Numerical experiments on two-dimensional foam

1992 ◽  
Vol 241 ◽  
pp. 233-260 ◽  
Author(s):  
Thomas Herdtle ◽  
Hassan Aref
Keyword(s):  
Initial Data ◽  
Analytical Solutions ◽  
Numerical Experiments ◽  
Laboratory Experiments ◽  
Evolution Equations ◽  
Two Dimensional ◽  
Von Neumann ◽  
Cpu Time ◽  
Statistical Evolution ◽  
Physical Principles

The statistical evolution of a two-dimensional polygonal, or ‘dry’, foam during diffusion of gas between bubbles lends itself to a very simple mathematical description by combining physical principles discovered by Young. Laplace, Plateau, and von Neumann over a period of a century and a half. Following a brief review of this ‘canonical’ theory, we report results of the largest numerical simulations of this system undertaken to date. In particular, we discuss the existence and properties of a scaling regime, conjectured on the basis of laboratory experiments on larger systems than ours by Glazier and coworkers, and corroborated in computations on smaller systems by Weaire and collaborators. While we find qualitative agreement with these earlier investigations, our results differ on important, quantitative details, and we find that the evolution of the foam, and the emergence of scaling, is very sensitive to correlations in the initial data. The largest computations we have performed follow the relaxation of a system with 1024 bubbles to one with O(10), and took about 30 hours of CPU time on a Cray-YMP supercomputer. The code used has been thoroughly tested, both by comparison with a set of essentially analytic results on the rheology of a monodisperse-hexagonal foam due to Kraynik & Hansen, and by verification of certain analytical solutions to the evolution equations that we found for a family of ‘fractal foams’.

scholarly journals An Efficient Explicit Decoupled Group Method for Solving Two–Dimensional Fractional Burgers’ Equation and Its Convergence Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
N. Abdi ◽  
H. Aminikhah ◽  
A. H. Refahi Sheikhani ◽  
J. Alavi ◽  
M. Taghipour
Keyword(s):  
Numerical Experiments ◽  
Taylor Expansion ◽  
Truncation Error ◽  
Burgers Equation ◽  
Stability And Convergence ◽  
Group Method ◽  
Two Dimensional ◽  
Cpu Time ◽  
The Stability ◽  
Crank Nicolson

In this paper, the Crank–Nicolson (CN) and rotated four-point fractional explicit decoupled group (EDG) methods are introduced to solve the two-dimensional time–fractional Burgers’ equation. The EDG method is derived by the Taylor expansion and 45° rotation of the Crank–Nicolson method around the x and y axes. The local truncation error of CN and EDG is presented. Also, the stability and convergence of the proposed methods are proved. Some numerical experiments are performed to show the efficiency of the presented methods in terms of accuracy and CPU time.

scholarly journals On the Convexity Preservation of a Quasi C3 Nonlinear Interpolatory Reconstruction Operator on σ Quasi-Uniform Grids

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 310 ◽  
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Initial Data ◽  
Numerical Experiments ◽  
Approximation Order ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Convexity Properties ◽  
Second Degree

This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for σ quasi-uniform grids with σ≤4, we get a quasi C3 reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.

scholarly journals Islands in linear dilaton black holes

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Georgios K. Karananas ◽  
Alex Kehagias ◽  
John Taskas
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Entanglement Entropy ◽  
Von Neumann Entropy ◽  
Two Dimensional ◽  
Extremal Case ◽  
Von Neumann ◽  
Dimensional Black Hole ◽  
Linear Dilaton ◽  
Linear Dilaton Black Hole

Abstract We derive a novel four-dimensional black hole with planar horizon that asymptotes to the linear dilaton background. The usual growth of its entanglement entropy before Page’s time is established. After that, emergent islands modify to a large extent the entropy, which becomes finite and is saturated by its Bekenstein-Hawking value in accordance with the finiteness of the von Neumann entropy of eternal black holes. We demonstrate that viewed from the string frame, our solution is the two-dimensional Witten black hole with two additional free bosons. We generalize our findings by considering a general class of linear dilaton black hole solutions at a generic point along the σ-model renormalization group (RG) equations. For those, we observe that the entanglement entropy is “running” i.e. it is changing along the RG flow with respect to the two-dimensional worldsheet length scale. At any fixed moment before Page’s time the aforementioned entropy increases towards the infrared (IR) domain, whereas the presence of islands leads the running entropy to decrease towards the IR at later times. Finally, we present a four-dimensional charged black hole that asymptotes to the linear dilaton background as well. We compute the associated entanglement entropy for the extremal case and we find that an island is needed in order for it to follow the Page curve.

