Uniform attractor for nonautonomous incompressible non-Newtonian fluid with a new class of external forces

This paper is joint with [27]. The authors prove in this article the existence and reveal its structure of uniform attractor for a two-dimensional nonautonomous incompressible non-Newtonian fluid with a new class of external forces.     

On the graph of large distances

For a setS of points in the plane, letd ₁>d ₂>... denote the different distances determined byS. Consider the graphG(S, k) whose vertices are the elements ofS, and two are joined by an edge iff their distance is at leastd _k . It is proved that the chromatic number ofG(S, k) is at most 7 if |S|constk ². IfS consists of the vertices of a convex polygon and |S|constk ², then the chromatic number ofG(S, k) is at most 3. Both bounds are best possible. IfS consists of the vertices of a convex polygon thenG(S, k) has a vertex of degree at most 3k – 1. This implies that in this case the chromatic number ofG(S, k) is at most 3k. The best bound here is probably 2k+1, which is tight for the regular (2k+1)-gon.     

On the stability of fluid motions

The fundamental ideas concerning the linear and nonlinear stability of fluid motions are presented. The thermal convection in a layer of a dielectric fluid subject to an alternative current is studied.     

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.     

The parallel volume at large distances

In this paper we examine the asymptotic behavior of the parallel volume of planar non-convex bodies as the distance tends to infinity. We show that the difference between the parallel volume of the convex hull of a body and the parallel volume of the body itself tends to 0. This yields a new proof for the fact that a planar body can only have polynomial parallel volume if it is convex. Extensions to Minkowski spaces and random sets are also discussed.     

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 theory of periodic layers in incompressible fluid

A conceptual error in the formally closed but physically incomplete Kim-Moin-Moser form of equations for viscous incompressible fluid in a horizontal periodic layer is corrected. This form, which has lately become popular, assumes that the vertical projections of the rotor and the second rotor of the field of accelerations vanish. This assumption considerably simplifies the calculations; however, it is insufficient for the equations of motion. In this paper, the fulfillment of these assumptions is ensured by the additional condition that the vector of the horizontal projection averaged over the period of the acceleration vorticity vanishes, which opens new possibilities. The resulting complete form of the equations with the rotors of three orders admits a reduction to two scalar equations (of the fourth and the sixth orders), which, however, are not less complicated than the equivalent Navier-Stokes equations.     

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.     

On MHD flow of an incompressible viscous fluid

In this paper, we apply Homotopy Perturbation Method (HPM) to find the analytical solutions of nonlinear MHD flow of an incompressible viscous fluid through convergent or divergent channels in presence of a high magnetic field. The flow of an incompressible electrically conducting viscous fluid in convergent or divergent channels under the influence of an externally applied homogeneous magnetic field is studied both analytically and numerically. The graphs are presented to reveal the physical characteristics of flow by changing angles of the channel, Hartmann and Reynolds numbers.     

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.     

Homogenization of boundary layer of pseudo-plastic fluid in the presence of rapidly oscillating external forces

In this paper, we study the homogenization problem for equations of magnetohydrodynamic boundary layer of pseudo-plastic fluid. It is assumed that the external flow velocity and the external magnetic field are described by oscillating functions and the frequency depends on a small parameter. In von Mises variables and in Cartesian variables, we construct the homogenized problem, establish strong convergence of solutions in a special norm, and estimate the rate of this convergence. We show that in von Mises variables the convergence rates in different norms are of different orders of smallness.     

On the asymptotics of motion of a viscous incompressible fluid for small viscosity

We study a nonstationary initial–boundary value problem on the motion of a viscous incompressible fluid in the case of small viscosity. We prove the convergence of solutions to the corresponding limit relations as the viscosity tends to zero.     

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.     

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.     

On the control of layered flow of a viscous incompressible fluid within MHD

The control of a viscous incompressible flow within MHD is considered. The phenomenon of internal heat release is explored as applied to a layered plane flow in which the kinetic energy dissipates strictly into heat. The influence exerted by the electromagnetic field of the flowing fluid on the surrounding medium is examined, and methods for flow control in the context of digital deposit field technology are considered.     

On the initial‐boundary value problem of the incompressible viscoelastic fluid system

In this paper, we shall establish the local well‐posedness of the initial‐boundary value problem of the viscoelastic fluid system of the Oldroyd model. We shall also prove that the local solutions can be extended globally and that the global solutions decay exponentially fast as time goes to infinity provided that the initial data are sufficiently close to the equilibrium state. © 2007 Wiley Periodicals, Inc.