On Dynamical Three-Dimensional Fluid-Solid Interaction Problem

2008 ◽  
Vol 15 (4) ◽  
pp. 601-618
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili ◽  
David Gordeziani
Keyword(s):  
Variational Formulation ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dimensional Problem ◽  
Energy Equality ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Fluid Solid Interaction ◽  
Interaction Problem

Abstract The present paper is devoted to the investigation of one dynamical three-dimensional mathematical model of the fluid-solid interaction. The variational formulation of the corresponding initial boundary problem is considered and a problem for abstract second order evolution equation is formulated, which is a generalization of the three-dimensional initial boundary value problem. For the stated abstract problem the existence and uniqueness of solution, and the energy equality are proved, which yield the corresponding result for the dynamical three-dimensional problem of fluid-solid interaction.

An investigation of the Green–Lindsay three-dimensional model

2017 ◽  
Vol 23 (7) ◽  
pp. 987-1003 ◽  
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili ◽  
Wolfgang H Müller
Keyword(s):  
Boundary Value Problem ◽  
Variational Formulation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Relaxation Times ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Three Dimensional Model ◽  
Initial Boundary Value

In this paper we consider the Green and Lindsay nonclassical model for inhomogeneous anisotropic thermoelastic bodies with two relaxation times, which depend on space variables. We obtain a variational formulation for the initial-boundary value problem corresponding to the Green–Lindsay model. On the basis of the variational formulation we define the spaces of vector-valued distributions corresponding to the initial-boundary value problem and by applying suitable a priori estimates we prove the existence and uniqueness of the solution, an energy equality, and the continuous dependence of the solution on given data.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

scholarly journals RELAXATION APPROXIMATION OF THE KERR MODEL FOR THE THREE-DIMENSIONAL INITIAL-BOUNDARY VALUE PROBLEM

2009 ◽  
Vol 06 (03) ◽  
pp. 577-614 ◽  
Author(s):  
GILLES CARBOU ◽  
BERNARD HANOUZET
Keyword(s):  
Boundary Value Problem ◽  
Response Time ◽  
Boundary Value ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Debye Model ◽  
Impedance Boundary Condition ◽  
Impedance Boundary ◽  
Initial Boundary Value

The electromagnetic wave propagation in a nonlinear medium is described by the Kerr model in the case of an instantaneous response of the material, or by the Kerr–Debye model if the material exhibits a finite response time. Both models are quasilinear hyperbolic and are endowed with a dissipative entropy. The initial-boundary value problem with a maximal-dissipative impedance boundary condition is considered here. When the response time is fixed, in both the one-dimensional and two-dimensional transverse electric cases, the global existence of smooth solutions for the Kerr–Debye system is established. When the response time tends to zero, the convergence of the Kerr–Debye model to the Kerr model is established in the general case, i.e. the Kerr model is the zero relaxation limit of the Kerr–Debye model.

scholarly journals On a third order initial boundary value problem in a plane domain

2012 ◽  
Vol 17 (3) ◽  
pp. 312-326
Author(s):  
Neringa Klovienė
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Plane Domain ◽  
Third Order ◽  
Initial Boundary Value

Third order initial boundary value problem is studied in a bounded plane domain σ with C4 smooth boundary ∂σ. The existence and uniqueness of the solution is proved using Galerkin approximations and a priory estimates. The problem under consideration appear as an auxiliary problem by studying a second grade fluid motion in an infinite three-dimensional pipe with noncircular cross-section.

