55 research outputs found
Cancellation of vorticity in steady-state non-isentropic flows of complex fluids
In steady-state non-isentropic flows of perfect fluids there is always
thermodynamic generation of vorticity when the difference between the product
of the temperature with the gradient of the entropy and the gradient of total
enthalpy is different from zero. We note that this property does not hold in
general for complex fluids for which the prominent influence of the material
substructure on the gross motion may cancel the thermodynamic vorticity. We
indicate the explicit condition for this cancellation (topological transition
from vortex sheet to shear flow) for general complex fluids described by
coarse-grained order parameters and extended forms of Ginzburg-Landau energies.
As a prominent sample case we treat first Korteweg's fluid, used commonly as a
model of capillary motion or phase transitions characterized by diffused
interfaces. Then we discuss general complex fluids. We show also that, when the
entropy and the total enthalpy are constant throughout the flow, vorticity may
be generated by the inhomogeneous character of the distribution of material
substructures, and indicate the explicit condition for such a generation. We
discuss also some aspects of unsteady motion and show that in two-dimensional
flows of incompressible perfect complex fluids the vorticity is in general not
conserved, due to a mechanism of transfer of energy between different levels.Comment: 12 page
The motion of whips and chains
We study the motion of an inextensible string (a whip) fixed at one point in
the absence of gravity, satisfying the equations with
boundary conditions and . We prove local existence
and uniqueness in the space defined by the weighted Sobolev energy when
. In addition we show persistence of smooth solutions as long as the
energy for remains bounded. We do this via the method of lines,
approximating with a discrete system of coupled pendula (a chain) for which the
same estimates hold.Comment: 47 pages, 8 figure
Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations
In 1922, Cartan introduced in differential geometry, besides the Riemannian
curvature, the new concept of torsion. He visualized a homogeneous and
isotropic distribution of torsion in three dimensions (3d) by the "helical
staircase", which he constructed by starting from a 3d Euclidean space and by
defining a new connection via helical motions. We describe this geometric
procedure in detail and define the corresponding connection and the torsion.
The interdisciplinary nature of this subject is already evident from Cartan's
discussion, since he argued - but never proved - that the helical staircase
should correspond to a continuum with constant pressure and constant internal
torque. We discuss where in physics the helical staircase is realized: (i) In
the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d
theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's
case of constant pressure and constant intrinsic torque - and b) in 3d Poincare
gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the
gauge field theory of dislocations of Lazar et al., as we prove for the first
time by arranging a suitable distribution of screw dislocations. Our main
emphasis is on the discussion of dislocation field theory.Comment: 31 pages, 8 figure
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