Abstract
A general formulation of the governing equations for the slow, steady, two-dimensional flow of a thixotropic or antithixotropic fluid in a channel of slowly varying width is described. These equations are equivalent to the equations of classical lubrication theory for a Newtonian fluid, but incorporate the evolving microstructure of the fluid, described in terms of a scalar structure parameter. We demonstrate how the lubrication equations can be further simplified in the weakly advective regime in which the advective Deborah number is comparable to the aspect ratio of the flow, and present illustrative analytical and semi-analytical solutions for particular choices of the constitutive and kinetic laws, including a purely viscous Moore–Mewis–Wagner model and a regularised viscoplastic Houška model. The lubrication results also allow the calibration and validation of cross-sectionally averaged, or otherwise reduced, descriptions of thixotropic channel flow which provide a first step towards models of thixotropic flow in porous media, and we employ them to explain why such descriptions may be inadequate.
Highlights
Recent years have seen increasing interest in thixotropic flow
When the Deborah number is very large, when D = O(1/δ2 ) or larger, the effects of buildup and breakdown are of the same order as, or smaller than, the neglected terms in classical lubrication theory, and so in this “very strongly advective” regime the structure parameter is entirely determined by the upstream boundary condition on λ to within the usual accuracy of lubrication theory
The minimal requirement that we can make of a reduced model of thixotropic flow is that in the weakly advective regime it captures both the dependence of G0 on h and the dependence of G1 on h and h to reasonable accuracy; in other words, that it captures the non-Newtonian nature of the leading-order flow and the first-order correction to this caused by thixotropy
Summary
Recent years have seen increasing interest in thixotropic flow. This interest stems both from applications, which include the flow of muds, processed foods, polymer solutions and waxy crude oils, and from the challenge that thixotropic fluids present to the modeller. The weakness of this approach is that the transverse variation must be obtained by numerical simulations of a non-reduced system, and there is no guarantee that the approximate profiles for λ obtained in this way will be applicable to different rheologies or to different problems With this in mind, our goal in this paper is to systematically develop the governing equations for lubrication flow of thixotropic and antithixotropic fluids in a slowly varying geometry.
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