Higher Symmetries in Lower-Dimensional Models

On lower-dimensional models in lubrication, Part B: Derivation of a Reynolds type of equation for incompressible piezo-viscous fluids

Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology ◽

10.1177/1350650120973800 ◽

2020 ◽

pp. 135065012097380

Author(s):

Andreas Almqvist ◽

Evgeniya Burtseva ◽

Kumbakonam Rajagopal ◽

Peter Wall

Keyword(s):

Film Thickness ◽

Reynolds Equation ◽

Constitutive Relations ◽

Viscous Fluids ◽

Dimensional Model ◽

Pressure Dependent Viscosity ◽

Constant Viscosity ◽

Dimensional Models ◽

Dependent Viscosity ◽

Lower Dimensional

The Reynolds equation is a lower-dimensional model for the pressure in a fluid confined between two adjacent surfaces that move relative to each other. It was originally derived under the assumption that the fluid is incompressible and has constant viscosity. In the existing literature, the lower-dimensional Reynolds equation is often employed as a model for the thin films, which lubricates interfaces in various machine components. For example, in the modelling of elastohydrodynamic lubrication (EHL) in gears and bearings, the pressure dependence of the viscosity is often considered by just replacing the constant viscosity in the Reynolds equation with a given viscosity-pressure relation. The arguments to justify this are heuristic, and in many cases, it is taken for granted that you can do so. This motivated us to make an attempt to formulate and present a rigorous derivation of a lower-dimensional model for the pressure when the fluid has pressure-dependent viscosity. The results of our study are presented in two parts. In Part A, we showed that for incompressible and piezo-viscous fluids it is not possible to obtain a lower-dimensional model for the pressure by just assuming that the film thickness is thin, as it is for incompressible fluids with constant viscosity. Here, in Part B, we present a method for deriving lower-dimensional models of thin-film flow, where the fluid has a pressure-dependent viscosity. The main idea is to rescale the generalised Navier-Stokes equation, which we obtained in Part A based on theory for implicit constitutive relations, so that we can pass to the limit as the film thickness goes to zero. If the scaling is correct, then the limit problem can be used as the dimensionally reduced model for the flow and it is possible to derive a type of Reynolds equation for the pressure.

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Lower-Dimensional Models

Encyclopedia of Continuum Mechanics ◽

10.1007/978-3-662-55771-6_300393 ◽

2020 ◽

pp. 1494-1494

Keyword(s):

Dimensional Models ◽

Lower Dimensional

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Relaxation Theorem and Lower-Dimensional Models in Micropolar Elasticity

Mathematics and Mechanics of Solids ◽

10.1177/1081286509342270 ◽

2009 ◽

Vol 15 (8) ◽

pp. 812-853 ◽

Cited By ~ 1

Author(s):

Igor Velčić ◽

Josip Tambača

Keyword(s):

Micropolar Elasticity ◽

Relaxation Theorem ◽

Dimensional Models ◽

Lower Dimensional

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Dynamic and quasi-dynamic modelling of earthquake sequences from zero to three dimensions: choose model complexity as needed

10.5194/egusphere-egu2020-10210 ◽

2020 ◽

Author(s):

Meng Li ◽

Casper Pranger ◽

Luca Dal Zilio ◽

Ylona van Dinther

Keyword(s):

Dynamic Models ◽

Model Complexity ◽

Computational Time ◽

Cycle Period ◽

Seismic Cycle ◽

Cycle Simulation ◽

Higher Dimensional ◽

Dimensional Models ◽

Earthquake Sequences ◽

Lower Dimensional

Earthquake sequences reflect the repetitive dynamic processes of stress accumulation and release on a fault. Understanding earthquake sequences is fundamental for the research of induced and natural earthquakes and may ultimately help to better assess long-term seismic hazard. Numerical models are well-suited to overcome limited spatiotemporal observations and improve our understanding on this topic. However, large models in 3D are still computational time and memory consuming. Moreover, this may not be optimal if the aspects of lateral or depth variations within the results are not needed to answer a particular objective. This motivated us to investigate the advantages and limitations of various dimensional models by simulating earthquake sequences in 0D, 1D, 2D and ultimately 3D. We applied a C++ numerical library GARNET [1] to deal with the various dimensional models in one simulator. This library uses a fully staggered finite difference scheme with a rectilinear adaptive grid. It also incorporates an automatic discretization algorithm and combines different physical ingredients such as visco-elasto-plastic rheology and quasi- and fully dynamic approaches into one algorithm.Here we present numerical experiments of a strike-slip fault under rate-and-state friction, surrounded by an elastic medium with constant tectonic loading and, test them under different parameters and initial conditions. By adding one dimension at a time, we simulate a more detailed structure of the seismic cycle. The higher dimensional models present both the validity and the limitations of the lower dimensional ones. For example, inertial waves are not possible to present in 0D while a quasi-dynamic radiation damping term can be added here instead. Another example is that due to lack of grid extension along the fault, both 0D and 1D model fail to reveal an earthquake nucleation phase. However, some important observables, such as the seismic cycle period, maximum/minimum stress and slip rates, are calculated accurately in lower dimensional models, which are much faster than higher dimensional models. We also implemented and compared quasi- and fully dynamic models in the same way. Our results indicate that both the size of simulated seismic events and their interval are reduced in quasi-dynamic models. This could provide us with guidance to identify the appropriate model complexity for various problems. We will also present 3D modeling results, which will be compared to their 2D equivalent. Finally, we present our results for the SCEC SEAS benchmarks [2] and compare them to other participating codes.[1] Pranger, C. C., L. Le Pourhiet, D. May, Y. van Dinther, and T. Gerya (2016). &#8220;Self- consistent seismic cycle simulation in a three-dimensional continuum model: method- ology and examples.&#8221; AGU Fall Meeting Abstracts.[2] Erickson, B. A., et al. (2019). The Community Code Verification Exercise for Simulating Sequences of Earthquakes and Aseismic Slip (SEAS). Poster Presentation at 2019 SCEC Annual Meeting.

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