Self-similar Solutions Research Articles

Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation

Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation

Rotational and self-similar solutions to the 2D Euler equations for Chaplygin gas

In this paper, we study the 2D Euler equations for Chaplygin gas, which is a model for dark energy of the universe in some of cosmology theories. By using an ansatz developed by Yuen, we present some rotational and self-similar analytical solutions. We obtain the global existence of the constructed solutions and investigate the qualitative properties of the gas density according to various parameters.

Thermal characteristics of MHD [formula omitted] water nanofluids flow past a stretching/shrinking wedge in the view of Cattaneo-Christov heat flux

Wedge flow has a variety of potential uses, including thermal systems, geothermal power plants, aerodynamics and cooling of nuclear power plants. In this framework, the modified Cattaneo-Christov heat flux is implemented to examine the thermal characteristics in the dissipative flow of a viscous nanoliquid over a shrinking/stretching wedge. The laminar and two-dimensional flow of CoFe2O4/ water over a wedge domain is scrutinized. The self-similar solutions using bvp4c together with the stability analysis are estimated to demonstrate the physical existence of first and second solutions under relevant parameters used in the study. The findings reveals that the reverse flow phenomena, the novel form of decreasing flow exhibits physical existence which is significantly different from those in the stretching flow scenario. Thermal relaxation has a substantial impact on energy flux, which results in oscillatory heat conduction with a narrower temperature. When the volumetric friction of nanoparticles grows from 0.0 to 0.02, the local skin friction coefficients rises. The upper branch elucidation is stable and physically realistic, while the second branch solution is imaginary and having no physical existence. The rise and fall in temperature and velocity is perceived with suction parameter. However, the dropping of velocity is seen with the nanoparticle volume percentage.

A perturbation approach to study the shock wave propagation in a non-ideal magnetogasdynamics under isothermal condition

The main goal of this paper is to obtain a global solution to the problem of imploding strong cylindrical shock waves (SWs) collapsing at the axis of symmetry in a non-ideal gas under the influence of an axial magnetic field using the perturbation series technique. This global solution is valid for the entire flow field up to the instant of collapse. Guderley's asymptotic solution, which is only applicable in the neighborhood of the axis of implosion, is properly reproduced by this global solution. Guderley's local self-similar solution allows for the determination of the first dominant similarity exponent only; however, this approach also enables the identification of additional, less dominant similarity exponents along with the corresponding amplitudes. Additionally, the computed values of the similarity exponents have been compared with the results drawn from Guderley's approach. The profiles of fluid variables and shock trajectory are shown graphically for different values of the non-ideal parameter, adiabatic index, and shock Cowling number. The “Mathematica” software has been used to do all numerical computations.

Unsteady flow of droplet liquid in hydraulic systems of aircrafts and helicopters: models and analytical solutions

The subject of this study is the unsteady flow of liquid in pipelines, which are part of the design of airplanes and helicopters. This name means, first of all, the phenomenon of a sharp increase in pressure in the pipeline, which is known as a hydraulic shock. Although we have already learned to deal with this phenomenon in some parts of the systems, in many structural elements (flexible pipelines), inside which the working pressure reaches several hundred atmospheres, this phenomenon is still quite dangerous. As you know, the best way to deal with an unwanted phenomenon is through theoretical study. To date, there has been a huge amount of work in the direction of hydraulic shock research. This article does not fully cover these studies. It is limited to references to reviews and relevant works. Because the phenomenon of hydraulic shock has a significantly nonlinear character, analytical solutions of systems of equations corresponding to the simplest models were unknown until recently. This work presents, as an overview, already known analytical solutions describing the process of shock wave propagation. Most importantly, new achievements are given, both for the inviscid approximation and for considering internal viscous friction. It is shown that the internal friction within the considered model is negligible almost everywhere, except for the thin shock layer. The asymptotic is proportional to the tangent function and inversely proportional to the square root of the product of the Reynolds number and the dimensionless parameter characterizing the convection effect. Convection of the velocity field significantly affects the distribution of characteristics in hydraulic shock. If the self-similar solutions that were obtained earlier have a power-law character for the velocity distribution in the shock wave, then the simultaneous consideration in the model of convection and friction on the pipeline walls (according to the Weisbach-Darcy model) made it possible to obtain a distribution in the form of an exponential function that decays with increasing distance from the shock wave front. In addition, the work includes an original approach to solving a nonlinear system of differential equations that describes the propagation of a shock wave without considering the friction on the walls. Analytical solutions were obtained in the form of a function of pressure versus the velocity of shock wave propagation. Research methods. This work uses purely theoretical approaches based on the use of well-known fluid flow models, methods of analytical solution of differential equations and their systems, asymptotic methods, derivation of self-model equations, and finding their solutions. Conclusions. Analytical solutions of systems of differential equations were obtained, which describe models of hydraulic shock without considering viscous effects. A comparison of the obtained results with the results of other studies is given.

