Vortex Motion in Two-Dimensional Viscous Fluid Flows

The diffusion and annihilation of vortices in axisymmetric and plane incompressible viscous fluid flows are considered. A formula relating the pressure with the velocity of the vortices in the viscous fluid is obtained.     

Regularity Results for Two-Dimensional Flows of Multiphase Viscous Fluids

Global regularity results for weak solutions of the Navier-Stokes equations for two-dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density. As an application, we deduce regularity properties for the integral curves of the corresponding velocity field. Finally, we prove regularity results “in the small” for strong solutions. (Accepted October 10, 1995)     

Asymptotic Theory of the Viscous Interaction Between a Vortex and a Plane

The problem of the viscous interaction between a flow induced by a vortex filament and an orthogonal rigid surface is solved for high Reynolds numbers using the method of matched asymptotic expansions. In view of the impossibility of matching the principal terms of the asymptotic expansions directly for the near-axial boundary layer and the main flow zone, the solution is obtained by introducing two intermediate zones. In this case a logarithmic singularity of the axial velocity arises inevitably on the vortex filament. In the near-axial and intermediate zones the solution is obtained numerically and analytically, respectively, while in the main zone the problem reduces to the problem of the flow induced by a line of weakly swirled vortex-sinks.     

Liouville Theorems for Stationary Flows of Shear Thickening Fluids in the Plane

We consider entire solutions of the equations for stationary flows of shear thickening fluids in 2D and prove Liouville results under conditions like global boundedness of the velocity field or finiteness of the energy.     

Liouville-Type Theorems for Steady Flows of Degenerate Power Law Fluids in the Plane

We extend the Liouville-type theorems of Gilbarg and Weinberger and of Koch, Nadirashvili, Seregin and Sverák valid for the stationary variant of the classical Navier–Stokes equations in 2D to the degenerate power law fluid model.     

A Note on Liouville Theorem for Stationary Flows of Shear Thickening Fluids in the Plane

In this paper we consider the entire weak solutions of the equations for stationary flows of shear thickening fluids in the plane and prove Liouville theorem under the global boundedness condition of velocity fields.     

On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities

We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids of the same density in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier–Stokes/Cahn–Hilliard system, which is capable of describing the evolution of droplet formation and collision during the flow. We prove the existence of weak solutions of the non-stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities, which ensures—in contrast to the usual case of a smooth free energy density—that the concentration stays in the physical reasonable interval. Furthermore, we find that unique “strong” solutions exist in two dimensions globally in time and in three dimensions locally in time. Moreover, we show that for any weak solution the concentration is uniformly continuous in space and time. Because of this regularity, we are able to show that any weak solution becomes regular for large times and converges as t → ∞ to a solution of the stationary system. These results are based on a regularity theory for the Cahn–Hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a Stokes system with variable viscosity and forces in L ²(0, ∞; H ^s (Ω)), ${s \in [0, \frac12)}$ , which are new themselves.     

Evolution of a Viscous Vortex Ring

The velocity field inside a viscous vortex ring and its overall velocity are obtained on the basis of an existing solution of the Stokes equation for the vorticity in a moving coordinate system. It is shown that these characteristics of the motion relate the results obtained earlier for limiting times and are confirmed by the experimental data. The effect of the initial Reynolds number on the development of the vortex ring is analyzed by constructing entrainment diagrams.     

Incompressible Viscous Flows in Borderline Besov Spaces

We establish two new estimates for a transport-diffusion equation. As an application we treat the problem of global persistence of the Besov regularity with , for the two-dimensional Navier–Stokes equations with uniform bounds on the viscosity. We provide also an inviscid global result.     

Harmonic Maps and Ideal Fluid Flows

Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in \mathbb Cⁿ{\mathbb {C^n}} with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.     

Spiral Waves in Three-Dimensional Helical Viscous Flows

A study is made of an unbounded viscous incompressible flow in the case in which the vortex lines of the absolute motion coincide with the streamlines of the relative motion; after Joukowsky, this flow is called helical flow. The vector potential of the flow is constructed and the notion of the stream function is generalized to include three-dimensional uniform helical flows in an arbitrary orthogonal coordinate system. It is shown that both axisymmetric and asymmetric waves may propagate in a rotating fluid; the amplitude of these waves decays exponentially with time, the decrement being proportional to the square of the rotational velocity of the fluid and the kinematic viscosity and inversely proportional to the square of the phase velocity.     

Dynamically Equilibrium Shape of Intrusive Vortex Formations in the Ocean

The problem of finding the dynamically equilibrium shape of a rotating mass of liquid with homogeneous density (lens) submerged in a stratified ocean at rest on the rotating Earth is formulated. An equation for the shape of the interface between water masses is derived. An exact solution of the problem for an anticyclonically rotating lens in a linearly stratified ocean in the neighborhood of the lens depth shows that the dynamically equilibrium shape of the interface is a triaxial ellipsoid inclined with respect to the horizon and similar to an ellipsoid of revolution for real parameters of the phenomenon. The limiting values of the latitudes at which these formations can exist are determined. Degeneration of the shape with decrease in the intrinsic lens angular rate is investigated.     

Radial Viscous Fingering in Case of Poorly Miscible Fluids

The displacement of a viscous fluid from an annular Hele-Shaw cell with a source of finite radius by a less viscous one is investigated. A special case of poorly miscible fluids is considered when corresponding dimensionless criteria—capillary and Peclet numbers—both tend to infinity. Brinkman model which additionally takes into account small viscous forces in a plane of the cell is used to describe the displacement process. Linear analysis shows a stabilizing effect of viscous forces and reveals a geometrical similarity criterion, namely the ratio of the interface’s radius to the gap between the cell’s plates. The displacement patterns, obtained numerically under Brinkman model, are very sensitive to the discovered criterion. The comparison with available experimental data is acceptable.     

Stability of Creeping Flows of Maxwell Fluids

We derive rigorous criteria for the linear stability of viscoelastic flows under periodic boundary conditions. The results are based on recent work of R. Shvydkoy (Commun Math Phys 265:507–545, 2006).     

平面粘性流体扰动与哈密顿体系

通过变分原理,将哈密顿体系的理论引入到平面粘性流体扰动的问题中,导出一套哈密顿算子矩阵的本征函数向量展开求解问题的方法。基于直接法求解流体力学基本方程,导出流场一般特征关系,通过本征值的求解及本征向量的叠加,得到波扰动解,继可分析流场端部效应。从而在该领域用在哈密顿体系下辛几何空间中研究问题的方法代替了传统在拉格朗日体系欧几里德空间分析问题的方法。为流体力学的研究提供一条新途径。     

Heat Convection of Compressible Viscous Fluids: I

The stationary problem for the heat convection of compressible fluid is considered around the equilibrium solution with the external forces in the horizontal strip domain z ₀ < z < z ₀ + 1 and it is proved that the solution exists uniformly with respect to z ₀ ≥ Z ₀. The limit system as z ₀ → + ∞ is the Oberbeck–Boussinesq equations.     

Heat Convection of Compressible Viscous Fluids. II

We consider steady heat convections of compressible viscous fluids in the horizontal strip domain ${z_0 < z < z_0 + 1}$ under the gravity. Pattern formations are shown uniformly for ${z_0 \geq Z_0}$ . The limit of them as ${Z_0 \rightarrow + \infty}$ is that of Oberbeck-Boussinesq equations.     

粘性流体力学的变分原理及其广义变分原理

本文通过引入Laplace变换,应用变积运算方法,建立了不可压缩粘性流体力学的变分原理及其广义变分原理.