40 research outputs found
Equation level matching: An extension of the method of matched asymptotic expansion for problems of wave propagation
We introduce an alternative to the method of matched asymptotic expansions.
In the "traditional" implementation, approximate solutions, valid in different
(but overlapping) regions are matched by using "intermediate" variables. Here
we propose to match at the level of the equations involved, via a "uniform
expansion" whose equations enfold those of the approximations to be matched.
This has the advantage that one does not need to explicitly solve the
asymptotic equations to do the matching, which can be quite impossible for some
problems. In addition, it allows matching to proceed in certain wave situations
where the traditional approach fails because the time behaviors differ (e.g.,
one of the expansions does not include dissipation). On the other hand, this
approach does not provide the fairly explicit approximations resulting from
standard matching. In fact, this is not even its aim, which to produce the
"simplest" set of equations that capture the behavior
Mathematical Models of Incompressible Fluids as Singular Limits of Complete Fluid Systems
A rigorous justification of several well-known mathematical models of incompressible fluid flows can be given in terms of singular limits of the scaled Navier-Stokes-Fourier system, where some of the characteristic numbers become small or large enough. We discuss the problem in the framework of global-in-time solutions for both the primitive and the target system. © 2010 Springer Basel AG
Optimal boundary conditions for the Navier-Stokes fluid in a bounded domain with a thin layer
Weak and strong solutions of equations of compressible magnetohydrodynamics
International audienceThis article proposes a review of the analysis of the system of magnetohydrodynamics (MHD). First, we give an account of the modelling asumptions. Then, the results of existence of weak solutions, using the notion of renormalized solutions. Then, existence of strong solutions in the neighbourhood of equilibrium states is reviewed, in particular with the method of Kawashima and Shizuta. Finally, the special case of dimension one is highlighted : the use of Lagrangian coordinates gives a simpler system, which is solved by standard techniques
Challenges in fluid dynamics: a new approach
This monograph presents a synopsis of fluid dynamics based on the personal scientific experience of the author who has contributed immensely to the field. The interested reader will also benefit from the general historical context in which the material is presented in the book. The book covers a wide range of relevant topics of the field, and the main tool being rational asymptotic modelling (RAM) approach. The target audience primarily comprises experts in the field of fluid dynamics, but the book may also be beneficial for graduate students