Bifurcation of periodic solutions of the Navier-Stokes equation in a thin domain (original) (raw)
On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations
2019
This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier-Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasiliner systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 000, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal LpL_pLp-$L_q$ regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, wh...
Periodic Structure of 2-D Navier-Stokes Equations
Journal of Nonlinear Science, 2005
We study in this article both the structure and the structural evolution of the solutions of 2-D Navier-Stokes equations with periodic boundary conditions and the evolution of their solutions. First the structure of all eigenvectors of the corresponding Stokes problem is classified using a block structure, and is linked to the typical structure of the Taylor vortices. Then the structure of the solutions of the Navier-Stokes equations forced either by eigenmodes or by potential forcing is classified.
Periodic Solutions in {\mathbb R}^n$$ for Stationary Anisotropic Stokes and Navier-Stokes Systems
Integral Methods in Science and Engineering
and references therein. In [KMW20, KMW21a, KMW21b, KMW21c] this field has been extended to the transmission and boundary-value problems for stationary Stokes and Navier-Stokes equations of anisotropic fluids, particularly, with relaxed ellipticity condition on the viscosity tensor. In this chapter, we present some further results in this direction considering periodic solutions to the stationary Stokes and Navier-Stokes equations of anisotropic fluids, with an emphasis on solution regularity. First, the solution uniqueness and existence of a stationary, anisotropic (linear) Stokes system with constant viscosity coefficients in a compressible framework are analysed on n-dimensional flat torus in a range of periodic Sobolev (Besselpotential) spaces. By employing the Leray-Schauder fixed point theorem, the linear results are employed to show existence of solution to the stationary anisotropic (nonlinear) Navier-Stokes incompressible system on torus in a periodic Sobolev space. Then the solution regularity results for stationary anisotropic Navier-Stokes system on torus are established.
On Bifurcating Time-Periodic Flow of a Navier-Stokes Liquid Past a Cylinder
Archive for Rational Mechanics and Analysis
We provide general sufficient conditions for branching out of a timeperiodic family of solutions from steady-state solutions to the twodimensional Navier-Stokes equations in the exterior of a cylinder. To this end, we first show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate function spaces. This is obtained by separating the time-independent averaged component of the velocity field from its "purely periodic" one. We then prove that time-periodic bifurcation occurs, provided the linearized time-independent operator of the parabolic problem possess a simple eigenvalue that crosses the imaginary axis when the Reynolds number passes through a (suitably defined) critical value. We also show that only supercritical or subcritical bifurcation may occur. Our approach is different and, we believe, more direct than those used by previous authors in similar, but distinct, context.
Global journal of research in engineering, 2023
It has recently been proposed by the author of the present work that the periodic NS equations (PNS) with high energy assumption can breakdown in finite time but with sufficient low energy scaling the equations may not exhibit finite time blowup. This article gives a general model using specific periodic special functions, that is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the PNS equations at the centers of cells of the 3-Torus. Satisfying a divergence free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term which is smooth and spatially periodic, the existence of solutions which blowup in finite time for PNS can occur starting with the first derivative and higher with respect to time. P. Isett ( ) has shown that the conservation of energy fails for the 3D incompressible Euler flows with H lder regularity below 1/3. (Onsager's second conjecture) The endpoint regularity in Onsager's conjecture is addressed, and it is found that conservation of energy occurs when the H lder regularity is exactly 1/3. The endpoint regularity problem has important connections with turbulence theory. Finally very recent developed new governing equations of fluid mechanics are proposed to have no finite time singularities.
Common periodic behavior in larger and larger truncations of the Navier-Stokes equations
Journal of Statistical Physics, 1988
The periodic behavior of N-mode truncations of the Navier-Stokes equations on a two-dimensional torus is studied for N= 44, 60, 80, and 98. Significant common features are found, particularly for not too high Reynolds numbers. In all models periodicity ends, giving rise, though at quite different parameter values, to quasiperiodicity.
Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations
Journal of Dynamics and Differential Equations, 2000
J. Dynamics and Diff. Eqns., in press, 2004. The article is devoted to the study of non-autonomous Navier-Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier-Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic,almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier-Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier-Stokes equations.
