Class Of Self-similar Solutions Research Articles

Self-similar solutions to a multidimensional singular heat equation with power nonlinearity

The paper deals with the construction of exact solutions to a singular heat equation with power nonlinearity in the case of numerous independent variables with spatial (e.g. axial or central) symmetry. A new class of self-similar solutions is proposed, which reduce to solving the Cauchy problem for a second-order nonlinear ordinary differential equation having singularities at the higher derivative with respect to the required function and/or the independent variable. The ordinary differential equation is studied in two ways: analytically and numerically. The analytical study uses a truncated Taylor series with recurrently computed coefficients, for which explicit formulas are obtained. The numerical solution to the problem uses an iteration algorithm based on the collocation method and radial basis functions. The numerical analysis shows the convergence of the proposed numerical algorithm and its sufficient accuracy enabling one to use the found self-similar solutions to verify approximate solutions to the original heat equation. Besides, the numerical analysis has allowed the radius of convergence of the constructed Taylor series to be evaluated. The form of the constructed self-similar solutions, namely their unboundedness near the symmetry center (axis), enables us to study the behavior and exactness of the numerical solutions to the nonlinear singular parabolic-type equation that have been obtained by the stepwise solution method proposed by us earlier and possess the same property.

Converging strong shock waves in magnetogasdynamics under isothermal condition

The similarity solutions to the problem of cylindrically symmetric strong shock waves in an ideal gas with a constant azimuthal magnetic field are presented. The flow behind the shock wave is assumed to spatially isothermal rather than adiabatic. We use the method of Lie group invariance to determine the possible class of self-similar solutions. Infinitesimal generators of Lie group transformations are determined by using the invariance surface conditions to the system and on the basis of arbitrary constants occurring in the expressions for the generators, four different possible cases of the solutions are reckoned and we observed that only two out of all possibilities hold self-similar solutions, one of which follows the power law and another follows the exponential law. To obtain the similarity exponents numerical calculations have been performed and comparison is made with the existing results in the literature. The flow patterns behind the shock are analyzed graphically.

The shape of an anisotropic plastic layer compressed by parallel planes with anisotropic friction

A nonlinear evolution equation describing the shape of a thin plastic layer compressed by rigid parallel planes is derived. The material of the layer and the contact friction are anisotropic. The similarity solutions and the classes of self-similar solutions are discussed. The large-time asymptotics is analyzed. The instability problem is considered for the spreading process for a strip.

Singular Limiting Induced from Continuum Solutions and the Problem of Dynamic Cavitation

In the works of Pericak-Spector and Spector (Arch Rational Mech Anal. 101:293–317, 1988, Proc. Royal Soc. Edinburgh Sect A 127:837–857, 1997) a class of self-similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.

On the shape of a plastic layer compressed by parallel planes

The problem of the shape of the contour which bounds a thin layer made of an ideally plastic material is investigated. The layer is compressed by parallel planes which approach one another along a common normal (a spreading problem). New formulations of the problems are presented together with the traditional formulation. In all cases, they reduce to a Cauchy problem for non-linear evolutionary equations and the study is carried out in the class of self-similar solutions. Using the method proposed in this paper, new solutions are obtained in both the traditional and new formulations.

Scalings for a Ballistic Aggregation Equation

We consider ballistic aggregation equation for gases in which each particle is iden- ti?ed either by its mass and impulsion or by its sole impulsion. For the constant aggregation rate we prove existence of self-similar solutions as well as convergence to the self-similarity for generic solutions. For some classes of mass and/or impulsion dependent rates we are also able to estimate the large time decay of some moments of generic solutions or to build some new classes of self-similar solutions.

Re-entrant corner flow for PTT fluids in the natural stress basis

We revisit the situation of steady planar flow of Phan–Thien–Tanner (PTT) fluids around re-entrant corners of angles π / α where 1 / 2 ≤ α < 1 . The model is considered in the absence of a solvent viscosity, under which a class of self-similar solutions has been identified with stress singularities of O ( r − 2 ( 1 − α ) ) and stream function behaviour O ( r α ( 1 + α ) ) ( r being the radial distance from the corner). The asymptotic analysis is completed by providing a solution for the downstream boundary layer using natural stress variables. We show that the matching of the outer (core) solution into the downstream boundary layer imposes a restriction on the range of α ∈ ( 2 / 3 , 1 ) for which these self-similar solutions are applicable, i.e. they only hold for corner angles between 180 ° and 270 ° .

