TY - JOUR
T1 - Unsteady evolution of slip and drag in surfactant-contaminated superhydrophobic channels
AU - Tomlinson, Samuel
AU - Gibou, Frédéric
AU - Luzzatto-Fegiz, Paolo
AU - Temprano-Coleto, Fernando
AU - Jensen, Oliver
AU - Landel, Julien
PY - 2024/12/10
Y1 - 2024/12/10
N2 - Recognising that surfactants can impede the drag reduction resulting from superhydrophobic surfaces (SHSs), we investigate the impact of spatio–temporal fluctuations in surfactant concentration on the drag-reduction properties of SHSs. We model the unsteady transport of soluble surfactant in a channel flow bounded by two SHSs. The flow is laminar, pressure-driven, and the SHSs are periodic in the streamwise and spanwise directions. We assume that the channel length is much longer than the streamwise period, the streamwise period is much longer than the channel height and spanwise period, and bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. By combining long-wave and homogenisation theories, we derive an unsteady advection– diffusion equation for surfactant-flux transport over the length of the channel, which is coupled to a quasi-steady advection–diffusion equation for surfactant transport over individual plastrons. As diffusion over the length of the channel is typically small, the surfactant flux is governed by a nonlinear wave equation. In the fundamental case of the transport of a bolus of surfactant, we predict its propagation speed and describe its nonlinear evolution via interaction with the SHS. The propagation speed can fall below the average streamwise velocity as the surfactant adsorbs and rigidifies the plastrons. Smaller concentrations of surfactant are advected faster than larger ones, so that wave-steepening effects can lead to shock formation in the surfactant-flux distribution. Our asymptotic results reveal how unsteady surfactant transport can affect the spatio–temporal evolution of the slip velocity, drag reduction and effective slip length in SHS channels.
AB - Recognising that surfactants can impede the drag reduction resulting from superhydrophobic surfaces (SHSs), we investigate the impact of spatio–temporal fluctuations in surfactant concentration on the drag-reduction properties of SHSs. We model the unsteady transport of soluble surfactant in a channel flow bounded by two SHSs. The flow is laminar, pressure-driven, and the SHSs are periodic in the streamwise and spanwise directions. We assume that the channel length is much longer than the streamwise period, the streamwise period is much longer than the channel height and spanwise period, and bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. By combining long-wave and homogenisation theories, we derive an unsteady advection– diffusion equation for surfactant-flux transport over the length of the channel, which is coupled to a quasi-steady advection–diffusion equation for surfactant transport over individual plastrons. As diffusion over the length of the channel is typically small, the surfactant flux is governed by a nonlinear wave equation. In the fundamental case of the transport of a bolus of surfactant, we predict its propagation speed and describe its nonlinear evolution via interaction with the SHS. The propagation speed can fall below the average streamwise velocity as the surfactant adsorbs and rigidifies the plastrons. Smaller concentrations of surfactant are advected faster than larger ones, so that wave-steepening effects can lead to shock formation in the surfactant-flux distribution. Our asymptotic results reveal how unsteady surfactant transport can affect the spatio–temporal evolution of the slip velocity, drag reduction and effective slip length in SHS channels.
KW - Marangoni convection
KW - drag reduction
KW - microfluidics
U2 - 10.1017/jfm.2024.676
DO - 10.1017/jfm.2024.676
M3 - Article
SN - 0022-1120
VL - 1000
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
M1 - A62
ER -