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A discrete geometric approach for simulating the dynamics of thin viscous threads. (English) Zbl 1349.76024
Summary: 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 kinematic 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 for thin threads is established formally. Our numerical method is validated against reference solutions for steady coiling. The method makes it possible to simulate the unsteady behavior of thin viscous threads in a robust and efficient way, including the combined effects of inertia, stretching, bending, twisting, large rotations and surface tension.

MSC:
76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
74K05 Strings
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Software:
AUTO; GENSMAC
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