Self-similar Ansatz Research Articles

Study of the Navier–Stokes regularity problem with critical norms

We study the basic problems of regularity of the Navier–Stokes equations. The blowup criteria on the basis of the critical -norm, is bounded from above by a logarithmic function, (Robinson et al 2012 J. Math. Phys. 53 115618). Assuming that the Cauchy–Schwarz inequality for the -norm is not an overestimate, we replace it by a square-root of a product of the energy and the enstrophy. We carry out a simple asymptotic analysis to determine the time evolution of the energy. This generalises the (already ruled-out) self-similar blowup ansatz. Some numerical results are also presented, which support the above-mentioned replacement. We carry out a similar analysis for the four-dimensional Navier–Stokes equations.

SELF-SIMILAR ANALYTIC SOLUTION OF THE TWO-DIMENSIONAL NAVIER-STOKES EQUATION WITH A NON-NEWTONIAN TYPE OF VISCOSITY

We investigate Navier-Stokes (NS) and the continuity equations in Cartesian coordinates and Eulerian description for the two dimensional incompressible nonNewtonian fluids. Due to the non-Newtonian viscosity we consider the Ladyzenskaya model with a non-linear velocity dependent stress tensor. The key idea is the multidimensional generalization of the well-known self-similar Ansatz, which has already been used for non-compressible and compressible viscous flow studies. Geometrical interpretations of the trial function are also discussed. Our recent results are compared to the former Newtonian ones.

Heat Conduction: Hyperbolic Self-similar Shock-waves in Solid Medium

Analytic solutions for cylindrical thermal waves in solid medium are given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation.We consider the original non-linear hyperbolic system itself with the self-similar Ansatz for the temperature distribution and for the heat flux. As results continuous.

Analytic self-similar solutions of the Oberbeck–Boussinesq equations

In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtonian–Navier–Stokes — with Boussinesq approximation — and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field shows a strongly damped single periodic oscillation which can mimic the appearance of Rayleigh–Bénard convection cells. Finally, it is discussed how our result may be related to nonlinear or chaotic dynamical regimes.

Remark on Luo-Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations

In this note we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile.

Self-similar shock wave solutions of the nonlinear Maxwell equations

In our study we consider nonlinear, power-law field-dependent electrical permittivity and magnetic permeability and investigate the time-dependent Maxwell equations with self-similar ansatz. This is a first-order hyperbolic partial differential equation system which can conserve non-continuous initial conditions describing electromagnetic shock. Such phenomena may happen in complex materials induced by the planned powerful Extreme Light Infrastructure laser pulses.

Analytic solutions for the three-dimensional compressible Navier–Stokes equation

We investigate the three-dimensional compressible Navier–Stokes (NS) and the continuity equations in Cartesian coordinates for Newtonian fluids. The problem has an importance in different fields of science and engineering like fluid, aerospace dynamics or transfer processes. Finding an analytic solution may bring considerable progress in understanding the transport phenomena and in the design of different equipments where the NS equation is applicable. For solving the equation the polytropic equation of state is used as a closing condition. The key idea is the three-dimensional generalization of the well-known self-similar ansatz which was already used for non-compressible viscous flow in our former study. The geometrical interpretations of the trial function is also discussed. We compared our recent results to the former non-compressible ones.

Absence of singular stretching of interacting vortex filaments

AbstractA promising mechanism for generating a finite-time singularity in the incompressible Euler equations is the stretching of vortex filaments. Here, we argue that interacting vortex filaments cannot generate a singularity by analysing the asymptotic dynamics of their collapse. We use the separation of the dynamics of the filament shape, from that of its core, to derive constraints that must be satisfied for a singular solution to remain self-consistent uniformly in time. Our only assumption is that the length scales characterizing filament shape obey scaling laws set by the dimension of circulation as the singularity is approached. The core radius necessarily evolves on a different length scale. We show that a self-similar ansatz for the filament shapes cannot induce singular stretching, due to the logarithmic prefactor in the self-interaction term for the filaments. More generally, there is an antagonistic relationship between the stretching rate of the filaments and the requirement that the radius of curvature of filament shape obeys the dimensional scaling laws. This suggests that it is unlikely that solutions in which the core radii vanish sufficiently fast to maintain the filament approximation exist.

Self-Similar Solutions of Three-Dimensional Navier—Stokes Equation

In this article we will present pure three dimensional analytic solutions for the Navier—Stokes and the continuity equations in Cartesian coordinates. The key idea is the three-dimensional generalization of the well-known self-similar Ansatz of Barenblatt. A geometrical interpretation of the Ansatz is given also. The results are the Rummer functions or strongly related. Our final formula is compared with other results obtained from group theoretical approaches.

