A Comparison of Homogenization and Direct Techniques for the Treatment of Roughness in Non-Newtonian-Modified Reynolds Equation: Numerical Results

Volume 1 ◽  
2004 ◽  
Author(s):  
Malal Kane ◽  
Benyebka Bousaid
Keyword(s):  
Exact Solution ◽  
Reynolds Equation ◽  
Direct Methods ◽  
Homogenization Method ◽  
Numerical Results ◽  
Homogenization Theory ◽  
Lubricated Contact ◽  
A New Technique ◽  
Direct Inspection ◽  
Theoretical Developments

This article is concerned with the simulation of a lubricated contact in severe running conditions considering the fluid as Non-Newtonian of Maxwell type. To overcome some limitations that become apparent at very small film thickness, notably when the roughness is two-dimensional, Jai in 1995 introduced a new technique based on a rigorous homogenization theory in the case of compressible fluid flow. This procedure was mathematically developed by Jai [1] and Buscaglia and Jai [2], and applied to tribological problems by Jai and Bou-Sai¨d [3]. The theoretical developments have been presented and discussed elsewhere [6] of this work and we present here some numerical results obtained from the homogenized technique. The obtained results were discussed and compared with the direct methods of calculation, and seem to us valid for a definitive validation of this method said about homogenization. These results have been compared to the exact solution obtained from a numerical simulation. By direct inspection it is clear that the symmetry predicted by the homogenization method is not present in the exact solution which qualitatively agrees with the homogenized solution.

A Study of Roughness and Non-Newtonian Effects in Lubricated Contacts

2005 ◽  
Vol 127 (3) ◽  
pp. 575-581 ◽  
Author(s):  
Malal Kane ◽  
Benyebka Bou-Said
Keyword(s):  
Film Thickness ◽  
Numerical Solutions ◽  
Operating Conditions ◽  
Homogenization Theory ◽  
Roughness Effects ◽  
Small Surface ◽  
Lubricated Contacts ◽  
Lubricated Contact ◽  
A New Technique ◽  
Numer Anal

This article is concerned with the simulation of a lubricated contact considering the fluid to be non-Newtonian of the Maxwell type. Severe operating conditions lead to very small surface-to-surface distances. In this situation it is necessary to take roughness effects into account. A popular method consists in averaging the film thickness following Patir and Cheng (ASME J. Lubr. Technol., 100, pp. 12–17, 1978), or more recently Wang et al. (Tribol. Trans., 45(1), pp. 1–10, 2000), with good reported results compared with experimental data. To overcome certain limitations that become apparent at very small film thickness, notably when the roughness is two-dimensional, in 1995 Jai (Math. Modell. Numer. Anal., 29(2), pp. 199–233, 1995) introduced a new technique based on a rigorous homogenization theory in the case of compressible fluid flow. This procedure was further mathematically developed by Buscaglia and Jai (Math. Probl. Eng., 7(4), pp. 355–377, 2001) and applied to tribological problems by Jai and Bou-Saı¨d (ASME J. Tribol., 124, pp. 327–355, 2002). In this paper, we propose a similar homogenized approach in the case of non-Newtonian fluids to avoid numerical problems which are often encountered in other approaches. Results in the homogenized roughness case are obtained and compared with direct numerical solutions.

Buckling of Rectangular Plates With Built-In Edges

1942 ◽  
Vol 9 (4) ◽  
pp. A171-A174
Author(s):  
Samuel Levy
Keyword(s):  
Exact Solution ◽  
Rectangular Plate ◽  
Infinite Series ◽  
Numerical Results ◽  
Buckling Load ◽  
Rectangular Plates

Abstract This paper presents an exact solution in terms of infinite series of the problem of buckling by compressive forces in one direction of a rectangular plate with built-in edges (zero slope, zero displacement in the direction normal to the plane of the plate). The buckling load is calculated for 14 ratios of length to width, ranging in steps of 0.25 from 0.75 to 4. On the basis of convergence, as the number of terms used in the infinite series is increased, it is estimated that the possible error in the numerical results presented is of the order of 0.1 per cent. A comparison is given with the work of other authors.

scholarly journals A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Xiong Liu ◽  
Wenming He
Keyword(s):  
Elliptic Problem ◽  
Homogenization Method ◽  
Second Order ◽  
Homogenization Theory ◽  
High Demand ◽  
Periodic Coefficients ◽  
Order Elliptic Problem ◽  
Multiscale Homogenization

