Nonlinear stability of laminar flows in an inclined heated layer with an imposed magnetic field

2020 ◽  
Vol 43 (8) ◽  
pp. 5441-5449
Author(s):  
Lanxi Xu
Keyword(s):  
Magnetic Field ◽  
Nonlinear Stability ◽  
Laminar Flows ◽  
Heated Layer

On the application of the Kelvin–Arnol'd energy principle to the stability of forced two-dimensional inviscid flows

1998 ◽  
Vol 356 ◽  
pp. 221-257 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Magnetic Field ◽  
Nonlinear Stability ◽  
Broad Class ◽  
Mhd Flow ◽  
Two Dimensional ◽  
Energy Principle ◽  
Inviscid Flows ◽  
Virtual Displacement ◽  
Euler Flows ◽  
Universal Procedure

Arnol'd developed two distinct yet closely related approaches to the linear stability of Euler flows. One is widely used for two-dimensional flows and involves constructing a conserved functional whose first variation vanishes and whose second variation determines the linear (and nonlinear) stability of the motion. The second method is a refinement of Kelvin's energy principle which states that stable steady Euler flows represent extremums in energy under a virtual displacement of the vorticity field. The conserved-functional (or energy-Casimir) method has been extended by several authors to more complex flows, such as planar MHD flow. In this paper we generalize the Kelvin–Arnol'd energy method to two-dimensional inviscid flows subject to a body force of the form −ϕ∇f. Here ϕ is a materially conserved quantity and f an arbitrary function of position and of ϕ. This encompasses a broad class of conservative flows, such as natural-convection planar and poloidal MHD flow with the magnetic field trapped in the plane of the motion, flows driven by electrostatic forces, swirling recirculating flow, self-gravitating flows and poloidal MHD flow subject to an azimuthal magnetic field. We show that stable steady motions represent extremums in energy under a virtual displacement of ϕ and of the vorticity field. That is, d1E=0 at equilibrium and whenever d2E is positive or negative definite the flow is (linearly) stable. We also show that unstable normal modes must have a spatial structure which satisfies d2E=0. This provides a single stability test for a broad class of flows, and we describe a simple universal procedure for implementing this test. In passing, a new test for linear stability is developed. That is, we demonstrate that stability is ensured (for flows of the type considered here) whenever the Lagrangian of the flow is a maximum under a virtual displacement of the particle trajectories, the displacement being of the type normally associated with Hamilton's principle. A simple universal procedure for applying this test is also given. We apply our general stability criteria to a range of flows and recover some familiar results. We also extend these ideas to flows which are subject to more than one type of body force. For example, a new stability criterion is obtained (without the use of Casimirs) for natural convection in the presence of a magnetic field. Nonlinear stability is also considered. Specifically, we develop a nonlinear stability criterion for planar MHD flows which are subject to isomagnetic perturbations. This differs from previous criteria in that we are able to extend the linear criterion into the nonlinear regime. We also show how to extend the Kelvin–Arnol'd method to finite-amplitude perturbations.

A nonlinear stability analysis in a double-diffusive magnetized ferrofluid with magnetic-field-dependent viscosity saturating a porous medium

2009 ◽  
Vol 87 (6) ◽  
pp. 659-673 ◽  
Author(s):  
Sunil ◽  
Amit Mahajan
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Nonlinear Stability ◽  
Stability Result ◽  
Darcy Number ◽  
Linear Instability ◽  
Nonlinear Stability Analysis ◽  
Field Dependent ◽  
Magnetized Ferrofluid ◽  
Mfd Viscosity

A rigorous nonlinear stability result is derived by introducing a suitable generalized energy functional for a magnetized ferrofluid layer heated and soluted from below with magnetic-field-dependent (MFD) viscosity saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by the magnetic-body and inertia forces. For ferrofluids, we find that there is possibility of existence of subcritical instabilities, however, it is noted that, in case of a non-ferrofluid, the global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of the magnetic parameter, M3; solute gradient, Sf; Darcy number, Da; and MFD viscosity parameter, δ; on the subcritical instability region has also been analyzed.

