Analytical Self-similar Solutions Research Articles

Comment on “Ported from Self-Similar Analytic Solutions of Ginzburg-Landau Equation with Varying Coefficients"

Comment on “Ported from Self-Similar Analytic Solutions of Ginzburg-Landau Equation with Varying Coefficients"

Reply to the “Comment on ‘Ported from Self-Similar Analytic Solutions of Ginzburg–Landau Equation with Varying Coefficients’"

Reply to the “Comment on ‘Ported from Self-Similar Analytic Solutions of Ginzburg–Landau Equation with Varying Coefficients’"

Inertia dominated drop collisions. II. An analytical solution of the Navier–Stokes equations for a spreading viscous film

This study is devoted to a theoretical description of an unsteady laminar viscous flow in a spreading film of a Newtonian fluid. Such flow is generated by normal drop impact onto a dry substrate with high Weber and Reynolds numbers. An analytical self-similar solution for the viscous flow in the spreading drop is obtained which satisfies the full Navier–Stokes equations. The characteristic thickness of a boundary layer developed near the wall uniformly increases as a square root of time. An expression for the thickness of the boundary layer is used for the estimation of the residual film thickness formed by normal drop impact and the maximum spreading diameter. The theoretical predictions agree well with the existing experimental data. A possible explanation of the mechanism of formation of an uprising liquid sheet leading to splash is also proposed.

Exact analytic self-similar solution of a wave attractor field

Stratified and rotating fluids support obliquely propagating internal waves. A symmetry-breaking shape of the fluid domain focuses them on a wave attractor. For a trapezoidal basin, it is here shown how to determine the internal wave field analytically. This requires solving the wave equation on a closed domain–an ill-posed Cauchy problem–whose solution exhibits a remarkable self-similar spatial structure. These results are relevant for mixing and mean flow generation in oceans, atmospheres and stars whose symmetry is generally broken and where internal waves are tidally forced.

Stability and structure of analytical MHD jet formation models with a finite outer disk radius

Context. Finite radius accretion disks are a strong candidate for launching astrophysical jets from their inner parts and disk-winds are considered as the basic component of such magnetically collimated outflows. Numerical simulations are usually employed to answer several open questions regarding the origin, stability and propagation of jets. The inherent uncertainties, however, of the various numerical codes, applied boundary conditions, grid resolution, etc., call for a parallel use of analytical methods as well, whenever they are available, as a tool to interpret and understand the outcome of the simulations. The only available analytical MHD solutions to describe disk-driven jets are those characterized by the symmetry of radial self-similarity. Those exact MHD solutions are used to guide the present numerical study of disk-winds. Aims. Radially self-similar MHD models, in general, have two geometrical shortcomings, a singularity at the jet axis and the nonexistence of an intrinsic radial scale, i.e. the jets formally extend to radial infinity. Hence, numerical simulations are necessary to extend the analytical solutions towards the axis and impose a physical boundary at finite radial distance. Methods. We focus here on studying the effects of imposing an outer radius of the underlying accreting disk (and thus also of the outflow) on the topology, structure and variability of a radially self-similar analytical MHD solution. The initial condition consists of a hybrid of an unchanged and a scaled-down analytical solution, one for the jet and the other for its environment. Results. In all studied cases, we find at the end steady two-component solutions. The boundary between both solutions is always shifted towards the solution with reduced quantities. Especially, the reduced thermal and magnetic pressures change the perpendicular force balance at the “surface” of the flow. In the models where the scaled-down analytical solution is outside the unchanged one, the inside solution converges to a solution with different parameters. In the models where the scaled-down analytical solution is inside the unchanged one, the whole two-component solution changes dramatically to stop the flow from collapsing totally to the symmetry axis. Conclusions. It is thus concluded that truncated exact MHD disk-wind solutions that may describe observed jets associated with finite radius accretion disks, are topologically stable.

Ported from Self-Similar Analytic Solutions of Ginzburg–Landau Equation with Varying Coefficients

Employing the technique of symmetry reduction of analytic method, we solve the Ginzburg–Landau equation with varying nonlinear, dispersion, gain coefficients, and gain dispersion which originates from the limiting effect of transition bandwidth in the realistic doped fibres. The parabolic asymptotic self-similar analytical solutions in gain medium of the normal GVD is found for the first time to our best knowledge. The evolution of pulse amplitude, strict linear phase chirp and effective temporal width are given with self-similarity results in longitudinal nonlinearity distribution and longitudinal gain fibre. These analytical solutions are in good agreement with the numerical simulations. Furthermore, we theoretically prove that pulse evolution has the characteristics of parabolic asymptotic self-similarity in doped ions dipole gain fibres.

