scholarly journals The effect of finite-conductivity Hartmann walls on the linear stability of Hunt’s flow

2017 ◽  
Vol 822 ◽  
pp. 880-891 ◽  
Author(s):  
Thomas Arlt ◽  
Jānis Priede ◽  
Leo Bühler
Keyword(s):  
Magnetic Field ◽  
Linear Stability ◽  
Metal Flow ◽  
Finite Conductivity ◽  
Asymptotic Results ◽  
Square Duct ◽  
Wall Conductivity ◽  
Side Walls ◽  
Conductance Ratio ◽  
The Stability

We analyse numerically the linear stability of fully developed liquid metal flow in a square duct with insulating side walls and thin, electrically conducting horizontal walls. The wall conductance ratio $c$ is in the range of 0.01 to 1 and the duct is subject to a vertical magnetic field with Hartmann numbers up to $\mathit{Ha}=10^{4}$. In a sufficiently strong magnetic field, the flow consists of two jets at the side walls and a near-stagnant core with relative velocity ${\sim}(c\mathit{Ha})^{-1}$. We find that for $\mathit{Ha}\gtrsim 300,$ the effect of wall conductivity on the stability of the flow is mainly determined by the effective Hartmann wall conductance ratio $c\mathit{Ha}.$ For $c\ll 1$, the increase of the magnetic field or that of the wall conductivity has a destabilizing effect on the flow. Maximal destabilization of the flow occurs at $\mathit{Ha}\approx 30/c$. In a stronger magnetic field with $c\mathit{Ha}\gtrsim 30$, the destabilizing effect vanishes and the asymptotic results of Priede et al. (J. Fluid Mech., vol. 649, 2010, pp. 115–134) for ideal Hunt’s flow with perfectly conducting Hartmann walls are recovered.

scholarly journals Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls

2015 ◽  
Vol 788 ◽  
pp. 129-146 ◽  
Author(s):  
Jānis Priede ◽  
Thomas Arlt ◽  
Leo Bühler
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Strong Magnetic Field ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Metal Flow ◽  
Three Dimensional ◽  
Transverse Magnetic ◽  
Base Flow ◽  
Square Duct

This study is concerned with the numerical linear stability analysis of liquid-metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios $c\ll 1$, an extremely strong magnetic field with Hartmann number $\mathit{Ha}\sim c^{-4}$ is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1$, 0.1 and 0.01 and Hartmann numbers up to $10^{4}$. As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness ${\it\delta}\sim \mathit{Ha}^{-1/2}$. This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $\mathit{Re}_{c}\sim 110\mathit{Ha}^{1/2}$ and $k_{c}\sim 0.5\mathit{Ha}^{1/2}$. The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll 1$ is $\overline{\mathit{Re}}_{c}\approx 520$. Although this value is somewhat larger than $\overline{\mathit{Re}}_{c}\approx 313$ found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.

Particle Transport in a Turbulent Square Duct Flow With an Imposed Magnetic Field

2013 ◽  
Author(s):  
Rui Liu ◽  
Surya P. Vanka ◽  
Brian G. Thomas
Keyword(s):  
Magnetic Field ◽  
Particle Transport ◽  
Continuous Flow ◽  
Duct Flow ◽  
Deposition Rates ◽  
Square Duct ◽  
Side Walls ◽  
Volume Method ◽  
The Magnetic Field ◽  
A Current

In this paper we study the particle transport and deposition in a turbulent square duct flow with an imposed magnetic field using Direct Numerical Simulations (DNS) of the continuous flow. A magnetic field induces a current and the interaction of this current with the magnetic field generates a Lorentz force which brakes the flow and modifies the flow structure. A second-order accurate finite volume method in time and space is used and implemented on a GPU. Particles are injected at the entrance to the duct continuously and their rates of deposition on the duct walls are computed for different magnetic field strengths. Because of the changes to the flow due to the magnetic field, the deposition rates are different on the top and bottom walls compared to the side walls. This is different than in a non-MHD square duct flow, where quadrant (and octant) symmetry is obtained.

scholarly journals Linear Stability Analysis of Liquid Metal Flow in an Insulating Rectangular Duct under External Uniform Magnetic Field