A Difference Method for Plane Problems in Magnetoelastodynamics

1972 ◽  
Vol 39 (3) ◽  
pp. 689-695 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Difference Method ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Symmetric Hyperbolic System ◽  
Von Neumann ◽  
First Order ◽  
Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Computational models of filtration gas combustion

2017 ◽  
Vol 32 (2) ◽  
Author(s):  
Yuri M. Laevsky ◽  
Tatyana A. Nosova
Keyword(s):  
Heat Flow ◽  
Numerical Experiments ◽  
Computational Models ◽  
Two Dimensional ◽  
Multidimensional Model ◽  
Gas Combustion ◽  
Total Enthalpy ◽  
Diffusive Flow ◽  
Temperature Heat ◽  
The One

AbstractA multidimensional model of filtration gas combustion is presented. The model is based on the system of conservation laws of ‘temperature – heat flow’, ‘mass–diffusive flow’ types with introducing the concept of total enthalpy flow. Results of numerical experiments are presented for the one- and two-dimensional problems for different conditions and parameters.

Dynamics research of Fangzhu’s nanoscale surface

2021 ◽  
pp. 146134842110527
Author(s):  
Pinxia Wu ◽  
Weiwei Ling ◽  
Xiumei Li ◽  
Xichun He ◽  
Liangjin Xie
Keyword(s):  
Energy Balance ◽  
Analytical Solutions ◽  
Numerical Experiments ◽  
Fractal Model ◽  
Balance Method ◽  
Transform Method ◽  
Energy Balance Method ◽  
Collection Rate ◽  
Fractal Derivative ◽  
Approximate Analytical Solutions

In this paper, we mainly focus on a fractal model of Fangzhu’s nanoscale surface for water collection which is established through He’s fractal derivative. Based on the fractal two-scale transform method, the approximate analytical solutions are obtained by the energy balance method and He’s frequency–amplitude formulation method with average residuals. Some specific numerical experiments of the model show that these two methods are simple and effective and can be adopted to other nonlinear fractal oscillators. In addition, these properties of the obtained solution reveal how to enhance the collection rate of Fangzhu by adjusting the smoothness of its surfaces.

Two-dimensional buoyant plumes in a uniform co-flow

2021 ◽  
Vol 932 ◽  
Author(s):  
Gary R. Hunt ◽  
Jamie P. Webb
Keyword(s):  
Theoretical Investigation ◽  
Analytical Solutions ◽  
Richardson Number ◽  
Relative Magnitude ◽  
Dynamical Behaviour ◽  
Finite Width ◽  
Two Dimensional ◽  
Buoyant Plumes ◽  
The Subject ◽  
Flow Case

The behaviour of turbulent, buoyant, planar plumes is fundamentally coupled to the environment within which they develop. The effect of a background stratification directly influences a plumes buoyancy and has been the subject of numerous studies. Conversely, the effect of an ambient co-flow, which directly influences the vertical momentum of a plume, has not previously been the subject of theoretical investigation. The governing conservation equations for the case of a uniform co-flow are derived and the local dynamical behaviour of the plume is shown to be characterised by the scaled source Richardson number and the relative magnitude of the co-flow and plume source velocities. For forced, pure and lazy plume release conditions the co-flow acts to narrow the plume and reduce both the dilution and the asymptotic Richardson number relative to the classic zero co-flow case. Analytical solutions are developed for pure plumes from line sources, and for highly forced and highly lazy releases from sources of finite width in a weak co-flow. Contrary to releases in quiescent surroundings, our solutions show that all classes of release can exhibit plume contraction and the associated necking. For entraining plumes, a dynamical invariance spatially only occurs for pure and forced releases and we derive the co-flow strengths that lead to this invariance.

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

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