In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form ∂ / ∂ x i a i j x / ε , x ∂ u ε x / ∂ x j = f x . Noticing the fact that the classic homogenization theory presented by Oleinik has a high demand for the smoothness of the homogenization solution u 0 , we present a new estimate for the homogenization method under the weaker smoothness that homogenization solution u 0 satisfies than the classical homogenization theory needs.

scholarly journals By using a new iterative method to the generalized system Zakharov-Kuznetsov and estimate the best parameters via applied the pso algorithm

2020 ◽  
Vol 19 (2) ◽  
pp. 1055 ◽  
Author(s):  
Abeer Aldabagh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Iterative Method ◽  
Numerical Method ◽  
Pso Algorithm ◽  
Accurate Solution ◽  
Generalized System ◽  
A New Technique ◽  
Best Parameters ◽  
New Iterative Method

In this paper, a new iterative method was applied to the Zakharov-Kuznetsov system to obtain the approximate solution and the results were close to the exact solution, A new technique has been proposed to reach the lowest possible error, and the closest accurate solution to the numerical method is to link the numerical method with the pso algorithm which is denoted by the symbol (NIM-PSO). The results of the proposed Technique showed that they are highly efficient and very close to the exact solution, and they are also of excellent effectiveness for treating partial differential equation systems.

scholarly journals Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1961
Author(s):  
Kanognudge Wuttanachamsri
Keyword(s):  
Mathematical Model ◽  
Exact Solution ◽  
Free Boundary ◽  
Horizontal Plane ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Brinkman Equation ◽  
Average Height ◽  
Free Interface ◽  
The Mathematical Model

Cilia on the surface of ciliated cells in the respiratory system are organelles that beat forward and backward to generate metachronal waves to propel mucus out of lungs. The layer that contains the cilia, coating the interior epithelial surface of the bronchi and bronchiolesis, is called the periciliary layer (PCL). With fluid nourishment, cilia can move efficiently. The fluid in this region is named the PCL fluid and is considered to be an incompressible, viscous, Newtonian fluid. We propose there to be a free boundary at the tips of cilia underlining a gas phase while the cilia are moving forward. The Brinkman equation on a macroscopic scale, in which bundles of cilia are considered rather than individuals, with the Stefan condition was used in the PCL to determine the velocity of the PCL fluid and the height/shape of the free boundary. Regarding the numerical methods, the boundary immobilization technique was applied to immobilize the moving boundaries using coordinate transformation (working with a fixed domain). A finite element method was employed to discretize the mathematical model and a finite difference approach was applied to the Stefan problem to determine the free interface. In this study, an effective stroke is assumed to start when the cilia make a 140∘ angle to the horizontal plane and the velocitiesof cilia increase until the cilia are perpendicular to the horizontal plane. Then, the velocities of the cilia decrease until the cilia make a 40∘ angle with the horizontal plane. From the numerical results, we can see that although the velocities of the cilia increase and then decrease, the free interface at the tips of the cilia continues increasing for the full forward phase. The numerical results are verified and compared with an exact solution and experimental data from the literature. Regarding the fixed boundary, the numerical results converge to the exact solution. Regarding the free interface, the numerical solutions were compared with the average height of the PCL in non-cystic fibrosis (CF) human tissues and were in excellent agreement. This research also proposes possible values of parameters in the mathematical model in order to determine the free interface. Applications of these fluid flows include animal hair, fibers and filter pads, and rice fields.

Averaged Reynolds Equation Based on the Homogenization Theory for Gas Film Lubrication in the Head/Disk Interface

2014 ◽  
Author(s):  
Ting-Yi Yang ◽  
Bao-Jun Shi ◽  
Pei-Qi Ge ◽  
Dong-Wei Shu
Keyword(s):  
Reynolds Equation ◽  
Hard Disk Drives ◽  
Groove Depth ◽  
Homogenization Theory ◽  
Disk Interface ◽  
Pressure Distributions ◽  
Pressure Profiles ◽  
Discrete Track ◽  
Lubrication Characteristics ◽  
Film Lubrication

