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Strong temperature-dependent viscosity effects on a rivulet draining down a uniformly heated or cooled slowly varying substrate

Wilson, S.K. (2003) Strong temperature-dependent viscosity effects on a rivulet draining down a uniformly heated or cooled slowly varying substrate. Physics of Fluids, 15 (4). pp. 827-840. ISSN 1070-6631

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We use the lubrication approximation to investigate the steady locally unidirectional gravity-driven draining of a thin rivulet of Newtonian fluid with temperature-dependent viscosity down a slowly varying substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere. We consider the situation in which the Biot number ~and hence the variation of temperature across the rivulet! is small, but in which the variation of viscosity with temperature is sufficiently strong that thermoviscosity effects appear at leading order in the limit of small Biot number. Three different models for the dependence of viscosity on temperature ~specifically, the linear, exponential and Eyring models! are considered, but our attention is concentrated on the more realistic exponential and Eyring models ~which coincide at leading order in the limit of small Biot number!. We show that the effect of cooling the atmosphere is always to widen and deepen the rivulet, while the effect of heating the atmosphere is always to narrow and shallow it. We interpret our results as describing a slowly varying rivulet draining in the azimuthal direction from the top to the bottom of a large horizontal circular cylinder, and find that the behavior of the rivulet is rather different on the upper and lower parts of the cylinder ~i.e., for sessile and pendent rivulets!. Specifically, the effect of strong cooling of the atmosphere is to produce a wide rivulet with finite uniform thickness on the upper part of the cylinder, but a deep rivulet with finite semi-width on the lower part of the cylinder. On the other hand, the effect of strong heating of the atmosphere is to produce a narrow and shallow rivulet everywhere except near the top and the bottom of the cylinder.