Control of the Macrosegregations during Solidification of a Binary Alloy by Means of a AC Magnetic Field

2006 ◽  
Vol 508 ◽  
pp. 163-168 ◽  
Author(s):  
Xiao Dong Wang ◽  
A. Ciobanas ◽  
Florin Baltaretu ◽  
Anne Marie Bianchi ◽  
Yves Fautrelle
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Mushy Zone ◽  
Binary Alloy ◽  
Solidification Front ◽  
Laminar Flows ◽  
Rotating Magnetic Field ◽  
Moderate Intensity ◽  
Mushy Region ◽  
Rotating Magnetic Fields

A numerical model aimed at simulating the segregations during the columnar solidification of a binary alloy is used to investigate the effects of a forced convection. Our objective is to study how the segregation characteristics in the mushy zone are influenced by laminar flows driven both by buoyancy and by AC fields of moderate intensity. Various types of magnetic fields have been tested, namely travelling, rotating magnetic field and slowly modulated electromagnetic forces. The calculations have been achieved on two types of alloys, namely tin-lead and aluminiumsilicon. It is shown that the flow configuration changes the segregation pattern. The change comes from the coupling between the liquid flow and the top of the mushy zone via the pressure distribution along the solidification front. The pressure difference along the front drives a mush flow, which transports the solute in the mushy region. Another interesting type of travelling magnetic field has been tested. It consists of a slowly modulated travelling magnetic field. It is shown that in a certain range of values of the modulation period, the channels are almost suppressed. The normal macrosegregation remains, but the averaged segregation in the mushy zone is weaker than in the natural convection case. The optimal period depends on the electromagnetic force strength as well as the cooling rate. The latter phenomenon cannot occur in the case of rotating magnetic fields, since in that configuration the sign of the pressure gradient along the solidification front remains unchanged. Recent solidification experiments with electromagnetic stirring confirm the predicted macrosegregation patterns.

Stability of hydromagnetic laminar flows in an inclined heated layer

2016 ◽  
Vol 66 (1) ◽  
pp. 125-140 ◽  
Author(s):  
Paolo Falsaperla ◽  
Andrea Giacobbe ◽  
Sebastiano Lombardo ◽  
Giuseppe Mulone
Keyword(s):  
Laminar Flows ◽  
Heated Layer

scholarly journals Magnetohydrodynamic Simulations of Hypersonic Flow over a Cylinder Using Axial- and Transverse-Oriented Magnetic Dipoles

2013 ◽  
Vol 2013 ◽  
pp. 1-15
Author(s):  
Andrew N. Guarendi ◽  
Abhilash J. Chandy
Keyword(s):  
Magnetic Field ◽  
Hypersonic Flow ◽  
Magnetic Reynolds Number ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Magnetic Dipoles ◽  
Quasistatic Approximation ◽  
Magnetohydrodynamic Simulations ◽  
Energy Equations ◽  
The Magnetic Field

Numerical simulations of magnetohydrodynamic (MHD) hypersonic flow over a cylinder are presented for axial- and transverse-oriented dipoles with different strengths. ANSYS CFX is used to carry out calculations for steady, laminar flows at a Mach number of 6.1, with a model for electrical conductivity as a function of temperature and pressure. The low magnetic Reynolds number (≪1) calculated based on the velocity and length scales in this problem justifies the quasistatic approximation, which assumes negligible effect of velocity on magnetic fields. Therefore, the governing equations employed in the simulations are the compressible Navier-Stokes and the energy equations with MHD-related source terms such as Lorentz force and Joule dissipation. The results demonstrate the ability of the magnetic field to affect the flowfield around the cylinder, which results in an increase in shock stand-off distance and reduction in overall temperature. Also, it is observed that there is a noticeable decrease in drag with the addition of the magnetic field.

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