Self-similar magnetic arcades

We study self-similar analytical solutions for force-free magnetic field in azimuthal symmetry and arcade topology. We assume the existence of a poloidal magnetic field, anchored on a heavy spherical conductor. The field is changed by shearing the foot points of the arcade due to differential rotation. This rotation gives rise to a toroidal component in the magnetic structure which reacts by expanding the poloidal flux outwards. This could be a slow process at the early stages; however, it becomes very fast at the final stages when the poloidal flux expands to infinity. We address the question of the pressure environment confining the arcade, a pressure profile proportional to r−4 is particularly interesting as it allows finite twist before the field expands to infinity. Finally, some time evolution estimates are made to demonstrate the limitations of this study.

High order dispersion effect of Ginzburg-Landau equation and its self-similar analytical solutions

Using the methods based on the technique of self-similar analyzing, we find the parabolic asymptotic self-similar analytical solutions with third-order dispersion effect of constant coefficient Ginzburg-Landau equation which considers both the influence of high order dispersion and gain dispersion on the evolution of self-similar pulse. The self-similar amplitude function, phase function, strict linear chirp function and effective temporal pulse width are given in the paper. The results show that self-similar pulses still have linear chirp and remarkable third-order dispersion effect. And these theoretical results are consistent with numerical simulations.

A self-similar unsteady flow with conjugated heat transfer

In this paper an exact analytical self-similar solution of the thermo-fluid dynamic field arising in an impulsively accelerated flow over a flat plate is proposed. The plate is considered of infinite thickness and the thermal field is computed both in the fluid and in the solid with the temperature and the heat flux unknown at the solid–fluid interface (conjugated heat transfer). The values of the initial temperatures in the solid and in the fluid are different constants. The solution, obtained in the incompressible case, is extended to compressible flows by the Stewartson–Dorodnitsin transformation. The influence of the non-dimensional parameters governing the phenomenon is discussed with particular emphasis to the simple expressions of the interface temperature and heat flux.

Note on Barenblatt power series solution to Boussinesq equation

To the self-similar analytical solution of the Boussinesq equation of groundwater flow in a semi-infinite porous medium, when the hydraulic head at the boundary behaved like a power of time, Barenblatt obtained a power series solution. However, he listed only the first three coefficients and did not give the recurrent formula among the coefficients. A formal proof of convergence of the series did not appear in his works. In this paper, the recurrent formula for the coefficients is obtained by using the method of power series expansion, and the convergence of the series is proven. The results can be easily understood and used by engineers in the catchment hydrology and baseflow studies as well as to solve agricultural drainage problems.

Analytical self-similar solutions of Ginzburg-Landau equation for the dispersion decreasing fiber

Using the method based on the technique of symmetry reduction, we find the general analytical parabolic asymptotic self-similar solutions for the varying coefficient of Ginzburg-Landau equation that take consideration of the influence of the doped fiber retarding time. The parabolic asymptotic amplitude function, change of strict linear phase chirp and the effective temporal pulse width of self-similar pulse with gain dispersion are given for the dispersion decreasing fibers with longitudinal exponential distribution and hyperbolic distribution. And these theoretical results have been confirmed by numerical simulation in this paper.

Optical vortex solitons in strongly nonlocal nonlinear media

We solved the equations of the Snyder-Mitchell model, which is a model describing the propagation of optical beams in nonlocal nonlinear media with strong nonlocality, in a rotating cylindrical coordinate system, and obtained a self-similar analytical solution of an optical vortex soliton. The result shows that the radial part of the solution is the product of the Whittaker and power function. The beam has a ring-shaped structure rotating around its core.

Transient temperature and thermal stress profiles in semi-transparent particles under high-flux irradiation

Transient temperature and thermal stress profiles in semi-transparent spherical particles heated by concentrated solar radiation are studied by means of a theoretical model. The analysis of radiative–conductive interaction is based on the spectral radiation transfer model in a refracting and absorbing particle. The stress–strain state of the particle is described by the thermoelastic approach. An analytical self-similar solution for the particle temperature profiles and thermal stresses during the quasi-steady period of the particle heating is derived. It is shown that the circumferential tensile stress near the particle surface is a non-monotonic function of the particle radius. The range of physical parameters corresponding to the maximal tensile stress near the particle surface is determined. The model is applied to ZnO and CaCO 3 particles, which are used as reactants in industrially-relevant high-temperature processes. It is shown that tensile stresses in the selected types of particles exposed to concentrated solar radiation cannot lead to their mechanical destruction. At the same time, the considerable temperature difference and thermal stresses in non-isothermal particles can be an interesting issue in a detailed analysis of the thermal decomposition of semi-transparent particles.

A self-similar axisymmetric pulson in rotating stratified fluid

Self-similar analytical nonlinear solutions to the hydrostatic Boussinesq equations are derived which describe unbalanced inertial pulsations of anticyclonic lens-like circular vortices in stably stratified rotating fluid. Any steady axisymmetric solution for a finite-volume anticyclonic vortex in the reduced-gravity approximation is shown to correspond to a set of time-periodic solutions with the amplitude of pulsations being within a range limited by the intensity of the stationary vortex. These solutions represent an extension of previous reduced-gravity analytical pulson solutions of particular forms with spatially uniform divergence of horizontal velocity oscillating in time within the vortex volume. In the self-similar form the pulson solution describes the expansion and contraction of a vortex which maintains the same spatial structure in the Lagrangian coordinates.

