On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

Opposing flows in a one dimensional convection-diffusion problem

In this paper, we examine a particular class of singularly perturbed convection-diffusion problems with a discontinuous coefficient of the convective term. The presence of a discontinuous convective coefficient generates a solution which mimics flow moving in opposing directions either side of some flow source. A particular transmission condition is imposed to ensure that the differential operator is stable. A piecewise-uniform Shishkin mesh is combined with a monotone finite difference operator to construct a parameter-uniform numerical method for this class of singularly perturbed problems.     

A blow-up criterion for two dimensional compressible viscous heat-conductive flows

We establish a blow-up criterion in terms of the upper bound of the density and temperature for the strong solution to 2D compressible viscous heat-conductive flows. The initial vacuum is allowed.     

Exponential stability for a one‐dimensional compressible viscous micropolar fluid

In this paper, we consider one‐dimensional compressible viscous and heat‐conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H¹(0,1))⁴, and then we establish the global existence and exponential stability of solutions in (H²(0,1))⁴ under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.     

On a finite element method for three‐dimensional unsteady compressible viscous flows

In this article we analyze a finite element method for three‐dimensional unsteady compressible Navier‐Stokes equations. We prove the existence and uniqueness of the numerical solution, and obtain a priori error estimates uniform in time. Numerical computations are carried out to test the orders of accuracy in the error estimates. Blend function interpolations are applied in the calculation of numerical integrations. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 432–449, 2004.     

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.     

Boundary tangency for density interfaces in compressible viscous fluid flows

We prove that, for solutions of the Navier-Stokes equations of two-dimensional, viscous, compressible flow, curves which are initially transverse to the spatial boundary and across which the fluid density is discontinuous become tangent to the boundary instantaneously in time. This effect is seen to result from the strong pressure gradient force, which in this case includes a vector measure supported on the curve, together with the fact that singularities in this system are convected with the fluid velocity.     

Periodic solutions for one dimensional wave equation with bounded nonlinearity

This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient $u(x)$ satisfies $\text{ess inf}{\eta }_u(x)>0$ with ${\eta }_u(x)=\frac{1}{2}\frac{u^{″}}{u}?\frac{1}{4}{\left(\frac{u^{\prime }}{u}\right)}^2$, which actually excludes the classical constant coefficient model. For the case ${\eta }_u(x)=0$, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form $T=\frac{2p?1}{q}$ ($p,q$ are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition $\text{ess inf}{\eta }_u(x)>0$.     

ALMOST PERIODIC SOLUTION OF ONE DIMENSIONAL VISCOUS CAMASSA-HOLM EQUATION

This article studies one dimensional viscous Camassa-Holm equation with a periodic boundary condition. The existence of the almost periodic solution is investigated by using the Galerkin method.     

Bifurcation of multimode flows of a viscous fluid in a plane diverging channel

The pattern of steady multimode flow of a viscous incompressible fluid in a plane diverging channel is constructed and investigated. It is shown that odd-mode flows have velocity profiles that are symmetrical about the axis of the channel and from one to three different flows with a fixed number of modes exist. The even-mode flows are asymmetric and exist as pairs. The existence of a denumerable set of finite ranges adjoining one another, in which a single-type of complex bifurcation of the flow occurs, is established in the case of an unbounded range of values of the Reynolds number. As the Reynolds number increases, transitions to flows with an increasing number of modes, containing domains of forward and backward flows, occur successively. Flow patterns with a smaller number of modes do not occur. An increase in the number of an range corresponding to an increase in the Reynolds number leads to an unlimited increase in the length of the range and the number of modes of permissible flows.     

Low Mach number limit of viscous polytropic fluid flows

This paper studies the singular limit of the non-isentropic Navier-Stokes equations with zero thermal coefficient in a two-dimensional bounded domain as the Mach number goes to zero. A uniform existence result is obtained in a time interval independent of the Mach number, provided that the initial data satisfy the “bounded derivative conditions”, that is, the time derivatives up to order two are bounded initially, and Navier?s slip boundary condition is imposed.     