Self-Similar Growth of Bose Stars.

We analytically solve the problem of Bose star growth in the bath of gravitationally interacting particles. We find that after nucleation of this object, the bath is described by a self-similar solution of the kinetic equation. Together with the conservation laws, this fixes mass evolution of the Bose star. Our theory explains, in particular, the slowdown of the star growth at a certain "core-halo" mass, but also predicts formation of heavier and lighter objects in magistral dark matter models. The developed "adiabatic" approach to self-similarity may be of interest for kinetic theory in general.

The (Self-Similar, Variational) Rolling Stones

Abstract The interplay between variational functionals and the Brunn–Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from Böröczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems à la Firey’s worn stone problem.

Metastability, adiabatic shear bands initiation and plastic strain localization in the AMg6 alloy under dynamic loading

New conception of adiabatic shear bands (ASB) and adiabatic shear failure mechanisms are proposed as special type of critical phenomen, structural-scaling transition, in the ensembles of microshears, governed by the characteristic non-linearity (metastability) of stored (free) energy of solid with mesodefects. Corresponding free energy release kinetics provides experimentally observed ASB induced staging of plastic strain localization and transition to adiabatic shear failure. ASB staging follows to collective properties of microshears ensemble given by the self-similar solutions of evolution equation providing spatial-temporal microshears localization, momentum transfer and damage localization. The criticality of ASB induced plastic strain localization and failure allows us to avoid the discrepancy in the interpretation of ASB effects as thermo-plastic instability in the balance of the stored energy and structural DRX transformation. The microshear ensemble is considered as the second phase and initiation of collective modes provide different staging according to the metastability decomposition and ASB scaling properties following to the self-similar solutions. Self-similar nature of microshears collective modes providing the ASB dynamics is analyzed as the mechanism of steady plastic wave front unversality in shocked materials. The dynamic split Hopkinson pressure bar tests were conducted with AlMg6 alloy combined with “in-situ” imaging of temperature kinetics by CEDIP Silver 450M high-speed infrared camera with conclusion of the secondary role of thermoplastic instability at the ASB staging. The microstructural study performed by an electron microscopy revealed the correlated behavior of the ensemble of defects, which can be classified as a structural transition and precursor of ASB induced strain localization and failure. The modeling reflecting the links of self-similar solutions in microshear ensembles with relaxation properies and damage localization was applied for the comparative analysis of ASB staging and temperature dynamics given be the infrared imaging.

Coalescence of Elastic Blisters Filled with a Viscous Fluid.

Pockets of viscous fluid coalescing beneath an elastic sheet are encountered in a wide range of natural phenomena and engineering processes, spanning across scales. As the pockets merge, a bridge is formed with a height increasing as the sheet relaxes. We study the spatiotemporal dynamics of such an elastohydrodynamic coalescence process by combining experiments, lubrication theory, and numerical simulations. The bridge height exhibits an exponential growth with time, which corresponds to a self-similar solution of the bending-driven thin-film equation. We address this unique self-similarity and the self-similar shape of the bridge, both of which are corroborated in numerical simulations and experiments.

Asymptotic behaviour for a parabolic p-Laplacian equation with an advection force

The rescaling method is presented to allow us establishing the interface functions and the nonnegative local solutions to the evolution of nonlinear degenerate parabolic p-Laplacian process with conservation laws that are posed in one-dimensional space. This equation has a self-similar solution which represents the main feature of this work. The Cauchy problem (CP) for this equation with specific restrictions in the range of parameters and the negative advection coefficient is considered. In this work, there are several regions to discuss the qualitative analysis for the local weak solutions and the asymptotic interfaces in the irregular domains. The dominating of the advection force over the p-Laplacian type diffusion will appear clearly in these regions. Moreover, the solutions of the CP for the degenerate parabolic p-Laplacian advection equations are asymptotically equal to the solutions of advection equations under some restrictions.

