24,285 research outputs found
Shape of an elastica under growth restricted by friction
We investigate the quasi-static growth of elastic fibers in the presence of
dry or viscous friction. An unusual form of destabilization beyond a critical
length is described. In order to characterize this phenomenon, a new definition
of stability against infinitesimal perturbations over finite time intervals is
proposed and a semi-analytical method for the determination of the critical
length is developed. The post-critical behavior of the system is studied by
using an appropriate numerical scheme based on variational methods. We find
post-critical shapes for uniformly distributed as well as for concentrated
growth and demonstrate convergence to a figure-8 shape for large lengths when
self-crossing is allowed. Comparison with simple physical experiments yields
reasonable accuracy of the theoretical predictions
Turing conditions for pattern forming systems on evolving manifolds
The study of pattern-forming instabilities in reaction-diffusion systems on
growing or otherwise time-dependent domains arises in a variety of settings,
including applications in developmental biology, spatial ecology, and
experimental chemistry. Analyzing such instabilities is complicated, as there
is a strong dependence of any spatially homogeneous base states on time, and
the resulting structure of the linearized perturbations used to determine the
onset of instability is inherently non-autonomous. We obtain general conditions
for the onset and structure of diffusion driven instabilities in
reaction-diffusion systems on domains which evolve in time, in terms of the
time-evolution of the Laplace-Beltrami spectrum for the domain and functions
which specify the domain evolution. Our results give sufficient conditions for
diffusive instabilities phrased in terms of differential inequalities which are
both versatile and straightforward to implement, despite the generality of the
studied problem. These conditions generalize a large number of results known in
the literature, such as the algebraic inequalities commonly used as a
sufficient criterion for the Turing instability on static domains, and
approximate asymptotic results valid for specific types of growth, or specific
domains. We demonstrate our general Turing conditions on a variety of domains
with different evolution laws, and in particular show how insight can be gained
even when the domain changes rapidly in time, or when the homogeneous state is
oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to
higher-order spatial systems are also included as a way of demonstrating the
generality of the approach
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