Coinciding local bifurcations in the Navier-Stokes equations
EPL (Europhysics Letters), 2021
Generically, a local bifurcation only affects a single solution branch. However, branches that are quite different may nonetheless share certain eigenvectors and eigenvalues, leading to coincident bifurcations. For the fluidic pinball, two supercritical pitchfork bifurcations, of the equilibrium and the periodic solutions, occur at nearly the same Reynolds number. The mechanism of this kind of non-generic coincidence is modelled and explained.
arXiv: Analysis of PDEs, 2021
A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R 3 /T 3 is hereby shown as well as results on the velocity-pressure distribution. 1 Introduction The global regularity of the Navier-Stokes equations remains to be an outstanding unsolved problem in fluid mechanics. The Clay Institute is offering a significant prize for those who are successful in solving either one of four proposed problems, that is either a periodic or non-periodic regular or finite time blowup problem for the full 3D Navier Stokes equations.[1] The motivation behind the author's present contribution and work was to pursue the periodic Navier Stokes equations problem and it was shown by corresponding author [2](using Geometric Algebra) and then the authors in [3] (also see references contained therein) who confirmed in an applied setting that there is no finite time blowup for the special case of the theoretical class of solutions G being multiplicative between spatial variable y and coordinates (x, y, t). (see[2] and Eq.(13,15) of [3]) In addition the plots shown in [3] (Figure 1 there) were obtained for the special case of F (x, y) = 1 in [3](Eq13); Saddle surface generation and saddle orbits were obtained where the initial condition was a min-max type function with a min-max point at the center of a typical cube in the periodic Lattice of R 3 /T 3. This was plotted in [3] in Fig 1 (a,c) there. These saddle orbits indicate that there are instabilities leading to turbulence or possibly even STC. [4] Also the form of the solution obtained showed no finite time blowup. However the problem remained unsolved as to the general class of the form of solutions G due to the arbitrariness of the function itself. It is the purpose of the following article to provide a rigorous mathematical proof that G cannot be a blowup itself with respect to t except at a finite number of points in any given bounded subset of R 3 /T 3 if and only if in the initial data the energy is infinite. It is hypothesized that energy "cascades" from large-scale structures to smaller scale structures by an inertial and inviscid mechanism. This process does continue, and there is the creation of smaller and smaller structures which produce a hierarchy of eddies. It is inevitable that this process produces structures that are so small that molecular diffusion becomes important and viscous dissipation of energy eventually takes place. The scale at which this happens is the Kolmogorov length scale. [5] At these fine scales there is potential finite time blowup. This is also addressed above and beyond what was done in [2] and [3]. The issue of stability of solutions to the forced Navier-Stokes and damped Euler systems in periodic boxes has been studied in [6].There it is shown that for large, but fixed, Grashoff (Reynolds) number the turbulent behavior of all Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations, in a periodic box, is suppressed, when viewed in the right frame of reference, by large enough average flow of the initial data. In the present work there exists such a frame of reference in the z direction where the average flow of the initial data is arbitrarily large. For the form of solution G which is multiplicative, a unique time-periodic solution exists when the average flow of the initial data is large or a fast oscillating forcing term occurs, with no blowup. Here there is a suppression of turbulence. However for a general form of G, it is shown that there exists infinite type singularities on sets of measure zero corresponding to the centres of periodic cells of the
2004
Poiseuille flows in infinite cylindrical pipes, in spite of it enormous simplicity, have a main role in many theoretical and applied problems. As is well known, the Poiseuille flow is a stationary solution to the Stokes and the Navier-Stokes equations with a given constant flux. Time-periodic flows in channels and pipes have a comparable importance. However, the problem of the existence of time-periodic flows in correspondence to any given, time-periodic, total flux, is still an open problem. A solution is known only in some very particular cases as, for instance, the Womersley flows. Our aim is to solve this problem in the general case. This existence result open the way to further investigations, in particular by following in the footsteps of the stationary case. As an application, we present the first steps to the study of Leray’s problem for the Stokes and Navier-Stokes equations. We leave to the interested reader, or to forthcoming papers, the adaptation to time-periodic flows ...
We study the bifurcation problem for periodic solutions of a nonautonomous damped wave equation deÿned in a thin domain. Here the bifurcation parameter is represented by the thinness ¿ 0 of the considered domain. This study has as starting point the existence result of periodic solutions already stated by the authors for this equation and it makes use of the condensivity properties of the associated Poincarà e map and its linearization around these solutions. We establish su cient conditions to guarantee that = 0 is or not a bifurcation point and a related multiplicity result. These results are in the spirit of those given by Krasnosel'skii and they are obtained by using the topological degree theory for k-condensing operators.
Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating Obstacle
Archive for Rational Mechanics and Analysis, 2014
We prove the existence and uniqueness of periodic motions to Stokes and Navier-Stokes flows around a rotating obstacle D ⊂ R 3 with the complement = R 3 \D being an exterior domain. In our strategy, we show the C b-regularity in time for the mild solutions to linearized equations in the Lorentz space L 3,∞ () (known as weak-L 3 spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L 3 spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier-Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.
