Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids

In a bounded domain of the space R ⁿ (n=2 or 3) we consider the initialand boundary-value problem regarding the determination of the velocity vector of the fluid, the pressure, and the density from the system of Navier-Stokes equation and the continuity equations, as well as from the initial conditions for the velocity and from the adherence boundary conditions. It is proved that the three-dimensional problem is uniquely solvable on some finite time interval and, in the case of a small initial velocity vector and a small volume force, also on an infinite interval; however, the two-dimensional problem is uniquely solvable for all t0 without any smallness restrictions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 52–109, 1975.     

Asymptotic expansions in time for rotating incompressible viscous fluids

We study the three-dimensional Navier–Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray–Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and without the zero spatial average of the solutions. Our method makes use of the Poincaré waves to rewrite the equations, and then implements the Gevrey norm techniques to deal with the resulting time-dependent bi-linear form. Special solutions are also found which form infinite dimensional invariant linear manifolds.     

Stability and bifurcation in viscous incompressible fluids

We consider heat-conducting viscous incompressible (not necessarily Newtonian) fluids under the general Stokesian constitutive hypotheses. Given a natural and mild condition on the stress tensor at vanishing velocity, which is satisfied for Newtonian fluids, we discuss the stability behavior of stationary states at which the fluid is at rest and at constant temperature. In particular, we prove the existence of global small strong solutions for rather general isothermal non-Newtonian fluids. We also study bifurcation problems and show that subcritical bifurcations can occur. This effect can be seen only if the full energy equation is taken into consideration, that is, if the energy dissipation term is not dropped, as is done in the usual Boussinesq approximation. Bibliography: 29 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 9–29.     

Generalized stochastic flows and applications to incompressible viscous fluids

We introduce a notion of generalized stochastic flows on manifolds, that extends to the viscous case the one defined by Brenier for perfect fluids. Their kinetic energy extends the classical kinetic energy to Brownian flows, defined as the L²$L^2$ norm of their drift. We prove that there exists a generalized flow which realizes the infimum of the kinetic energy among all generalized flows with prescribed initial and final configuration. We also construct generalized flows with prescribed drift and kinetic energy smaller than the L²$L^2$ norm of the drift.     

Large-eddy-simulation of 3-dimensional Rayleigh-Taylor instability in incompressible fluids

The 3-dimensional incompressible Rayleigh-Taylor instability is numerically studied through the large-eddy-simulation ( LES) approach based on the passive scalar transport model. Both the instantaneous velocity and the passive scalar fields excited by sinusoidal perturbation and random perturbation are simulated. A full treatment of the whole evolution process of the instability is addressed. To verify the reliability of the LES code, the averaged turbulent energy as well as the flux of passive scalar are calculated at both the resolved scale and the subgrid scale. Our results show good agreement with the experimental and other numerical work. The LES method has proved to be an effective approach to the Rayleigh-Taylor instability.     

On the variational inequalities related to viscous density-dependent incompressible fluids

We consider some variational inequality formulations related to density-dependent incompressible fluids. Firstly, we state the density-dependent micropolar model, which let us to introduce a generic (vectorial) differential inequality formulation. Then, two relaxations of this differential inequality will be considered, driving to concepts of weak and generalized solutions (observing that the weak solutions are generalized solutions but the contrary is not clear). Afterwards, under similar conditions imposed to prove the existence of generalized solutions for density-dependent Navier–Stokes equations (see Salvi, Riv Mat Parma 4:453–466, 1982), we prove the existence of weak solutions for this generic problem, which involves several variational inequality problems for viscous density-dependent incompressible fluids.     

Nonlinear instability of a viscous liquid jet

We investigate the weakly nonlinear temporal instability of an axisymmetric Newtonian liquid jet. Early nonlinear studies on the capillary instability of inviscid liquid jets were carried up to the third order contributions to the jet deformation and showed the nonlinear interaction between different modes. A recent study on the weakly nonlinear instability of planar Newtonian liquid sheets revealed the role of the liquid viscosity in the sheet stability behavior and showed a complicated influence [1]. Here, the instability of a liquid jet is examined as the axisymmetric counterpart of the sheet, in search for corresponding insight into the role of the liquid viscosity in the jet instability mechanism. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

On the stationary flows of viscous,incompressible and heat-conducting fluids

We consider a boundary-value problem describing the motion of viscous, incompressible and heat-conducting fluids in a bounded domain in ?³. We admit non-homogeneous boundary conditions, the appearance of exterior forces and heat sources. Our aim is to prove the existence of a solution of the problem in Sobolev spaces.     