Fluids ◽  
2019 ◽  
Vol 4 (4) ◽  
pp. 177 ◽  
Author(s):  
Tagawa
Keyword(s):  
Magnetic Field ◽  
Stability Analysis ◽  
Liquid Metal ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Uniform Magnetic Field ◽  
Metal Flow ◽  
Rectangular Duct ◽  
Liquid Metal Flow ◽  
External Uniform Magnetic Field

Linear stability analysis of liquid metal flow driven by a constant pressure gradient in an insulating rectangular duct under an external uniform magnetic field was carried out. In the present analysis, since the Joule heating and induced magnetic field were neglected, the governing equations consisted of the continuity of mass, momentum equation, Ohm’s law, and conservation of electric charge. A set of linearized disturbance equations for the complex amplitude was decomposed into real and imaginary parts and solved numerically with a finite difference method using the highly simplified marker and cell (HSMAC) algorithm on a two-dimensional staggered mesh system. The difficulty of the complex eigenvalue problem was circumvented with a Newton—Raphson method during which its corresponding eigenfunction was simultaneously obtained by using an iterative procedure. The relation among the Reynolds number, the wavenumber, the growth rate, and the angular frequency was successfully obtained for a given value of the Hartmann number as well as for a direction of external uniform magnetic field.

scholarly journals Linear stability of Hunt's flow

2010 ◽  
Vol 649 ◽  
pp. 115-134 ◽  
Author(s):  
JĀNIS PRIEDE ◽  
SVETLANA ALEKSANDROVA ◽  
SERGEI MOLOKOV
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Transverse Magnetic ◽  
Base Flow ◽  
Maximal Velocity ◽  
Square Duct ◽  
Stream Function Formulation ◽  
The Magnetic Field

We analyse numerically the linear stability of the fully developed flow of a liquid metal in a square duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow the disturbances with four different parity combinations over the duct cross-section decouple from each other. Magnetic field renders the flow in a square duct linearly unstable at the Hartmann number Ha ≈ 5.7 with respect to a disturbance whose vorticity component along the magnetic field is even across the field and odd along it. For this mode, the minimum of the critical Reynolds number Rec ≈ 2018, based on the maximal velocity, is attained at Ha ≈ 10. Further increase of the magnetic field stabilizes this mode with Rec growing approximately as Ha. For Ha > 40, the spanwise parity of the most dangerous disturbance reverses across the magnetic field. At Ha ≈ 46 a new pair of most dangerous disturbances appears with the parity along the magnetic field being opposite to that of the previous two modes. The critical Reynolds number, which is very close for both of these modes, attains a minimum, Rec ≈ 1130, at Ha ≈ 70 and increases as Rec ≈ 91Ha1/2 for Ha ≫ 1. The asymptotics of the critical wavenumber is kc ≈ 0.525Ha1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.

scholarly journals Linear stability of thermocapillary convection in cylindrical liquid bridges under axial magnetic fields

1999 ◽  
Vol 394 ◽  
pp. 281-302 ◽  
Author(s):  
M. PRANGE ◽  
M. WANSCHURA ◽  
H. C. KUHLMANN ◽  
H. J. RATH
Keyword(s):  
Magnetic Field ◽  
Free Surface ◽  
Magnetic Fields ◽  
Prandtl Number ◽  
Linear Stability ◽  
Thermocapillary Convection ◽  
Axial Magnetic Field ◽  
Unstable Mode ◽  
Linear Stability Theory ◽  
The Stability

The stability of axisymmetric steady thermocapillary convection of electrically conducting fluids in half-zones under the influence of a static axial magnetic field is investigated numerically by linear stability theory. In addition, the energy transfer between the basic state and a disturbance is considered in order to elucidate the mechanics of the most unstable mode. Axial magnetic fields cause a concentration of the thermocapillary flow near the free surface of the liquid bridge. For the low Prandtl number fluids considered, the most dangerous disturbance is a non-axisymmetric steady mode. It is found that axial magnetic fields act to stabilize the basic state. The stabilizing effect increases with the Prandtl number and decreases with the zone height, the heat transfer rate at the free surface and buoyancy when the heating is from below. The magnetic field also influences the azimuthal symmetry of the most unstable mode.