Group theoretic method for analyzing interaction of a weak discontinuity wave with a bore in shallow water waves¶

A group theoretic method is used to establish the entire class of self-similar solutions to the problem of a bore as it approaches the shoreline on a sloping beach. Necessary conditions for the existence of similarity solutions for a bore are obtained. The results obtained compare well with the earlier results. A particular solution to the problem has been found, and used to study the interaction of the bore with a weak discontinuity.

Envelope expansion with core collapse - I. Spherical isothermal similarity solutions

We investigate self-similar dynamical processes in an isothermal self-gravitational fluid with spherical symmetry. With reference to the earlier complementary solution results of Larson, Penston, Shu, Hunter and Whitworth & Summers, we further explore the ‘semi-complete solution space’ from an initial instant t→ 0+ to a final stage t→+∞. These similarity solutions can describe and accommodate physical processes of radial inflow, core collapse, oscillations and envelope expansion (namely, outflow or wind) or contraction as well as shocks. In particular, we present new classes of self-similar solutions, referred to as ‘envelope expansion with core collapse’ (EECC) solutions, that feature concurrent interior core collapse and exterior envelope expansion. The interior collapse towards the central core approaches a free-fall state as the radius r→ 0, while the exterior envelope expansion gradually approaches a constant radial flow speed as r→+∞. There exists at least one spherical stagnation surface of zero flow speed that separates the core collapse and the envelope outflow and that travels outward at constant speed, either subsonically or supersonically, in a self-similar manner. Without crossing the sonic critical line the travel speed of non-linear disturbances relative to the radial flow is equal to the sound speed, there exists a continuous band of infinitely many EECC solutions with only one supersonic stagnation point as well as a continuous band of infinitely many similarity solutions for ‘envelope contraction with core collapse’ (ECCC) without stagnation point. Crossing the sonic critical line twice analytically, there are infinitely many discrete EECC solutions with one or more subsonic stagnation points. Such discrete EECC similarity solutions generally allow radial oscillations in the subsonic region between the central core collapse and the outer envelope expansion. In addition, we obtained complementary discrete ECCC similarity solutions that cross the sonic critical line twice with subsonic oscillations. In all these discrete solutions, subsonic spherical stagnation surfaces resulting from similarity oscillations travel outward at constant yet different speeds in a self-similar manner. With specified initial boundary or shock conditions, it is possible to construct an infinite number of such EECC similarity solutions, which are conceptually applicable to various astrophysical problems involving gravitational collapses and outflows. We mention potential applications of EECC similarity solutions to the formation process of protoplanetary nebulae connecting the asymptotic giant branch phase and the planetary nebula phase to H ii clouds surrounding star formation regions, and to a certain evolution phase of galaxy clusters.

A self-similar solution of expanding cylindrical flux ropes for any polytropic index value

We found a new class of solutions for MHD equations that satisfies the condition that cylindrical flux ropes can expand self-similarly even when the polytropic index γ is larger than 1. We achieved this by including the effects of elongation along the symmetry axis as well as radial expansion and assuming that the radial expansion rate is the same as the elongation rate. In previous studies (Osherovich et al., 1993a, 1995), a class of self-similar solutions was described for which cylindrical flux ropes expand only in the medium where γ is less than 1. We compare the models including elongation and excluding elongation observationally by using the WIND key parameters. The difference in the fitting results of the magnetic field between these two models is slight. However the fitting of the velocity is improved when elongation is included and when new geometric parameters that are necessary to represent the elongation are introduced. The values of these parameters are almost the same scale as the radius of flux ropes, which is consistent with the assumption of the isotropic expansion. This new exact solution to a time-dependent two-dimensional MHD problem can also be used to test numerical codes.

Laser beams with axially symmetric polarization

An analysis of the vector wave equation was conducted. A classof self-similar solutions with inhomogeneous polarization corresponding tothe resonator modes is deduced. The modes with inhomogeneous polarizationcan be selected by a diffraction element with polarization selectivityused as one of the resonator mirrors. Diffraction elements with highpolarization selectivity of about 100% are necessary for generating`pure' radially polarized modes. The radially polarized beam provides ahigher energy efficiency (the product of the depth of the cut by thecutting velocity) for laser cutting metals than the circularly polarizedmain mode does under the same conditions. The two limiting cases ofresonance absorption on the spherical plasma target could be realizedusing axially polarized beams: the resonance absorption is maximum in thecase of radial polarization and does not occur in the case of azimuthalpolarization.

Amplification of a magnetic field in systems with a finite electric conductivity

In this study we investigated the kinematics of electromagnetic fields with various configurations of the magnetic and electric fields in systems with spherical and cylindrical symmetry using the exact solutions to the diffusion equation for the electromagnetic field in systems with moving boundaries. In the class of self-similar solutions we determined analytically conditions for the amplification of the electromagnetic field as functions of material velocity and electric conductivity. We investigated also the effects of the amplification of an electromagnetic field during propagation of a spherical shock wave in a dielectric material whereby the shock wave causes transformation of a material into a conductor.

