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Modeling mass flows is classically based on hydrostatic balance equations. However, if momentum transfers scale similarly in slope parallel and flow depth directions, then the gravity and acceleration can have the same order of magnitude…
For $\mathbb{R}^2$, the stability of smooth solutions of 2D anisotropic Boussinesq equations with horizontal dissipation is an open problem. In this work, we present a partial answer to this problem in a rougher function space…
Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion which govern the evolution of the barycenter of the landslide mass include various dissipative…
We analyze a diffuse interface model for multi-phase flows of $N$ incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space…
In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the…
In this paper, we construct an example of temperature patch solutions for the two-dimensional, incompressible Boussinesq system with kinematic viscosity such that both the curvature and perimeter grow to infinity over time. The presented…
The goal of this work is to study waves interacting with partially immersed objects allowed to move freely in the vertical direction, and in a regime in which the propagation of the waves is described by the one dimensional…
The inviscid 2D Boussinesq system with thermal diffusivity and multiplicative noise of transport type is studied in the $L^2$-setting. It is shown that, under a suitable scaling of the noise, weak solutions to the stochastic 2D Boussinesq…
In this work, a novel Boussinesq system is put forward. The system is naturally nonlinearly entropy/energy-stable, and is designed for problems with sharply varying bathymetric features. The system is flexible and allows tuning of the…
A number of new closed-form fundamental solutions for the generalized unsteady Oseen and Stokes flows associated with arbitrary time-dependent translational and rotational motions have been developed. These solutions are decomposed into two…
A physical model of a three-dimensional flow of a viscous bubbly fluid in an intermediate regime between bubble formation and breakage is presented. The model is based on mechanics and thermodynamics of a single bubble coupled to the…
In the present work we propose and analyze a fully coupled virtual element method of high order for solving the two dimensional nonstationary Boussinesq system in terms of the stream-function and temperature fields. The discretization for…
The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq systems helps gain insight into certain weather phenomena.…
The theory of internal waves between two layers of immiscible fluids is important both for its applications in oceanography and engineering, and as a source of interesting mathematical model equations that exhibit nonlinearity and…
In this article, we derive a viscous Boussinesq system for surface water waves from Navier-Stokes equations. We use neither the irrotationality assumption, nor the Zakharov-Craig-Sulem formulation. During the derivation, we find the bottom…
This study deals with higher-ordered asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and…
The subject of moving curves (and surfaces) in three dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a…
We propose a method to obtain iterative schemes guarantee unique solutions for systems of partial differential equations that are not symmetric with respect to the time by generalizing He variational iteration method and using Banach fixed…
We derive hydrodynamic equations from Vicsek-style dry active matter models in three dimensions (3D), building on our experience on the 2D case using the Boltzmann-Ginzburg-Landau approach. The hydrodynamic equations are obtained from a…
In this paper, we rigorously derive the governed equations describing the motion of stable stratified fluid, from the mathematical point of view. Specially, we prove that the scaled Boussinesq equations strongly converge to the viscous…