Approximate Self-similar Solution Research Articles

Modal stability analysis of toroidal pipe flow approaching zero curvature

The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature $\delta$ (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for $10^{-7} \leq \delta \leq ~10^{-2}$ is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below $1.5 \times 10^{-6}$ , the critical Reynolds number ${{Re}}_{cr}$ is proportional to $\delta ^{-1/2}$ . Hence, the critical Dean number is constant, ${{De}}_{cr} = 2\,{{Re}}_{cr}\,\sqrt {\delta } \approx 113$ . This behaviour confirms that the Hagen–Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.

Supersonic–subsonic transition region in radiative heat flow via self-similar solutions

We study the radiative hydrodynamics flow of radiation-driven heat waves in hot dense plasmas, using approximate self-similar solutions. Specifically, we have focused on the intermediate regime between the pure radiative supersonic flow and the pure subsonic regime. These two regimes were investigated using both exact self-similar solutions and numerical simulations; however, most of the study used numerical simulations, mainly because the radiative heat wave and the shock regions are not self-similar altogether. In a milestone work [Garnier et al., “Self-similar solutions for a nonlinear radiation diffusion equation,” Phys. Plasmas 13, 092703 (2006)], it was found that for a specific power law dependency temperature profile, a unique exact self-similar solution exists that is valid for all physical regimes. In this work, we approximate Garnier's exact solution for a general power-law temperature-dependency, using simple analytical considerations. This approximate solution yields a good agreement compared to numerical simulations for the different thermodynamic profiles within the expected range of validity. In addition, we offer an approximate solution for the energies absorbed in the matter, again, for a general power-law temperature profile. Our approximate self-similar solution for the energy yields very good results compared to exact numerical simulations for both gold and Ta2O5. We also set a comparison of our self-similar solutions with the results of an experiment for radiation temperature measurement in a Hohlraum in low-density foams that is addressed directly to the intermediate regime, yielding a good agreement and similar trends. The different models as well as the numerical simulations are powerful tools to analyze the supersonic–subsonic transition region.

Thermal boundary layer of laminar flow of dilute polymer solution

The thermal boundary layer flow is a canonical flow with characteristics that are present in most natural and industrial convection flows. An approximate self-similar solution is proposed for the first time for the thermal boundary layer of steady laminar flow of viscoelastic fluids, described by the finitely extensible nonlinear elastic constitutive equation with Peterlin's closure (FENE-P model). This semi-analytical thermal solution is obtained by performing an order of magnitude analysis and ensuing simplifications of the governing equations by assuming that the fluid properties are independent of temperature therefore decoupling the flow governing equations from the energy equation. The effects of viscoelasticity quantified with the Weissenberg number based on the streamwise coordinate (x) (Wix) up to Wix=1 and viscous dissipation (results are presented for Brinkman numbers between -40 and +40) on thermal boundary layer characteristics are investigated comprehensively for both constant wall temperature and constant wall heat flux. At low elasticity levels (Wix<0.01) the solution exhibits a global self-similar behavior in which flow and thermal quantities collapse on the corresponding Newtonian curves, and the polymer characteristics show a unique behavior if adequately normalized. However, by increasing flow elasticity the unique self-similar behavior of the approximate solution is lost, with the elasticity dependent results exhibiting local variations. In addition, the effects of elasticity are intensified by viscous dissipation. For the present study cases, it is observed that elasticity may change Nusselt numbers by more than 8%, and the thermal boundary layer thickens by up to 10%.

Self-Similar Solutions in the Theory of Nonstationary Radiative Transfer in Spectral Lines in Plasmas and Gases

Radiative transfer (RT) in spectral lines in plasmas and gases under complete redistribution of the photon frequency in the emission-absorption act is known as a superdiffusion transport characterized by the irreducibility of the integral (in the space coordinates) equation for the atomic excitation density to a diffusion-type differential equation. The dominant role of distant rare flights (Lévy flights, introduced by Mandelbrot for trajectories generated by the Lévy stable distribution) is well known and is used to construct approximate analytic solutions in the theory of stationary RT (the escape probability method is the best example). In the theory of nonstationary RT, progress based on similar principles has been made recently. This includes approximate self-similar solutions for the Green’s function (i) at an infinite velocity of carriers (no retardation effects) to cover the Biberman–Holstein equation for various spectral line shapes; (ii) for a finite fixed velocity of carriers to cover a wide class of superdiffusion equations dominated by Lévy walks with rests; (iii) verification of the accuracy of above solutions by comparison with direct numerical solutions obtained using distributed computing. The article provides an overview of the above results with an emphasis on the role of distant rare flights in the discovery of nonstationary self-similar solutions.

