20 research outputs found
Thermocapillary Motion in an Emulsion
The phenomenological model for the motion of an emulsion or a gas-liquid mixture exposed to thermocapillary forces and micro-acceleration is formulated. The analytical and numerical investigation of one-dimensional flows for these media is fulfilled, the structure of discontinuous motion is studied. The stability conditions of a space-uniform state and of the interface between an emulsion and a pure liquid are obtained
Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials
We study initial boundary value problems for the convective Cahn-Hilliard
equation \Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0. It is well-known that without
the convective term, the solutions of this equation may blow up in finite time
for any . In contrast to that, we show that the presence of the convective
term u\px u in the Cahn-Hilliard equation prevents blow up at least for
. We also show that the blowing up solutions still exist if is
large enough (). The related equations like
Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard
equation, are also considered
Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem
The quasi-geostrophic two-layer model is of superior interest in dynamic
meteorology since it is one of the easiest ways to study baroclinic processes
in geophysical fluid dynamics. The complete set of point symmetries of the
two-layer equations is determined. An optimal set of one- and two-dimensional
inequivalent subalgebras of the maximal Lie invariance algebra is constructed.
On the basis of these subalgebras we exhaustively carry out group-invariant
reduction and compute various classes of exact solutions. Where possible,
reference to the physical meaning of the exact solutions is given. In
particular, the well-known baroclinic Rossby wave solutions in the two-layer
model are rediscovered.Comment: Extended version, 24 pages, 1 figur
The evolution operator of the Hartree-type equation with a quadratic potential
Based on the ideology of the Maslov's complex germ theory, a method has been
developed for finding an exact solution of the Cauchy problem for a
Hartree-type equation with a quadratic potential in the class of
semiclassically concentrated functions. The nonlinear evolution operator has
been obtained in explicit form in the class of semiclassically concentrated
functions. Parametric families of symmetry operators have been found for the
Hartree-type equation. With the help of symmetry operators, families of exact
solutions of the equation have been constructed. Exact expressions are obtained
for the quasi-energies and their respective states. The Aharonov-Anandan
geometric phases are found in explicit form for the quasi-energy states.Comment: 23 pege
Long-Wave Instability of Advective Flows in Inclined Layer with Solid Heat Conductive Boundaries
We investigate the stability of the steady convective flow in a plane tilted
layer with ideal thermal conductivity of solid boundaries in the presence of
uniform longitudinal temperature gradient. Analytically found the stability
boundary with respect to the long-wave perturbations, find the critical Grashof
number for the most dangerous among them of even spiral perturbation.Comment: in Russian, 18 pages, 5 figures, submited to Appl. mechanics and
physics, RAS Siberian brunch, Novosibirsk, Russia; Key words: advective flow,
oblique layer, a longitudinal temperature gradient, long-wave instabilit