Patterns and Their Large-Scale Distortions in Marangoni Convection with Insoluble Surfactant

Alexander B Mikishev,Alexander A Nepomnyashchy

doi:10.3390/fluids6080282

Abstract

Nonlinear dynamics of patterns near the threshold of long-wave monotonic Marangoni instability of conductive state in a heated thin layer of liquid covered by insoluble surfactant is considered. Pattern selection between roll and square planforms is analyzed. The dependence of pattern stability on the heat transfer from the free surface of the liquid characterized by Biot number and the gravity described by Galileo number at different surfactant concentrations is studied. Using weakly nonlinear analysis, we derive a set of amplitude equations governing the large-scale roll distortions in the presence of the surface deformation and the surfactant redistribution. These equations are used for the linear analysis of modulational instability of stationary rolls.

Highlights

Marangoni convection is a type of non-equilibrium process, which creates a variety of spatiotemporal periodic patterns, see [1,2], and is crucial in thin liquid layers, where the interfacial effects prevail over the bulk effects
It is known that when the free surface of the liquid is covered by surface-active agent, the formation of the convective patterns significantly changes
In the present work we investigate the large-scale Marangoni convection in a liquid layer with insoluble surfactant spread over a deformable free surface, in the interval of wavenumbers O(Bi1/2)

Summary

Introduction

Marangoni convection is a type of non-equilibrium process, which creates a variety of spatiotemporal periodic patterns, see [1,2], and is crucial in thin liquid layers, where the interfacial effects prevail over the bulk effects. The formation of large-scale convective patterns is a result of long-wavelength instability. It is known that when the free surface of the liquid is covered by surface-active agent (surfactant), the formation of the convective patterns significantly changes. In the present work we investigate the large-scale Marangoni convection in a liquid layer with insoluble surfactant spread over a deformable free surface, in the interval of wavenumbers O(Bi1/2). We consider the modulation of rolls by long-wave disturbances near the instability threshold and derive the system of the amplitude equations. The modulation of convective patterns in the neighborhood of the instability threshold is explained by Newell, Whitehead, and Segel [11,12] as interaction of disturbances with different wavenumbers close to the critical one. The analysis of that phenomenon has not been previously carried out for Marangoni patterns in the presence of a surfactant

Statement of the Problem

Equation for the Growth Rate

Stability in the Absence of Surfactant

Influence of Insoluble Surfactant