Superdiffusive Transport Based on Lévy Walks in a Homogeneous Medium: General and Approximate Self-Similar Solutions

The general and approximate self-similar solutions for the Green’s function have been obtained for a wide class of integrodifferential equations for the two- and three-dimensional (in spatial coordinates) nonstationary superdiffusive transport of a perturbation of a homogeneous medium for a finite fixed velocity of carriers. This problem concerns the transport of resonance radiation in astrophysical gases and plasma, migration of animals, and the transfer of energy of electromagnetic waves in the plasma. The case of a model free path distribution function decreasing by a power law with increasing distance has been considered. Numerical calculations have been performed for two particular types of free path distribution functions, including the case with a Lorentzian shape of wings of the spectral line profile for the emission of photons by atoms or ions. The method developed in [A.B. Kukushkin and P.A. Sdvizhenskii, J. Phys. A: Math. Theor. 49, 255002 (2016)] for an infinite velocity of carriers has been used to construct the approximate self-similar solution. The inclusion of a finite velocity corresponds to the generalization of transport based on Levy flights to transport based on “Levy walks with stops.” The self-similar solution for arbitrary one-dimensional superdiffusive transport obtained in [A.B. Kukushkin and A.A. Kulichenko, Phys. Scripta 94, 115009 (2019)] has been applied to the case of the two- and three-dimensional transports. The accuracy of the self-similar solution has been tested by comparing it to the numerically calculated general solution.

Self-similar azimuthons in strongly nonlocal nonlinear media with $${\varvec{{\mathcal {PT}}}}$$ PT -symmetry

A (2+1)-dimensional coupled nonlocal nonlinear Schrodinger equation is studied in \({\mathcal {PT}}\)-symmetric nonlocal nonlinear couplers with gain and loss, and approximate self-similar solution of vector rotating azimuthon is obtained. Various structures of rotating azimuthon and soliton cluster are constructed. Moreover, dynamical behaviors of azimuthon and soliton cluster are discussed in the constant diffraction system and the exponential diffraction decreasing waveguide.

Unsteady Flow of Shear-Thinning Fluids in Porous Media with Pressure-Dependent Properties

Inthispaper,amodelforinjectionofapower-lawshear-thinningfluidinamedium withpressure-dependentpropertiesisdevelopedinageneralizedgeometry(plane,radial,and spherical). Permeability and porosity are taken to be power functions of the pressure incre- ment with respect to the ambient value. The model mimics the injection of non-Newtonian fluids in fractured systems, in which fractures are already present and are enlarged and even- tually extended and opened by the fluid pressure, as typical of fracing technology. Empiric equations are combined with the fundamental mass balance equation. A reduced model is adopted, where the medium permeability resides mainly in the fractures; thefluid and porous medium compressibility coefficients are neglected with respect to the effects induced by pressure variations. At early and intermediate time, the flow interests only the fractures. In these conditions, the problem admits a self-similar solution, derived in closed form for an instantaneous injection (or drop-off) of the fluid, and obtained numerically for a generic monomial law of injection. At late times, the leak-off of the fluid towards the porous matrix is taken into accountvia a sink term in the mass balance equation. In this case, the original set of governing equations needs to be solved numerically; an approximate self-similar solution valid for a special combination of parameters is developed by rescaling time. An example of application in a radial geometry is provided without and with leak-off. The system behaviour isanalysedconsideringthespeedofthepressurefrontandthevariationofthepressurewithin

Approximate self-similar solutions to a nonlinear diffusion equation with time-fractional derivative

In this paper, we consider a fractional nonlinear problem for anomalous diffusion. The diffusion coefficient we use is of power type, and hence the investigated problem generalizes the porous-medium equation. A generalization is made by introducing a fractional time derivative. We look for self-similar solutions for which the fractional setting introduces other than classical space–time scaling. The resulting similarity equations are of nonlinear integro-differential type. We approximate these equations by an expansion of the integral operator and by looking for solutions in a power function form. Our method can be easily adapted to solve various problems in self-similar diffusion. The approximations obtained give very good results in numerical analysis. Their simplicity allows for easy use in applications, as our fitting with experimental data shows. Moreover, our derivation justifies theoretically some previously used empirical models for anomalous diffusion.

