Self-similar Solutions Research Articles

Analytical solutions to the pressureless Navier–Stokes equations with density-dependent viscosity coefficients

In this paper, we construct a class of spherically symmetric and self-similar analytical solutions to the pressureless Navier–Stokes equations with density-dependent viscosity coefficients satisfying [Formula: see text], [Formula: see text] for all [Formula: see text]. Under the continuous density free boundary conditions imposed on the free surface, we investigate the large-time behavior of the solutions according to various [Formula: see text] and [Formula: see text]. When the time grows up, such solutions exhibit interesting information: Case (i) If the free surface initially moves inward, then the free surface infinitely approaches to the symmetry center and the fluid density blows up at the symmetry center, or the free surface tends to an equilibrium state; Case (ii) If the free surface initially moves outward, then the free surface infinitely expands outward and the fluid density decays and tends to zero almost everywhere away from the symmetry center, or the free surface tends to an equilibrium state. We also study the large-time behavior of the solutions for [Formula: see text] without any boundary conditions.

On the rigidity of self-shrinkers of the r-mean curvature flow

In this paper, we consider self-similar solutions of the [Formula: see text]-mean curvature flow into the [Formula: see text]-dimensional Euclidean space. By employing some general maximum principles as the main tool, we characterize some self-similar solutions of the [Formula: see text]-mean curvature flow under some suitable geometric constraints.

Universal gravothermal evolution of isolated self-interacting dark matter halos for velocity-dependent cross-sections

ABSTRACT We study the evolution of isolated self-interacting dark matter halos using spherically symmetric gravothermal equations allowing for the scattering cross-section to be velocity dependent. We focus our attention on the large class of models where the core is in the long mean free path regime for a substantial time. We find that the temporal evolution exhibits an approximate universality that allows velocity-dependent models to be mapped onto velocity-independent models in a well-defined way using the scattering time-scale computed when the halo achieves its minimum central density. We show how this time-scale depends on the halo parameters and an average cross-section computed at the central velocity dispersion when the central density is minimum. The predicted collapse time is fully defined by the scattering time-scale, with negligible variation due to the velocity dependence of the cross-section. We derive new self-similar solutions that provide an analytic understanding of the numerical results.

Global transonic solution of hot accretion flow with thermal conduction

ABSTRACT We examine the effect of thermal conduction on the low-angular momentum hot accretion flow (HAF) around non-rotating black holes accreting mass at very low rate. While doing so, we adopt the conductive heat flux in the saturated form, and solve the set of dynamical equations corresponding to a steady, axisymmetric, viscous, advective accretion flow using numerical methods. We study the dynamical and thermodynamical properties of accreting matter in terms of the input parameters, namely energy (ε0), angular momentum (ℓ0), viscosity parameter (α), and saturation constant (Φs) regulating the effect of thermal conduction. We find that Φs plays a pivotal role in deciding the transonic properties of the global accretion solutions. In general, when Φs is increased, the critical point (rc) is receded away from the black hole, and flow variables are altered particularly in the outer part of the disc. To quantify the physically acceptable range of Φs, we compare the global transonic solutions with the self-similar solutions, and observe that the maximum saturation constant ($\Phi ^{\rm max}_{\rm s}$) estimated from the global solutions exceeds the saturated thermal conduction limit (Φsc) derived from the self-similar formalism. Moreover, we calculate the correlation between α and $\Phi ^{\rm max}_{\rm s}$ and find ample disagreement between global solutions and self-similar solutions. Further, using the global flow variables, we compute the Bernoulli parameter (Be) which remains positive all throughout the disc, although flow becomes loosely unbound for higher Φs. Finally, we indicate the relevance of this work in the astrophysical context in explaining the possibility of mass-loss/outflows from the unbound disc.

Quasi self-similarity and its application to the global in time solvability of a superlinear heat equation

This paper concerns the global in time existence of solutions for a semilinear heat equation (P)∂tu=Δu+f(u),x∈RN,t>0,u(x,0)=u0(x)≥0,x∈RN,where N≥1, u0 is a nonnegative initial function and f∈C1([0,∞))∩C2((0,∞)) denotes superlinear nonlinearity of the problem. We consider the global in time existence and nonexistence of solutions for problem (P). The main purpose of this paper is to determine the critical decay rate of initial functions for the global existence of solutions. In particular, we show that it is characterized by quasi self-similar solutions which are solutions W of ΔW+y2⋅∇W+f(W)F(W)+f(W)+|∇W|2f(W)F(W)[q−f′(W)F(W)]=0in y∈RN, where F(s)≔∫s∞1f(η)dη and q≥1.

