In this paper, we consider the Cauchy problem for (abcd)-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona et al. (J Nonlinear Sci 12:283–318, 2002, Nonlinearity 17:925–952, 2004), describes a small-amplitude waves surface of an inviscid fluid, is derived as first order approximation incompressible, irrotational Euler equations. We ma...

In this paper we further investigate some applications of Nambu mechanics in hydrodynamical systems. Using the Euler equations for a rotating rigid body Névir and Blender [J. Phys. A 26 (1993), L1189–L1193] had demonstrated the connection between Nambu mechanics and noncanonical Hamiltonian mechanics. Nambu mechanics is extended to incompressible ideal hydrodynamical fields using energy and hel...

We consider the n-dimensional Boussinesq equations with fractional dissipation, and establish a regularity criterion in terms of the velocity gradient in Besov spaces with negative order.

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter a 1⁄4 a=h0 and wavelength parameter b 1⁄4 ðh0=lÞ2, where a and l are the actual amplitude and wavelength of the sur...

We investigate the linear stability of unstably strati ed Poiseuille ow between two horizontal parallel plates under non-Boussinesq conditions. It is shown, that Squire's transformation can be used to reduce the three-dimensional stability problem to an equivalent two-dimensional one. The eigenvalue problem, consisting of the generalized Orr-Sommerfeld equations, is solved numerically using an ...

This paper presents a parametric finite-difference scheme concerning the numerical solution of theone-dimensional Boussinesq-type set of equations, as they were introduced byPeregrine, in the case of wavesrelatively long with small amplitudes in water of constant depth. The method which is used can be considered asa generalization of the Crank-Nickolson method and it has been ap...

We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion...

We study the three-dimensional, incompressible, non-hydrostatic Boussinesq fluid equations, which are applicable to the dynamics of the oceans and atmosphere. These equations describe the interplay between velocity shear and buoyancy gradients in a rotating frame, which are represented in a hierarchy of dynamical variables Ωm(t) for 1 ≤ m < ∞, identified as the sum of the L-norms of these varia...

We use the Cahn-Hilliard approach to model the slow dissolution dynamics of binary mixtures. An important peculiarity of the Cahn-Hilliard-Navier-Stokes equations is the necessity to use the full continuity equation even for a binary mixture of two incompressible liquids due to dependence of mixture density on concentration. The quasicompressibility of the governing equations brings a short tim...

For flows with small density variations, it is convenient to approximate them as incompressible whilst retaining the leading order effects due to the density variations, thus avoiding issues associated with acoustic waves. The classical approach is the Boussinesq approximation which was originally motivated by the desire to account for gravitational buoyancy effects. There are many situations o...