Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T08:49:12.440Z Has data issue: false hasContentIssue false

Thin-film modelling of poroviscous free surface flows

Published online by Cambridge University Press:  09 November 2005

J. R. KING
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK email: [email protected], [email protected]
J. M. OLIVER
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK email: [email protected], [email protected]

Abstract

Thin-film models for the flow of a low reduced-Reynolds-number poroviscous droplet over a planar substrate are developed. One of the formulations is used to develop a minimal model for active animal cell motion in which the microscopic mechanisms of polymerisation and depolymerisation near the outer cell periphery are modelled by specifying the rate of mass transfer between the phases at the contact-line in terms of the velocity of the latter. An asymptotic analysis in the limit corresponding to strong cell-substrate adhesion is shown to lead to a novel class of multi-valued contact-line laws, a qualitative analysis of which leads in two dimensions to some intriguing behaviour, including (i) periodic contraction and expansion (pulsation), (ii) steady propagation at a contant speed, (iii) an unsteady combination of pulsation and propagation, and (iv) a bistable regime in which both non-motile and motile solutions are admissible, each of them being stable to sufficiently small perturbations, but with sufficiently large perturbations being able to ‘prod’ a stationary cell into motion or halt a moving one; these qualitative predictions are where possible compared with experiment. The contact-line behaviour is likely to be highly sensitive to environmental signals; the formulation may, therefore, provide a useful ‘minimal’ modelling framework for investigation of chemotactic effects at the cell scale. The corresponding extensional flow formulations are also noted.

Type
Papers
Copyright
2005 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)