Expansion dynamics and equilibrium conditions in a laser ablation plume of lithium: Modeling and experiment

The gas dynamics and atomic kinetics of a laser ablation plume of lithium, expanding adiabatically in vacuum, are included in a numerical model, using isothermal and isentropic self-similar analytical solutions and steady-state collisional radiative equations, respectively. Measurements of plume expansion dynamics using ultrafast imaging for various laser wavelengths (266–1064nm), fluences (2–6.5Jcm−2), and spot sizes (50–1000μm) are performed to provide input parameters for the model and, thereby, study the influence of laser spot size, wavelength, and fluence, respectively, on both the plume expansion dynamics and atomic kinetics. Target recoil pressure, which clearly affects plume dynamics, is included in the model. The effects of laser wavelength and spot size on plume dynamics are discussed in terms of plasma absorption of laser light. A transition from isothermal to isentropic behavior for spot sizes greater than 50μm is clearly evidenced. Equilibrium conditions are found to exist only up to 300ns after the plume creation, while complete local thermodynamic equilibrium is found to be confined to the very early parts of the expansion.

SELF-SIMILAR COLLAPSE OF A SCALAR FIELD IN DILATON GRAVITY AND CRITICAL BEHAVIOR

We present new analytical self-similar solutions describing a collapse of a massless scalar field in scalar–tensor theories. The solutions exhibit a type of critical behavior. The black hole mass for the near critical evolution is analytically obtained for several scalar–tensor theories and the critical exponent is calculated. Within the framework of the analytical models we consider it is found that the black hole mass law for some scalar–tensor theories is of the form M BH =f(p-p cr )(p-p cr )γ which is slightly different from the general relativistic law M BH = const (p-p cr )γ. The explicit form of the function f depends on the particular scalar–tensor theory.

Self-similar analytical solutions for blast waves in inhomogeneous atmospheres with frozen-in-magnetic field

In the present paper, we have made an attempt to study the effects of the presence of a magnetic field on the cavity formation inside a blast wave propagating into a perfectly conducting gas with density varying as some power of distance from the plane or line of explosion. In order to obtain the closed form solutions for the flow variables inside the blast wave and to solve the problem of cavity formation analytically, a relation is taken between the ordinary pressure and the total pressure. It i s found that if the value of the inhomogeneity index $\alpha$ is greater than a critical value $\alpha_c$ (a function of MA - Alfven Mach Number, $\gamma$ - adiabatic index and i - wave geometry index), a contact discontinuity appears at some point inside the blast wave and the cavity formation occurs. The effect of the presence of magnetic field is found to increase the tendency of cavity formation.

On the quasi-current-free electrodynamics of current-carrying hot space plasma

A reduced self-consistent kinetic approach is developed for the study of nonlinear plasma electrodynamics near weakly magnetized regions in a current carrying collisionless plasma. In these regions in addition to quasineutral we have quasi-current-free dynamics. Quasi-current free nonlinear plasma dynamics is realized when the diamagnetic spatial dispersion parameter originating from the diamagnetic current is small compared to the plasma and field dynamic scales. In this case, the eddy currents consisting of diamagnetic and resistive parts are mutually compensated. We illustrate this effect in a liner approach to plasma electrodynamics. A quasi MHD beam-like approach to the accelerated electrons forming resistivity and a kinetic approach to the diamagnetic plasma particles allows us to develop an analytical self-similar solution to the one dimensional magnetic annihilation problem.

Dynamics of relativistic magnetized blast waves

The dynamics of a relativistic blast wave propagating through a magnetized medium is considered, taking into account possible inhomogeneities of density and magnetic field and additional energy supply. Under the simplifying assumption of a spherically symmetric explosion in a medium with toroidal magnetic field self-similar solutions for the internal dynamics of the flow are derived. In the weakly magnetized case, when the bulk of the flow may be described by the unmagnetized solutions, there is a strongly magnetized sheath near the contact discontinuity (when it exists). Self-similar solutions inside the sheath are investigated. In the opposite limit of strongly magnetized upstream plasma new analytical self-similar solutions are found. Possible application to the physics of gamma-ray bursts is discussed.

Transition to Self-Similarity of Diffusion of Tracer in Turbulent Patch

A mixing of a passive tracer inside a turbulent patch generated by a localized short-time perturbation is studied numerically and analytically. Two kinds of an initial distribution of a tracer are considered: two-layer and continuous with constant gradient. For the turbulent patch shaped as a layer, it is shown that, regardless of details of initial distributions of a turbulent energy and dissipation, a tracer concentration evolves to self-similar regimes as time elapses. Analytical self-similar solutions to turbulent diffusion equations are found for three symmetric shapes of a turbulent patch: layer, cylinder, and sphere. Distributions of the concentration inside a patch are found to be substantially nonuniform, with a typical ratio of a concentration gradient in the middle of a patch to its initial value of about 0.5.