Linear stability analysis of non-Newtonian blood flow with magnetic nanoparticles: application to controlled drug delivery

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Pascalin Tiam Kapen ◽  
Cédric Gervais Njingang Ketchate ◽  
Didier Fokwa ◽  
Ghislain Tchuen
Keyword(s):  
Magnetic Field ◽  
Drug Delivery ◽  
Stability Analysis ◽  
Magnetic Nanoparticles ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Controlled Drug Delivery ◽  
Darcy Number ◽  
Content Type ◽  
The Stability

Purpose For this purpose, a linear stability analysis based on the Navier–Stokes and Maxwell equations is made leading to an eigenvalue differential equation of the modified Orr–Sommerfeld type which is solved numerically by the spectral collocation method based on Chebyshev polynomials. Unlike previous studies, blood is considered as a non-Newtonian fluid. The effects of various parameters such as volume fraction of nanoparticles, Casson parameter, Darcy number, Hartmann number on flow stability were examined and presented. This paper aims to investigate a linear stability analysis of non-Newtonian blood flow with magnetic nanoparticles with an application to controlled drug delivery. Design/methodology/approach Targeted delivery of therapeutic agents such as stem cells and drugs using magnetic nanoparticles with the help of external magnetic fields is an emerging treatment modality for many diseases. To this end, controlling the movement of nanoparticles in the human body is of great importance. This study investigates controlled drug delivery by using magnetic nanoparticles in a porous artery under the influence of a magnetic field. Findings It was found the following: the Casson parameter affects the stability of the flow by amplifying the amplitude of the disturbance which reflects its destabilizing effect. It emerges from this study that the taking into account of the non-Newtonian character is essential in the modeling of such a system, and that the results can be very different from those obtained by supposing that the blood is a Newtonian fluid. The presence of iron oxide nanoparticles in the blood increases the inertia of the fluid, which dampens the disturbances. The Strouhal number has a stabilizing effect on the flow which makes it possible to say that the oscillating circulation mechanisms dampen the disturbances. The Darcy number affects the stability of the flow and has a stabilizing effect, which makes it possible to increase the contact surface between the nanoparticles and the fluid allowing very high heat transfer rates to be obtained. It also emerges from this study that the presence of the porosity prevents the sedimentation of the nanoparticles. By studying the effect of the magnetic field on the stability of the flow, it is observed that the Hartmann number keeps the flow completely stable. This allows saying that the magnetic field makes the dissipations very important because the kinetic energy of the electrically conductive ferrofluid is absorbed by the Lorentz force. Originality/value The originality of this paper resides on the application of the linear stability analysis for controlled drug delivery.

scholarly journals Linear stability of magnetohydrodynamic flow in a perfectly conducting rectangular duct

2012 ◽  
Vol 708 ◽  
pp. 111-127 ◽  
Author(s):  
Jānis Priede ◽  
Svetlana Aleksandrova ◽  
Sergei Molokov
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Strong Magnetic Field ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Metal Flow ◽  
Transverse Magnetic ◽  
Rectangular Duct ◽  
Instability Modes ◽  
The Magnetic Field

AbstractWe analyse numerically the linear stability of a liquid-metal flow in a rectangular duct with perfectly electrically conducting walls subject to a uniform transverse magnetic field. A non-standard three-dimensional vector stream-function/vorticity formulation is used with a Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. A relatively weak magnetic field is found to render the flow linearly unstable as two weak jets appear close to the centre of the duct at the Hartmann number $\mathit{Ha}\approx 9. 6. $ In a sufficiently strong magnetic field, the instability following the jets becomes confined in the layers of characteristic thickness $\delta \ensuremath{\sim} {\mathit{Ha}}^{\ensuremath{-} 1/ 2} $ located at the walls parallel to the magnetic field. In this case the instability is determined by $\delta , $ which results in both the critical Reynolds number and wavenumber scaling as ${\ensuremath{\sim} }{\delta }^{\ensuremath{-} 1} . $ Instability modes can have one of the four different symmetry combinations along and across the magnetic field. The most unstable is a pair of modes with an even distribution of vorticity along the magnetic field. These two modes represent strongly non-uniform vortices aligned with the magnetic field, which rotate either in the same or opposite senses across the magnetic field. The former enhance while the latter weaken one another provided that the magnetic field is not too strong or the walls parallel to the field are not too far apart. In a strong magnetic field, when the vortices at the opposite walls are well separated by the core flow, the critical Reynolds number and wavenumber for both of these instability modes are the same: ${\mathit{Re}}_{c} \approx 642{\mathit{Ha}}^{1/ 2} + 8. 9\ensuremath{\times} 1{0}^{3} {\mathit{Ha}}^{\ensuremath{-} 1/ 2} $ and ${k}_{c} \approx 0. 477{\mathit{Ha}}^{1/ 2} . $ The other pair of modes, which differs from the previous one by an odd distribution of vorticity along the magnetic field, is more stable with an approximately four times higher critical Reynolds number.

