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A proof of existence, uniqueness and smoothness of the Navier-Stokes equations is an actual problem, which solution is important for different branches of science. The subject of this study is obtaining the smooth and unique solutions of…
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…
In the present note, we show that the uni-directional flows in a rectangular channel and in a circular pipe are exact spatio-temporal solutions of the Navier-Stokes equations over a short time interval. We assert that the classical plane…
The study solves the general solution to 2D steady Navier-Stokes equation for incompressible flow without vorticity diffusion, which is more general than Stokes flow. In order to obtain the general solution, two potential functions are…
We propose a method to stabilise a solution to equations describing the interface of thin liquid films falling under gravity with a finite number of actuators and restricted observations. As for many complex systems, full observation of the…
The Navier-Stokes equations, which govern fluid motions, are not resolved yet. This investigation relates to the application of the power series method to the incompressible Navier-Stokes equations. This method involves replacing variables…
This paper concerns the instability and stability of the trivial steady states of the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the stability…
New classes of exact solutions of the three-dimensional unsteady Navier-Stokes equations containing arbitrary functions and parameters are described. Various periodic and other solutions, which are expressed through elementary functions are…
Existence, uniqueness, and regularity of time-periodic solutions to the Navier-Stokes equations in the three-dimensional whole-space are investigated. We consider the Navier-Stokes equations with a non-zero drift term corresponding to the…
The Navier-Stokes motions in cylindrical domain with Navier boundary conditions are considered. First the existence of global regular two-dimensional solutions are proved. The solutions are bounded by the same constant for all time.…
We study the two-dimensional stationary Navier-Stokes equations describing the flows around a rotating obstacle. The unique existence of solutions and their asymptotic behavior at spatial infinity are established when the rotation speed of…
The large deformations and break up of circular 2D liquid patches in a high Reynolds number (Re=1000) gas flow are investigated numerically. The 2D, plane flow Navier--Stokes equations are directly solved with explicit tracking of the…
In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier--Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically…
In this paper we establish a mathematical framework which may be used to design Monte-Carlo simulations for a class of time irreversible dynamic systems, such as incompressible fluid flows, including turbulent flows in wall-bounded regions,…
We consider the stability of a system of equations which are a singular perturbation of the incompressible rigid-plastic flow equations used to model granular flow. A linear stability analysis shows that solutions of these equations are…
Consider the Navier-Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity $-h(t)u_\infty$ with constant vector $u_\infty\in \mathbb R^3\setminus\{0\}$. Finn raised the…
Recent studies suggest that unstable, non-chaotic solutions of the Navier-Stokes equation may provide deep insights into fluid turbulence. In this article, we present a combined experimental and numerical study exploring the dynamical role…
A well-known unsolved problem (in the classical theory of fluid mechanics) is to identify a set of initial velocities, which may depend on the viscosity, the body forces and possibly the boundary of the fluid that will allow global in time…
Through a discussion of some typical unsteady hydrodynamic flows, we argue that the time averaged hydrodynamic functions at each point give a rather sparse filling of the local jet space. This situation then suggests a set of time dependent…
Exploring the general analytical solutions to the Euler equations for ideal fluids holds significant theoretical and practical importance. The steady flows in two-dimensional spaces are considered whether there is an analytical solution in…