Dunn, G.J. and Duffy, B.R. and Wilson, S.K. and Holland, D. (2009) Quasisteady spreading of a thin ridge of fluid with temperaturedependent surface tension on a heated or cooled substrate. Quarterly Journal of Mechanics and Applied Mathematics, 62 (4). pp. 365402. ISSN 00335614

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Abstract
We investigate theoretically the problem of the quasisteady spreading or contraction of a thin twodimensional sessile or pendent ridge of viscous fluid with temperaturedependent surface tension on a planar horizontal substrate that is uniformly heated or cooled relative to the atmosphere. We derive an implicit solution of the leadingorder thinfilm equation for the freesurface profile of the ridge and use this to examine the quasisteady evolution of the ridge, the dynamics of the moving contact lines being modelled by a 'Tanner law' relating the velocity of the contact line to the contact angle; in particular, we obtain a complete description of the possible forms that the evolution may take. In both the case of a (sessile or pendent) ridge on a heated substrate and the case of a pendent ridge on a cooled substrate when gravitational effects are relatively weak, there is one stable final state to which the ridge may evolve. In the case of a pendent ridge on a cooled substrate when gravitational effects are stronger, there may be one or two stable final states; moreover, the contact angles may vary nonmonotonically with time during the evolution to one of these states. In the case of a pendent ridge on a cooled substrate when gravitational effects are even stronger, there may be up to three stable final states with qualitatively different solutions; moreover, the ridge may evolve via an intermediate state from which quasisteady motion cannot persist, and so there will be a transient nonquasisteady adjustment (in which the contact angles change rapidly, with the positions of the contact lines unaffected), after which quasisteady motion is resumed. Lastly, we consider the behaviour of the ridge in the asymptotic limits of strong heating or cooling of the substrate and of strong or weak gravitational effects.
Item type:  Article 

ID code:  20628 
Keywords:  volatile liquid droplets, similarity solutions, solid surface, contact line, films, evaporation, drops, thermocapillarity, instability, Mathematics, Mechanics of Materials, Mechanical Engineering, Applied Mathematics, Condensed Matter Physics 
Subjects:  Science > Mathematics 
Department:  Faculty of Science > Mathematics and Statistics 
Depositing user:  Mrs Carolynne Westwood 
Date Deposited:  16 Jun 2010 09:17 
Last modified:  12 Dec 2015 14:05 
Related URLs:  
URI:  http://strathprints.strath.ac.uk/id/eprint/20628 
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