Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The dynamics of simple fluids in steady circular shear

Author: J. D. Goddard
Journal: Quart. Appl. Math. 41 (1983), 107-118
MSC: Primary 76A10
DOI: https://doi.org/10.1090/qam/700665
MathSciNet review: 700665
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Abstract: An isotropic simple fluid of constant density $ \rho $ is confined between two infinite horizontal planes which rotate steadily about separate vertical $ \left( z \right)$ axes with common angular velocity $ \omega $. We show that at least one solution to the exact equations of motion is determined by the differential equation

$\displaystyle \frac{d}{{dz}}\left( {\eta \frac{{du}}{{dz}}} \right) - i\rho \omega u = 0$

where $ u\left( z \right)$ is a complex variable representing the horizontal velocity and $ \eta \left( {\left\vert {du/dz} \right\vert,\omega } \right)$ is a complex shear modulus. This equation represents the extension to nonlinear viscoelasticity of the previous works of Berker and of Abbott and Walters on linear viscoelastic fluids, for which $ \eta $ reduces to the usual dynamic viscosity $ \eta *\left( \omega \right)$

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DOI: https://doi.org/10.1090/qam/700665
Article copyright: © Copyright 1983 American Mathematical Society

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