Related papers: Tracking complex singularities of fluids on log-la…
We deal with the incompressible Navier-Stokes equations, in two and three dimensions, when some vortex patches are prescribed as initial data i.e. when there is an internal boundary across which the vorticity is discontinuous. We show…
Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial…
We analyze the Vlasov equation coupled with the compressible Navier--Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative…
We explore the utility of the recently proposed alpha equations in providing a subgrid model for fluid turbulence. Our principal results are comparisons of direct numerical simulations of fluid turbulence using several values of the…
The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several CFD…
Numerical simulation of fluids plays an essential role in modeling many physical phenomena, such as weather, climate, aerodynamics and plasma physics. Fluids are well described by the Navier-Stokes equations, but solving these equations at…
The purpose of this note is to offer a birds-eye view on the history and the state-of-the-art in the research surrounding the Millenium Prize problem for the Navier-Stokes equations, the general problem of singularities in fluid dynamics…
We show how a complete mathematical description of a complicated physical phenomenon can be learned from observational data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality,…
On one hand, classical Monte Carlo and molecular dynamics (MD) simulations have been very useful in the study of liquids in nanotubes, enabling a wide variety of properties to be calculated in intuitive agreement with experiments. On the…
We present a study by computer simulations of a class of complex-valued solutions of the three-dimensional Navier-Stokes equations in the whole space, which, according to Li and Sinai, present a blow-up (singularity) at a finite time. The…
This paper discussed the existence and uniqueness of the smoothing solution of the Navier-Stokes equations. At first, we construct the theory of the linear equations which is about the unknown four variables functions with constant…
In this paper, we study the global Cauchy problem for a two-phase fluid model consisting of the pressureless Euler equations and the incompressible Navier-Stokes equations where the coupling of two equations is through the drag force. We…
Singularity of Navier-Stokes equations is uncovered for the first time which explains the mechanism of transition of a smooth laminar flow to turbulence. It is found that when an inflection point is formed on the velocity profile in…
In this paper we consider the r\^ole that numerical computations -- in particular Galerkin approximations -- can play in problems modelled by the 3d Navier-Stokes equations, for which no rigorous proof of the existence of unique solutions…
The Navier-Stokes (NS) equations as a turbulence model have been widely applied in lots of fields. The NS equations contain such a fundamental assumption that all small physical/artificial disturbances could be neglected. Is this assumption…
In this paper we give a review on the computational methods used to characterize the complex singularities developed by some relevant PDEs. We begin by reviewing the singularity tracking method based on the analysis of the Fourier spectrum.…
We show that, for two-dimensional space-periodic incompressible flow, the solution can be evaluated numerically in Lagrangian coordinates with the same accuracy achieved in standard Eulerian spectral methods. This allows the determination…
We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier-Stokes equations are solved using a finite-difference projection method coupled with a…
We investigate a one dimensional flow described with the non-compressible coupled Euler and non-compressible Navier-Stokes equations in Cartesian coordinate systems. We couple the two fluids through the continuity equation where different…
A new formulation of the Navier-Stokes equation, in terms of the gradient of the total mechanical energy, is derived for the time-averaged flows, and the singular point possibly existing in the Navier-Stokes equation is exactly found.…