Dynamics of Hydrodynamics

Abstract

The mathematical description of the states of a fluid is based on the study of three fields defined on the domain occupied by the fluid: the velocity field $\vec{V }$, the density ρ, and the pressure field P. These three “unknowns” are determined by integrating other five scalar equations, namely the mass conservation (continuity equation), the three components of the equation of momentum balance (Euler or Navier–Stokes), and the energy balance. This last equation needs in addition information about the thermodynamics of the fluid, so it may need to be supplied with some equation of state. In addition to these five equations, we request regularity, asymptotic and, if it is the case, boundary conditions, to provide a unique solution. When we study the dynamics of the fluid confined in a compact domain with free boundaries, the system is slightly more complicated, and we have to add the kinematical equation of the free surface, as well as equations of momentum balance at the surface. If we take into account the nonlinear terms in the dynamical equations, and in the associated curved geometry, some interesting solutions occur. Special nonlinear effects related to fluids on compact domains with free surface could be Gibbs–Marangoni effect, dividing the flow in cells (Bènard effect), couplings between different modes, collective effects, separation of flow in layer (boundary layer, turbulence), standing traveling surface waves, etc. In this chapter, we introduce some elements of general hydrodynamics which we will use later on in the book, boundary conditions especially at free surfaces, surface pressure theory, and representation theorems.