In order to improve the magnetic recording density of hard disk drives, discrete track disks and/or bit patterned disks are being considered. The gas film lubrication characteristics of a disk with microscale geometric surface features are different from those of traditional “smooth” disks. In this paper, an averaged Reynolds equation suitable for the analysis of gas film lubrication with discrete track recording (DTR) disks is derived based on the homogenization theory and a simplified model of the Reynolds equation with linearized flow rate (LFR). The averaged Reynolds equation and the LFR model are solved simultaneously using the finite volume method. Numerical results show that the pressure solution of the averaged Reynolds equation agrees well with the LFR model for DTR disks. The exact pressure values fluctuate in the neighborhood of those of the averaged pressure distribution curve. The pressure distributions of a complex slider for different groove depths are presented to investigate the effects of groove depth on pressure profiles. The proposed approach is shown to have a high computational efficiency.

scholarly journals New Solutions for System of Fractional Integro-Differential Equations and Abel’s Integral Equations by Chebyshev Spectral Method

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Hassan A. Zedan ◽  
Seham Sh. Tantawy ◽  
Yara M. Sayed
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Spectral Method ◽  
Operational Matrix ◽  
Numerical Results ◽  
Efficient Technique ◽  
Test Problems ◽  
Chebyshev Spectral Method

Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.

A Mass-Conserving Algorithm for Dynamical Lubrication Problems With Cavitation

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Roberto F. Ausas ◽  
Mohammed Jai ◽  
Gustavo C. Buscaglia
Keyword(s):  
Exact Solution ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Source Code ◽  
Squeeze Flow ◽  
Complementarity Conditions ◽  
Relaxation Scheme ◽  
Fluid Fraction ◽  
Oscillatory Squeeze Flow

A numerical algorithm for fully dynamical lubrication problems based on the Elrod–Adams formulation of the Reynolds equation with mass-conserving boundary conditions is described. A simple but effective relaxation scheme is used to update the solution maintaining the complementarity conditions on the variables that represent the pressure and fluid fraction. The equations of motion are discretized in time using Newmark’s scheme, and the dynamical variables are updated within the same relaxation process just mentioned. The good behavior of the proposed algorithm is illustrated in two examples: an oscillatory squeeze flow (for which the exact solution is available) and a dynamically loaded journal bearing. This article is accompanied by the ready-to-compile source code with the implementation of the proposed algorithm.

The Asymptotic Homogenization Method of the Settlement Calculation of the Cement-Soil Pile Composite Foundation

2011 ◽  
Vol 217-218 ◽  
pp. 390-395
Author(s):  
Ming Qin Guo ◽  
Jie Li ◽  
Shi Cheng Ma ◽  
Zhi Lin Liu
Keyword(s):  
Elastic Foundation ◽  
Three Dimensional ◽  
Elastic Parameter ◽  
Homogenization Method ◽  
Calculation Model ◽  
Composite Foundation ◽  
Homogenization Theory ◽  
Asymptotic Homogenization ◽  
Asymptotic Homogenization Method ◽  
Cement Soil

The finite element calculation model of the cement-soil pile composite foundation based on asymptotic homogenization theory has been built, under the state of three-dimensional stress, the equivalent elastic parameter of the composite elastic foundation model has been calculated, and depending on which, the settlement quantity has been calculated. Analysis through comparison with the testing result shows that it is feasible to use the homogenization method for calculating the settlement quantity of the composite elastic foundation, which sets up a theoretical basis for further analysis of homogenization method about composite foundation.

Reply to N. C. Steenland by the authors

Geophysics ◽  
1987 ◽  
Vol 52 (10) ◽  
pp. 1437-1438
Author(s):  
Mark Pilkington ◽  
D. J. Crossley
Keyword(s):  
Spectral Density ◽  
Potential Field ◽  
Point Of View ◽  
The Other ◽  
Theoretical Studies ◽  
Research Development ◽  
Future Studies ◽  
Analytical Point ◽  
A New Technique ◽  
Theoretical Developments

We welcome the chance to respond to the comments by Steenland on our two recent papers. Let us first acknowledge the gap that exists between geophysical practice, discussed by Steenland, and geophysics research development of the type presented in our papers. It is correct, healthy, and inevitable that such a gap should exist and yet understandable that not all techniques get transferred (as a successful example, take use of log spectral density which, while problematical from an analytical point of view, is widely used to ascertain depth to sources; Spector and Grant, 1970). On the other hand, we are reluctant to concede defeat in our case (and this applies to a host of similar theoretical studies of potential‐field interpretation) simply on the basis of there being limitations to the theory. Where theoretical demonstrations of a new technique are found to be practically useful, they will be adopted (often with appropriate modifications) by the industry; a good example is the use of Werner deconvolution, quoted by Steenland. Even where their limitations prevent this, theoretical developments often show the way for future studies.

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