Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles. (22nd April 2021)

• Record Type: Journal Article
• Authors: Yariv, Ehud; Kirk, Toby L
• Is Part Of: IMA journal of applied mathematics. Volume 86:Number 3(2021)
• Page Start: 490
• Page End: 501
• Publication Date: 2021-04-22
• Subjects: superhydrophobic surfaces -- thermocapillary flows -- matched asymptotic expansions; Mathematics -- Periodicals; Mathematics; Periodicals; 519
• DOI: 10.1093/imamat/hxab004
• Languages: English
• ISSNs: 0272-4960

Abstract

Abstract: A common realization of superhydrophobic surfaces comprises of a periodic array of cylindrical bubbles which are trapped in a periodically grooved solid substrate. We consider the thermocapillary animation of liquid motion by a macroscopic temperature gradient which is longitudinally applied over such a bubble mattress. Assuming a linear variation of the interfacial tension with the temperature, at slope $\sigma _T$, we seek the effective velocity slip attained by the liquid at large distances away from the mattress. We focus upon the dilute limit, where the groove width $2c$ is small compared with the array period $2l$ . The requisite velocity slip in the applied-gradient direction, determined by a local analysis about a single bubble, is provided by the approximation $$\begin{align*}& \pi \frac{G\sigma_T c^2}{\mu l} I(\alpha), \end{align*}$$ wherein $G$ is the applied-gradient magnitude, $\mu $ is the liquid viscosity and $I(\alpha )$, a non-monotonic function of the protrusion angle $\alpha $, is provided by the quadrature, $$\begin{align*}& I(\alpha) = \frac{2}{\sin\alpha} \int_0^\infty\frac{\sinh s\alpha}{ \cosh s(\pi-\alpha) \sinh s \pi} \, \textrm{d} s. \end{align*}$$