4,260 research outputs found
A singular limit for compressible rotating fluids
We consider a singular limit problem for the Navier-Stokes system of a
rotating compressible fluid, where the Rossby and Mach numbers tend
simultaneously to zero. The limit problem is identified as the 2-D
Navier-Stokes system in the ``horizontal'' variables containing an extra term
that accounts for compressibility in the original system
Scale interactions in compressible rotating fluids
We study a triple singular limit for the scaled barotropic Navier-Stokes
system modeling the motion of a rotating, compressible, and viscous fluid,
where the Mach and Rossby numbers are proportional to a small parameter, while
the Reynolds number becomes infinite. If the fluid is confined to an infinite
slab bounded above and below by two parallel planes, the limit behavior is
identified as a purely horizontal motion of an incompressible inviscid fluid,
the evolution of which is described by an analogue of the Euler system
Dispersive effects of weakly compressible and fast rotating inviscid fluids
We consider a system describing the motion of an isentropic, inviscid, weakly
com-pressible, fast rotating fluid in the whole space R^3 , with initial data
belonging to H^s(R^3) , s \textgreater{} 5/2. We prove that the system admits a
unique local strong solution in L^([0, T ]; H^s(R^3)) , where T is
independent of the Rossby and Mach numbers. Moreover, using Strichartz-type
estimates, we prove that the solution is almost global, i.e. its lifespan is of
the order of ^(--) , \textgreater{} 0, without any
smallness assumption on the initial data (the initial data can even go to
infinity in some sense), provided that the rotation is fast enough.Comment: Revised versio
Multi-scale analysis of compressible viscous and rotating fluids
We study a singular limit for the compressible Navier-Stokes system when the
Mach and Rossby numbers are proportional to certain powers of a small parameter
\ep. If the Rossby number dominates the Mach number, the limit problem is
represented by the 2-D incompressible Navier-Stokes system describing the
horizontal motion of vertical averages of the velocity field. If they are of
the same order then the limit problem turns out to be a linear, 2-D equation
with a unique radially symmetric solution. The effect of the centrifugal force
is taken into account
The Inviscid, Compressible and Rotational, 2D Isotropic Burgers and Pressureless Euler-Coriolis Fluids; Solvable models with illustrations
The coupling between dilatation and vorticity, two coexisting and fundamental
processes in fluid dynamics is investigated here, in the simplest cases of
inviscid 2D isotropic Burgers and pressureless Euler-Coriolis fluids
respectively modeled by single vortices confined in compressible, local,
inertial and global, rotating, environments. The field equations are
established, inductively, starting from the equations of the characteristics
solved with an initial Helmholtz decomposition of the velocity fields namely a
vorticity free and a divergence free part and, deductively, by means of a
canonical Hamiltonian Clebsch like formalism, implying two pairs of conjugate
variables. Two vector valued fields are constants of the motion: the velocity
field in the Burgers case and the momentum field per unit mass in the
Euler-Coriolis one. Taking advantage of this property, a class of solutions for
the mass densities of the fluids is given by the Jacobian of their sum with
respect to the actual coordinates. Implementation of the isotropy hypothesis
results in the cancellation of the dilatation-rotational cross terms in the
Jacobian. A simple expression is obtained for all the radially symmetric
Jacobians occurring in the theory. Representative examples of regular and
singular solutions are shown and the competition between dilatation and
vorticity is illustrated. Inspired by thermodynamical, mean field theoretical
analogies, a genuine variational formula is proposed which yields unique
measure solutions for the radially symmetric fluid densities investigated. We
stress that this variational formula, unlike the Hopf-Lax formula, enables us
to treat systems which are both compressible and rotational. Moreover in the
one-dimensional case, we show for an interesting application that both
variational formulas are equivalent
Highly rotating viscous compressible fluids in presence of capillarity effects
In this paper we study a singular limit problem for a Navier-Stokes-Korteweg
system with Coriolis force, in the domain and for general
ill-prepared initial data. Taking the Mach and the Rossby numbers to be
proportional to a small parameter \veps going to , we perform the
incompressible and high rotation limits simultaneously. Moreover, we consider
both the constant capillarity and vanishing capillarity regimes. In this last
case, the limit problem is identified as a -D incompressible Navier-Stokes
equation in the variables orthogonal to the rotation axis. If the capillarity
is constant, instead, the limit equation slightly changes, keeping however a
similar structure. Various rates at which the capillarity coefficient can
vanish are also considered: in most cases this will produce an anisotropic
scaling in the system, for which a different analysis is needed. The proof of
the results is based on suitable applications of the RAGE theorem.Comment: Version 2 includes a corrigendum, which fixes errors contained in the
proofs to Theorems 6.5 and 6.
Energy spectra of vortex distributions in two-dimensional quantum turbulence
We theoretically explore key concepts of two-dimensional turbulence in a
homogeneous compressible superfluid described by a dissipative two-dimensional
Gross-Pitaeveskii equation. Such a fluid supports quantized vortices that have
a size characterized by the healing length . We show that for the
divergence-free portion of the superfluid velocity field, the kinetic energy
spectrum over wavenumber may be decomposed into an ultraviolet regime
() having a universal scaling arising from the vortex
core structure, and an infrared regime () with a spectrum that
arises purely from the configuration of the vortices. The Novikov power-law
distribution of intervortex distances with exponent -1/3 for vortices of the
same sign of circulation leads to an infrared kinetic energy spectrum with a
Kolmogorov power law, consistent with the existence of an inertial
range. The presence of these and power laws, together with
the constraint of continuity at the smallest configurational scale
, allows us to derive a new analytical expression for the
Kolmogorov constant that we test against a numerical simulation of a forced
homogeneous compressible two-dimensional superfluid. The numerical simulation
corroborates our analysis of the spectral features of the kinetic energy
distribution, once we introduce the concept of a {\em clustered fraction}
consisting of the fraction of vortices that have the same sign of circulation
as their nearest neighboring vortices. Our analysis presents a new approach to
understanding two-dimensional quantum turbulence and interpreting similarities
and differences with classical two-dimensional turbulence, and suggests new
methods to characterize vortex turbulence in two-dimensional quantum fluids via
vortex position and circulation measurements.Comment: 19 pages, 8 figure
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