Phase fluctuations of linearly chirped solitons in a noisy optical fiber channel with varying dispersion, nonlinearity, and gain

The phase fluctuations of arbitrarily nonlinearity- and dispersion-managed solitons propagating in a noisy fiber channel are studied both analytically and numerically. We begin by discussing the stability problem of such linearly chirped solitons with a full linear stability analysis. It is shown that these sophisticated solitons possess an enhanced stability against perturbations and therefore hold promise for applications in optical telecommunications. We then make an approach to the phase statistics of these solitons, which stems from an inevitable random walk in phase evolutions due to amplified spontaneous emission noise. By using the variational approach together with impulse-response (Green) functions, an elegant closed-form expression for the phase variance is derived based on an unconstrained self-similar soliton ansatz in which the effect of chirp fluctuations has been critically taken into account as well as the dispersive and nonlinear effects. An inspection of the intriguing subtleties of the interplay among these effects reveals that the chirp fluctuations effect does play an important role in the control of nonlinear phase noise via fiber dispersion, independently of whether the input solitons are initially chirped or not. Our analytical result also offers many possibilities of optimally manipulating nonlinear phase noise with engineered fiber parameters that may lead to the steady pulse propagation, broadening, or compression under favorable parametric conditions. Last, we demonstrate our result by several convincible examples and show an excellent agreement between analytical predictions and numerical simulations.

Diffusion of super-Gaussian profiles

The present analysis describes an analytically simple and systematic approximation procedure for modelling the free diffusive spreading of initially super-Gaussian profiles. The approach is based on a self-similar ansatz for the evolution of the diffusion profile, and the parameter functions involved in the modelling are determined by suitable moments of the diffusion equation. The resulting approximate solutions provide an accurate description of the evolution of the initial profile into the Gaussian profile, which is the asymptotically exact solution of the diffusion problem for, in fact, arbitrary initial conditions.

An averaging method for nonlinear laminar Ekman layers

We study steady laminar Ekman boundary layers in rotating systems using an averaging method similar to the technique of von Kármán and Pohlhausen. The method allows us to explore nonlinear corrections to the standard Ekman theory even at large Rossby numbers. We consider both the standard self-similar ansatz for the velocity profile, which assumes that a single length scale describes the boundary layer structure, and a new non-self-similar ansatz in which the decay and the oscillations of the boundary layer are described by two different length scales. For both profiles we calculate the up-flow in a vortex core in solid-body rotation analytically. We compare the quantitative predictions of the model with the family of exact similarity solutions due to von Kármán and find that the results for the non-self-similar profile are in almost perfect quantitative agreement with the exact solutions and that it performs markedly better than the self-similar profile.

Self-similar tip growth in filamentary organisms.

The growth of a family of filamentary microorganisms is described in terms of self-similar growth at the tip which is driven by pressure and sustained by a wall-building growth process. The cell wall is modeled biomechanically as a stretchable elastic membrane using large-deformation elasticity theory. Incorporation of geometry dependent elastic moduli and a self-similar ansatz shows how these equations can generate realistic tip shapes corresponding to a self-similar expansion process.

Critical phenomena in perfect fluids

We investigate the gravitational collapse of a spherically symmetric, perfect fluid with equation of state P = (-1). We restrict attention to the ultrarelativistic (`kinetic-energy-dominated', `scale-free') limit where black-hole formation is anticipated to turn on at infinitesimal black-hole mass (type II behaviour). Critical solutions (those which sit at the threshold of black-hole formation in parametrized families of collapse) are found by solving the system of ordinary differential equations which result from a self-similar ansatz, and by solving the full Einstein/fluid partial differential equations (PDEs) in spherical symmetry. These latter PDE solutions (`simulations') extend the pioneering work of Evans and Coleman ( = (4/3)) and verify that the continuously self-similar solutions previously found by Maison and Hara et al for 1.051.89 are (locally) unique critical solutions. In addition, we find strong evidence that globally regular critical solutions do exist for 1.892, that the sonic point for dn1.889 6244 is a degenerate node, and that the sonic points for >dn are nodal points, rather than focal points as previously reported. We also find a critical solution for = 2, and present evidence that it is continuously self-similar and type II. Mass-scaling exponents for all of the critical solutions are calculated by evolving near-critical initial data, with results which confirm and extend previous calculations based on linear perturbation theory. Finally, we comment on critical solutions generated with an ideal-gas equation of state.

Jet collimation by turbulent viscosity. I

In this paper it is assumed that the subscale turbulent eddies induced in an ambient medium by the emergence of a (already collimated) jet from a galactic nucleus (VLBI jet) are the source of the viscosity which causes material to be entrained into the large-scale (VLA) jet. New analytic solutions are derived by a generalization of the self-similar Ansatz used in the Landau-Squires solution to include variable density and viscosity. It is shown that such a process of viscous collimation of the VLA jets can account for the observed collimation-luminosity correlation, the magnetic flux, and the inferred mass flux of these jets. Order of magnitude comparisons of velocity and density fields with recently observed emission-line flow regions near radio jets are made. All of the viscosity-dependent observational checks imply roughly the same plausible value for the eddy viscosity. It is emphasized that storing the initial VLBI jet energy in the intermediate scales occupied by the turbulent eddies allows this energy to be largely undetected. 35 references.