3,057 research outputs found
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
The Steady Boltzmann and Navier-Stokes Equations
The paper discusses the similarities and the differences in the mathematical
theories of the steady Boltzmann and incompressible Navier-Stokes equations
posed in a bounded domain. First we discuss two different scaling limits in
which solutions of the steady Boltzmann equation have an asymptotic behavior
described by the steady Navier-Stokes Fourier system. Whether this system
includes the viscous heating term depends on the ratio of the Froude number to
the Mach number of the gas flow. While the steady Navier-Stokes equations with
smooth divergence-free external force always have at least one smooth
solutions, the Boltzmann equation with the same external force set in the
torus, or in a bounded domain with specular reflection of gas molecules at the
boundary may fail to have any solution, unless the force field is identically
zero. Viscous heating seems to be of key importance in this situation. The
nonexistence of any steady solution of the Boltzmann equation in this context
seems related to the increase of temperature for the evolution problem, a
phenomenon that we have established with the help of numerical simulations on
the Boltzmann equation and the BGK model.Comment: 55 pages, 4 multiple figure
Observers for compressible Navier-Stokes equation
We consider a multi-dimensional model of a compressible fluid in a bounded
domain. We want to estimate the density and velocity of the fluid, based on the
observations for only velocity. We build an observer exploiting the symmetries
of the fluid dynamics laws. Our main result is that for the linearised system
with full observations of the velocity field, we can find an observer which
converges to the true state of the system at any desired convergence rate for
finitely many but arbitrarily large number of Fourier modes. Our
one-dimensional numerical results corroborate the results for the linearised,
fully observed system, and also show similar convergence for the full nonlinear
system and also for the case when the velocity field is observed only over a
subdomain
Institute for Computational Mechanics in Propulsion (ICOMP)
The Institute for Computational Mechanics in Propulsion (ICOMP) is a combined activity of Case Western Reserve University, Ohio Aerospace Institute (OAI) and NASA Lewis. The purpose of ICOMP is to develop techniques to improve problem solving capabilities in all aspects of computational mechanics related to propulsion. The activities at ICOMP during 1991 are described
Mathematical Aspects of Hydrodynamics
The workshop dealt with the partial differential equations that describe fluid motion and related topics.
These topics included both inviscid and viscous fluids in two and three dimensions. Some talks addressed
aspects of fluid dynamics such as the construction of wild weak solutions, compressible shock formation,
inviscid limit and behavior of boundary layers, as well as both polymer/fluid and structure/fluid interaction
Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow
A technical obstruction preventing the conclusion of nonlinear stability of
large-Froude number roll waves of the St. Venant equations for inclined thin
film flow is the "slope condition" of Johnson-Noble-Zumbrun, used to obtain
pointwise symmetrizability of the linearized equations and thereby
high-frequency resolvent bounds and a crucial H s nonlinear damping estimate.
Numerically, this condition is seen to hold for Froude numbers 2 \textless{} F
3.5, but to fail for 3.5 F. As hydraulic engineering applications typically
involve Froude number 3 F 5, this issue is indeed relevant to practical
considerations. Here, we show that the pointwise slope condition can be
replaced by an averaged version which holds always, thereby completing the
nonlinear theory in the large-F case. The analysis has potentially larger
interest as an extension to the periodic case of a type of weighted
"Kawashima-type" damping estimate introduced in the asymptotically-constant
coefficient case for the study of stability of large-amplitude viscous shock
waves
Geophysical Fluid Dynamics
The workshop “Geophysical Fluid Dynamics” addressed recent advances in analytical, stochastic, modeling and computational studies of geophysical fluid models. Of central interest were the reduced geophysical models, that are derived by means of asymptotic and scaling techniques, and their investigations by methods from the above disciplines. In particular, contributions concerning the viscous and inviscid geostrophic models, the primitive equations of oceanic and atmospheric dynamics, tropical atmospheric models and their coupling to nonlinear dynamics of phase changes moisture, thermodynamical effects, stratifying effects, as well as boundary layers were presented and discussed
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