International conference "Geometrical Methods in Mathematical Physics"
December 13, 2011 14:45–15:30, Moscow, Lomonosov Moscow State University
Nonlinear hyperbolic integrodifferential equations and their
applications in hydrodynamics
A. A. Chesnokov
M. A. Lavrent'ev Institute of Hydrodynamics, Novosibirsk
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We study the nonlinear mathematical models of shear flows of ideal liquid in
open channels and the kinetic equations of quasineutral collisionless
plasma. Emphasis is placed on the characteristic properties of the equations
of motion, the construction of exact solutions and their physical
interpretation. Some results of numerical modeling are also presented.
Theoretical analysis of these models is based on the proposed by V. M.
Teshukov concept of hyperbolicity and characteristics for systems of
equations with operator coefficients. A distinctive feature of
integrodifferential models is the presence of both discrete and continuous
spectrum of characteristic velocities. This is due to the fact that
disturbances in shear flows are transmitted through the surface and internal
Open channel flows of ideal incompressible fluid with velocity shear are
considered in the long wave approximation. Nonlinear integrodifferential
models of shallow flow with continuous vertical or horizontal velocity
distribution are derived. It is shown that mathematically the models are
equivalent and, consequently, the obtained early results for two-dimensional
open channel flows with a vertical shear can be applied to the spatial flows
with horizontally nonhomogeneous velocity field. Necessary and sufficient
conditions of generalized hyperbolicity for the equations of motion are
formulated, and the characteristic form of the system is calculated. In the
case of a channel of constant width, the model reduces to the Riemann
integral invariants which are conserved along the characteristics. Stability
of shear flows in terms of hyperbolicity of the governing equations is
studied. It is shown that the type of the equations of motion can change
during the evolution of the flow, which corresponds to the long wave
instability for a certain velocity field.
The concepts of sub- and supercritical flows are introduced for the long
wave approximation model describing the steady-state horizontal-shear
motions of an ideal incompressible fluid with a free boundary in a channel
of variable cross-section. Fluid layer flows developed in a local channel
contraction or expansion are analyzed. Continuous and discontinuous exact
solutions describing different flow regimes are constructed and their
properties are studied. Analytical solutions for flows with the formation of
recirculation zones are obtained.
For the nonlinear kinetic equation describing the one-dimensional motion of
quasineutral collisionless plasma, perturbation velocities are determined
and conditions of generalized hyperbolicity are formulated. An example of
verification of the hyperbolicity conditions is given, and an analogy with
the well-known stability criterion for shear flows is noted. Exact (in
particular, periodical) solutions of the model are constructed and
interpreted physically for the class of traveling waves. It is shown that
traveling waves are stable in the linear approximation only in the case of
an insignificant change in the electric potential. Differential conservation
laws approximating the basic integrodifferential equation are proposed.
These laws are used to perform numerical calculations of wave propagation,
which show the possibility of turnover of the kinetic distribution function.
These results are obtained jointly with V.Yu. Liapidevskii and A.K. Khe.
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