field, the flow is driven to move clockwise and counter-clockwise periodically, by the Lorentz force. As a result, the interface between the fluids stretches rapidly, and therefore good mixing can be achieved. Better mixing performance was observed for larger values of the sinusoidal potential wave period. Furthermore, for the cases considered, increasing the total operation time beyond where the mixing quality vs. time plot plateaus, the marginal increase in the mixing quality is small. Mixing performance can be enhanced by increasing the current or the magnetic field strength, thereby increasing the Lorentz force. By introducing two flow structures having opposite sense of rotation with the use of two electrode pairs, one can enhance mixing even better. By applying sine wave potentials with opposite signs on two working electrodes, two counter-rotational flows can be generated and better mixing quality obtained with a shorter mixing time (Scheme A). This idea was then extended to 4 electrode pairs in which two pairs were active at a given time by switching between two schemes identical to Scheme A. This scheme (Scheme B) in which the electrode pairs are made active or inactive during a half period according to the switching schedule, allows more complex chaotic advection that enhances mixing performance even with a smaller magnetic field intensity. In Scheme B, the unmixed regions near the wall found in previous cases are mostly eliminated. Acknowledgement We acknowledge support under the National Science Foundation grant award number CBET-1336722. Reference [1] Isaac, K.M.; Gonzales, C.; Sen, D. Modeling of redox electrochemical MHD and three-dimensional CFD simulations of transient phenomena in microfluidic cells. Microfluid Nano-fluid., 2014, DOI: 10.1007/s10404-014-1370-6. Figure 1

Influence of Newtonian heating on three dimensional MHD flow of couple stress nanofluid with viscous dissipation and Joule heating.

The present exploration discusses the influence of Newtonian heating on the magnetohydrodynamic (MHD) three dimensional couple stress nanofluid past a stretching surface. Viscous dissipation and Joule heating effects are also considered. Moreover, the nanofluid model includes the combined effects of thermophoresis and Brownian motion. Using an appropriate transformation, the governing non linear partial differential equations are converted into nonlinear ordinary differential equations. Series solutions using Homotopy Analysis method (HAM) are computed. Plots are presented to portrait the arising parameters in the problem. It is seen that an increase in conjugate heating parameter results in considerable increase in the temperature profile of the stretching wall. Skin friction coefficient, local Nusselt and local Sherwood numbers tabulated and analyzed. Higher values of conjugate parameter, Thermophoresis parameter and Brownian motion parameter result in enhancement of temperature distribution.