A self-similar solution for the two-dimensional Broadwell system via the Bateman equation

A self-similar solution of the Broadwell system is found. Here the solution is sought using a reduction that transforms the given system into a system of differential equations. Further, the solution is constructed using the Painlev\'e series. Here the system already passes the Painlev\'e test and it is possible to find the solution if the equations in resonance satisfy the solution of the two-dimensional Bateman equation. Exact solution of the Bateman equation is established, allowing to find new explicit solution for the original system. In the process of calculations, we use the Wolfram Mathematica program. The proof of these results is carried out at a rigorous mathematical level.

Separation Phenomenon in a Forced Convection Non-Similar Externally Retarded Nanofluid Flow

The study of heat transfer phenomena in non-similar flow of nanofluid is the subject of this investigation. The external retarded flow past a flat plate is considered which does not allow the self-similarity solution. To enhance the heat transfer rate nanofluid has been considered instead of the pure fluid. The nanoparticles of Aluminum Oxide are disseminated in the Water, being base fluid, to form the nanofluid. The consideration of nanofluid results in a substantial heat transfer augmentation along with the skin friction coefficient and both are observed to be further enhanced with higher concentration of nanoparticles. Almost 48% and 36% of gain in heat transfer rate and skin friction coefficient, respectively, have been observed in the 20% nanoparticle concentration at the downstream location where separation is occurring. However, a 67% gain in skin friction coefficient is observed for other downstream locations. The effect of nanoparticle concentration on the separation phenomena has also been investigated carefully and it is found that the concentration of nanoparticle does not delay the flow separation in this case. The effect of nanoparticle concentration on velocity and temperature profiles and their gradients is depicted and discussed through several graphs.

Solutions of the converging and diverging shock problem in a medium with varying density

We consider the solutions of the Guderley problem, consisting of a converging and diverging hydrodynamic shock wave in an ideal gas with a power law initial density profile. The self-similar solutions and specifically the reflected shock coefficient, which determines the path of the reflected shock, are studied in detail for cylindrical and spherical symmetries and for a wide range of values of the adiabatic index and the spatial density exponent. Finally, we perform a comprehensive comparison between the analytic solutions and Lagrangian hydrodynamic simulations by setting proper initial and boundary conditions. A very good agreement between the analytical solutions and the numerical simulations is obtained. This demonstrates the usefulness of the analytic solutions as a code verification test problem.

Universal Multistream Radial Structures of Cold Dark Matter Halos

Virialized halos of cold dark matter generically exhibit multistream structures of accreted dark matter within an outermost radial caustic known as the splashback radius. By tracking the particle trajectories that accrete onto the halos in cosmological N-body simulations, we count their number of apocenter passages (p) and use them to characterize the multistream structure of dark matter particles. We find that the radial density profile for each stream, classified by the number of apocenter passages, exhibits universal features and can be described by a double power-law function comprising shallow inner slopes and steep outer slopes of indices of −1 and −8, respectively. Surprisingly, these properties hold over a wide range of halo masses. The double power-law feature is persistent when dividing the sample by concentration or accretion rate. The dependence of the characteristic scale and amplitude of the profile on p cannot be replicated by known self-similar solutions, requiring consideration of complexities such as the distribution of angular momentum or mergers.

Gravity currents of viscous fluids in a vertically widening and converging fracture

We are investigating flows in the viscous-buoyancy balance regime in a converging channel with an upward increase in the width, with the gap of the channel varying according to a xkzr power function, being x and z the horizontal and vertical coordinate, respectively, and with 0<k<1 and 0<r<1 in order to be consistent with the model. The fluid rheology is described according to the Ostwald–de Waele model, with a power-law relationship between shear stress and shear rate and with application for shear-thinning, shear-thickening and, as a special case, Newtonian fluids. While for the case of flow in the direction of widening of the horizontal channel, a self-similar solution of the first kind can be expected, for flow toward the origin, with channel narrowing horizontally, the solution is self-similar of the second kind, with the space and a reduced time coupled in a self-similar independent variable but with an unknown parameter of the transformation group that makes the differential problem invariant. The solution is found in the phase plane by numerical integration of the paths connecting pairs of singular points, separately for the pre-closure phase, in which the current front advances invading the channel, and for the post-closure or leveling phase, in which the fluid has reached the origin and the front no longer propagates, while the level progressively increases canceling the pressure gradient. Integration is performed with a trial and error procedure by modifying the unknown parameter, generally named eigenvalue and specifically critical eigenvalue when the path has been successfully integrated. The overall effect of an increasing permeability upward is that of an increase in the front speed, with the current profile also becoming locally steeper. The effect of an increase in the fluid-behavior index is mixed, as it reduces the speed of the front but still increases the steepness of the local current profile. In any case, the model implies that the eigenvalue tends to infinity for k→1 even in the presence of an increase in the vertical permeability (r > 0).

