## Publication Date:

22 April 2021

## Journal:

IMA Journal of Applied Mathematics

## Last Updated:

2021-07-28T02:03:24.967+01:00

## Issue:

3

## Volume:

86

## DOI:

10.1093/imamat/hxab004

## page:

490-501

## abstract:

<jats:title>Abstract</jats:title>

<jats:p>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 $$\begin{align*}&amp; \pi \frac{G\sigma_T c^2}{\mu l} I(\alpha), \end{align*}$$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, $$\begin{align*}&amp; I(\alpha) = \frac{2}{\sin\alpha} \int_0^\infty\frac{\sinh s\alpha}{ \cosh s(\pi-\alpha) \sinh s \pi} \, \textrm{d} s. \end{align*}$$</jats:p>

<jats:p>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 $$\begin{align*}&amp; \pi \frac{G\sigma_T c^2}{\mu l} I(\alpha), \end{align*}$$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, $$\begin{align*}&amp; I(\alpha) = \frac{2}{\sin\alpha} \int_0^\infty\frac{\sinh s\alpha}{ \cosh s(\pi-\alpha) \sinh s \pi} \, \textrm{d} s. \end{align*}$$</jats:p>

## Symplectic id:

1183442

## Submitted to ORA:

Submitted

## Publication Type:

Journal Article