Practical error estimates for Reynolds' lubrication approximation and its higher order corrections

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Reynolds lubrication approximation is used extensively to study flows between moving machine parts, in narrow channels, and in thin films. The solution of Reynolds equation may be thought of as the zeroth order term in an expansion of the solution of the Stokes equations in powers of the aspect ratio {var_epsilon} of the domain. In this paper, we show how to compute the terms in this expansion to arbitrary order on a two-dimensional, x-periodic domain and derive rigorous, a-priori error bounds for the difference between the exact solution and the truncated expansion solution. Unlike previous studies of this sort, the … continued below

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Wilkening, Jon December 10, 2008.

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Reynolds lubrication approximation is used extensively to study flows between moving machine parts, in narrow channels, and in thin films. The solution of Reynolds equation may be thought of as the zeroth order term in an expansion of the solution of the Stokes equations in powers of the aspect ratio {var_epsilon} of the domain. In this paper, we show how to compute the terms in this expansion to arbitrary order on a two-dimensional, x-periodic domain and derive rigorous, a-priori error bounds for the difference between the exact solution and the truncated expansion solution. Unlike previous studies of this sort, the constants in our error bounds are either independent of the function h(x) describing the geometry, or depend on h and its derivatives in an explicit, intuitive way. Specifically, if the expansion is truncated at order 2k, the error is O({var_epsilon}{sup 2k+2}) and h enters into the error bound only through its first and third inverse moments {integral}{sub 0}{sup 1} h(x){sup -m} dx, m = 1,3 and via the max norms {parallel} 1/{ell}! h{sup {ell}-1}{partial_derivative}{sub x}{sup {ell}}h{parallel}{sub {infinity}}, 1 {le} {ell} {le} 2k + 2. We validate our estimates by comparing with finite element solutions and present numerical evidence that suggests that even when h is real analytic and periodic, the expansion solution forms an asymptotic series rather than a convergent series.

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  • Journal Name: SIAM Journal of Mathmatical Analysis

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  • Report No.: LBNL-1303E
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 945052
  • Archival Resource Key: ark:/67531/metadc900789

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  • December 10, 2008

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  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 8, 2016, 1:21 p.m.

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Wilkening, Jon. Practical error estimates for Reynolds' lubrication approximation and its higher order corrections, article, December 10, 2008; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc900789/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.

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