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Tractions, Balances, and Boundary Conditions for Nonsimple Materials with Application to Liquid Flow at Small-Length Scales

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Abstract

Using a nonstandard version of the principle of virtual power, we develop general balance equations and boundary conditions for second-grade materials. Our results apply to both solids and fluids as they are independent of constitutive equations. As an application of our results, we discuss flows of incompressible fluids at small-length scales. In addition to giving a generalization of the Navier–Stokes equations involving higher-order spatial derivatives, our theory provides conditions on free and fixed boundaries. The free boundary conditions involve the curvature of the free surface; among the conditions for a fixed boundary are generalized adherence and slip conditions, each of which involves a material length scale. We reconsider the classical problem of plane Poiseuille flow for generalized adherence and slip conditions.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of Washington in St. Louis, St. Louis, MO, 63130-4899, USA

    Eliot Fried

  2. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213-3190, USA

    Morton E. Gurtin

Authors
  1. Eliot Fried
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  2. Morton E. Gurtin
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Correspondence to Eliot Fried.

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Communicated by S.S. Antman

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Fried, E., Gurtin, M.E. Tractions, Balances, and Boundary Conditions for Nonsimple Materials with Application to Liquid Flow at Small-Length Scales. Arch Rational Mech Anal 182, 513–554 (2006). https://doi.org/10.1007/s00205-006-0015-7

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  • DOI: https://doi.org/10.1007/s00205-006-0015-7

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