scholarly journals Nonlinear stability of two-layer shallow water flows with a free surface

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190594
Author(s):  
Francisco de Melo Viríssimo ◽  
Paul A. Milewski
Keyword(s):  
Free Surface ◽  
Initial Data ◽  
Nonlinear Stability ◽  
Mathematical Proof ◽  
Passive Layer ◽  
Physical Parameters ◽  
Immiscible Fluid ◽  
Dimensional System ◽  
The Cauchy Problem ◽  
The Difference

The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two ‘baroclinic’ parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability.

Nonlinear Stability Of The Difference Schemes For Equations Of Isentropic Gas Dynamics

2008 ◽  
Vol 8 (2) ◽  
pp. 155-170 ◽  
Author(s):  
P. MATUS ◽  
A. KOLODYNSKA
Keyword(s):  
Initial Data ◽  
Numerical Experiment ◽  
Difference Scheme ◽  
Nonlinear Stability ◽  
Gas Dynamics ◽  
Differential Problem ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas ◽  
The Difference ◽  
Theoretical Results

AbstractFor the difference scheme approximating the gas dynamics problem in Riemann invariants a priory estimates with respect to the initial data have been obtained. These estimates are proved without any assumptions about the solution of the differential problem using only limitations for the initial and boundary conditions. Estimates of stability in the general case have been obtained only for the finite instant of time. The uniqueness and convergence of the difference solution are also considered. The results of the numerical experiment confirming theoretical results are given.

scholarly journals Acoustic Characterization of Some Steel Industry Waste Materials

2021 ◽  
Vol 11 (13) ◽  
pp. 5924
Author(s):  
Elisa Levi ◽  
Simona Sgarbi ◽  
Edoardo Alessio Piana
Keyword(s):  
Thermal Characteristic ◽  
Environmental Benefits ◽  
Acoustic Properties ◽  
Physical Parameters ◽  
Industry Waste ◽  
Acoustic Characterization ◽  
Minimization Procedure ◽  
The Difference ◽  
By Products

From a circular economy perspective, the acoustic characterization of steelwork by-products is a topic worth investigating, especially because little or no literature can be found on this subject. The possibility to reuse and add value to a large amount of this kind of waste material can lead to significant economic and environmental benefits. Once properly analyzed and optimized, these by-products can become a valuable alternative to conventional materials for noise control applications. The main acoustic properties of these materials can be investigated by means of a four-microphone impedance tube. Through an inverse technique, it is then possible to derive some non-acoustic properties of interest, useful to physically characterize the structure of the materials. The inverse method adopted in this paper is founded on the Johnson–Champoux–Allard model and uses a standard minimization procedure based on the difference between the sound absorption coefficients obtained experimentally and predicted by the Johnson–Champoux–Allard model. The results obtained are consistent with other literature data for similar materials. The knowledge of the physical parameters retrieved applying this technique (porosity, airflow resistivity, tortuosity, viscous and thermal characteristic length) is fundamental for the acoustic optimization of the porous materials in the case of future applications.

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

Analytical Limit Load Procedure for The Axial Complex Shaped Defect in a Pressurized Pipe

2021 ◽  
Author(s):  
Igor Orynyak ◽  
Julia Bai ◽  
Roman Mazuryk
Keyword(s):  
General Procedure ◽  
Limit Load ◽  
Admissible Solution ◽  
Experimental Comparison ◽  
Physical Parameters ◽  
Equilibrium Equations ◽  
Axial Complex ◽  
The Difference ◽  
Full Scale Tests ◽  
Distribution Of Stress

Abstract The paper is devoted to elaboration of the analytical O-procedure for limit load analysis of complex shaped axial defect in a pressurized pipe. It is based on the classical lower bound theorem of the theory of plasticity, and consists in construction of the statically admissible solution, where distribution of stress satisfies to the equilibrium equations and strength conditions. O-procedure is an optimization process to get the most favorable stress distribution for providing the maximal pressure. It allows to explicitly account for the variable geometrical and physical parameters. Contrary to other approaches, the derived formula for rectangular defect is only a particular case of the general procedure application. Four different methods for the complex defects are compared. They are: first, ASME, A-, rectangular defect formula combined with RSTRENG, R-, procedure, i.e., A-R approach; second, PCORRC, P-, formula with R-procedure, P-R approach; third, Orynyak's, O-, formula with R-procedure, O-R approach; and fourth, our universal O-procedure. The verification begins for rectangular defects where both theoretical and experimental comparison is performed for A-, P-, and O- formulas. The difference between them is small, provided that all three employ the same characteristic of material, here the ultimate strength. Then theoretical comparison for A-R, P-R, O-R approaches and O-procedure is performed for the artificial complex defects, for two symmetrical rectangular defects, for triangular defect. Experimental comparison between four methods is made based on the well-known University of Waterloo full scale tests.

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].

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

Forces Exerted on a Hovercraft by a Moving Pressure Distribution: Robustness of Mathematical Models

2006 ◽  
Vol 50 (01) ◽  
pp. 38-48 ◽  
Author(s):  
Gregory Zilman
Keyword(s):  
Free Surface ◽  
Mathematical Models ◽  
Pressure Distribution ◽  
Constant Pressure ◽  
Wave Resistance ◽  
Excess Pressure ◽  
Physical Parameters ◽  
Rectangular Region ◽  
Heavy Fluid ◽  
Side Force

The wave resistance, side force, and yawing moment acting on a hovercraft moving on the free surface of a heavy fluid is studied. The hovercraft is represented by a distributed excess pressure. Various types of pressure and bounding contours are considered. The sensitivity of the results to numerous uncertainties in the problem's physical parameters is investigated. It is found that constant pressure over a rectangular region moving with an angle of drift results in peculiar side force values. Several robust mathematical models of a moving hovercraft are proposed and analyzed.

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

2021 ◽  
Vol 76 (5) ◽  
pp. 745-819
Author(s):  
S. Yu. Dobrokhotov ◽  
V. E. Nazaikinskii ◽  
A. I. Shafarevich
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Water Waves ◽  
Theory Of Elasticity ◽  
Extensive Literature ◽  
Pseudodifferential Equations ◽  
Tsunami Waves ◽  
The Cauchy Problem ◽  
Linear Differential ◽  
Localized Initial Data

Abstract We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.

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