Dynamics of thin viscous layers

Abstract

As Isaac Newton discovered in Mechanics, and published in his Principia Mathematica in 1687, inertial effects are essential to describe natural phenomena.In the case of Fluid Dynamics, inertia was included by Claude-Louis Navier andGeorge Gabriel Stokes almost two centuries after in 1822 when calculatingthe mass and momentum conservation of au infinitesimal fluid elemeut. AlthoughNavier-Stokes equations are very useful for describing any event of any liquidor gas and there are thousands of numerical studies about it, they are not very compact and in most cases, difficult to handle. In the last century, the scientific community has developed many different approaches to simplify the N avier-Stokes equations with the aim of obtaining physical information in different cases and approximations.Sorne examples are the Euler equations, the averaged Navier-Stokes equations (RANS) , Shallow Water Theory, Boussinesq approximation, Boundary Layer Theory, etc [1, 4, 4 7, 70]. One of these approaches is Lubrication Theory which describes the flow of fluids (liquids or gases) in a geometry in which one dimension is siguificautly srnaller than the others. This approach is useful in many problems but in sorne cases, when the smaller length scale in a fluid layer isits thickness, inertial effects can be important. We have derived the minimal set of equations containing inertial effects in this strongly dissipative regime, that we named Inertial Lubrication Theory (ILT). This novel result is a tool that allow us to study diverse nonlinear problems in fluid dynamics involving thin fluid layerswith a free surface at atmospherical pressure and a hard wall at the bottom. Sorne of these problems are famous for being not well understood for decades such as the Faraday instability, the hydraulic jump, the teapot effect or dropmotion . The key point of all these problems is the inclusion of inertia as aprincipal component. The first chapter addresses to a problem known as the Faraday instability. Faraday waves, named after Michael Faraday in 1831, are nonlinear standing waves that appear on a liquid embedded in a vibrating receptacle. When the vibration amplitude exceeds a critical value, the flat hydrostatic surface becomes unstable and waves with a determined wavelength appear at the fluid-air interface. This is knowu as the Faraday iustability. The first theoretical descriptiou was proposed by Benjamín and Ursell [G] in the 20th century for non-viscous fluids. In this limit, the authors derived a Mathieu equation to describe the thicknessof the vibrated fluid in the shallow water limit nevertheless the result obtained does not predict the experimental wavelength, as viscosity is fundamental il the selection of the instability wavelengths. From a theoretical point of viewl, theFaraday instability has been studied in the context of amplitude equations and in the linear case the formal inclusion of the viscosity was proposed only at thE end of the nineties, where a systematic way to obtain a non local Mathieu equation successfully predicted the instability thresholds measured in experiments. We have been interested in the nonlinear aspects related to this instability appllying the Inertial Lubrication Theory and obtaining good agreements with experimental measures. Our results involve the thresholds needed to observf the instability, critical wavelengths, numerical 3D simulations of the free surfac , the velocity field in the bulk, spatiotemporal diagrams to analyze the harmoic or sub-harmonic nature of the waves, Fourier spectra of the patterns observed and phase diagram analysis in a region of the parameter space where harmonid and subharmonic responses co-exist [TQ]. The second chapter is related to the well known hydraulic jump which is an intriguing and interesting phenomenon that has caught the imagination of any research workers sin ce its first description by Leonardo da Vinci. HydJaulic jumps are frequently observed in open channel fiows such as rivers and spil~ways or even in daily situations when the faucet flux spreads at the kitchen sink. fhen liquid at high velocity discharges into a zone of lower velocity, a rather al:J>rupt 1 rise occurs in the liquid surface. The rapidly fiowing liquid is abruptly slowed andincreases in height, converting sorne of the initial kinetic energy of the fiow into an 1 increase in potential energy, with sorne energy irreversibly lost through turbulence to ,heat. , In an open channel fiow, this manifests as the fast fiow rapidly, sldwing and pilirig up on top of itself, similar to how a shockwave forms. Hydraulic jbmps can be seen in both a stationary form, called a hydraulic jump, and a dynarrlic ormoving form, called a positive surge or tidal bore. We will focus our attentidn on the circular stationary hydraulic jump. In this field we have applied the In~rtial Lubrication Theory obtaining numerical profile solutions. Our approach also apows to perform direct calculations of the velocity field. We found the vortex associated to the energy dissipation cascade process. In the axi-symmetric regimel, the hydraulic jump forms a circular ring. The jump radii were calculated numeribally, in good agreement with experimental results. A scaling law for the Jump length was also found in a particular limit and compared with previous experiJents. The third chapter concerns the study of the fern spore ejection mecharilism. Fern reproduction has been studied for centuries and it has been known for a\ long time that sporangia act as a catapult. Almost a century ago, it hasbeen shown that high negative pressures were reached in the annulus cells.Later the only theoretical study of the mechanism was performed by King, whowas able to predict the ejection speed for the spores based on an energy balance. Ritman and Milburn [G9] have confirmed that cavitation vent triggers the sbring release by using an ultrasound method to detect cavitation bubbles. Spore ejection is the basis of the reproduction mechanism and fertilization in ferns. This process consists of three basic steps: opening of the sporangium, ejection of the spores and closing of the sporangia. Based on previous works these stages will be explained and a viscoelastic theory developed for the opening and ejection process.