A Highly Accurate Approach That Resolves the Pressure Spike of Elastohydrodynamic Lubrication

1988 ◽  
Vol 110 (2) ◽  
pp. 241-246 ◽  
Author(s):  
E. J. Bissett ◽  
D. W. Glander
Keyword(s):  
Contact Problem ◽  
Numerical Method ◽  
Mesh Refinement ◽  
Elastohydrodynamic Lubrication ◽  
Length Scale ◽  
Line Contact ◽  
Small Length ◽  
Small Length Scale ◽  
Pressure Spike ◽  
Refinement Procedure

We propose an accurate numerical method to solve the classical line contact problem of elastohydrodynamic lubrication. The method incorporates a second order accurate discretization and a straightforward automatic local mesh refinement procedure. Using these elements, we remove discretization errors which have produced significant inaccuracies in previously published results, and we completely resolve the pressure spike which is shown to be smooth on a sufficiently small length scale.

The line contact problem of elastohydrodynamic lubrication - I. Asymptotic structure for low speeds

1989 ◽  
Vol 424 (1867) ◽  
pp. 393-407 ◽  
Keyword(s):  
Asymptotic Solution ◽  
Contact Problem ◽  
Film Thickness ◽  
Numerical Solutions ◽  
Elastohydrodynamic Lubrication ◽  
Thin Layers ◽  
Line Contact ◽  
Asymptotic Regime ◽  
Lubricating Film ◽  
Pressure Spike

This paper reports the first formal asymptotic solution to the line contact problem of elastohydrodynamic lubrication (EHL), a fundamental problem describing the elastic deformation of lubricated rolling elements such as roller bearings, gear teeth and other contacts of similar geometry. The asymptotic régime considered is that of small λ , a dimensionless parameter proportional to rolling speed, viscosity and the elastic modulus. The solution is shown to possess four regions: a zone where the lubricating film is both thin and slowly narrowing and which is closely related to the contact area that occurs in the absence of lubricant, an upstream inlet zone of low pressure, and two thin layers on either side of the contact zone. The solutions in the first two just-mentioned zones are given by simple analytical expressions. The solutions in the two thin layers are obtained from two universal functions obtained by Bissett & Spence ( Proc. R. Soc. Lond . A 424, 409 (1989)). Although these two functions, related to the local film thickness, are obtained by numerical techniques by Bissett & Spence, it should be emphasized that all cases in the asymptotic régime considered are hereby solved definitively without recourse to further computation. Although some features of this structure have been suggested by other solution approaches, generally, these are numerical or ad hoc approximations. See the texts by Johnson ( Contact Mechanics , pp. 328 (1985)) and Dowson & Higginson ( Elasto-hydrodynamic lubrication (1977)), this work provides a formal mathematical basis for understanding most of the principal features of EHL. The solution provides a simple formula for minimum film thickness and displays the sharp narrowing of the lubricating film in the thin layer near the exit. In the basic asymptotic solution provided here, the dimensionless pressure-viscosity coefficient, α , is assumed to be O (1), and in this parameter régime, no pressure spike will occur. By comparing with the work of Hooke ( J. mech. Engng Sci . 19(4), 149 (1977)), we can show that an incipient pressure spike occurs when α becomes as large as O ( λ -1/5 ). However, asymptotic solutions in this latter parameter régime require new numerical solutions for each case of interest and are not pursued here.

Visualization of buoyancy opposing flow structures in a small length scale fluidic junction

2005 ◽  
Vol 8 (4) ◽  
pp. 288-288 ◽  
Author(s):  
P. A. Walsh ◽  
M. R. D. Davies ◽  
T. Dalton
Keyword(s):  
Flow Structures ◽  
Length Scale ◽  
Small Length ◽  
Small Length Scale

Nonlocal Approaches for the Vibration of Lattice Plates Including Both Shear and Bending Interactions

2018 ◽  
Vol 18 (07) ◽  
pp. 1850094 ◽  
Author(s):  
F. Hache ◽  
N. Challamel ◽  
I. Elishakoff
Keyword(s):  
Natural Frequencies ◽  
Scale Effects ◽  
Dynamical Behavior ◽  
Mindlin Plate ◽  
Length Scale ◽  
Small Length ◽  
Simply Supported ◽  
Small Length Scale ◽  
Plate Models ◽  
Scale Coefficient

