On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities

2003 ◽  
Vol 476 ◽  
pp. 389-409 ◽  
Author(s):  
GORDON I. OGILVIE ◽  
MICHAEL R. E. PROCTOR
Keyword(s):  
Viscoelastic Medium ◽  
Magnetic Reynolds Number ◽  
Deborah Number ◽  
Elastic Instability ◽  
Magnetorotational Instability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Magnetohydrodynamic Flows ◽  
Elastic Instabilities ◽  
The Stability

We demonstrate a close analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field. Specifically, the dynamics of the Oldroyd-B fluid in the limit of large Deborah number corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. As a definite example of this analogy, we compare the stability properties of differentially rotating viscoelastic and MHD flows. We show that there is an instability of the Oldroyd-B fluid that is physically distinct from both the inertial and elastic instabilities described previously in the literature, but is directly equivalent to the magnetorotational instability in MHD. It occurs even when the specific angular momentum increases outwards, provided that the angular velocity decreases outwards; it derives from the kinetic energy of the shear flow and does not depend on the curvature of the streamlines. However, we argue that the elastic instability of viscoelastic Couette flow has no direct equivalent in MHD.

Rayleigh–Bénard convection and pattern formation in magnetohydrodynamics

1998 ◽  
Vol 60 (3) ◽  
pp. 529-539 ◽  
Author(s):  
RENU BAJAJ ◽  
S. K. MALIK
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Thermal Instability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
The Magnetic Field

A nonlinear thermal instability in a layer of electrically conducting fluid in the presence of a magnetic field is discussed. Steady-state bifurcation results in the formation of patterns: rolls, squares and hexagons. The stability of various patterns is also investigated. It is found that in the absence of a magnetic field only rolls are stable, but when the magnetic field strength exceeds a certain finite value, squares and hexagons also become stable.

scholarly journals The influence of stratification upon small-scale convectively-driven dynamos

2010 ◽  
Vol 6 (S271) ◽  
pp. 197-204 ◽  
Author(s):  
Paul J. Bushby ◽  
Michael R. E. Proctor ◽  
Nigel O. Weiss
Keyword(s):  
Magnetic Energy ◽  
Magnetic Reynolds Number ◽  
Small Scale ◽  
Initial Growth ◽  
Dynamo Action ◽  
Quiet Sun ◽  
Electrically Conducting ◽  
Solar Observations ◽  
Electrically Conducting Fluid ◽  
Convective Motions

AbstractIn the quiet Sun, convective motions form a characteristic granular pattern, with broad upflows enclosed by a network of narrow downflows. Magnetic fields tend to accumulate in the intergranular lanes, forming localised flux concentrations. One of the most plausible explanations for the appearance of these quiet Sun magnetic features is that they are generated and maintained by dynamo action resulting from the local convective motions at the surface of the Sun. Motivated by this idea, we describe high resolution numerical simulations of nonlinear dynamo action in a (fully) compressible, non-rotating layer of electrically-conducting fluid. The dynamo properties depend crucially upon various aspects of the fluid. For example, the magnetic Reynolds number (Rm) determines the initial growth rate of the magnetic energy, as well as the final saturation level of the dynamo in the nonlinear regime. We focus particularly upon the ways in which the Rm-dependence of the dynamo is influenced by the level of stratification within the domain. Our results can be related, in a qualitative sense, to solar observations.

THE STAEILITY OF COUETTE FLOW IN AN AXIAL MAGNETIC FIELD

1958 ◽  
Vol 36 (11) ◽  
pp. 1509-1525 ◽  
Author(s):  
E. R. Niblett
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Theoretical Prediction ◽  
Viscous Flow ◽  
Rotation Speed ◽  
Axial Magnetic Field ◽  
Radioactive Isotopes ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
The Stability

Chandrasekhar's theory of the stability of viscous flow of an electrically conducting fluid between coaxial rotating cylinders with perfectly conducting walls is extended to include the case of non-conducting walls, and it is found that their effect is to reduce the critical Taylor numbers and increase the wavelength of the instability patterns by considerable amounts. An experiment designed to measure the values of magnetic field and rotation speed at the onset of instability in mercury between perspex cylinders is described. The radioactive isotopes Hg197 and Hg203 were used to trace the flow. The results support the theoretical prediction that the boundary conditions can have a large effect on the motion.