Approximate self-similar solutions for the boundary value problem arising in modeling of gas flow in porous media with pressure dependent permeability

The flow of a gas through porous medium is considered in the case of pressure dependent permeability and viscosity. Approximate self-similar solutions of the boundary-value problems are found.

Convergent flow in a two-layer system and mountain building

With the purpose of modeling the process of mountain building, we investigate the evolution of the ridge produced by the convergent motion of a system consisting of two layers of liquids that differ in density and viscosity to simulate the crust and the upper mantle that form a lithospheric plate. We assume that the motion is driven by basal traction. Assuming isostasy, we derive a nonlinear differential equation for the evolution of the thickness of the crust. We solve this equation numerically to obtain the profile of the range. We find an approximate self-similar solution that describes reasonably well the process and predicts simple scaling laws for the height and width of the range as well as the shape of the transversal profile. We compare the theoretical results with the profiles of real mountain belts and find an excellent agreement.

Self-similar optical pulses in competing cubic-quintic nonlinear media with distributed coefficients

We present a systematic analysis of the self-similar propagation of optical pulses within the framework of the generalized cubic-quintic nonlinear Schr\"odinger equation with distributed coefficients. By appropriately choosing the relations between the distributed coefficients, we not only retrieve the exact self-similar solitonic solutions, but also find both the approximate self-similar Gaussian-Hermite solutions and compact solutions. Our analytical and numerical considerations reveal that proper choices of the distributed coefficients could make the unstable solitons stable and could restrict the nonlinear interaction between the neighboring solitons.

Large-scale ECOKATS experiments: Spreading of oxide melt on ceramic and concrete surfaces

Large-scale ECOKATS experiments are performed to study spreading of an oxide melt on ceramic and concrete surfaces. The oxide melt generated by a thermite reaction was composed of 41 wt.% Al 2O 3, 24 wt.% FeO, 19 wt.% CaO and 16 wt.% SiO 2. This melt was selected as the most appropriate simulation of a corium melt because of its wide freezing range of approx. 450 K. Despite a rather low liquidus temperature, the attempt to measure melt viscosity failed. As spreading of high-temperature oxide melts is nearly isothermal during the early phase of motion, i.e., only thin thermal boundary layers will develop, the melt viscosity can be estimated from a two-dimensional spreading experiment, ECOKATS-V1, on a ceramic substrate by approximate self-similar solutions. To further study the influence of the gas release from the substrate caused by thermal erosion of the underlying concrete by a corium melt on spreading, a large amount of the oxide melt was released into a 2.6 m long and 0.29 m wide channel leading into a 3 m × 4 m rectangular surface. Spreading on a concrete substrate is influenced by the gas release from the decomposed concrete, which changes viscosity. A viscosity increase by a factor of 3.6 was estimated from spreading in the concrete channel.

Self-similar boundary blow-up for higher-order quasilinear parabolic equations

We study evolution properties of boundary blow-up for 2mth-order quasilinear parabolic equations in the case where, for homogeneous power nonlinearities, the typical asymptotic behaviour is described by exact or approximate self-similar solutions. Existence and asymptotic stability of such similarity solutions are established by energy estimates and contractivity properties of the rescaled flows.Further asymptotic results are proved for more general equations by using energy estimates related to Saint-Venant's principle. The established estimates of propagation of singularities generated by boundary blow-up regimes are shown to be sharp by comparing with various self-similar patterns.

Modulational instability dynamics in a spatial focusing and temporal defocusing medium.

