Equivalent Inclusion Method for the Stokes Flow of Drops Moving in a Viscous Fluid

2014 ◽  
Vol 81 (7) ◽  
Author(s):  
H. M. Yin ◽  
P.-H. Lee ◽  
Y. J. Liu
Keyword(s):  
Viscous Fluid ◽  
Stokes Flow ◽  
The Body ◽  
Function Technique ◽  
Elongation Ratio ◽  
Spherical Drop ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix

The equivalent inclusion method is presented to derive the Stokes flow of multiple drops moving in a viscous fluid at a small Reynolds number. The drops are replaced by inclusions with the same viscosity as the fluid, but an eigenstrain rate field that is a fictitious nonmechanical strain rate field is introduced to represent the viscosity mismatch between each drop and the matrix fluid. The velocity and pressure fields can be solved by considering the body force and eigenstrain rate on the inclusions with the Green's function technique. When one spherical drop is considered, the solution recovers the closed-form classic solution. This method is versatile and can be used in the simulation of a many-body system with different drop size, elongation ratio, and viscosity. Numerical examples demonstrate the capability and accuracy of the proposed formulation and illustrate particles' rotation and motion caused by particle interactions.

Two-Ellipsoidal Inhomogeneities by the Equivalent Inclusion Method

1975 ◽  
Vol 42 (4) ◽  
pp. 847-852 ◽  
Author(s):  
Z. A. Moschovidis ◽  
T. Mura
Keyword(s):  
Computer Program ◽  
Strain Field ◽  
Applied Strain ◽  
Stress And Strain ◽  
Isotropic Matrix ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix ◽  
Final Stress

The problem of two ellipsoidal inhomogeneities in an infinitely extended isotropic matrix is treated by the equivalent inclusion method. The matrix is subjected to an applied strain field in the form of a polynomial of degree M in the position coordinates xi. The final stress and strain states are calculated for two isotropic ellipsoidal inhomogeneities both in the interior and the exterior (in the matrix) by using a computer program developed. The method can be extended to more than two inhomogeneities.

Numerical methods for solving the equivalent inclusion equation in semi-analytical models

2021 ◽  
pp. 135065012110001
Author(s):  
Zhiqiang Yan ◽  
Mengqi Zhang ◽  
Shulan Jiang
Keyword(s):  
Matrix Material ◽  
Current Method ◽  
Equation System ◽  
Analytical Models ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix ◽  
Convergence Mechanism ◽  
Computing Accuracy

Equivalent inclusion method is the basis for semi-analytical models in tackling inhomogeneity problems. Equivalent eigenstrains are obtained by solving the consistency equation system of the equivalent inclusion method and then stress disturbances caused by inhomogeneities are determined. The equivalent inclusion method equation system can only be solved numerically, but the current fixed-point iteration method may not be able to achieve deep convergence when the Young's modulus of inhomogeneity is lower than that of the matrix material. The most significant innovation of this paper is to reveal the non-convergence mechanism of the current method. Considering the limitation, the Jacobian-free Newton Krylov algorithm is selected to solve the equivalent inclusion method equation. Results indicate that the new algorithm has significant advantages of computing accuracy and efficiency compared with the classic method.

Elastic Fields in Double Inhomogeneity by the Equivalent Inclusion Method

2000 ◽  
Vol 68 (1) ◽  
pp. 3-10 ◽  
Author(s):  
H. M. Shodja ◽  
A. S. Sarvestani
Keyword(s):  
Elastic Constants ◽  
Present Method ◽  
Present Theory ◽  
Actual Distribution ◽  
Equivalent Inclusion Method ◽  
Inclusion Method ◽  
Ellipsoidal Inhomogeneity ◽  
Equivalent Inclusion ◽  
Comparison Results ◽  
Special Cases

Consider a double-inhomogeneity system whose microstructural configuration is composed of an ellipsoidal inhomogeneity of arbitrary elastic constants, size, and orientation encapsulated in another ellipsoidal inhomogeneity, which in turn is surrounded by an infinite medium. Each of these three constituents in general possesses elastic constants different from one another. The double-inhomogeneity system under consideration is subjected to far-field strain (stress). Using the equivalent inclusion method (EIM), the double inhomogeneity is replaced by an equivalent double-inclusion (EDI) problem with proper polynomial eigenstrains. The double inclusion is subsequently broken down to single-inclusion problems by means of superposition. The present theory is the first to obtain the actual distribution rather than the averages of the field quantities over the double inhomogeneity using Eshelby’s EIM. The present method is precise and is valid for thin as well as thick layers of coatings, and accommodates eccentric heterogeneity of arbitrary size and orientation. To establish the accuracy and robustness of the present method and for the sake of comparison, results on some of the previously reported problems, which are special cases encompassed by the present theory, will be re-examined. The formulations are easily extended to treat multi-inhomogeneity cases, where an inhomogeneity is surrounded by many layers of coatings. Employing an averaging scheme to the present theory, the average consistency conditions reported by Hori and Nemat-Nasser for the evaluation of average strains and stresses are recovered.

scholarly journals Analysis of Line Contact Elastohydrodynamic Lubrication with the Particles under Rough Contact Surface

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Keying Chen ◽  
Liangcai Zeng ◽  
Juan Chen ◽  
Xianzhong Ding
Keyword(s):  
Film Thickness ◽  
Rough Surface ◽  
Thickness Distribution ◽  
Traction Force ◽  
Elastohydrodynamic Lubrication ◽  
Equation System ◽  
Line Contact ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
The Matrix

A numerical solution for line contact elastohydrodynamic lubrication (EHL) occurring on the rough surface of heterogeneous materials with a group of particles is presented in this study. The film thickness disturbance caused by particles and roughness is considered into the solution system, and the film pressure between the contact gap generated by the particles and the surface roughness is obtained through a unified Reynold equation system. The inclusions buried in the matrix are made equivalent to areas with the same material as that of the matrix through Eshelby’s equivalent inclusion method and the roughness is characterized by related functions. The results present the effects of different rough topographies combined with the related parameters of the particles on the EHL performance, and the minimum film thickness distribution under different loads, running speeds, and initial viscosities are also investigated. The results show that the roughness morphology and the particles can affect the behavior of the EHL, the traction force on a square rough surface is smaller, and the soft particles have more advantages for improving the EHL performance.

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