A class of densely invertible parabolic operator equations

1969 ◽  
Vol 1 (3) ◽  
pp. 363-374 ◽  
Author(s):  
R.S. Anderssen
Keyword(s):  
Variational Methods ◽  
Variational Formulation ◽  
Lagrange Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Parabolic Differential Equations ◽  
Operator Equations ◽  
Initial Boundary Value

Before variational methods can be applied to the solution of an initial boundary value problem for a parabolic differential equation, it is first necessary to derive an appropriate variational formulation for the problem. The required solution is then the function which minimises this variational formulation, and can be constructed using variational methods. Formulations for K-p.d. operators have been given by Petryshyn. Here, we show that a wide class of initial boundary value problems for parabolic differential equations can be related to operators which are densely invertible, and hence, K-p.d.; and develop a method which can be used to prove dense invertibility for an even wider class. In this way, the result of Adler on the non-existence of a functional for which the Euler-Lagrange equation is the simple parabolic is circumvented.

scholarly journals Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 181
Author(s):  
Evgenii S. Baranovskii
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Evolutionary Equation ◽  
Long Time ◽  
Special Basis ◽  
External Forces ◽  
Initial Boundary Value

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

Linear potential theory of steady internal supersonic flow with quasi-cylindrical geometry. Part 1. Flow in ducts

1994 ◽  
Vol 281 ◽  
pp. 159-191 ◽  
Author(s):  
Andreas Dillmann
Keyword(s):  
Supersonic Flow ◽  
Potential Theory ◽  
Broad Class ◽  
Three Dimensional ◽  
Double Series ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Potential ◽  
Linear Potential Theory ◽  
Initial Boundary Value

Based on linear potential theory, the general three-dimensional problem of steady supersonic flow inside quasi-cylindrical ducts is formulated as an initial-boundary-value problem for the wave equation, whose general solution arises as an infinite double series of the Fourier–Bessel type. For a broad class of solutions including the general axisymmetric case, it is shown that the presence of a discontinuity in wall slope leads to a periodic singularity pattern associated with non-uniform convergence of the corresponding series solutions, which thus are unsuitable for direct numerical computation. This practical difficulty is overcome by extending a classical analytical method, viz. Kummer's series transformation. A variety of elementary flow fields is presented, whose complex cellular structure can be qualitatively explained by asymptotic laws governing the propagation of small perturbations on characteristic surfaces.

On convergence and error estimates for the horizontal line method applied to Maxwell's initial boundary problem in anisotropic, inhomogeneous media

1980 ◽  
Vol 86 (1-2) ◽  
pp. 53-59 ◽  
Author(s):  
Rainer Picard
Keyword(s):  
Perturbation Theory ◽  
Error Estimates ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Horizontal Line ◽  
Line Method ◽  
Initial Boundary Problem ◽  
Convergence Results ◽  
Rothe Method ◽  
Initial Boundary Value

SynopsisIn the following paper, the horizontal line method (the Rothe method) is applied to Maxwell's initial boundary value problem. By means of results from abstract perturbation theory, convergence results and error estimates are established.

Strain localization in polycarbonates deformed at high strain rates

2011 ◽  
Vol 31 (6-7) ◽  
Author(s):  
Anoop G. Varghese ◽  
Romesh C. Batra
Keyword(s):  
Strain Softening ◽  
Effective Strain ◽  
Three Dimensional ◽  
Strain Curve ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Strain Rates ◽  
The Finite Element Method ◽  
Maximum Shear Strain ◽  
Initial Boundary Value

Abstract We studied three-dimensional transient large coupled thermomechanical deformations of a polycarbonate (PC) plate with a through-the-thickness inhomogeneity at its centroid. The PC exhibits strain softening followed by strain hardening and its elastic moduli are taken to be functions of strain rate and temperature. The inhomogeneity is either a void or a region of initial temperature higher than that of the rest of the plate. The nonlinear initial-boundary-value problem is solved numerically by the finite element method. It is found that deformations localize into narrow regions that we call bands. For a plate deformed in tension, the maximum principal stretch within the band is almost twice that of the maximum shear strain and for a plate deformed in shear the two have approximately the same magnitude. For the PC deformed in uniaxial compression, we call the minimum slope of the effective stress vs. the effective strain curve in the strain softening regime as the softening modulus, E s , and find values of E s and the defect strength needed for the deformations to localize. These values are found to be different for the plate deformed in shear from that deformed in tension and the minimum value of E s for the localization of deformation also depends upon the defect type and the defect strength (e.g., the ratio of the major to the minor axes of the elliptic void).

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