Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface
File(s)
Published version
Author(s)
Crowdy, Darren
Type
Journal Article
Abstract
A class of exact solutions is presented describing the time evolution
of insoluble surfactant to a stagnant-cap equilibrium on the surface of deep
water in the Stokes flow regime at zero capillary number and infinite surface
P´eclet number. This is done by demonstrating, in a two-dimensional model
setting, the relevance of the forced complex Burgers equation to this problem
when a linear equation of state relates the surface tension to the surfactant
density. A complex-variable version of the method of characteristics can then
be deployed to find an implicit representation of the general solution. A special
class of initial conditions is considered for which the associated solutions can
be given explicitly. The new exact solutions, which include both spreading and
compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order
reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads
to a forced complex Burgers equation with linear damping. Generalized exact
solutions to the latter equation at infinite surface P´eclet number are also found
and used to study how reaction effects destroy the surfactant cap equilibrium.
of insoluble surfactant to a stagnant-cap equilibrium on the surface of deep
water in the Stokes flow regime at zero capillary number and infinite surface
P´eclet number. This is done by demonstrating, in a two-dimensional model
setting, the relevance of the forced complex Burgers equation to this problem
when a linear equation of state relates the surface tension to the surfactant
density. A complex-variable version of the method of characteristics can then
be deployed to find an implicit representation of the general solution. A special
class of initial conditions is considered for which the associated solutions can
be given explicitly. The new exact solutions, which include both spreading and
compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order
reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads
to a forced complex Burgers equation with linear damping. Generalized exact
solutions to the latter equation at infinite surface P´eclet number are also found
and used to study how reaction effects destroy the surfactant cap equilibrium.
Date Issued
2021-10-20
Date Acceptance
2021-08-30
Citation
Journal of Engineering Mathematics, 2021, 133 (10)
ISSN
0022-0833
Publisher
Springer
Journal / Book Title
Journal of Engineering Mathematics
Volume
133
Issue
10
Copyright Statement
Copyright reserved
License URL
Identifier
https://link.springer.com/article/10.1007%2Fs10665-021-10180-w
Subjects
Science & Technology
Technology
Physical Sciences
Engineering, Multidisciplinary
Mathematics, Interdisciplinary Applications
Engineering
Mathematics
Complex Burgers equation
Insoluble surfactant
Marangoni flow
Method of characteristics
Stagnant cap
INTERFACIAL FLOWS
MARANGONI NUMBER
DROP DEFORMATION
STOKES-FLOW
BUBBLES
STABILITY
FILM
EMULSIONS
EVOLUTION
RHEOLOGY
Applied Mathematics
0102 Applied Mathematics
0103 Numerical and Computational Mathematics
0913 Mechanical Engineering
Publication Status
Published
Date Publish Online
2021-10-20