On a self-similar solution for the decay of turbulent bursts

1992 ◽  
Vol 3 (4) ◽  
pp. 319-341 ◽  
Author(s):  
S. P. Hastings ◽  
L. A. Peletier
Keyword(s):  
Asymptotic Behaviour ◽  
Physical Parameter ◽  
Dimensional Analysis ◽  
Existence And Uniqueness ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Eigenvalue Parameter ◽  
Self Similar ◽  
Similar Solutions

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

The self-similar solutions to full compressible Navier–Stokes equations without heat conductivity

2019 ◽  
Vol 29 (12) ◽  
pp. 2271-2320
Author(s):  
Xin Liu ◽  
Yuan Yuan
Keyword(s):  
Small Perturbation ◽  
Heat Conductivity ◽  
Stokes Equations ◽  
The Self ◽  
Similar Solution ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar ◽  
Similar Solutions

In this work, we establish a class of globally defined large solutions to the free boundary problem of compressible full Navier–Stokes equations with constant shear viscosity, vanishing bulk viscosity and heat conductivity. We establish such solutions with initial data perturbed around the self-similar solutions when [Formula: see text]. In the case when [Formula: see text], solutions with bounded entropy can be constructed. It should be pointed out that the solutions we obtain in this fashion do not in general keep being a small perturbation of the self-similar solution due to the second law of thermodynamics, i.e. the growth of entropy. If, in addition, in the case when [Formula: see text], we can construct a solution as a global-in-time small perturbation of the self-similar solution and the entropy is uniformly bounded in time. Our result extends the one of Hadžić and Jang [Expanding large global solutions of the equations of compressible fluid mechanics, J. Invent. Math. 214 (2018) 1205.] from the isentropic inviscid case to the non-isentropic viscous case.

Freely draining gravity currents in porous media: Dipole self-similar solutions with and without capillary retention

2007 ◽  
Vol 18 (3) ◽  
pp. 337-362 ◽  
Author(s):  
JOCHONIA S. MATHUNJWA ◽  
ANDREW J. HOGG
Keyword(s):  
Unsaturated Flow ◽  
Multiple Scales ◽  
Initial Conditions ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Theoretical Predictions ◽  
Analytic Expressions ◽  
Self Similar ◽  
Similar Solutions

We analyse the two-dimensional, gravitationally-driven spreading of fluid through a porous medium overlying a horizontal impermeable boundary from which fluid can drain freely at one end. Under the assumption that none of the intruding fluid is retained within the pores in the trail of the current, the motion of the current is described by the dipole self-similar solution of the first kind derived by Barenblatt and Zel'dovich (1957). We show that small perturbations of arbitrary shape imposed on this solution decay in time, indicating that the self-similar solution is linearly stable. We use the connection between the perturbation eigenfunctions and symmetry transformations of the self-similar solution to demonstrate that variables can always be specified in terms of which the rate of decay of the perturbations is maximised. Unsaturated flow can be modelled by assuming that a constant fraction of the fluid is retained within the pores by capillary action in the trail of the current. It has been shown (Barenblatt and Zel'dovich, 1998; Ingerman and Shvets, 1999) that in this case, the motion of the current is described by a self-similar solution of the second kind characterised by an anomalous exponent. We derive leading-order analytic expressions for the anomalous exponent and the self-similar quantities valid for small values of the fraction of fluid retained using direct asymptotic analysis and by using a novel application of the method of multiple scales. The latter offers a number of advantages and permits the evolution of the current to be clearly connected with its initial conditions in a way not possible with conventional approaches. We demonstrate that the theoretical predictions provided by these expressions are in excellent agreement with results from the numerical integration of the governing equations.

On the correctness of a model for phase transitions in binary alloys

1990 ◽  
Vol 1 (4) ◽  
pp. 327-338 ◽  
Author(s):  
I. G. Götz
Keyword(s):  
Phase Transitions ◽  
Heat Conduction ◽  
Stability Criterion ◽  
Binary Alloys ◽  
The Self ◽  
Self Similar Solution ◽  
Self Similar ◽  
The Stability ◽  
Similar Solutions ◽  
And Diffusion

The main result of this paper is a non-uniqueness theorem for the self-similar solutions of a model for phase transitions in binary alloys. The reason for this non-uniqueness is the discontinuity in the coefficients of heat conduction and diffusion at the inter-phase. Also the existence of a self-similar solution and the stability criterion are discussed.

Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In 𝒟 ⊂ ℝ3

2004 ◽  
Vol 47 (1) ◽  
pp. 30-37
Author(s):  
Xinyu He
Keyword(s):  
Boundary Condition ◽  
Stokes Equations ◽  
Local Existence ◽  
Similar Solution ◽  
Navier Stokes ◽  
Smooth Bounded Domain ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar ◽  
Similar Solutions

AbstractLeray's self-similar solution of the Navier-Stokes equations is defined bywhere . Consider the equation for U(y) in a smooth bounded domain D of with non-zero boundary condition:We prove an existence theorem for the Dirichlet problem in Sobolev space W1,2(D). This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at t = t* with t* < +∞, provided the function is permissible.