Localized shear as a quasi-plastic mechanism of momentum transfer in liquids

Two types of dispersion relations (DRs) in condensed media and associated momentum transfer mechanisms are discussed: gapless DRs corresponding to acoustic modes and DRs with energy (or frequency) gap. Studying the viscoelastic effects in condensed media based on the generalization of the Maxwell-Frenkel approach, the emphasis was placed on the gap states in the momentum transfer mechanisms (GMS), when the k-gap was accompanied by qualitative changes in the DRs type. In this paper, the gap states in the mechanisms of momentum transfer in liquids are associated with the effects of shear elasticity and the formation of collective modes of quasi-plastic shears under conditions of the established type of critical phenomena in the ensembles of shear defects — structural-scaling transitions. It has been found that the formation of collective shear modes, having the character of self-similar solutions of the autosoliton type, accounts for the qualitative change in the dispersion properties corresponding to the quasi-plastic momentum transfer mechanisms operating in the characteristic range of loading times. The results of comparison with the data of original experiments confirm the initiation of quasi-plastic mechanisms of momentum transfer in liquids during the formation of collective shear modes and corresponding change in dissipative properties.

Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon

In the description of transport phenomena, diffusion represents an important aspect. In certain cases, the diffusion may appear together with convection. In this paper, we study the diffusion equation with the self-similar Ansatz. With an appropriate change of variables, we have found an original new type of solution of the diffusion equation for infinite horizon. We derive novel even solutions of diffusion equation for the boundary conditions presented. For completeness, the odd solutions are also mentioned as well, as part of the previous works. We have found a countable set of even and odd solutions, of which linear combinations also fulfill the diffusion equation. Finally, the diffusion equation with a constant source term is discussed, which also has even and odd solutions.

Stability of self-similar solutions to geometric flows

We show that self-similar solutions for the mean curvature flow, surface diffusion, and Willmore flow of entire graphs are stable upon perturbations of initial data with small Lipschitz norm. Roughly speaking, the perturbed solutions are asymptotically self-similar as time tends to infinity. Our results are built upon the global analytic solutions constructed by Koch and Lamm in 2012, the compactness arguments adapted by Asai and Giga in 2014, and the spatial equi-decay properties on certain weighted function spaces. The proof for all of the above flows are achieved in a unified framework by utilizing the estimates of the linearized operator.

Axisymmetric self-similar finite-time singularity solution of the Euler equations

Self-similar finite-time singularity solutions of the axisymmetric Euler equations in an infinite system with a swirl are provided. Using the Elgindi approximation of the Biot–Savart kernel for the velocity in terms of vorticity, we show that an axisymmetric incompressible and inviscid flow presents a self-similar finite-time singularity of second specie, with a critical exponent ν. Contrary to the recent findings by Hou and collaborators, the current singularity solution occurs at the origin of the coordinate system, not at the system’s boundaries or on an annular rim at a finite distance. Finally, assisted by a numerical calculation, we sketch an approximate solution and find the respective values of ν. These solutions may be a starting point for rigorous mathematical proofs.

Self-similar solution for Hardy operator

We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup.

Anisotropic fast diffusion equations

We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The combination of self-similarity and anisotropy is essential in our analysis and creates serious mathematical difficulties that are addressed by means of novel methods.

A framework for crossover of scaling law as a self-similar solution: dynamical impact of viscoelastic board.

In this paper, a new framework for crossover of scaling law is proposed: a crossover of scaling law can be described by a self-similar solution. A crossover emerges as a result of the interference from similarity parameters of the higher class of the self-similarity. This framework was verified for the dynamical impact of a solid sphere onto a viscoelastic board. All the physical factors including the size of spheres and the impact of velocity are successfully summarized using primal dimensionless numbers which construct a self-similar solution of the second kind, which represents the balance between dynamical elements involved in the problem. The self-similar solution gives two different scaling laws by the perturbation method describing the crossover. These theoretical predictions are compared with experimental results to show good agreement. It was suggested that a hierarchical structure of similarity plays a fundamental role in crossover, which offers a fundamental insight into self-similarity in general.

Self-Similar Solution of the Problem of Superheated Water Vapor Injection into a Porous Reservoir

Self-Similar Solution of the Problem of Superheated Water Vapor Injection into a Porous Reservoir

On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow

Abstract The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m \lambda +\mu {e}^{-\phi /m} for some constants μ \mu and λ \lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

Self-similar Solution of Population Balance Equation for Aggregation with Constant Kernel

Self-similar solution of population balance equation includes the number density function which remains invariant or contains a part that is invariant. This paper describes self-similar solution of aggregation population balance model with constant kernel. Using this constant kernel, moment of the population balance system achieves the form μ_i (t)∝t^(i-1) and the aggregation population balance equation with constant kernel reduces to a first order linear ordinary integro-differential equation whose solution is exponential function with negative power.