COMPLEX AND POTENTIAL ANALYSIS IN HELE-SHAW CELLS ...

COMPLEX AND POTENTIAL ANALYSIS IN HELE-SHAW CELLS
(OVERVIEW)
BJÖRN GUSTAFSSON AND ALEXANDER VASIL’EV
H.S.Hele-Shaw wrote his short note in 1898 where he proposed a cell consisting of two
parallel plates with a small gap between them and first considered a viscous flow in this cell.
Almost 50 years later P.Ya.Polubarinova-Kochina, L.A.Galin, P.P.Kufarev made the next
important step applying complex variable methods to describe the shape of time-dependent
phase domain in two dimensions. Their works remained unknown for many western researches
for approximately 30 years after appearance. In 1958 a remarkable contribution
was made by Sir G.I.Taylor and P.G.Saffman who discovered the viscous fingering phenomenon
named after them ”Saffman-Taylor fingering” which became a powerful tool in many
fields of natural and engineering sciences. The first modern description of the complex variable
approach and the study of the complex moments was made by S.Richardson in 1972.
Since then, step by step, Hele-Shaw problem has been converted into a modern challenging
branch of applied mathematics. Contributions made by scientists from Great Britain
(J.R.Ockendon, S.D.Howison, C.M.Elliott, S.Richardson, J.R.King, L.J.Cummings) are to
be emphasized. The last couple of decades the interest to Hele-Shaw flows has increased considerably
and such problems are now studied from different aspects all over the world. Thus,
a systematical mathematical up-to-date treatment of the subject in the book form turns
out to be quite necessary at the moment. Based on our experience and previous works on
Hele-Shaw flows we took this risk to write such a book which is a cross between a monograph
and lecture notes.
The book is intended for a wide group of readers from graduate students to applied
scientists and mathematicians. In particular, it contains a comprehensive overview of the
Navier-Stokes equations and deduction of the Hele-Shaw equation and further development
of the Hele-Shaw flows. We have tried to keep the book as self-contained as possible. For
most parts of this book we assume the background provided by graduate courses in real and
complex analysis, in particular, the theory of conformal mappings and in fluid mechanics.
We also try to make some historical remarks concerning the persons that have contributed
to the topic.
In our book, we aim at giving a presentation of recent and new ideas that arise from the
problems of planar fluid dynamics and which are interesting from the point of view of geometric
function theory and potential theory as well as various applications. In particular, we
are concerned with geometric problems for Hele-Shaw flows. Several known and new explicit
solutions are constructed. We also view Hele-Shaw flows on modelling spaces (Teichmueller
spaces) what is proved to be useful in mathematical physics (I.Krichever, P.Wiegmann,
1

2 BJÖRN GUSTAFSSON AND ALEXANDER VASIL’EV
A.Zabrodin et al.). Ultimately, we see the interaction between several branches of complex
and potential analysis, and planar fluid mechanics.
The book contains 6 chapters. The first is ”Introduction and background”. It contains
a necessary quintessence of the information on several topics of Fluid Mechanics and Conformal
Maps. All basic equations are deduced using Reinolds’ Transport Theorem, such as
the continuity equation, the Euler equation, the Navier-Stokes equation. Then we obtain
the Hele-Shaw and Polubarinova-Galin equations and discuss the problems of solvability,
uniqueness and regularization. We also consider the Polubarinova-Galin equation from different
points of view, in particular, Richardson’s complex moments, the Schwarz function,
relations with integrable systems.
Chapter ”Explicit solutions” is dedicated to those of the Polubarinova-Galin equation.
We start with the famous Polubarinova-Galin’s cardioid and Saffman-Taylor’s fingers, then
continue with rational solutions. The existence and uniqueness theorem is proved for rational
solutions as well as their several properties are deduced. We proceed with corner flows
and derive explicit solutions in terms of hypergeometric functions. They are analogous to
Saffman-Taylor’s fingers and their particular cases are got earlier by Kadanoff, Ben Amar,
Couder and others.
In the previous chapter we discussed strong solutions, which for their definition required
smooth analytic boundaries. ”Weak solutions and balayage” chapter is devoted to weak
solutions and their relations to potential theory. Weak solutions are closer to the physical
model and the existence-uniqueness theorem is easier to prove. We give this proof and
discuss the geometry of weak solutions. The connections with quadrature domains and with
balayage are revealed. We prove a theorem on the uniqueness of the backward in time
solutions under some assumptions on their geometry.
In the next chapter we deal with geometric properties of general Hele-Shaw flows. Special
classes of univalent functions that admit explicit geometric interpretations are considered to
characterize the shape of the free interface under injection. In particular, we are concerned
with the following question: which geometrical properties are preserved during the time
evolution of the moving boundary. We also discuss the geometry of weak solutions. Here we
prove a general ”inner normal theorem” which has various applications.
The chapter ”Capacities and isoperimetric inequalities” presents several results on connections
between the rate of the area growth and some characteristics of the moving boundary
in several models of Hele-Shaw flows. In particular, we get connections between Hele-Shaw
problem and the estimates of integral means which is now a topical problem in conformal
maps (Brennan’s Conjecture). We also obtain several such estimates for phase domains of
special geometry.
The last chapter is dedicated to general evolution equations motivated by the Polubarinova-
Galin equation. In fact, the solutions to the Polubarinova-Galin equation form subordination
chains that was a subject of deep investigation in conformal maps and what is called the

COMPLEX AND POTENTIAL ANALYSIS IN HELE-SHAW CELLS (OVERVIEW) 3
Loewner-Kufarev theory. We study projections of such chains to the modeling space (Teichmueller
space) and derive general evolution equations for quasiconformal and smooth curves.
This leads to investigation of a manifold that appeared earlier in string theory and which
is connected with so called Virasoro-Bott group. As a result we obtain a projection of the
Hele-Shaw dynamics onto the Teichmueller space. This is closely related to recent investigations
of integrable systems and their visualization on the moduli space of Riemann surfaces.
Finally we discuss the Diffusion-Limited Aggregation model and fractal growth.