Free boundary and turbulence for incompressible viscous fluids

Description

The mathematical bases of the dynamics of viscous fluids are given by the classical Navier- Stokes equations, which model the motion of a viscous incompressible fluid. We can consider a wide variety of scenarios involving these type of fluids. In particular, one can classify the motion of fluids in two general regimes: laminar and turbulent. The Reynolds number is a constant intimately related to this behavior, which associates the viscosity forces acting on the fluid (causing friction between particles), with the inertial forces (causing acceleration of the fluid). In this thesis, we study the dynamics of viscous fluids from two very different perspectives. On the one hand, we study the scenario where the Reynolds number is vanishingly small, giving rise to the Stokes system. We describe the behavior of two different two-dimensional fluids which evolve in time, and we analyze the properties of the interface between them. This problem lies in the class of free boundary problems. On the other hand, we consider a drastically different scenario, where the Reynolds number is large and turbulence is developed. We study the motion of a two or three-dimensional fully developed homogeneous isotropic turbulent fluid, through the Kolmogorov two-equation model of turbulence. This thesis is divided into two parts, each of them devoted to one of the problems. The first part of the thesis contains an introduction and two chapters. In the first chapter, we present the model which describes the dynamics of two incompressible immiscible viscous fluids in the Stokes regime, filling a 2D horizontally periodic strip. We assume that the fluids are subject to the gravity force and they have different densities. This framework is chosen motivated by the lack of results in the density jump setting with an infinitely deep geometry and a non-integrable density. In this scenario, the density jump induces the dynamics of the free interface arising between the two fluids. One of the classical methods to deal with free boundary problems is to use potential theory to furnish explicit solutions for the system. Using this approach, we derive a contour dynamics formulation for this problem, through a x1-periodic version of the Stokeslet. This technique yields explicit solutions of the system, even for more general forcing terms than the one used in our analysis (the gravity force). Furthermore, this formulation of the velocity consist of a non-local and strongly non-linear equation. As a first approach, we analyze the linear operator inside the explicit solution, which shows what we call a weak damping effect in the stable stratification of the densities, when the lighter fluid lies above the denser one. This type of operator shows a contrast between this and other related free boundary problems, whose linear operators are of parabolic type. In our case, the weak damping effect suggests that the solutions do not gain regularization in time, hence the nature of the problem is hyperbolic. Having this in mind, we study the full non-linear equation and we show local-in-time well-posedness when the initial interface is described by a curve with no self-intersections and C1+γ H¨older regularity, with 0 < γ < 1. According to the expected hyperbolic behavior, the solution does not gain any regularity, it is C1+γ in space. This well-posedness result holds regardless of the Rayleigh-Taylor stability of the physical system, i.e., the system is well-posed even when the denser fluid lies above the lighter one. This behavior is due to the viscosity of the fluids. In the second chapter, we study the long time behavior of solutions when the initial data is small and described by the graph of a function. The techniques used exploit the properties of the linear semi-group and the so-called weak damping effect. With these techniques, we prove the global-in-time existence for the Rayleigh-Taylor stable case of the densities (the lighter fluid lies above the denser fluid). The proof relies on a priori energy estimates on suitable Sobolev spaces and the careful study of the singular kernels appearing. We also prove stability of the flat interface, i.e., the decay of the free interface to the flat steady state. In particular, we prove existence and uniqueness of global interfaces with H3 regularity and polynomial decay of the interface. Moreover, we can extend this global-in-time existence result to analytic solutions in suitable Wiener spaces. We use Fourier techniques of the contour dynamics equation and the properties of the linear semi-group in Wiener algebras to obtain global-in-time existence and exponential decay to the flat interface. Finally, in the Rayleigh-Taylor unstable regime, we construct a wide family of smooth solutions with exponential in time growth for an arbitrarily large interval of existence, showing that the free boundaries can grow exponentially. The second part of the thesis contains an introduction and one chapter. In this chapter, we establish a local well-posedness result for the Kolmogorov two-equation model of turbulence. This model belongs to the k-ε models, and describes the dynamics of an homogeneous and isotropic fully-developed turbulent flow. We generalize the previous results letting the turbulent kinetic energy vanish, in order to cover a wider range of phenomena. Consequently, we lose the parabolicity of the system, and an accurate analysis is needed to find the existence and uniqueness of solutions. We prove local well-posedness in critical Sobolev spaces Hs for s > 1 + d/2, for the cases of two and three dimensional fluids, in a periodic box Td. We consider fractional regularity, and consequently, our study involves paradifferential calculus, passing through Littlewood-Paley decomposition, in order to have a priori high order energy bounds.

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