Dimensional Navier-Stokes Flow Research Articles

Space-time asymptotics of the two dimensional Navier–Stokes flow in the whole plane

We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in L1(R2)∩Lσ2(R2). Moreover, we characterize the necessary and sufficient condition for the rapid energy decay ‖u(t)‖2=o(t−1) as t→∞ motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow u(t) lies in the Hardy space H1(R2) for t>0, we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay ‖u(t)‖2=o(t−32) as t→∞ with cyclic symmetry introduced by Brandolese [2].

The generalized Proudman–Johnson equation and its singular perturbation problems

We consider the generalized Proudman–Johnson equation with an external force. By varying the Reynolds number $$R$$ and another nondimensional parameter $$\alpha $$ , branching stationary solutions are computed numerically for the global picture of bifurcations of the equation. Asymptotic behavior of solutions as the Reynolds number tends to zero or infinity is also studied by a combination of heuristic analysis and the asymptotic expansion. In doing so, singular perturbation problems of new type are derived and analyzed. As a consequence, through the asymptotic analysis argument, the peculiarity of two dimensional Navier–Stokes flows related to the unimodality is re-confirmed.

On the form of the viscous term for two dimensional Navier-Stokes flows

The form of the viscous term is discussed for incompressible flow on a two-dimensional curved surface S and for the shallow water equations. In the case of flow on a surface three versions are considered. These correspond to taking curl twice, to applying the Laplacian defined in terms of a metric, and to taking the divergence of a symmetric stress tensor. These differ on a curved surface, for example a sphere. The three terms are related and their properties discussed, in particular energy and angular momentum conservation. In the case of the shallow water equations again three forms of dissipation are considered, the last of which involves the divergence of a stress tensor. Their properties are discussed, including energy conservation and whether the rotating bucket solution of the three-dimensional Navier–Stokes equation is reproduced. A derivation of the viscous term is also given based on shallow water equations as a truncation of the Navier–Stokes equation, with forces on a column determined by integration over the vertical. For both incompressible flow on a surface and for the shallow water equations, it is argued that a viscous term based on a symmetric stress tensor should be used as this leads to correct treatment of angular momentum.

Oseen vortex as a maximum entropy state of a two dimensional fluid

During the last four decades, a considerable number of investigations has been carried out into the evolution of turbulence in two dimensional Navier-Stokes flows. Much of the information has come from numerical solution of the (otherwise insoluble) dynamical equations and thus has necessarily required some kind of boundary conditions: spatially periodic, no-slip, stress-free, or free-slip. The theoretical framework that has proved to be of the most predictive value has been one employing an entropy functional (sometimes called the Boltzmann entropy) whose maximization has been correlated well in several cases with the late-time configurations into which the computed turbulence has relaxed. More recently, flow in the unbounded domain has been addressed by Gallay and Wayne who have shown a late-time relaxation to the classical Oseen vortex (also sometimes called the Lamb-Oseen vortex) for situations involving a finite net circulation or non-zero total integrated vorticity. Their proof involves powerful but difficult mathematics that might be thought to be beyond the preparation of many practicing fluid dynamicists. The purpose of this present paper is to remark that relaxation to the Oseen vortex can also be predicted in the more intuitive framework that has previously proved useful in predicting computational results with boundary conditions: that of an appropriate entropy maximization. The results make no assumption about the size of the Reynolds numbers, as long as they are finite, and the viscosity is treated as finite throughout.

On Vorticity Directions near Singularities for the Navier-Stokes Flows with Infinite Energy

We give a geometric nonblow-up criterion on the direction of the vorticity for the three dimensional Navier-Stokes flow whose initial data is just bounded and may have infinite energy. We prove that under a restriction on behavior in time (type I condition) the solution does not blow up if the vorticity direction is uniformly continuous at the place where the vorticity magnitude is large. This improves the regularity condition for the vorticity direction first introduced by P. Constantin and C. Fefferman (1993) for finite energy weak solution. Our method is based on a simple blow-up argument which says that the situation looks like two-dimensional under continuity of the vorticity direction. We also discuss boundary value problems.