Dynamics of plasma bunches in slowly varying external fields

We use the quasineutrality approximation and the method of moments to analyze a system of kinetic equations that describes the expansion into vacuum of a plasma bunch generally containing several species of charged particles. For a two-component collisionless plasma in slowly varying external potential fields, we obtain a complete description of the dynamics of the matrices of centered second moments of the particle velocity distribution functions. We construct a new class of self-similar solutions of the kinetic equations in which the moments of the distribution functions act as parameters. These solutions are found to be valid for any mass and energy ratios of the constituent particles and generally describe the dynamics of a plasma bunch that is asymmetric in space. For a symmetric bunch we also find an analytical solution corresponding to the presence of eddy electric currents in the plasma, while for an asymmetric bunch we find that interparticle collisions, which give rise to anisotropy in the process of expansion of plasma into vacuum, play an important role. The method developed in the paper is used to study the acceleration and compression of a plasma bunch in time-dependent magnetic fields with a mirror configuration.

Charged jet of an incompressible liquid in an electric field

The problem of the jet flow of an incompressible liquid with free boundaries in an electric field is solved in the approximation of a laminar boundary layer. An exact solution for a round jet is found in the class of self-similar solutions. In the case of a flat slit jet, a solution is constructed in the form of a series in powers of the coordinate transverse to the plane of symmetry. The dependence of the radius (half-width) on the longitudinal coordinate is given.

Self-similarity and nonlinear dynamics of thermally unstable media

We apply the theory of continuous group transformations to equations governing optically thin gaseous media in the isobaric limit in order to find the class of self-similar solutions admitted of such equations. Depending on the character of the diffusivity (linear or nonlinear) three types of self-similar variables are found generating (i) the well-known root-mean-square law of heat transfer, (ii) a power-law enhanced or de-pressed by nonlinearity of diffusivity or radiation losses, and (iii) progressive wave type of solutions, respectively. The constraints on the form of the source (i.e., cooling) function have been determined for each solution. These results have been applied to the study of the nonlinear evolution of a thermal instability of a plasma as an example of astrophysical interest. We find that the nonlinear phase of the thermal instability leads to the formation of conductive/cooling fronts in the medium if a thermally unstable equilibrium point does exist. The specific dependence of diffusivity on temperature was shown to determine the shape of cooling/conductive fronts, while radiative losses influence the temporal behavior of perturbations. The transition between different types of self-similar solutions are discussed along with their stability.

Self-similar solutions for displacement of non-Newtonian fluids through porous media

This paper presents a class of self-similar solutions describing piston-like displacement (single-phase flow is included as a special case) of one slightly compressible non-Newtonian, power-law, dilatant fluid by another through a homogeneous, isotropic porous medium. These solutions can be used to evaluate the validity and accuracy of existing approximate solutions, such as the assumption of constant flow rate at each radial distance that Ikoku and Ramey use to linearize the partial differential equation for the flow of non-Newtonian, power-law fluid through a porous medium.

Auto-waveguide propagation and the collapse of spiral light beams in non-linear media

The equation governing the propagation of spiral beams in a non-linear medium is analysed. It is shown that, taking into account the saturation effect, spiral beams have a tube-like structure with a periodicity along the axis of a non-linear autoguide. A critical power is found which it is necessary to exceed in order to give rise to the indicated autoguided regime of the beam propagation. Furthermore, in this work a strict self-similar solution is obtained of a non-linear Schrodinger-type equation that describes the collapse of spiral beams, and a new class of self-similar solutions is found for the case of spherical symmetry.

Model solutions of the Wood-Kirkwood equations

The Wood-Kirkwood equations model ZND detonation waves in cylindrical geometry where the flow is restricted to the central stream tube. The entire class of self-similar solutions is obtained and in a special limit the governing equations are reduced to a single first-order non-linear equation. Solutions are obtained in the case in which the radial divergence of the flow is constant.

Class of self-similar solutions for a high-temperature axisymmetric jet

Class of self-similar solutions for a high-temperature axisymmetric jet

Convective plumes associated with local sources of a heat releasing impurity in a stably stratified medium

A class of self-similar solutions is found for steady convective plumes (axisymmetric and flat, laminar and turbulent) in a stably stratified medium associated with local sources of a weightless heat releasing impurity. The boundary layer approximation is used. Characteristic of such convection is the “regime with neutral buoyancy,” in which the growth in the buoyancy associated with the heat release is compensated by a reduction due to the ascent to less dense layers of the medium.