Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid

1990 ◽  
Vol 115 (1-2) ◽  
pp. 39-59 ◽  
Author(s):  
John A. Nohel ◽  
Robert L. Pego ◽  
Athanasios E. Tzavaras
Keyword(s):  
Shear Stress ◽  
Strain Rate ◽  
Newtonian Fluid ◽  
Space Dimension ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Steady States ◽  
Constitutive Law ◽  
Shear Strain Rate ◽  
Initial Boundary Value

SynopsisWe study the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain-rate that results in steady states having, in general, discontinuities in the strain rate. We show that every solution tends to a steady state as t → ∞, and we identify steady states that are stable.

A Comprehensive Energy Formulation for General Nonlinear Material Continua

1997 ◽  
Vol 64 (2) ◽  
pp. 353-360 ◽  
Author(s):  
A. Carini ◽  
O. De Donato
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Small Strain ◽  
Constitutive Law ◽  
Nonlinear Material ◽  
Total Potential ◽  
Initial Boundary Value ◽  
Energy Formulation ◽  
The Continuum

By specialization to the continuum problem of a general formulation of the initial/boundary value problem for every nonpotential operator (Tonti, 1984) and by virtue of a suitable choice of the “integrating operator,” a comprehensive energy formulation is established. Referring to the small strain and displacement case in the presence of any inelastic generally nonlinear constitutive law, provided that it is differentiable, this formulation allows us to derive extensions of well-known principles of elasticity (Hu-Washizu, Hellinger-Reissner, total potential energy, and complementary energy). An illustrative example is given. Peculiar properties of the formulation are the energy characterization of the functional and the use of Green functions of the same problem in the elastic range for every inelastic, generally nonlinear material considered.

Application and Prospect of the Non-Newtonian Fluid in Industrial Field

2013 ◽  
Vol 770 ◽  
pp. 396-401 ◽  
Author(s):  
Yan Peng ◽  
Bing Hai Lv ◽  
Ju Long Yuan ◽  
Hong Bo Ji ◽  
Lei Sun ◽  
...  
Keyword(s):  
Shear Stress ◽  
Strain Rate ◽  
Shear Strain ◽  
Linear Relationship ◽  
Rheological Properties ◽  
Newtonian Fluid ◽  
Shear Strain Rate ◽  
Mechanical Processing ◽  
Different Types ◽  
Individual Protection

Non-Newtonian fluid is a kind of fluid that its shear stress is not always keeps a linear relationship with the shear strain rate. An overview of its applications was made here. Based on the special rheological properties, non-Newtonian fluids are divided into different types and used as additives, mediums and protective materials in many fields. The paper focuses on its applications in fluid rheological properties improving, damping devices, individual protection equipments and mechanical processing. The main achievements in application of the non-Newtonian fluid were introduced and a further prospect was also summarized.

Trusted frequency region of convergence for the enclosure method in thermal imaging

2017 ◽  
Vol 25 (1) ◽  
Author(s):  
Masaru Ikehata ◽  
Kiwoon Kwon
Keyword(s):  
Thermal Imaging ◽  
Space Dimension ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Frequency Region ◽  
Time Interval ◽  
Enclosure Method ◽  
Initial Boundary Value ◽  
Region Of Convergence ◽  
Large Frequency

AbstractThis study deals with the numerical implementation of a formula in the enclosure method as applied to a prototype inverse initial boundary value problem for thermal imaging in a one-space dimension. A precise error estimate of the formula is given and the effect on the discretization of the used integral of the measured data in the formula is studied. The formula requires a large frequency to converge; however, the number of time interval divisions grows exponentially as the frequency increases. Therefore, for a given number of divisions, we fixed the trusted frequency region of convergence with some given error bound. The trusted frequency region is computed theoretically using the theorems provided in this paper and is numerically implemented for various cases.

Internal Variable Formulations of Problems in Elastoplasticity: Constitutive and Algorithmic Aspects

1994 ◽  
Vol 47 (9) ◽  
pp. 429-456 ◽  
Author(s):  
B. D. Reddy ◽  
J. B. Martin
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Internal Variable ◽  
Solution Algorithm ◽  
Strain Increment ◽  
Constitutive Law ◽  
Discrete Problem ◽  
Initial Boundary Value ◽  
Continuous Problem ◽  
Entire Treatment

This work surveys a broad range of related issues in quasistatic elastoplasticity, beginning with a development of an internal variable constitutive theory. The initial-boundary value problem is then considered, and the remainder of the work is concerned with the properties of the time-discrete problem. It is shown how this discrete problem has associated with it a holonomic constitutive law (that is, one relating stress to strain or strain increment), and this holonomic law in turn forms the basis of a solution algorithm. Conditions for the convergence of the algorithm are discussed. The entire treatment applies to the spatially continuous problem.

The Rothe method for the wave equation in several space dimensions

1979 ◽  
Vol 84 (1-2) ◽  
pp. 1-18 ◽  
Author(s):  
Erich Martensen
Keyword(s):  
Wave Equation ◽  
Space Dimension ◽  
Continuous Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
Rothe Method ◽  
Initial Boundary Value ◽  
The Given ◽  
Continuous Problem

SynopsisThe interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.

scholarly journals On the existence of weak solutions for the initial-boundary value problem in the Jeffreys model of motion of a viscoelastic medium

2004 ◽  
Vol 2004 (10) ◽  
pp. 815-829 ◽  
Author(s):  
D. A. Vorotnikov ◽  
V. G. Zvyagin
Keyword(s):  
Boundary Value Problem ◽  
Weak Solutions ◽  
Viscoelastic Medium ◽  
Dimensional Space ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Constitutive Law ◽  
Existence Of Weak Solutions ◽  
Initial Boundary Value

This paper deals with the initial-boundary value problem for the system of motion equations of an incompressible viscoelastic medium with Jeffreys constitutive law in an arbitrary domain of two-dimensional or three-dimensional space. The existence of weak solutions of this problem is obtained.

scholarly journals The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential

2018 ◽  
Vol 16 (1) ◽  
pp. 1466-1477
Author(s):  
Yukun Song ◽  
Shuai Chen ◽  
Fengming Liu
Keyword(s):  
Newtonian Fluid ◽  
Gas Flow ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Of Solutions ◽  
Potential Term ◽  
Viscosity Term ◽  
Isentropic Gas ◽  
Initial Boundary Value

AbstractWe study the initial boundary value problem of a compressible non-Newtonian fluid. The system describes the motion of the compressible viscous isentropic gas flow driven by the non-Newtonian self-gravitational force. The existence of strong solutions are derived in one dimensional bounded intervals by constructing a semi-discrete Galerkin scheme. Moreover, the uniqueness of solutions are also investigated. The main point of the study is that the viscosity term and potential term are fully nonlinear, and the initial vacuum is allowed.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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.

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