On the continuity equation for electrons in microwave-afterglow plasmas

1992 ◽  
Vol 47 (2) ◽  
pp. 193-195 ◽  
Author(s):  
H. I. Abdel-Gawad
Keyword(s):  
Continuity Equation ◽  
Spherical Geometry ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Self Similar

We construct a continuity equation for electrons in microwave-afterglow plasmas. The self-similar solution of the equation is obtained for a plasma with plane, cylindrical or spherical geometry.

scholarly journals Phase-space structure of cold dark matter haloes inside splashback: multistream flows and self-similar solution

2020 ◽  
Vol 493 (2) ◽  
pp. 2765-2781 ◽  
Author(s):  
Hiromu Sugiura ◽  
Takahiro Nishimichi ◽  
Yann Rasera ◽  
Atsushi Taruya
Keyword(s):  
Dark Matter ◽  
Phase Space ◽  
Cold Dark Matter ◽  
The Self ◽  
Similar Solution ◽  
Sharp Drop ◽  
Self Similar Solution ◽  
Self Similar ◽  
Fitting Parameters ◽  
Radial Phase

ABSTRACT Using the motion of accreting particles on to haloes in cosmological N-body simulations, we study the radial phase-space structures of cold dark matter (CDM) haloes. In CDM cosmology, formation of virialized haloes generically produces radial caustics, followed by multistream flows of accreted dark matter inside the haloes. In particular, the radius of the outermost caustic called the splashback radius exhibits a sharp drop in the slope of the density profile. Here, we focus on the multistream structure of CDM haloes inside the splashback radius. To analyse this, we use and extend the SPARTA algorithm developed by Diemer. By tracking the particle trajectories accreting on to the haloes, we count their number of apocentre passages, which is then used to reveal the multistream flows of the dark matter particles. The resultant multistream structure in radial phase space is compared with the prediction of the self-similar solution by Fillmore & Goldreich for each halo. We find that $\sim \!30{{\ \rm per\ cent}}$ of the simulated haloes satisfy our criteria to be regarded as being well fitted to the self-similar solution. The fitting parameters in the self-similar solution characterize physical properties of the haloes, including the mass accretion rate and the size of the outermost caustic (i.e. the splashback radius). We discuss in detail the correlation of these fitting parameters and other measures directly extracted from the N-body simulation.

scholarly journals Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents

2014 ◽  
Vol 747 ◽  
pp. 218-246 ◽  
Author(s):  
Zhong Zheng ◽  
Ivan C. Christov ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Numerical Solutions ◽  
Gravity Currents ◽  
Similar Solution ◽  
Phase Plane Analysis ◽  
Plane Analysis ◽  
Self Similar Solution ◽  
Theoretical Predictions ◽  
Self Similar ◽  
Similar Solutions

AbstractWe report experimental, theoretical and numerical results on the effects of horizontal heterogeneities on the propagation of viscous gravity currents. We use two geometries to highlight these effects: (a) a horizontal channel (or crack) whose gap thickness varies as a power-law function of the streamwise coordinate; (b) a heterogeneous porous medium whose permeability and porosity have power-law variations. We demonstrate that two types of self-similar behaviours emerge as a result of horizontal heterogeneity: (a) a first-kind self-similar solution is found using dimensional analysis (scaling) for viscous gravity currents that propagate away from the origin (a point of zero permeability); (b) a second-kind self-similar solution is found using a phase-plane analysis for viscous gravity currents that propagate toward the origin. These theoretical predictions, obtained using the ideas of self-similar intermediate asymptotics, are compared with experimental results and numerical solutions of the governing partial differential equation developed under the lubrication approximation. All three results are found to be in good agreement.

scholarly journals Far Field of a Three-Dimensional Laminar Wall Jet

2021 ◽  
Vol 56 (6) ◽  
pp. 812-823
Author(s):  
I. I. But ◽  
A. M. Gailfullin ◽  
V. V. Zhvick
Keyword(s):  
Numerical Solution ◽  
Stokes Equations ◽  
Plane Surface ◽  
Three Dimensional ◽  
The Self ◽  
Similar Solution ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar

Abstract We consider a steady submerged laminar jet of viscous incompressible fluid flowing out of a tube and propagating along a solid plane surface. The numerical solution of Navier–Stokes equations is obtained in the stationary three-dimensional formulation. The hypothesis that at large distances from the tube exit the flowfield is described by the self-similar solution of the parabolized Navier–Stokes equations is confirmed. The asymptotic expansions of the self-similar solution are obtained for small and large values of the coordinate in the jet cross-section. Using the numerical solution the self-similarity exponent is determined. An explicit dependence of the self-similar solution on the Reynolds number and the conditions in the jet source is determined.

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