A discrete geometric approach for simulating the dynamics of thin viscous threads

Abstract

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.

A Geometric Model for the Coiling of an Elastic Rod Deployed Onto a Moving Substrate

“…The FMSM has been investigated more extensively than the ESM, with numerous experimental [6][7][8][9][10], theoretical [11][12][13], and numerical [14][15][16][17][18] studies. Studies of the ESM, on the other hand, are more recent [1][2][3].…”

Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities

“…1), as well as various resonant patterns such as double coils and double meanders [1][2][3]. This system has been extensively studied [1][2][3][4][5] but has lacked a simple explanation until now. The resemblance of these patterns to the stitch patterns of a sewing machine led Ref.…”

“…Accordingly, our geometric model uses the position of this point as a state variable, as well as the direction of the tangent to the thread. Before deriving the model we perform direct simulations of the FMSM with the Discrete Viscous Rods (DVR) algorithm [4,5] to propose a rationalization of the FMSM phase diagram when inertia is negligible, i.e., for moderate fall heights. Since this DVR algorithm is known to accurately predict the experimental FMSM patterns [4,5], we will not repeat a detailed comparison with experiments here.…”

Fluid dynamic instabilities: theory and application to pattern forming in complex media

“…the coiling radius, and the pattern shape cannot be captured by traditional means of stability analysis. Instead, a full fledged nonlinear model is needed to account for those shapes, which are produced far from equilibrium [76]. In particular, some nonlinearities are rooted in geometry.…”