Dynamic interaction of gravity currents in a confined porous layer

We study the dynamic interaction of two gravity currents in a confined porous layer, one heavier and one lighter, partly inspired by the practice of geological $\mathrm {CO}_2$ sequestration in oil fields. Two coupled nonlinear advective-diffusive equations are derived to describe the time evolution of the profile shape of both the upper (lighter) and lower (heavier) currents. At early times, the upper and lower currents remain separated and propagate independently. As time progresses, the currents approach each other and start to interact. We have identified eight different regimes of gravity current interaction at late times, impacted by four dimensionless parameters, representing the flow rate partition, ratio of buoyancy over the injection force, and the viscosity contrasts between the two injecting and displaced fluids. By defining appropriate similarity variables at either the early or late times, the governing partial differential equations (PDEs) reduce to different ordinary differential equations (ODEs), corresponding to the classic nonlinear diffusion solutions at early times and eight different self-similar solutions at late times when the currents attach to each other. It is of interest to note that in four of the eight regimes of late-time interaction (regimes 2, 6–8), self-similar solutions can be constructed by combining appropriately the three basic solutions (i.e. shock, rarefaction and travelling wave solutions) identified during single fluid injection in confined porous layers. In the four other regimes (regimes 1, 3–5), implicit solutions in the form of logarithm or error functions are obtained due to the influence of flow confinement and interaction of gravity currents. Potential implications of the model and solutions are also briefly discussed in the context of ${\rm CO}_2$ -water co-flooding for simultaneous ${\rm CO}_2$ sequestration and oil recovery.

Discontinuous initial value and Whitham modulation for the generalized Gerdjikov-Ivanov equation

This paper mainly focuses on the possible wave patterns for the discontinuous initial value problem of the generalized Gerdjikov-Ivanov equation utilizing the Whitham modulation theory. The zero-phase, one-phase and two-phase solutions of the generalized Gerdjikov-Ivanov equation along with the corresponding Whitham equations are constructed by the approach of finite-gap integration. Considering the self-similar solution of the Whitham equations, the elementary wave patterns of rarefaction wave, cnoidal dispersive shock wave, contact dispersive shock wave and combined shocks are found. For the general step-like condition of initial discontinuity, the evolution wave patterns consist of plateau, rarefaction wave and cnoidal dispersive shock wave. The classification of all possible wave patterns in the monotonic region is presented graphically. Furthermore, the results of the Whitham modulation theory are verified by full numerical simulations.

On anisotropic parabolic equation with nonstandard growth order

ABSTRACT In this paper, the existence and the uniqueness of an evolutionary anisotropic p i ( x ) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic p ( x ) − Laplace equation with convection term via L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

Stable Singularity Formation for the Keller–Segel System in Three Dimensions

We consider the parabolic–elliptic Keller–Segel system in dimensions d≧3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\geqq 3$$\\end{document}, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example was found more than two decades ago by Brenner et al. (Nonlinearity 12(4):1071–1098, 1999), and is conjectured to be nonlinearly radially stable. We prove this conjecture for d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document}. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we use a technique we recently established in Glogić and Schörkhuber (Comm Part Differ Equ 45(8):887–912, 2020). To the best of our knowledge, this provides the first result on stable self-similar blowup for the Keller–Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.

Capillary rise in sharp corners: not quite universal

We study the capillary rise of viscous liquids into sharp corners formed by two surfaces whose geometry is described by power laws $h_i(x) = c_i x^n$ , $i = 1,2$ , where $c_2 > c_1$ for $n \geq 1$ . Prior investigations of capillary rise in sharp corners have shown that the meniscus altitude increases with time as $t^{1/3}$ , a result that is universal, i.e. applies to all corner geometries. The universality of the phenomenon of capillary rise in sharp corners is revisited in this work through the analysis of a partial differential equation for the evolution of a liquid column rising into power-law-shaped corners, which is derived using lubrication theory. Despite the lack of geometric similarity of the liquid column cross-section for $n>1$ , there exist a scaling and a similarity transformation that are independent of $c_i$ and $n$ , which gives rise to the universal $t^{1/3}$ power law for capillary rise. However, the prefactor, which corresponds to the tip altitude of the self-similar solution, is a function of $n$ , and it is shown to be bounded and monotonically decreasing as $n\to \infty$ . Accordingly, the profile of the interface radius as a function of altitude is also independent of $c_i$ and exhibits slight variations with $n$ . Theoretical results are compared against experimental measurements of the time evolution of the tip altitude and of profiles of the interface radius as a function of altitude.

