Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid

In the analysis of the long-time behavior of two-dimensional incompressible viscous fluids, Oseen vortices play a major role as attractors of any homogeneous solution with integrable initial vorticity [T. Gallay, C.E. Wayne, Global stability of vortex solutions of the two-dimensional Navier–Stokes equation, Commun. Math. Phys. 255 (1) (2005) 97–129]. As a first step in the study of the density-dependent case, the present paper establishes the asymptotic stability of Oseen vortices for slightly inhomogeneous fluids with respect to localized perturbations.     

Investigation of the stability of periodic flows of a viscous fluid

An exact expression is obtained for the critical Reynolds number (R_*) for loss of stability in a wide class of one-dimensional periodic flows. An evolutionary equation is derived in the case of a small subcritically (R − R_* 1) which describes the dynamics of the secondary vortex structure.     

On the interface boundary conditions between two interacting incompressible viscous fluid flows

We consider two incompressible viscous fluid flows interacting through thin non-Newtonian boundary layers of higher Reynolds? number. We study the asymptotic behaviour of the problem, with respect to the vanishing thickness of the layers, using Γ-convergence methods. We derive general interfacial boundary conditions between the two fluid flows. These boundary conditions are specified for some particular cases including periodic or fractal structures of layers.     

Suitable weak solutions: from compressible viscous to incompressible inviscid fluid flows

We establish the asymptotic limit of the compressible Navier–Stokes system in the regime of low Mach and high Reynolds number on unbounded spatial domains with slip boundary condition. The result holds in the class of suitable weak solutions satisfying a relative entropy inequality.     

Eigenvalue estimates for a preconditioned Galerkin matrix arising from mixed finite element discretizations of viscous incompressible flows

The article deals with Galerkin matrices arising with finite element discretizations of the Navier–Stokes system. Usually these matrices are indefinite and nonsymmetric. They have to be preconditioned if a related linear system is to be solved efficiently by an iterative method. We consider preconditioning by a pressure mass matrix. It is shown how upper and lower bounds of the eigenvalues of a preconditioned Galerkin matrix may be found by variational arguments.     

Leray's problem for a viscous incompressible micropolar fluid

This work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity. We prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain. We also study the uniqueness, the regularity and the asymptotic behavior of the solution.     

Estimation of the error of approximation methods for calculating flows of a viscous incompressible fluid with a free boundary

Under the assumption of a sufficient smoothness of the solutions, one investigates the error produced by the approximation methods in the computation of Navier-Stokes equations and in the restoration of the surface from its mean curvature in the course of the method of successive approximations, used for obtaining the solution of the problem on the motion of a viscous fluid with a free boundary.     

Solutions to the mixed problem of viscous incompressible flows in a channel

We study regularity of viscous incompressible fluid flows in a 2D channel with “do nothing” outflow boundary condition on the output for the steady Stokes and Navier–Stokes equations.     

Friction boundary conditions on thermal incompressible viscous flows

This work adresses an unsteady heat flow problem involving friction and convective heat transfer behaviors on a part of the boundary. The problem is constituted by a variational motion inequality with energy dependent coefficients, and the energy equation in the framework of L ¹-theory for the dissipative term. Using the duality theory of convex analysis, it also envolves the existence of Lagrange multipliers. Weak solutions of an approximate coupled system are proven by a fixed point argument for multivalued mappings and compactness methods. Then the existence result for the initial coupled system is proven by the passage to the limit. This work was partially supported by FCT research program POCTI (Portugal/FEDER-EU).     

Towards the problem of hydrodynamic stability of plane-parallel dusty-gas flows

Several modifications of the classical Saffman formulation of the dusty-gas flow linear stability problem are considered. Dispersed flows are described by a two-fluid model. Linear stability problems are reduced to the solution of modified Orr-Sommerfeld equations which are solved by the orthogonalization method. It is shown, that the additional factors taken into account in the problem formulation affect significantly the flow stability limits. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the motion of rigid bodies in a viscous incompressible fluid

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.AMS Subject Classification: 49Q10m, 35P15, 49J20.     

On the stability of a uniformly rotating viscous incompressible self-gravitating liquid

The paper is devoted to justification of the potential energy minimum principle in the problem of stability of a uniformly rotating viscous incompressible self-gravitating liquid. The capillary forces on the free boundary of the liquid are not taken into account. It is proved that the regime of rigid rotation is stable if the second variation of the energy functional is positive. The proof is based on the analysis of the evolution free boundary problem for perturbations in the velocity and pressure of the rotating liquid. Bibliography: 15 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 165–208.     

Spectral element solution of steady incompressible viscous free-surface flows

In this paper we present a new approach for the solution of the steady incompressible Navier-Stokes equations in a domain bounded in part by a free surface. In the spatial discretization procedure, a Legendre spectral element method is used to generate the discrete equations. For effective solution of the set of algebraic equations, the geometry is decoupled from the fluid velocity and pressure. In addition, two different algorithms are proposed depending on the importance of surface tension effects. Numerical results are presented to demonstrate the effectiveness of the proposed algorithms.     

Application of lattice Boltzmann method for incompressible viscous flows

Because of the presence of corner eddies that change in number and pattern the lid-driven cavity problem has been found suitable to study various aspects of the performance of solution algorithms for incompressible viscous flows. It retains all the difficult flow physics and is characterized by a large primary eddy at the centre and secondary eddies located near the cavity corners. In this work, lid-driven cavity flow is simulated by lattice Boltzmann method with single-relaxation-time and it is compared with those by lattice Boltzmann method with multi-relaxation-time and finite difference method. The effects of the Reynolds number on the size, centre position and number of vortices are studied in detail together with the flow pattern in the cavity. The close agreement of the results bears testimony to the validity of this relatively new approach. However lattice Boltzmann method with multi-relaxation-time model is seen to remove the difficulties faces by the lattice Boltzmann method with single-relaxation-time at higher Reynolds numbers.     

On the optimal control of viscous incompressible fluid flow

The framework of the Navier-Stokes (N-S) equations is used to study flow past an arbitrary body on whose surface the tangential or normal velocity is under control. The necessary conditions are obtained for the minimum rate of energy dissipation. Exact analytical solutions of the corresponding problems are found for the case of flow past an ellipsoid in the Stokes approximation.     

Dynamics of a rotor partially filled with a viscous incompressible fluid

A rotor partially filled with a viscous incompressible fluid is modeled as planar system. Its structural part, i. e. the rotor, is assumed to be rigid, circular, elastically supported and running with a prescribed time-dependent angular velocity. Both parts, structure and fluid, interact via the no-slip condition and the pressure. The point of departure for the mathematical formulation of the fluid filling is the Navier-Stokes equation, which is complemented by an additional equation for the evolution of its free inner boundary. Further, rotor and fluid are subjected to volume forces, namely gravitation. Trial functions are chosen for the fluid velocity field, the pressure field and the moving boundary, which fulfill the incompressibility constraint as well as the boundary conditions. Inserting these trial functions into the partial differential equations of the fluid motion, and applying the method of weighted residuals yields equations with time derivatives only. Finally, in combination with the rotor equations, a nonlinear system of 12 differential-algebraic equations results, which sufficiently describes solutions near the circular symmetric state and which may indicate the loss of its stability. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)