On the solutions in the large of the two-dimensional flow of a nonviscous incompressible fluid

The Euler equations (1.1) for the motion of a nonviscous imcompressible fluid in a plane domain Ω are studied. Let E be the Banach space defined in (1.4), let the initial data v₀ belong to E, and let the external forces f(t) belong to L_loc¹(R; E). In Theorem 1.1 the strong continuity and the global boundedness of the (unique) solution v(t) are proved, and in Theorem 1.2 the strong-continuous dependence of v on the data v₀ and f is proved. In particular the vorticity rot v(t) is a continuous function in $\text{}\text{?}$gW, for every t ? R, if and only if this property holds for one value of t. In Theorem 1.3 some properties for the associated group of nonlinear operators S(t). are stated. Finally, in Theorem 1.4 a quite general sufficient condition is given on the data in order to get classical solutions.     

On stability of steady-state plane–parallel shearing flows in a homogeneous in density ideal incompressible fluid

A problem on linear stability of stationary plane–parallel shearing flows in a homogeneous in density inviscid incompressible fluid between two immovable impermeable solid parallel infinite plates is studied. With the use of the direct Lyapunov method it is shown that all sufficient conditions (by Rayleigh, Fjørtoft, Arnol’d) known to this moment for stability of these flows with respect to small plane perturbations are the necessary ones as well. An a priori lower estimate is constructed; the estimate displays exponential in time growth of the considered perturbations if these conditions are not affected. An analytical example of steady-state plane–parallel shearing flows and superimposed small plane perturbations growing in time in accordance with the constructed estimate is given.     

On the search for singularities in incompressible flows

In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities.     

On existence of two-dimensional nonstationary flows of an ideal incompressible liquid admitting a curl nonsummable to any power greater than 1

Rostov-on-Don. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 209–212, September–October, 1992.     

On the finite element approximation of incompressible flows of an electrically conducting fluid

We consider the finite element approximation of incompressible flows field of an electrically conducing fluid in the presence of a magnetic where it is assumed that this field is prescribed. A weak form is chosen that is similar in some respects to a weak form used by many authors for the Navier-Stokes equations. Existence and uniqueness results are presented for the weak problem. A finite element Algorithm is given for the approximate solution of the weak problem and error estimates are derived.     

Free surface motion of an incompressible ideal fluid

We consider the free boundary problem for an incompressible ideal fluid in the two-dimensional space. We show the unique existence of the solution, locally in time, even if the initial surface and the bottom are uneven.     

On non-stationary flows of incompressible asymmetric fluids

We consider an initial boundary value problem for the system of equations describing non-stationary flows of incompressible asymmetric fluids. We prove the existence of a local in time, weak solution of the problem in the case when the initial density is not separated from zero by a positive constant.     

On the incompressible limits for the full magnetohydrodynamics flows

In this article the incompressible limits of weak solutions to the governing equations for magnetohydrodynamics flows on both bounded and unbounded domains are established. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and with an additional equation which describes the evolution of the magnetic field. The scaled analogues of the governing equations for magnetohydrodynamic flows involve the Mach number, Froude number and Alfven number. In the case of bounded domains the establishment of the singular limit relies on a detail analysis of the eigenvalues of the acoustic operator, whereas the case of unbounded domains is being treated by their suitable approximation by a family of bounded domains and the derivation of uniform bounds.     

On accelerated flows of an Oldroyd-B fluid in a porous medium

In this work, the problems dealing with unsteady unidirectional flows of an Oldroyd-B fluid in a porous medium are investigated. By using modified Darcy's law of an Oldroyd-B fluid, the equations governing the flow are modelled. Employing Fourier sine transform, the analytic solutions of the modelled equations are developed for the following two problems: (i) constant accelerated flow, (ii) variable accelerated flow. Explicit expressions for the velocity field and adequate tangential stress are obtained in each case. The solutions for Newtonian, second grade and Maxwell fluids in a porous medium appear as the limiting cases of the present analysis.     

Hyperbolic submodels of an incompressible viscoelastic Maxwell medium

A two-dimensional motion of an incompressible viscoelastic Maxwell continuum is considered. The system of quasilinear equations describing this motion has both real and complex characteristics. A class of effectively one-dimensionalmotions is analyzed for which the original system of equations is decomposed into a hyperbolic subsystem and a quadrature. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. When one chooses the Jaumann corotational derivative as the invariant derivative, the equations of the submodel remain quasilinear. They can be represented in the form of conservation laws, which allows one to analyze discontinuous solutions to these equations. When one chooses the upper or lower convected derivative, the equations of one-dimensional hyperbolic submodels turn out to be linear. The problem of shear motion between parallel plates and the problem of interaction between the stress field that does not depend on one of the coordinates and a transverse shear flow with initially constant vorticity are studied in detail. It is established that a plane Couette flow in the model with the corotational derivative is unstable in the linear approximation in the class of shear flows if the Weissenberg number is greater than one. The development of small perturbations gives rise to discontinuities in tangential velocities and stresses. The hysteresis phenomenon is observed when the Weissenberg number successively increases and decreases while passing through a critical value. The Couette flow in models with the upper and lower convected derivative remains stable with respect to one-dimensional perturbations.     

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 the form of a closed cavity in which there exist uniform vortex motions of an ideal incompressible fluid

The paper of S.V. Jacques [1] deals with the problem of finding forms of cavities in which there exist uniform vortex motions of an ideal incompressible fluid. In [1] the surface of the cavity was assumed to be a surface of revolution. The present work solves this problem without resorting to this assumption.     

On the stabilizing effect of convection in three‐dimensional incompressible flows

We investigate the stabilizing effect of convection in three‐dimensional incompressible Euler and Navier‐Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three‐dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three‐dimensional Euler or Navier‐Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier‐Stokes equations. We will present numerical evidence that seems to support that the three‐dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three‐dimensional model and how the convection term in the full Euler and Navier‐Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. © 2008 Wiley Periodicals, Inc.     

On the linear stability study of zonal incompressible flows on a sphere

The normal mode (linear) stability of zonal flows of a nondivergent fluid on a rotating sphere is considered. The spherical harmonics are used as the basic functions on the sphere. The stability matrix representing in this basis the vorticity equation operator linearized about a zonal flow is analyzed in detail using the recurrent formula derived for the nonlinear triad interaction coefficients. It is shown that the zonal flow having the form of a Legendre polynomial P_n(μ) of degree n is stable to infinitesimal perturbations of every invariant set I_m with |m| ≥ n. For each zonal number m, I_m is here the span of all the spherical harmonics $Y^{m}_{k}(x)$, whose degree k is greater than or equal to m. It is also shown that such small-scale perturbations are stable not only exponentially, but also algebraically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 649–665, 1998     

Dynamics of a rotating layer of an ideal electrically conducting incompressible fluid

A system of nonlinear partial differential equations is considered that models perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces. The system is reduced to a scalar equation. The solvability of initial boundary value problems arising in the theory of waves in conducting rotating fluids can be established by analyzing this equation. Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in an infinite horizontal layer and a long narrow channel.     

On the definition of variations in the mechanics of continuous media

The basic forms of variations used in the mechanics of continuous media are presented, and relations between various types of variations of vectors and tensors are established.     

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.