Prsentation PowerPoint

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Summary of the last episode Or what you should
know if youve been sleeping last week

• Instabilities occur when perturbations grow
• Stability depends on the nature of the
  perturbation (AMPLITUDE and SHAPE).
• The same system can be stable or unstable
  depending on the values taken by its CONTROL
  PARAMETERS
• Instabilities appear in NON LINEAR systems,
  which can have NON-UNIQUE solutions

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• Chapter 2 The linear stability
• theory
• 1. General principles
• 2. Electromagnetic pinch effect
• 3. More complex examples
• 4. Hartmann layers, or where it all goes wrong

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1.1. The problem

• Chapter 1 systems can be stable or unstable.
• For which values of the relevant parameters are
  they unstable?
• What will be the shape of the growing
  perturbations, if any?
• How quickly do they develop?
• Need to have a go at the full non-linear
  equations governing the system dynamics. !!

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1.2. The basic ideas
Sytsem determined by n relevant parameters
Equilibrium
ASSUMPTION At the onset, the perturbation grows
from zero
infinitesimal
linearisation around equilibrium
(Methdod can be generalised when )
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1.3.Normal modes analysis
A, B Linear operators / ABBA ?A and B have a
common basis of eigenvectors

• TIME
• If then eigenvect From

Can be decomposed into a countable sum of normal
modes

• SPACE
• If then

Rem

• ? is complex !
• k is imaginary (no local instability)
• a vanishes NO INFORMATION ON THE PETURBATIONS
  AMPLITUDE CAN BE OBTAINED

Dispersion relation
gives the time evolution of each mode k
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Anatomy of a Normal mode
In 2 dimensions x and Y at time t
At one point (x,y)
Phase velocityspeed of wave crests
Group velocitySpeed of energy
Exponential Growth rate For fixed
Neutral stability
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Consequence on stability
Marginal stability ? Depends on ( the
dispertion relation)
the Marginal stability curve
Example one parameter R (convection in a
container, Rayleigh number) RltRc ?? for all
values of k ?? flow is linearly stable RgtRc
?? for at least one value of k ?? flow is
unstable
Exchange of stabilities
Perturbations either grow or decay, but dont
keep their amplitude (not always the case)
Real-life growing perturbations They are a SUM
of normal modes ? exponential growth never seen
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2. Electromagnetic pinch effect

Goal to maintain a jet of electrically
conducting fluid within a given radius R0 ,
without walls (contactless confinement).
0
Uniform current density j0 Fluid density ?
(incompressible) magnetic permeability ? Surface
tension ? Assumption Quantities depend only on z
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2.2. Stability analysis
Base equations (see appendix)
(Momentum budget) (Mass conservation)
(1)
Base solution
Linearised sytem
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(2)
Form of the perturbation
(3)
Rem amplitudes are complex, (phase shifts
between variables )
injecting (3) into (2)
Eliminating R, we get the dissipation equation
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Growth rate a given k
Marginally stable, oscillating modes
Unstable modes
Most unstable mode
All k are present there are always unstable
perturbations The system is always unstable
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2.3. Stabilisation with a longitudinal magnetic
field
Modified Base equations (see appendix)
Base solution unchanged
Linearised sytem
Using the same perturbations as previously
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3. Some more complex stability diagrams
The Taylor- Couette problem of 2 rotating
cylinders
Fluid viscosity ? Narrow gap
Relevant parameters

• Radii ratio
• Angular velocities ratio
• Taylor number

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Taylor-couette instability
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Taylor-couette instability with azimuthal
pressure gradient One more relevant parameter
Stability diagrams without viscosity (Tinfinity)
Stability diagrams with viscosity
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4.The Hartmann layer
Fluid viscosity ? Density ? Electrical
conductivity ?
Magnetic field
B
U
Hartmann layer is destabilised by finite
amplitude perturbations
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Summary Or what you should know if youve been
sleeping during the past hour

• The linear theories cannot give the amplitude of
  the perturbation.
• Use of normal modes only justfied when DLLD
• LS only gives information on the stability of
  the system for CERTAIN TYPES of INFINITESIMAL
  perturbations
• LS Only returns a NECESSARY CONDITION for
  stability, Not always SUFFICIENT.
• Very difficult to perform a linear stability
  analysis for ALL TYPES of perturbations.
• Overlooks transcient growth

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Appendix basic equations of the
electromagnetically confined jet
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Hydrostatic balance

Goal to maintain a jet of electrically
conducting fluid within a given radius R0 ,
without walls (contactless confinement).
0
Uniform current density j0 Fluid density ?
(incompressible) magnetic permeability ?
Calculation of the electromagnetic force in as
section of radius R with current j Ampère
theorem, on surface S(r) surrounded by curve C(r)

Electromagnetic force per unit of volume
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Pressure p in the jet Fluid is in hydrostatic
balance
Along er
Integrating
(surface tension)
The current is conserved between two sections of
the jet of radius R and R0
Rem in fact, one could also account for the
effect of surface tension due to the curvature of
the surface in the azimuthal direction by setting
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Equation governing the radius of the jet

Mass budget on a slice of the jet between times t
and tdt
Volume variationinwards mass flux-outwards mass
flux
assuming
using Taylor formula, for any function g(z)
expanding, and dividing by 2?Rdzdt
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Equation for the axial momentum
Budget of momentum on a cynlinder of axis ez
small radius r, in the referencial moving with
the jet
Variation of axial momentum during dt forces
appied to V ?? dt
Using Taylor formula and dividing by ???r²dzdt
()
Budget of mass on the same cylinder
()
Eliminating ur between () and () and assuming
radial hydrostatic balance
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Addition of a longitudinal magnetic field
Magnetic field is constant, Additional
electromagnetic force
Mass conservation
On a cylinder of small enough radius
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Additional force only modifies the hydrostatic
pressure
Modified motion equation Momentum budget still
valid
using the new expression for the pressure