Dynamics of complex capillary flows: stability, rupture, and influence of surfactants

Abstract

Fluid-fluid interfaces are ubiquitous in nature and everyday life, where they are found across scales and material properties, as for instance in many engineering, biological, and physiological applications and processes. In particular, cylindrical interfaces and, in general, the spontaneous tendency of surface-tension-driven flows to break up into drops, have fascinated naturalists and scientists throughout history; a fascination that lasts to date due to its crucial relevance in many phenomena of fundamental and applied interest. This is the reason for the huge research effort devoted to understand the behavior and dynamics of these filaments, namely elongated vesicles and membranes which are frequent in biological environments, or liquid jets that are routinely used for additive manufacturing applications. In most of these scenarios, the interface is usually populated with surface-active molecules, macromolecules, proteins, or contaminated with particles, which eventually form a complex microstructure that endows the interface with a rheologically complex behavior. The interaction between this structure and the hydrodynamic forces is traduced macroscopically into nonlinear interfacial rheological properties and nontrivial constitutive equations relating the surface stress with the deformation of the surface. An interface that possesses these kinds of properties is usually referred to as a complex interface, and the particular field of study is typically denoted by interfacial rheology. Nonetheless, despite of this complexity, these material cylinders share the same intrinsic instability induced by the interfacial tension known as the Plateau-Rayleigh instability, where disturbances of sufficiently long wavelength trigger the instability by decreasing the surface energy at constant volume. The complex interactions between the bulk fluids and the surface layer complicate the theoretical modelling and the experimental protocols and measurements of the material properties associated with the interface. A vast number of issues regarding the behavior and dynamics of such complex fluid threads are yet not understood. In particular, this thesis aims to unravel fundamental aspects of the linear and nonlinear dynamics of liquid filaments whose interface is endowed with complex surface rheology, which can be elastic and/or viscous. We first deduce the components of the surface stress balance modified by interfacial elastic and viscous forces, which is necessary for the derivation of leading-order and second-order one-dimensional models. The performance of these approximations is then evaluated in the linear regime by comparing their associated growth rate of small perturbations with the one obtained from the complete conservation equations. To this end, we use Rayleigh’s temporal linear stability analysis to deduce the corresponding dispersion relation of a liquid filament with interfacial rheology. Additionally, by performing simulations of the full conservation equations, we then investigate the nonlinear dynamics of these complex filaments. In particular we study the effect of Marangoni and surface viscous stresses on the natural breakup and thinning of threads, and the subsequent formation of satellite droplets. Finally, we study the linear and nonlinear dynamics of a capillary jet injected in the direction of gravity and confined between the nozzle and a bath of the same fluid.