The modulational instability of a wave in media where the nonlinearity has a focusing-in-space and defocusing-in-time character is investigated. An approximate self-similar solution of the nonlinear Schr\"odinger equation is derived for this nonlinearity and is used as background envelope of the wave. We show that this background wave envelope can initially enhance exponential growth of a transverse spatial modulation. The analysis is extended to the case of a modulation in both time and space, and it is found that such a spatiotemporal modulation can be unstable as well.

Rheological effects of non-Newtonian fluids on natural convection in a porous medium

The question of the rheological effects of non-Newtonian fluids on the natural convection mechanism in a porous medium is addressed. These effects are illustrated for a power-law fluid with a yield stress, in which the yield stress is temperature dependent. The case of a constant-temperature boundary along a heated vertical plate embedded in a porous medium is investigated. For this case the approximate self-similar solutions in closed form of steady and transient states are presented and discussed. The rheological effects on the convection pattern are shown.

On blow-up and degeneracy for the semilinear heat equation with source

SynopsisThe asymptotic behaviour of the solution of the semilinear parabolic equation ut = uxx + (1 + u)ln2(l + u) for t &gt; 0, x ∊[−π, π ], ux(t, ± π) = 0 for t &gt; 0 and u(0, x) = u0(x) ≧ 0 in [−π, π], which blows up at a finite time T0, is investigated. It is proved that for some two-parametric set of initial functions u0 the behaviour of u(t, x) near t = T0 is described by the approximate self-similar solution va(t, x) = exp {(T0 −t)−1 cos2 (x/2)} − 1, satisfying the first order nonlinear Hamilton–Jacobi equation vt, = (vx)2 /(1 + v) + (1 + v) ln2 (1 + v). Some open problems of degeneracy near a finite blow-up time for other semilinear or quasilinear parabolic equations with source ut, = Δu + (1 + u) lnβ (1 + u) (β &gt;1), ut, = Δu + uβ(β &gt; l), ut = Δu + eu; ut = ∇. (lnσ(1 + u)∇u)+ (1 + u)lnβ(1 + u) (σ &gt; 0, β &gt; 1) are discussed.

Self-similar solutions of the nonlinear problem of nonsteady, induced forward scattering of light

Regimes of saturation of nonsteady, induced forward scattering of light in media with different types of nonlinearity (thermal, ionization, or striction) are investigated within the framework of the three-wave model. Approximate analytic solutions of the nonlinear equations are found. An approximate self-similar solution of the Gordon sine equation can be constructed similarly.

ON ASYMPTOTIC “EIGENFUNCTIONS” OF THE CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION

The asymptotic () behavior of solutions of the Cauchy problem is studied for the semilinear parabolic equation where and as . The existence is established of an infinite collection (a continuum) of distinct self-similar solutions of the form , , where the function satisfies an ordinary differential equation. Conditions for the asymptotic stability of these solutions are established. It is shown that for there exist solutions of the problem whose behavior as is described by approximate self-similar solutions (ap.s.-s.s.'s) which in the case coincide with a family of self-similar solutions of the heat equation , while for and the ap.s.-s.s. has the form where .Figures: 2. Bibliography: 78 titles.

ON APPROXIMATE SELF-SIMILAR SOLUTIONS OF A CLASS OF QUASILINEAR HEAT EQUATIONS WITH A SOURCE

Quasilinear parabolic equations of the form are considered; here and are sufficiently smooth given functions (respectively, the coefficient of thermal conductivity and the power of heat sources depending on the temperature ). A family of coefficients and corresponding functions is distinguished for which the properties of the solution of the boundary value problem for the equation in question is described by invariant solutions of a first-order equation of Hamilton-Jacobi type The function is an approximate self-similar solution of the original equation.Tables: 1. Figures: 1. Bibliography: 70 titles.

Turbulent heat?Concentration plume in the case of high viscosity in a nonstratified medium

In this paper, an approximate self-similar solution for the structure of the heat—concentration plume produced by an instantaneous point source of heat in the presence of multicomponent admixture and when the coefficient of turbulence is large is found on the basis of the turbulent transfer equations of vorticity, energy, and matter. Analytical expressions are obtained for the propagation velocity of the buoyancy “core” and the toroidal vortex formed. The influence of the source parameters and of the coefficient of turbulence on the structure and dynamics of thermal lift is investigated. A comparison is made with the laminar regime of motion of similar formations.