Catenaries in viscous fluid

Abstract

Slender structures in fluid flow exhibit a variety of rich behaviors. Here we study the equilibrium shapes of perfectly flexible strings that are moving with a uniform velocity and axial flow in viscous fluid. The string is acted upon by local, anisotropic, linear drag forces and a uniform body force. Generically, the configurations of the string are planar, and we provide analytical expressions for the equilibrium shapes of the string as a first order five parameter dynamical system for the tangential angle of the body (θ). Phase portraits in the anglecurvature (θ, ∂ s θ) plane are generated, that can be shown to be π periodic after appropriate scaling and reflection operations. The rich parameter space allows for different kinds of phase portraits that give rise to a variety of curve geometries. Some of these solutions are unstable due to the presence of compressive stresses. Special cases of the problem include sedimenting filaments, dynamic catenaries, and towed strings. We also discuss equilibrium configurations of towed cables and other relevant problems with fixed boundary conditions. Special cases of the boundary value problem involve towing of neutrally buoyant cables and strings with pure axial flow between two fixed points. AcknowledgmentLet me start by thanking my parents and kaka. Like always, they have been very supportive over the last two years. They have been with me through some of my difficult phases and have always kept their faith in me. It would not have been possible to finish this work without their love and encouragement.

Possible mechanism for aligning microscopic flexible filaments predicted using “caterpillar” hydrodynamics

“…Given the complexity of the interplay between varying hydrodynamic forces and the deformation of even a single flexible filament, analytic solutions are few [13]. This is particularly true outside the linear regime.…”

Variational approaches to the elasticity of deformable strings with and without mass redistribution

“…Interestingly enough, we remark that the hyperbolic cosine reappears in the explicit formula of the true catenary. Another interesting generalization concerns the viscous catenary, which consists in a filament of a highly viscous incompressible fluid, supported at its ends, relaxing under the influence of gravity [30][31][32][33][34][35]. This matter has also been investigated for filaments composed of soft materials, described by an arbitrary constitutive equation [36], or exhibiting a viscoelastic behavior [37,38].…”