15:00
15:00
Mathematical aspects of invisibility
Abstract
Rotational Elasticity
Abstract
We consider a 3-dimensional elastic continuum whose material points
can experience no displacements, only rotations. This framework is a
special case of the Cosserat theory of elasticity. Rotations of
material points of the continuum are described mathematically by
attaching to each geometric point an orthonormal basis which gives a
field of orthonormal bases called the coframe. As the dynamical
variables (unknowns) of our theory we choose the coframe and a
density.
In the first part of the talk we write down the general dynamic
variational functional of our problem. In doing this we follow the
logic of classical linear elasticity with displacements replaced by
rotations and strain replaced by torsion. The corresponding
Euler-Lagrange equations turn out to be nonlinear, with the source
of this nonlinearity being purely geometric: unlike displacements,
rotations in 3D do not commute.
In the second part of the talk we present a class of explicit
solutions of our Euler-Lagrange equations. We call these solutions
plane waves. We identify two types of plane waves and calculate
their velocities.
In the third part of the talk we consider a particular case of our
theory when only one of the three rotational elastic moduli, that
corresponding to axial torsion, is nonzero. We examine this case in
detail and seek solutions which oscillate harmonically in time but
depend on the space coordinates in an arbitrary manner (this is a
far more general setting than with plane waves). We show [1] that
our second order nonlinear Euler-Lagrange equations are equivalent
to a pair of linear first order massless Dirac equations. The
crucial element of the proof is the observation that our Lagrangian
admits a factorisation.
[1] Olga Chervova and Dmitri Vassiliev, "The stationary Weyl
equation and Cosserat elasticity", preprint http://arxiv.org/abs/1001.4726
14:00
Disappearing bodies and ghost vortices
Abstract
In many dispersed multiphase flows droplets, bubbles and particles move and disappear due to a phase change. Practical examples include fuel droplets evaporating in a hot gas, vapour bubbles condensing in subcooled liquids and ice crystals melting in water. After these `bodies' have disappeared, they leave behind a remnant `ghost' vortex as an expression of momentum conservation.
A general framework is developed to analyse how a ghost vortex is generated. A study of these processes is incomplete without a detailed discussion of the concept of momentum for unbounded flows. We show how momentum can be defined unambiguously for unbounded flows and show its connection with other expressions, particularly that of Lighthill (1986). We apply our analysis to interpret new observations of condensing vapour bubble and discuss droplet evaporation. We show that the use of integral invariants, widely applied in turbulence, introduces a new perspective to dispersed multiphase flows
16:15