We present a mathematical investigation of the oscillatory boundary layer in a semi-infinite fluid bounded by an oscillating wall (the so-called 'Stokes problem'), when the fluid has a thixotropic or antithixotropic rheology. We obtain asymptotic solutions in the limit of small-amplitude oscillations, and we use numerical integration to validate the asymptotic solutions and to explore the behaviour of the system for larger-amplitude oscillations. The solutions that we obtain differ significantly from the classical solution for a Newtonian fluid. In particular, for antithixotropic fluids the velocity reaches zero at a finite distance from the wall, in contrast to the exponential decay for a thixotropic or a Newtonian fluid.For small amplitudes of oscillation, three regimes of behaviour are possible: the structure parameter may take values defined instantaneously by the shear rate, or by a long-term average; or it may behave hysteretically. The regime boundaries depend on the precise specification of structure build-up and breakdown rates in the rheological model, illustrating the subtleties of complex fluid models in non-rheometric settings. For larger amplitudes of oscillation the dominant behaviour is hysteretic. We discuss in particular the relationship between the shear stress and the shear rate at the oscillating wall. © 2012 Elsevier B.V.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: C.R.McA. is supported by a Doctoral Training Award funded by the Engineering and Physical Sciences Research Council. Part of this work was undertaken while S.K.W. was a Visiting Fellow in the Department of Mechanical and Aerospace Engineering in the School of Engineering and Applied Science at Princeton University, USA, and part was undertaken while he was a Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM) at the University of Oxford, England. This publication was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). We are also grateful to Dr. Michele Taroni (formerly of OCCAM, University of Oxford) and Prof. Iain W. Stewart (University of Strathclyde) for their valuable suggestions on aspects of this work, and to two anonymous reviewers for their comments on the original version.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.