Archive for Rational Mechanics and Analysis, 2005
Poiseuille flows in infinite cylindrical pipes, in spite of their enormous simplicity, have a main role in many theoretical and applied problems. As is well known, the Poiseuille flow is a stationary solution of the Stokes and the Navier-Stokes equations with a given constant flux. Time-periodic flows in channels and pipes have a comparable importance. However, the problem of the existence of timeperiodic flows in correspondence to any given time-periodic total flux, is still an open problem. A solution is known only in some very particular cases, for instance, the Womersley flows. Our aim is to solve this problem in the general case. The above existence result opens the way to further investigations. As an example of this possibility we consider the extension of the classical Leray's problem for Poiseuille flows to arbitrary time-periodic flows. z 0, for each cross section (z) = {(x, z) : x ∈ } and at any time t 0. We call the flux g(t), in the cross sections of the pipe, the total flux. It may be possible that after a long time, in a very long pipe, the outflow velocity "forgets" the particular point wise distribution of the inflow velocity v 0 , and merely "remembers" the total flux g(t). If we assume that a unique limit solution exists, in correspondence to a given g, than the solution must be independent of z. In spite of the recognized, theoretical and applied, significance of this very basic problem, a positive answer is known only in a very few cases: for instance, the classical Poiseuille steady flow, when the flux is constant; and the Womersley flow, which corresponds to a quite particular but important class of periodic sinusoidal fluxes in circular pipes, see [26]. The central position occupied by periodic flows in pipes leads us to consider the possibility of replacing Poiseuille and Womersley flows by flows with an arbitrary time-periodic total flux g(t). Now, the basic open problem is to prove the existence of a time-periodic flow with a given time-periodic flux g(t). As we will see, this leads to a non-standard variational problem. Contrary to the stationary case, the main open problem is now whether there exists, in an infinite pipe = × R, a periodic flow with a given time-periodic flux g(t). As in the Womersley paper, we also have in mind flows in large arteries. Here, the heart beat gives rise to a periodic variation, the pulsatility, and hence to a time periodic total flux g(t). However, this flux is far from being of sinusoidal type. Nevertheless, in many blood flow simulations, the Womersley model is used; this may be due to the lack of information on more general periodic solutions. Concerning blood flow problems see, for instance, [22]. Another motivation for our study is the extension of the famous Leray problem to periodic flows. In the classical formulation, two cylindrical semi-infinite pipes, 1 and 2 , are connected by a reservoir 0. We consider the problem of the existence of a viscous, incompressible fluid flow, subjected to convergence to Poiseuille flows, in both pipes, as the distance goes to infinity. A constant flux g is assigned. A fundamental contribution to Leray's problem is that given by Amick in [1], dedicated to Leray himself, and in [2], to which we refer the interested reader. Leray's problem seems to have been proposed, see [1], by Leray himself to Ladyzhenskaya, who in [13] attempted an existence proof under no restrictions on the viscosity. As referenced in [1], this problem is also mentioned by Finn in the review paper [7]. For the Leray's and related problems we refer, in particular, to [
Global Existence of Strong Solution to 3D Periodic Navier-Stokes Equations
2020
The purpose of this paper is to bring to light a method through which the global in time existence for arbitrary large in H1 initial data of a strong solution to 3D periodic Navier-Stokes equations follows. The method consists of subdividing the time interval of existence into smaller sub-intervals carefully chosen. These sub-intervals are chosen based on the hypothesis that for any wavenumber m, one can find an interval of time on which the energy quantized in low-frequency components (up to m) of the solution u is lesser than the energy quantized in high-frequency components (down to m) or otherwise the opposite. We associate then a suitable number m to each one of the intervals and we prove that the norm ‖u(t)‖H1 is bounded in both mentioned cases. The process can be continued until reaching the maximal time of existence Tmax which yields the global in time existence of strong solution.
DISCRETE AND CONTINUOUS NAVIER–STOKES EQUATIONS ON THE β-PLANE
We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier-Stokes equations on the periodic β-plane (i.e. with the Coriolis force varying as f 0 + βy) will become nearly zonal: with the vorticity ω(x, y, t) =ω(y, t) +ω(x, y, t), one has |ω| 2 H s ≤ β −1 Ms(· · · ) as t → ∞. We use this show that, for sufficiently large β, the global attractor of this system reduces to a point.