1,058 research outputs found
Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes
We formulate hydrodynamic equations and spectrally accurate numerical methods
for investigating the role of geometry in flows within two-dimensional fluid
interfaces. To achieve numerical approximations having high precision and level
of symmetry for radial manifold shapes, we develop spectral Galerkin methods
based on hyperinterpolation with Lebedev quadratures for -projection to
spherical harmonics. We demonstrate our methods by investigating hydrodynamic
responses as the surface geometry is varied. Relative to the case of a sphere,
we find significant changes can occur in the observed hydrodynamic flow
responses as exhibited by quantitative and topological transitions in the
structure of the flow. We present numerical results based on the
Rayleigh-Dissipation principle to gain further insights into these flow
responses. We investigate the roles played by the geometry especially
concerning the positive and negative Gaussian curvature of the interface. We
provide general approaches for taking geometric effects into account for
investigations of hydrodynamic phenomena within curved fluid interfaces.Comment: 14 figure
The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
In this paper, we use the Weil-Petersson gradient flow for renormalized
volume to study the space of convex cocompact hyperbolic structures
on the relatively acylindrical 3-manifold . Among the cases of interest
are the deformation space of an acylindrical manifold and the Bers slice of
quasi-Fuchsian space associated to a fixed surface. To treat the possibility of
degeneration along flow-lines to peripherally cusped structures, we introduce a
surgery procedure to yield a surgered gradient flow that limits to the unique
structure with totally geodesic convex core
boundary facing . Analyzing the geometry of structures along a flow line, we
show that if is the renormalized volume of , then
is bounded below by a linear function of the
Weil-Petersson distance ,
with constants depending only on the topology of . The surgered flow gives a
unified approach to a number of problems in the study of hyperbolic
3-manifolds, providing new proofs and generalizations of well-known theorems
such as Storm's result that has minimal volume for
acylindrical and the second author's result comparing convex core volume and
Weil-Petersson distance for quasifuchsian manifolds
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