The role of microstructure in taming the Rayleigh capillary instability of cylindrical jets

Abstract

It is observed both in nature and in technological processes that filaments with anisotropic molecular-scale structure are less susceptible to breakup due to capillary instability than homogeneous, isotropic fluids in similar filament flows. Here we provide rigorous evidence that the strong coupling of microstructure to the hydrodynamics of cylindrical axisymmetric free surface filaments, indeed fundamentally alters the linearized stability of cylindrical jets. We extend Rayleigh's classical inviscid analysis of cylindrical jets to the three-dimensional (3D), macroscopic flow-orientation equations derived from the Doi kinetic theory for liquid crystalline polymers (LCPs). These equations assume rigid rod-like molecules and incorporate LCP effects of molecular relaxation, anisotropic drag, polymer kinetic energy, LCP density, and an intermolecular potential which couple orientation dynamics to standard free surface fluid equations. Depending on the LCP density, there are between one and three flow-independent orientation equilibria which persist in a constant-velocity, cylindrical free surface flow: an isotropic phase exists at all concentrations, whereas two anisotropic phases exist at sufficiently high LCP density. These equilibrium LCP cylindrical jets have two independent sources of instability, hydrodynamic and orientational, each identified within the coupled flow/orientation free surface equations. For this paper we restrict to equilibria free of orientational instabilities. All streamwise perturbations of wavelength greater than the jet circumference are unstable to capillary instability; only the strength of the instability and most dominant wavelength are affected by LCP microstructure. The degree to which microstructure reduces the capillary instability depends on two critical scaling parameters: an LCP capillary number Ca lcp (a ratio of LCP-induced surface stress to interfacial capillary stress); and the anisotropic drag/friction parameter σ. The most striking result is: for sufficiently large Ca lcp and highly anisotropic drag ( σ ∼ 0) the capillary growthrate can be uniformly lowered, arbitrarily close to zero. For sufficiently small Ca lcp, all capillary-dominated growthrates are reduced, but are bounded below in terms of an explicit, sharp estimate and bounded above by the Rayleigh formula. The upshot is: inviscid LCP jets are predicted to yield bigger drops which form on longer timescales than an inviscid isotropic fluid with the same surface tension.