Pattern Formation

In den nächsten Wochen bis zum 20.2.2020 möchte Anna Hein,
Studentin der Wissenschaftskommunikation am KIT, eine Studie im
Rahmen ihrer Masterarbeit über den Podcast Modellansatz
durchführen. Dazu möchte sie gerne einige Interviews mit Ihnen,
den Hörerinnen und Hörern des Podcast Modellansatz führen, um
herauszufinden, wer den Podcast hört und wie und wofür er genutzt
wird. Die Interviews werden anonymisiert und werden jeweils circa
15 Minuten in Anspruch nehmen. Für die Teilnahme an der Studie
können Sie sich bis zum 20.2.2020 unter der Emailadresse
studie.modellansatz@web.de bei Anna Hein melden. Wir würden uns
sehr freuen, wenn sich viele Interessenten melden würden.


In the coming weeks until February 20, 2020, Anna Hein,
student of science communication at KIT, intends to conduct a
study on the Modellansatz Podcast within her master's thesis. For
this purpose, she would like to conduct some interviews with you,
the listeners of the Modellansatz Podcast, to find out who
listens to the podcast and how and for what purpose it is used.
The interviews will be anonymous and will take about 15 minutes
each. To participate in the study, you can register with Anna
Hein until 20.2.2020 at studie.modellansatz@web.de . We would be
very pleased if many interested parties would contact us.



This is the second of three conversation recorded Conference on
mathematics of wave phenomena 23-27 July 2018 in Karlsruhe.

Gudrun is in conversation with Mariana Haragus about
Benard-Rayleigh problems. On the one hand this is a much studied
model problem in Partial Differential Equations. There it has
connections to different fields of research due to the different
ways to derive and read the stability properties and to work with
nonlinearity. On the other hand it is a model for various
applications where we observe an interplay between boyancy and
gravity and for pattern formation in general.


An everyday application is the following: If one puts a pan with
a layer of oil on the hot oven (in order to heat it up) one
observes different flow patterns over time. In the beginning it
is easy to see that the oil is at rest and not moving at all. But
if one waits long enough the still layer breaks up into small
cells which makes it more difficult to see the bottom clearly.
This is due to the fact that the oil starts to move in circular
patterns in these cells.


For the problem this means that the system has more than one
solutions and depending on physical parameters one solution is
stable (and observed in real life) while the others are unstable.
In our example the temperature difference between bottom and top
of the oil gets bigger as the pan is heating up. For a while the
viscosity and the weight of the oil keep it still. But if the
temperature difference is too big it is easier to redistribute
the different temperature levels with the help of convection of
the oil.


The question for engineers as well as mathematicians is to find
the point where these convection cells evolve in theory in order
to keep processes on either side of this switch. In theory (not
for real oil because it would start to burn) for even bigger
temperature differences the original cells would break up into
even smaller cells to make the exchange of energy faster.


In 1903 Benard did experiments similar to the one described in
the conversation which fascinated a lot of his colleagues at the
time. The equations where derived a bit later and already in 1916
Lord Rayleigh found the 'switch', which nowadays is called the
critical Rayleigh number. Its size depends on the thickness of
the configuration, the viscositiy of the fluid, the gravity force
and the temperature difference.


Only in the 1980th it became clear that Benards' experiments and
Rayleigh's analysis did not really cover the same problem since
in the experiment the upper boundary is a free boundary to the
surrounding air while Rayleigh considered fixed boundaries. And
this changes the size of the critical Rayleigh number. For each
person doing experiments it is also an observation that the shape
of the container with small perturbations in the ideal shape
changes the convection patterns.


Maria does study the dynamics of nonlinear waves and patterns.
This means she is interested in understanding processes which
change over time. Her main questions are:


Existence of observed waves as solutions of the equations

The stability of certain types of solutions

How is the interaction of different waves



She treats her problems with the theory of dynamical systems and
bifurcations. The simplest tools go back to Poincaré when
understanding ordinary differential equations. One could consider
the partial differential equations to be the evolution in an
infinite dimensional phase space. Here, in the 1980s, Klaus
Kirchgässner had a few crucial ideas how to construct special
solutions to nonlinear partial differential equations.


It is possible to investigate waterwave problems which are
dispersive equations as well as flow problems which are
dissipative. Together with her colleagues in Besancon she is also
very keen to match experiments for optical waves with her
mathematical analysis. There Mariana is working with a variant of
the Nonlinear Schrödinger equation called Lugiato-Lefever
Equation. It has many different solutions, e.g. periodic
solutions and solitons.


Since 2002 Mariana has been Professor in Besancon (University of
Franche-Comté, France). Before that she studied and worked in a
lot of different places, namely in Bordeaux, Stuttgart,
Bucharest, Nice, and Timisoara.


References

V.A. Getling: Rayleigh-Bénard Convection Structures and
Dynamics, Advanced Series in Nonlinear Dynamics, Volume 11, World
Scientific, Oxford (1998)

P. H. Rabinowitz: Existence and nonuniqueness of rectangular
solutions of the Bénard problem. Arch. Rational Mech. Anal.
(1968) 29: 32.

M. Haragus and G. Iooss: Local bifurcations, center
manifolds, and normal forms in infinite-dimensional dynamical
systems. Universitext. Springer-Verlag London, Ltd., London; EDP
Sciences, Les Ulis, 2011.

Newell, Alan C. Solitons in mathematics and physics. CBMS-NSF
Regional Conference Series in Applied Mathematics, 48. Society
for Industrial and Applied Mathematics (SIAM), Philadelphia, PA,
1985.

Y. K. Chembo, D. Gomila, M. Tlidi, C. R. Menyuk: Topical
Issue: Theory and Applications of the Lugiato-Lefever Equation.
Eur. Phys. J. D 71 (2017).



Podcasts

S. Fliss, G. Thäter: Transparent Boundaries. Conversation in
the Modellansatz Podcast episode 75, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.

M. Kray, G. Thäter: Splitting Waves. Conversation in the
Modellansatz Podcast episode 62, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.

F. Sayas, G. Thäter: Acoustic scattering. Conversation in the
Modellansatz Podcast episode 58, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.