On viscous incompressible flows of nonsymmetric fluids with mixed boundary conditions

In this paper we are concerned with the initial boundary value problem for the micropolar fluid system in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The Dirichlet boundary condition for the microrotation of the fluid is commonly used in practice. However, the well-posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present paper is devoted to the proof of the existence, regularity and uniqueness of the solution in distribution spaces.     

Electrohydrodynamic instability of two superposed viscous miscible streaming fluids

The linear stability of two dielectric viscous fluids separated by a horizontal interface is investigated. The interface admits heat and mass transfer. The system is stressed by a normal periodic electric field producing surface charges at the interface. The effect of surface tension, small viscosity, velocity streaming and gravity on the critical surface charges density and on corresponding electric field are analyzed. The contribution of viscosity with the existence of surface charges and streaming are discussed. The investigation includes the stability analysis of the presence of the periodic electric field as well as the constant one. It is found that the presence of the surface charges made by the normal electric field play a dual role in the stability criterion, which shows some analogy with the nonlinear theory of stability. Some previous studies are compared using appropriate data. The marginal state of stability is also considered. It is found that the surface charges vanish under certain conditions. This study shows that the mass and heat transfer parameter has a destabilizing effect whether the electric field is static or periodic. Parametric excitation of the electrohydrodynamic (EHD) surface waves is analyzed in the case of Rayleigh–Taylor (R–T) instability. The transition curves are obtained by means of Whittaker's technique. The analytical results are numerically confirmed.     

Stability and instability in adiabatic shearing motions of incompressible fluids

We investigate the effect of temperature dependence of the viscosity on the stability of the adiabatic shearing flows of an incompressible Newtonian viscous fluid between two parallel plates. When the viscosity strongly decreases with temperature, the shearing flow caused by a steady motion of the upper plate (steady shearing) becomes unstable, while the shearing flow caused by a time-dependent body force is found to be stable.     

Nonlinear Marangoni instability in dielectric superposed fluids

Summary The nonlinear Marangoni instability of two dielectric superposed fluids is investigated. The system is stressed by a normal electric field such that it allows for the presence of surface charges at the interface. The method of multiple scale perturbations is used in order to obtain uniformly valid expansions. Two nonlinear Schrödinger equations describing the perturbed system are obtained. One of these equations is used to describe analytically and numerically the necessary conditions for stability and instability near the marginal state, while the other equation is used to obtain the nonlinear electrohydrodynamic cutoff wavenumber separating stable and unstable disturbances for the system.     

A numerical method for the motion of a mixture of two viscous incompressible fluids

A finite difference technique for the simulation of the motion of a mixture of two viscous incompressible fluids in a closed basin is presented. The mathematical model which has been discretized is the closed system deduced from the general equations, governing the motion of the mixture. The numerical scheme is based on the marker and cell method [4] extended to consider the molecular diffusion process. Computational examples are described and discussed at the end of the paper.     

Stochastic nonhomogeneous incompressible Navier-Stokes equations

We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier-Stokes equations with general multiplicative noise. These equations model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic equations were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331-332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240-1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, Equations de Navier-Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992].The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier-Stokes equations with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier-Stokes equations, Applicandae Math. 25 (1991) 59-85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov.     

Nonlinear instability of Hele-Shaw flows with smooth viscous profiles

We rigorously derive nonlinear instability of Hele-Shaw flows moving with a constant velocity in the presence of smooth viscosity profiles where the viscosity upstream is lower than the viscosity downstream. This is a single-layer problem without any material interface. The instability of the basic flow is driven by a viscosity gradient as opposed to conventional interfacial Saffman-Taylor instability where the instability is driven by a viscosity jump across the interface. Existing analytical techniques are used in this paper to establish nonlinear instability.