The present study investigates the dynamical behavior of lattice plates, including both bending and shear interactions. The exact natural frequencies of this lattice plate are calculated for simply supported boundary conditions. These exact solutions are compared with some continuous nonlocal plate solutions that account for some scale effects due to the lattice spacing. Two continualized and one phenomenological nonlocal UflyandMindlin plate models that take into account both the rotary inertia and the shear effects are developed for capturing the small length scale effect of microstructured (or lattice) thick plates by associating the small length scale coefficient introduced in the nonlocal approach to some length scale coefficients given in a Taylor or a rational series expansion. The nonlocal phenomenological model constitutes the stress gradient Eringen’s model applied at the plate scale. The continualization process constructs continuous equation from the one of the discrete lattice models. The governing partial differential equations are solved in displacement for each nonlocal plate model. An exact analytical vibration solution is obtained for the natural frequencies of the simply supported rectangular nonlocal plate. As expected, it is found that the continualized models lead to a constant small length scale coefficient, whereas for the phenomenological nonlocal approaches, the coefficient, calibrated with respect to the element size of the microstructured plate, is structure-dependent. Moreover, comparing the natural frequencies of the continuous models with the exact discrete one, it is concluded that the continualized models provide much more accurate results than the nonlocal Uflyand–Mindlin plate models.

On the Stability and Uniqueness of Elastohydrodynamic Lubrication Solutions

1988 ◽  
Vol 110 (4) ◽  
pp. 628-631 ◽  
Author(s):  
D. W. Glander ◽  
E. J. Bissett
Keyword(s):  
Standard Model ◽  
Contact Problem ◽  
Elastohydrodynamic Lubrication ◽  
Line Contact ◽  
Multiplicity Of Solutions ◽  
The Standard Model ◽  
Unstable Solutions ◽  
Model Equations ◽  
The Stability

It was suggested in [1] that solutions of the line contact problem of elastohydrodynamic lubrication (EHL) are unstable in a certain parameter regime, and that both stable and unstable solutions can coexist in another regime. The author also suggested that these regimes limit the applicability of the standard model equations. In this work, the present authors repeat this calculation using the highly accurate solutions described in our previous work [3]. In all cases we have considered, we find no evidence of instability or multiplicity of solutions. We conclude that the existence of regions of instability or multiplicity were based on numerical artifacts and that considerations of stability or multiplicity do not limit the applicability of the standard model equations of EHL.

Small Length-Scale Analysis of Transport Phenomena in Oil-Wells Formation with Drilling Fluid Losses

2009 ◽  
Vol 27 (3) ◽  
pp. 345-356 ◽  
Author(s):  
G. Espinosa-Paredes ◽  
R. Vázquez-Rodriguez ◽  
R. Ramos-Alcantara ◽  
R. Varela-Ham ◽  
H. Romero-Paredes ◽  
...  
Keyword(s):  
Drilling Fluid ◽  
Transport Phenomena ◽  
Oil Wells ◽  
Length Scale ◽  
Scale Analysis ◽  
Small Length ◽  
Small Length Scale ◽  
Fluid Losses

POSTBUCKLING OF NANO RODS/TUBES BASED ON NONLOCAL BEAM THEORY

2009 ◽  
Vol 01 (02) ◽  
pp. 259-266 ◽  
Author(s):  
C. M. WANG ◽  
Y. XIANG ◽  
S. KITIPORNCHAI
Keyword(s):  
Differential Equation ◽  
Scale Effect ◽  
Shooting Method ◽  
Beam Theory ◽  
Length Scale ◽  
Small Length ◽  
Concentrated Load ◽  
Small Length Scale ◽  
Nano Rods ◽  
Nonlocal Beam

This paper is concerned with the postbuckling problem of cantilevered nano rods/tubes under an end concentrated load. Eringen's nonlocal beam theory is used to account for the small length scale effect. The governing equation is derived from statical and geometrical considerations and Eringen's nonlocal constitutive relation. The nonlinear differential equation is solved using the shooting method for the postbuckling load and the buckled shape. By comparing with the classical postbuckling solutions, the sensitivity of the small length scale effect on the buckling load and buckled shape may be observed. It is found that the small length scale effect decreases the postbuckling load and increases the deflection of the rod.

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