13,961 research outputs found
A discrete geometric approach for simulating the dynamics of thin viscous threads
We present a numerical model for the dynamics of thin viscous threads based
on a discrete, Lagrangian formulation of the smooth equations. The model makes
use of a condensed set of coordinates, called the centerline/spin
representation: the kinematical constraints linking the centerline's tangent to
the orientation of the material frame is used to eliminate two out of three
degrees of freedom associated with rotations. Based on a description of twist
inspired from discrete differential geometry and from variational principles,
we build a full-fledged discrete viscous thread model, which includes in
particular a discrete representation of the internal viscous stress.
Consistency of the discrete model with the classical, smooth equations is
established formally in the limit of a vanishing discretization length. The
discrete models lends itself naturally to numerical implementation. Our
numerical method is validated against reference solutions for steady coiling.
The method makes it possible to simulate the unsteady behavior of thin viscous
jets in a robust and efficient way, including the combined effects of inertia,
stretching, bending, twisting, large rotations and surface tension
Finite volume approach for the instationary Cosserat rod model describing the spinning of viscous jets
The spinning of slender viscous jets can be described asymptotically by
one-dimensional models that consist of systems of partial and ordinary
differential equations. Whereas the well-established string models possess only
solutions for certain choices of parameters and set-ups, the more sophisticated
rod model that can be considered as -regularized string is generally
applicable. But containing the slenderness ratio explicitely in the
equations complicates the numerical treatment. In this paper we present the
first instationary simulations of a rod in a rotational spinning process for
arbitrary parameter ranges with free and fixed jet end, for which the hitherto
investigations longed. So we close an existing gap in literature. The numerics
is based on a finite volume approach with mixed central, up- and down-winded
differences, the time integration is performed by stiff accurate Radau methods
Spectral Dynamics of the Velocity Gradient Field in Restricted Flows
We study the velocity gradients of the fundamental Eulerian equation,
, which shows up in different contexts
dictated by the different modeling of 's. To this end we utilize a basic
description for the spectral dynamics of , expressed in terms of the
(possibly complex) eigenvalues, , which are shown to
be governed by the Ricatti-like equation .
We address the question of the time regularity of four prototype models
associated with different forcing . Using the spectral dynamics as our
essential tool in these investigations, we obtain a simple form of a critical
threshold for the linear damping model and we identify the 2D vanishing
viscosity limit for the viscous irrotational dusty medium model. Moreover, for
the -dimensional restricted Euler equations we obtain global
invariants, interesting for their own sake, which enable us to precisely
characterize the local topology at breakdown time, extending previous studies
in the -dimensional case. Finally, as a forth model we introduce the
-dimensional restricted Euler-Poisson (REP)system, identifying a set of
global invariants, which in turn yield (i) sufficient conditions for
finite time breakdown, and (ii) characterization of a large class of
2-dimensional initial configurations leading to global smooth solutions.
Consequently, the 2D restricted Euler-Poisson equations are shown to admit a
critical threshold
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