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For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $\psi$ and its vorticity $\omega$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if…
For the water-air system, the bulk density ratio is as high as about 1000; no model can fully tackle such a high density ratio system. In the Navier-Stokes and Euler equations, the density $\rho$ within the water-air interface is assumed to…
Equilibrium statistical mechanics predicts that inviscid, two-dimensional, incompressible flow on the sphere eventually reaches a state in which spherical harmonic modes of degrees $n=1$ and $n=2$ hold all the energy. By a separate theory,…
A numerical study of the problem of laminar infinite flow of viscous incompressible fluid around a rotating circular cylinder at Reynolds number $ 50 \le {\rm Re} \le 500 $ and dimensionless rotation rate $ 0 \le \alpha \le 7 $ has been…
A method is proposed to estimate the velocity field of an unsteady flow using a limited number of flow measurements. The method is based on a non-linear low-dimensional model of the flow and on expanding the velocity field in terms of…
In this paper, we investigate a system coupled by nonhomogeneous incompressible Navier-Stokes equations and Allen-Cahn equations describing a diffuse interface for two-phase flow of viscous fluids with different densities in a bounded…
An alternative form of the general solution of the linearized stationary Navier-Stokes equations for an incompressible fluid in spherical coordinates is obtained by the vector potential method. A previously published solution to this…
Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of \emph{incompressible} fluids is nowadays available not only for Navier--Stokes fluids but also for various…
A geometric flow on $(2,2)$-forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established,…
We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible…
This paper studies the two-dimensional inhomogeneous Navier--Stokes equations governing stratified flows in a bounded domain under a gravitational potential \(f\). Our main results are as follows. First, we provide a rigorous…
Three-dimensional direct numerical simulations of an incompressible open square cavity flow are conducted. Features of the permanent (non-linear) regime together with the linear stability analysis of a two-dimensional steady base flow are…
This paper presents a streamfunction-vorticity formulation for the Navier--Stokes and Euler equations on general surfaces. Notably, this includes non-simply connected surfaces, on which the harmonic components of the velocity field play a…
Turbulent flows driven by a vertically invariant body force were proven to become exactly two-dimensional above a critical rotation rate, using upper bound theory. This transition in dimensionality of a turbulent flow has key consequences…
This paper investigates the nature of the development of two-dimensional steady flow of an incompressible fluid at the rear stagnation-point.
We show the existence and the regularity properties of the weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient, by analyzing a fourth-order nonlinear…
We are concerned with the vortex sheet solutions for the inviscid two-phase flow in two dimensions. In particular, the nonlinear stability and existence of compressible vortex sheet solutions under small perturbations are established by…
In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of Bernoulli's function in the upstream is…
In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier-Stokes equations, even for 3D case of compressible gas flow or 3D case of non-stationary flow of incompressible fluid.…
We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform…