scholarly journals Exact tensor closures for the three-dimensional Jeffery's equation

2011 ◽  
Vol 680 ◽  
pp. 321-335 ◽  
Author(s):  
STEPHEN MONTGOMERY-SMITH ◽  
WEI HE ◽  
DAVID A. JACK ◽  
DOUGLAS E. SMITH
Keyword(s):  
Fibre Orientation ◽  
Elliptic Integral ◽  
Moment Tensor ◽  
Three Dimensional ◽  
Exact Formula ◽  
Fourth Moment ◽  
Newtonian Flow ◽  
Curve Fit ◽  
Explicit Formulae ◽  
Jeffery’S Equation

This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three-dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of Cintra & Tucker (J. Rheol., vol. 39, 1995, p. 1095), or equivalently, a natural closure in the sense of Verleye & Dupret (Developments in Non-Newtonian Flow, 1993, p. 139). The existence of these explicit formulae has been asserted previously, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever fibre orientation was isotropic at some point in time. Finally, this paper presents the fast exact closure, a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures nor the elliptic integral computation.

The development of asymmetry and period doubling for oscillatory flow in baffled channels

1996 ◽  
Vol 328 ◽  
pp. 19-48 ◽  
Author(s):  
E. P. L. Roberts ◽  
M. R. Mackley
Keyword(s):  
Reynolds Number ◽  
Oscillatory Flow ◽  
Three Dimensional ◽  
Period Doubling ◽  
Progressive Increase ◽  
Newtonian Flow ◽  
Chaotic Flow ◽  
Spatially Periodic ◽  
Dimensional Simulation ◽  
Time Periodic

We report experimental and numerical observations on the way initially symmetric and time-periodic fluid oscillations in baffled channels develop in complexity. Experiments are carried out in a spatially periodic baffled channel with a sinusoidal oscillatory flow. At modest Reynolds number the observed vortex structure is symmetric and time periodic. At higher values the flow progressively becomes three-dimensional, asymmetric and aperiodic. A two-dimensional simulation of incompressible Newtonian flow is able to follow the flow pattern at modest oscillatory Reynolds number. At higher values we report the development of both asymmetry and a period-doubling cascade leading to a chaotic flow regime. A bifurcation diagram is constructed that can describe the progressive increase in complexity of the flow.

Three-dimensional coating flow of nematic liquid crystal on an inclined substrate

2015 ◽  
Vol 26 (5) ◽  
pp. 647-669 ◽  
Author(s):  
M. A. LAM ◽  
L. J. CUMMINGS ◽  
T.-S. LIN ◽  
L. KONDIC
Keyword(s):  
Free Surface ◽  
Liquid Crystal ◽  
Nematic Liquid Crystal ◽  
Nonlinear Partial Differential Equation ◽  
Three Dimensional ◽  
Travelling Wave ◽  
Linear Stability Theory ◽  
Newtonian Flow ◽  
Surface Evolution ◽  
Coating Flow

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

Theoretical and Experimental Investigation of Frictional and Thermal Behavior in Oscillatory Sliding Line Contact

2005 ◽  
Author(s):  
M. Mansouri ◽  
M. M. Khonsari ◽  
D. Y. Hua
Keyword(s):  
Thermal Behavior ◽  
Three Dimensional ◽  
Axial Direction ◽  
Contact Theory ◽  
Design Stage ◽  
Heat Equations ◽  
Line Contact ◽  
Test Rig ◽  
Curve Fit ◽  
Future Extension

A model is developed to predict the thermal behavior of two sliding bodies undergoing oscillatory relative motion. The thermal model is capable of predicting the temperature rise distribution within the pin-bushing pair and the housing. The bodies geometrically form a pin-bushing configuration and the Hertzian line contact theory is used to approximate the contact pressure and the width. A quasi- three dimensional temperature model is developed by averaging the temperature in the axial direction. The resulting dimensionless heat equations and proper boundary conditions are solved by the finite element method. A series of dimensionless equations for use at the design stage is presented. A test rig capable of inducing oscillatory motion under heavy loading condition is used for measuring friction and temperature. The measured coefficient of friction history, which is curve fit as a function of time, is used in the simulations. The description of the test rig, modeling aspects, and the future extension of the research comprise the contents of this paper.

A Parametric Study on Fatigue Life for Mixed Elastohydrodynamic Lubrication Point Contacts

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Xiao-Liang Yan ◽  
Xiao-Li Wang ◽  
Yu-Yan Zhang
Keyword(s):  
Shear Stress ◽  
Fatigue Life ◽  
Three Dimensional ◽  
Elastohydrodynamic Lubrication ◽  
Point Contacts ◽  
Newtonian Flow ◽  
Lubrication Regimes ◽  
Close Relationship ◽  
Lubrication Characteristics ◽  
Effect Of Surface

The lubrication characteristics and fatigue life are numerically analyzed under full film and mixed lubrication regimes, in which the three-dimensional sinusoidal surfaces with changeable wavelengths in x and y directions are used, the geometry changes of the contact areas are described by the various ellipticity, and the non-Newtonian flow of lubricant is described by the sinh-law rheology model. The results show that the influences of characteristic shear stress, wavelength ratio, and ellipticity on lubrication characteristics and fatigue life are remarkable. The effect of surface topography on lubrication characteristics has a close relationship with speed. Increasing the ellipticity and decreasing wavelength ratio and characteristic shear stress can prolong the fatigue life.

scholarly journals Long-Period Ground Motion Simulation Based on Three-dimensional Centroid Moment Tensor Inversion Solutions in the Kanto Region, Japan

2020 ◽  
Author(s):  
Shunsuke Takemura ◽  
Kazuo Yoshimoto ◽  
Katsuhiko Shiomi
Keyword(s):  
Ground Motion ◽  
Ground Motions ◽  
Moment Tensor ◽  
Three Dimensional ◽  
Centroid Moment Tensor ◽  
Velocity Models ◽  
Long Period ◽  
Ground Motion Simulations ◽  
Kanto Basin ◽  
Long Period Ground Motion

Abstract We conducted centroid moment tensor (CMT) inversions of moderate (Mw 4.5–6.5) earthquakes in the Kanto region, Japan, using a local three-dimensional (3D) model. We then investigated the effects of our 3D CMT solutions on long-period ground motion simulations. Grid search CMT inversions were conducted using displacement seismograms for periods of 25–100 s. By comparing our 3D CMT solutions with those from the local one-dimensional (1D) catalog, we found that our 3D CMT inversion systematically provides magnitudes smaller than those in the 1D catalog. The Mw differences between 3D and 1D catalogs tend to be significant for earthquakes within the oceanic slab. By comparing ground motion simulations between 1D and 3D velocity models, we confirmed that observed Mw differences could be explained by differences in the rigidity structures around the source regions between 3D and 1D velocity models. The 3D velocity structures (especially oceanic crust and mantle) are important for estimating seismic moments in intraslab earthquakes. The seismic moments directly affect the amplitudes of ground motions. Thus, 3D CMT solutions are essential for the precise forward and inverse modeling of long-period ground motion. We also conducted long-period ground motion simulations using our 3D CMT solutions to evaluate reproducibility of long-period ground motions at stations within the Kanto Basin. The simulations of our 3D CMT inversion well-reproduced observed ground motions for periods longer than 10 s, even at stations within the Kanto Basin.

Three-dimensional thermo-electro-elastic field in one-dimensional hexagonal piezoelectric quasi-crystal weakened by an elliptical crack

2021 ◽  
pp. 108128652110592
Author(s):  
Yuwei Liu ◽  
Xuesong Tang ◽  
Peiliang Duan ◽  
Tao Wang ◽  
Peidong Li
Keyword(s):  
Analytical Solution ◽  
Elliptic Integral ◽  
Electric Displacement ◽  
Three Dimensional ◽  
Elastic Field ◽  
Elliptical Crack ◽  
Elementary Functions ◽  
Surface Displacements ◽  
Electric Displacement Intensity Factor ◽  
Generalized Potential Theory

In this paper, an analytical solution is developed for the problem of an infinite 1D hexagonal piezoelectric quasi-crystal medium weakened by an elliptical crack and subjected to mixed loads on the crack surfaces. The mixed loads comprise the phonon pressure, phason pressure, electric displacement, and temperature increment, and the crack surfaces can be electrically permeable or impermeable. Based on a general solution, combined with the generalized potential theory, the steady-state 3D thermo-electro-elastic field variables in the quasi-crystal are obtained in terms of elliptic integral functions and elementary functions. Several important physical quantities on the cracked plane, such as the generalized crack surface displacements, normal stresses, and stress intensity factors, are derived in closed forms. An illustrative numerical calculation verifies the presented analytical solution and shows the distribution of the 3D thermo-electro-elastic field. It is indicated that the influence of the phason field on the result is pronounced, especially for the electric field variables, and the electric permeability of crack surfaces has a significant effect on the electric displacement intensity factor at the crack tip.

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