219 research outputs found
Competition between Diffusion and Fragmentation: An Important Evolutionary Process of Nature
We investigate systems of nature where the common physical processes
diffusion and fragmentation compete. We derive a rate equation for the size
distribution of fragments. The equation leads to a third order differential
equation which we solve exactly in terms of Bessel functions. The stationary
state is a universal Bessel distribution described by one parameter, which fits
perfectly experimental data from two very different system of nature, namely,
the distribution of ice crystal sizes from the Greenland ice sheet and the
length distribution of alpha-helices in proteins.Comment: 4 pages, 3 figures, (minor changes
Exact Periodic Solutions of Shells Models of Turbulence
We derive exact analytical solutions of the GOY shell model of turbulence. In
the absence of forcing and viscosity we obtain closed form solutions in terms
of Jacobi elliptic functions. With three shells the model is integrable. In the
case of many shells, we derive exact recursion relations for the amplitudes of
the Jacobi functions relating the different shells and we obtain a Kolmogorov
solution in the limit of infinitely many shells. For the special case of six
and nine shells, these recursions relations are solved giving specific analytic
solutions. Some of these solutions are stable whereas others are unstable. All
our predictions are substantiated by numerical simulations of the GOY shell
model. From these simulations we also identify cases where the models exhibits
transitions to chaotic states lying on strange attractors or ergodic energy
surfaces.Comment: 25 pages, 7 figure
Turbulent Binary Fluids: A Shell Model Study
We introduce a shell (``GOY'') model for turbulent binary fluids. The
variation in the concentration between the two fluids acts as an active scalar
leading to a redefined conservation law for the energy, which is incorporated
into the model together with a conservation law for the scalar. The model is
studied numerically at very high values of the Prandtl and Reynolds numbers and
we investigate the properties close to the critical point of the miscibility
gap where the diffusivity vanishes. A peak develops in the spectrum of the
scalar, showing that a strongly turbulent flow leads to an increase in the
mixing time. The peak is, however, not very pronounced. The mixing time
diverges with the Prandtl number as a power law with an exponent of
approximately 0.9. The continuum limit of the shell equations leads to a set of
equations which can be solved by a scaling ansatz, consistent with an exact
scaling of the Navier-Stokes equations in the inertial range. In this case a
weak peak also persists for a certain time in the spectrum of the scalar. Exact
analytic solutions of the continuous shell equations are derived in the
inertial range. Starting with fluids at rest, from an initial variation of the
concentration difference, one can provoke a ``spontaneous'' generation of a
velocity field, analogous to MHD in the early universe.Comment: 25 pages including 9 figures in epsf. Some figures have been revised,
and a few comments have been inserted in the text. To be published in Physic
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