Imperial College London
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ProfessorDarrenCrowdy

Faculty of Natural Sciences, Department of Mathematics

Professor in Applied Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 8587d.crowdy Website

 
 
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Location

 

735Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Crowdy:2021:10.1137/21M1400316,
author = {Crowdy, D},
doi = {10.1137/21M1400316},
journal = {SIAM Journal on Applied Mathematics},
pages = {2526--2546},
title = {Viscous Marangoni flow driven by insoluble surfactant and the complex Burgers equation},
url = {http://dx.doi.org/10.1137/21M1400316},
volume = {81},
year = {2021}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - A new mathematical connection is established between a class of two dimensional viscous Marangoni flows driven by insoluble surfactant and the complex Burgers equation. It is shown that the Marangoni-driven dynamics of a bath of viscous fluid at zero Reynolds and capillary number, and with a linear equation of state, is described by the evolution of a lower-analytic function with positive imaginary part on the real line satisfying the complex Burgers equation.  Surface diffusion of  surfactant  plays  the  role  of  viscosity  in  the  more  familiar  real-valued  Burgers  equation  arising in gas dynamics.  Using this mathematical connection it is shown that, at arbitrary surface P eclet number, the Marangoni dynamics is linearizable, and integrable, via a transformation of Cole-Hopf type.   A  new  class  of  time-evolving  exact  solutions  is  identified  for  the  Marangoni-induced  fluid motion at any finite surface P eclet number.  These are shown to be given by a class of evolvingN- pole solutions which differ from, and generalize, known pole dynamics solutions to the real Burgers equation.   Analogous  meromorphic  solutions  describing  spatially  singly-periodic  Marangoni  flows are  also  reported.   For  infinite  surface  P eclet  number  it  is  shown  how  a  generalized  method  of characteristics leads to an implicit form of the general solution. For a special choice of initial condition it is demonstrated that this implicit solution can be made explicit and,  from it,  the formation at finite time of an instantaneous weak singularity is observed.  Together these new solutions afford a mathematical view of the effect of surface diffusion on Marangoni flows via the evolution of complex singularities  in  a  non-physical  region  of  the  complex  plane.   The  observations  open  up  valuable new mathematical connections between viscous Marangoni flows and the theory of caloric functions, Calogero-Moser systems, random matrices and Dyson diffusi
AU  - Crowdy,D
DO  - 10.1137/21M1400316
EP  - 2546
PY  - 2021///
SN  - 0036-1399
SP  - 2526
TI  - Viscous Marangoni flow driven by insoluble surfactant and the complex Burgers equation
T2  - SIAM Journal on Applied Mathematics
UR  - http://dx.doi.org/10.1137/21M1400316
UR  - https://epubs.siam.org/doi/10.1137/21M1400316
UR  - http://hdl.handle.net/10044/1/92218
VL  - 81
ER  -
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