Toroidal equilibrium states with reversed magnetic shear and parallel flow in connection with the formation of Internal Transport Barriers

2015 ◽  
Vol 81 (4) ◽  
Author(s):  
Ap. Kuiroukidis ◽  
G. N. Throumoulopoulos
Keyword(s):  
Magnetic Field ◽  
Linear Stability ◽  
Current Density ◽  
Sufficient Condition ◽  
Magnetic Shear ◽  
Transport Barriers ◽  
Internal Transport Barriers ◽  
The Stability ◽  
The Magnetic Field ◽  
Internal Transport

We construct nonlinear toroidal equilibria of fixed diverted boundary shaping with reversed magnetic shear and flows parallel to the magnetic field. The equilibria have hole-like current density and the reversed magnetic shear increases as the equilibrium nonlinearity becomes stronger. Also, application of a sufficient condition for linear stability implies that the stability is improved as the equilibrium nonlinearity correlated to the reversed magnetic shear gets stronger with a weaker stabilizing contribution from the flow. These results indicate synergetic stabilizing effects of reversed magnetic shear, equilibrium nonlinearity and flow in the establishment of Internal Transport Barriers (ITBs).

Electromagnetic Pump With Thin Metal Walls

1994 ◽  
Vol 116 (2) ◽  
pp. 298-302
Author(s):  
N. Ma ◽  
T. J. Moon ◽  
J. S. Walker
Keyword(s):  
Magnetic Field ◽  
Boundary Layers ◽  
Metal Flow ◽  
Side Wall ◽  
Transverse Magnetic ◽  
Thin Metal ◽  
Total Flow ◽  
Electromagnetic Pump ◽  
Liquid Metal Flow ◽  
Side Walls

This paper treats a liquid-metal flow in a rectangular duct with a strong, uniform, transverse magnetic field and with thin metal walls, except for two finite-length, perfectly conducting electrodes in the side walls, which are parallel to the magnetic field. There are large velocities inside the boundary layers adjacent to the thin metal side walls, but not inside the layers adjacent to the electrodes. Upstream and downstream of the electrodes, a significant fraction of the total flow leaves and enters the side-wall boundary layers, respectively. For the particular duct treated here, the fully developed side layers, which carry 38.8 percent of the total flow, are realized at a distance of three characteristic lengths from the ends of the electrodes.

Stability of axially symmetric flow driven by a rotating magnetic field in a cylindrical cavity

2001 ◽  
Vol 431 ◽  
pp. 407-426 ◽  
Author(s):  
I. GRANTS ◽  
G. GERBETH
Keyword(s):  
Magnetic Field ◽  
Metal Flow ◽  
Finite Amplitude ◽  
Rotating Magnetic Field ◽  
Axially Symmetric ◽  
Stable Regime ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
Steady Solutions ◽  
The Stability

This paper deals with the stability analysis of an axially symmetric liquid metal flow driven by a rotating magnetic field in a cylinder of finite dimensions. The limit of linear stability with respect to axially symmetric perturbations is found for diameter-to-height ratios between 0.4 and 1. This oscillatory instability is shown to be different from the expected Taylor–Görtler vortices. Several linearly unstable steady solutions are found close to the stable basic state. It is shown that small finite-amplitude perturbations in the form of Taylor–Görtler vortices give rise to instability in the linearly stable regime.

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