Scrutinization using both numerical and analytical techniques for Darcy Forchheimer flow in the gyrotactic microorganism of nanofluid over a rotating disk

The current investigation explores the impact of Darcy Forchheimer’s flow of nanofluid due to a spinning disk with slip conditions and microorganisms. The mechanism of fluid flow over a rotating disk incorporates the theoretical as well as the practical interest of it in applied sciences and engineering. Energy and mass transport equations are produced with thermal radiation and chemical reactions. The modeled couple of nonlinear PDEs are simplified to the system of ODEs using the self-similar solution. Analytical solutions are found for the resulting nonlinear ODEs using the homotopy analysis method (HAM) technique, while numerical solutions are found using ND-solve and bvp4c. The convergent of the series solution is investigated. The graphical evaluation of prospective parameters encompasses the examination of their consequences on several factors, including the velocity profile, temperature profile, volume percentage of nanoparticles, and profile of microorganisms. The occurrence of radiation parameter, thermophoresis, and Brownian motion effects is commonly observed to contribute to the augmentation of heat transfer. We computed the percentage result in numerical as well as analytical study. The utilization of the provided observations can effectively contribute to the enhancement of heat transfer devices, centrifugal compressors, internal combustion engines, and microbial fuel cells.

On the absence of shock waves and vacuum birefringence in Born–Infeld electrodynamics

We study the interaction of two counter–propagating electromagnetic waves in vacuum in the Born–Infeld electrodynamics. First we investigate the Born case for linearly polarized beams, E · B = 0, i.e., G2=0 (crossed field configuration), which is identical for Born–Infeld and Born electrodynamics; subsequently we study the general Born–Infeld case for beams which are nonlinearly polarized, G2≠0. In both cases, we show that the nonlinear field equations decouple using self-similar solutions and investigate the shock wave formation. We show that the only nonlinear solutions are exceptional traveling wave solutions which propagate with constant speed and which do not turn into shocks for our approximation. We obtain two types of exceptional wave solutions, then we numerically analyze which phase velocities correspond to the counter- or co-propagating beams and subsequently we determine the direction of propagation of the exceptional waves.

Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

Stability of selfsimilar solutions to the fragmentation equation with polynomial daughter fragments distribution

We study fragmentation equations with power-law fragmentation rates and polynomial daughter fragments distribution function $ p(s) $. The corresponding selfsimillar solutions are analysed and their exponentially decaying asymptotic behaviour and $ C^{\infty } $ regularity deduced. Stability of selfsimilar solutions (under smooth exponentially decaying perturbations), with sharp exponential decay rates in time are proved, as well as $ C^{\infty } $ regularity of solutions for $ t>0 $. The results are based on explicit expansion in terms of generalized Laguerre polynomials and the analysis of such expansions. For perturbations with power-law decay at infinity stability is also proved. Finally, we consider real analytic $ p(s) $.

Local correlation, compressibility, and crossflow corrections of γ-Reθ transition model for high-speed flows

Based on the original γ-Reθ transition model framework, in this work, an improved local correlation-based transition closure model is developed for high-speed flows. The local correlation between the vorticity Reynolds number and the momentum thickness Reynolds number obtained by compressible boundary-layer self-similar solutions, local compressibility correction including the pressure gradient parameter and momentum thickness Reynolds number, and local crossflow correlation are applied to improve the original γ-Reθ model for hypersonic transition predictions. The function Fonset1 used to control the transition onset and several relevant model parameters are also modified to make the improved model suitable for high-speed flow. The improved transition model is validated through several basic test cases under a wide range of flow conditions, including high-speed flat plates, sharp cones, double ramp, Hypersonic International Flight Research Experimentation, and complex hypersonic configuration X-33 vehicles. The numerical results show that the transition onset locations and the changes of heat transfer rate predicted by the present improved transition model are reasonably consistent with experimental results. The proposed model predicts the high-speed boundary layer transition behaviors induced by streamwise and crossflow instabilities with reasonable precision.