On the hydrodynamic interaction of two spheres oscillating in a viscous fluid. II. Three dimensional case

Asymptotic solution of the Navier-Stokes equations is derived for the problem of the viscous flow induced by two spheres oscillating in a direction perpendicular to their central line.The main purpose of the paper is to study the effect of the hydrodynamic interaction of the spheres on the forces acting upon them in this complicated three dimensional flow.It is shown that the drag on each of the two spheres decreases as the distance between them increases and also that each sphere experiences a repulsive force in a direction perpendicular to the oscillating stream.

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Zusammenfassung Das asymptotische Verfahren für die Lösung der Navier-Stokesschen Gleichungen wird angewendet für die Strömung, die bei der Schwingung von zwei Kugeln senkrecht zur Verbindungslinie ihrer Mittelpunkte entsteht.Das wichtigste Ziel der Arbeit ist die Untersuchung der hydrodynamischen Beeinflussung der Kugeln durch die Kräfte, die bei dieser komplizierten dreidimensionalen Strömung entstehen.Es wurde gefunden, daß der Widerstand der Kugeln sich mit wachsendem Abstand zwischen ihnen verringert. In Richtung senkrecht zur Schwingung wirkt auf jede Kugel eine abstoßende Kraft.

     

New estimates of the stability of one-dimensional plane-parallel flows of a viscous incompressible fluid

The stability of a number of one-dimensional plane-parallel steady flows of a viscous incompressible fluid is investigated analytically using the method of integral relations. The mathematical formulation is reduced to eigenvalue problems for the Orr–Sommerfeld equation. One of three versions is chosen as the boundary conditions: all the components of the velocity perturbation are equal to zero on both boundaries of the layer (in this case we have the classical Orr–Sommerfeld problem), all the components of the velocity perturbation on one of the boundaries are equal to zero and the perturbations of the shear component of the stress vector and of the normal component of the velocity are equal to zero on the other, and all the components of the velocity perturbation are equal to zero on one boundary and the other boundary should be free. The boundary conditions derived in the latter case, are characterized by the occurrence of a spectral parameter in them. For kinematic conditions the lower estimates of the critical Reynolds number – the Joseph–Yih estimates, are improved. In the remaining cases the technique of the integral-relations method is developed, leading to new estimates of the stability. Analogs of Squire's theorem are derived for the boundary conditions of all the types mentioned above. Upper estimates of the increment of the increase in perturbations in eigenvalue problems for the Rayleigh equation with two types of boundary conditions are given.     

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.     

The frozen-in condition for a direction field,small denominators and chaotization of steady flows of a viscous fluid

A classical theorem of Helmholtz states that vortex lines are frozen into a flow of barotropic ideal fluid in a potential force field. This result leads to the following general problem: it is required to find conditions under which a given dynamical system admits of a direction field frozen into its phase flow. By the rectification theorem for trajectories, a whole family of frozen direction fields always exists locally. It turns out that the problem of the existence of non-trivial frozen direction fields defined in the whole phase space is closely related to the well-known problem of small denominators. Results of a general nature are applied to Hamiltonian systems, and also to steady flows of a viscous fluid.     

Stirring of a viscous fluid

Zusammenfassung Untersucht wird die stationäre strömende Geschwindigkeit, die durch eine kreisförmige rührende Bewegung eines starren Zylinders induziert wird. Die Amplitude dieser kreisförmigen Bewegung wird im Vergleich zum Durchmesser des Zylinders als klein angenommen. Es wird gezeigt, dass bei genügend hohen Frequenzen der nicht-stationäre Wirbeleffekt auf eine dünne Stokes-Scherwellenschicht beschränkt ist, dass aber die induzierte Strömung ausserhalb dieser Region bestehen bleibt. Für den besonderen Fall eines runden Zylinders ergibt sich, dass diese stationäre Strömung von der Reynolds-Zahl unabhängig ist.     

On a decay rate for 1D-viscous compressible barotropic fluid equations

The Navier-Stokes equations of a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H ¹ as well as the mass force such that the stationary density is positive. The uniform lower bound for the density is proved. By constructing suitable Lyapunov functionals, decay rate estimates in L ²-norm and H ¹-norm are given. The decay rate is exponential if so the decay rate of the nonstationary part of the mass force is. The results are proved in the Eulerian coordinates for a wide class of increasing state functions including with any γ > 0 as well as functions of arbitrarily fast growth. We also extend the results for equations of a multicomponent compressible barotropic mixture (in the absence of chemical reactions). Received December 20, 2000; accepted February 27, 2001.