Hydromagnetic stability of a thin electrically conducting fluid film flowing down the outside surface of a long vertically aligned column

2009 ◽  
Vol 223 (2) ◽  
pp. 415-426 ◽  
Author(s):  
P-J Cheng
Keyword(s):  
Magnetic Field ◽  
Multiple Scales ◽  
Vertical Cylinder ◽  
Conducting Fluid ◽  
Fluid Film ◽  
Electrically Conducting ◽  
Free Film ◽  
Electrically Conducting Fluid ◽  
Non Linear ◽  
The Stability

This article considers the stability of a thin electrically conducting fluid film flowing down the outer surface of a long vertical cylinder in the presence of an applied magnetic field. Using the long-wave perturbation method to solve the generalized non-linear kinematic equations with free film interface, the normal mode approach is first used to compute the linear stability solution. The method of multiple scales is then used to obtain the weak non-linear dynamics. The results indicate that both subcritical instability and supercritical stability conditions are possible. The degree of instability in the film flow is intensified by the lateral curvature of the cylinder. The results also show that increasing the strength of the magnetic field tends to enhance the stability.

A simple scenario of the laminar breakdown in liquid metal flows

2021 ◽  
Vol 57 (2) ◽  
pp. 191-210
Keyword(s):  
Reynolds Number ◽  
Electrical Potential ◽  
Finite Amplitude ◽  
Magnetic Reynolds Number ◽  
Small Length ◽  
Turbulent Transition ◽  
Local Perturbations ◽  
Magnetohydrodynamic Flows ◽  
The Stability ◽  
Velocity Pressure

In the article, authors present a numerical method for modelling a laminar-turbulent transition in magnetohydrodynamic flows. The small magnetic Reynolds number approach is considered. Velocity, pressure and electrical potential are decomposed to the sum of state values and finite amplitude perturbations. A solver based on the Nektar++ framework is described. The authors suggest using small-length local perturbations as a transition trigger. They can be imposed by blowing or by electrical enforcing. The stability of the Hartmann flow and the flow in the bend are considered as examples. Tables 4, Figs 19, Refs 28.

Evolution of wave packets in magnetohydrodynamics

1995 ◽  
Vol 53 (2) ◽  
pp. 145-167 ◽  
Author(s):  
Anju Pusri ◽  
S. K. Malik
Keyword(s):  
Magnetic Field ◽  
Wave Packets ◽  
Stability Diagram ◽  
Envelope Soliton ◽  
Electrically Conducting ◽  
Self Focusing ◽  
Electrically Conducting Fluid ◽  
Regions Of Stability ◽  
Taylor Problem ◽  
The Stability

The propagation of wave packets on the surface of an electrically conducting fluid of uniform depth in the presence of a tangential magnetic field is investigated in (2 + 1) dimensions. The evolution of wave envelope is governed by two coupled partial differential equations with cubic nonlinearity. The stability analysis reveals the existence of different regions of instability. The effect of the applied magnetic field is not only significant but also different for different regions of stability. ‘Envelope soliton’ and ‘waveguide’ solutions of the amplitude equation are also discussed. The self-focusing phenomenon that arises when the amplitude of the wave becomes infinite in finite time is also examined. It is found that in a certain region of the stability diagram it may be easier to observe this phenomenon in the presence of a magnetic field. The Rayleigh-Taylor problem is also studied and various criteria for the existence of instability are obtained.

scholarly journals Convective Instability in a Hele Shaw Cell with the Effect of Through Flow and Magnetic Field

2018 ◽  
Author(s):  
Dhananjay Yadav
Keyword(s):  
Magnetic Field ◽  
Convective Instability ◽  
Fluid Layer ◽  
Analytical Investigation ◽  
Linear Stability Theory ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Through Flow ◽  
The Stability ◽  
Hele Shaw Cell

In this paper, an analytical investigation of the combined effect of through flow and magnetic field on the convective instability in an electrically conducting fluid layer, bounded in a Hele-Shaw cell is presented within the context of linear stability theory. The Galarkin method is utilized to solve the eigenvalue problem. The outcome of the important parameters on the stability of the system is examined analytically as well as graphically. It is observed that the through flow and magnetic field have both stabilizing effects, while the Hele-Shaw number has destabilizing effect on the stability of system. It is also found that the oscillatory mode of convection possible only when the magnetic Prandtl number takes the values less than unity.

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