We describe a class of inhomogeneous two-dimensional porous medium flows, driven by a finite number of multipole sources; the free boundary dynamics can be parametrized by polynomial conformal maps

Howison, S. D.

Loutsenko, I.

Ockendon, J. R.

English

AbstractWe describe a class of inhomogeneous two-dimensional porous medium flows, driven by a finite number of multipole sources; the free boundary dynamics can be parametrized by polynomial conformal maps

Howison, S.D.

Ockendon, J.R.

Elsevier - Publisher Connector 

Applied Mathematics Letters 20 (2007) 93–97
www.elsevier.com/locate/aml
A class of exactly solvable free-boundary inhomogeneous porous
medium flows
S.D. Howison, I. Loutsenko, J.R. Ockendon∗
OCIAM, Mathematical Institute, University of Oxford, 24-29 St. Gilles, OX1 3LB, UK
Received 27 February 2006; accepted 6 March 2006
Abstract
We describe a class of inhomogeneous two-dimensional porous medium flows, driven by a finite number of multipole sources;
the free boundary dynamics can be parametrized by polynomial conformal maps.
c© 2006 Elsevier Ltd. All rights reserved.
Keywords: Inhomogeneous porous medium flow; Hele-Shaw flow; Intertwining operator
1. Inhomogeneous porous medium flows
A class of two-phase porous medium flows in two dimensions involves the dynamics of the boundary ∂Ω(t) in the
(x, y) plane separating two disjoint, open regions, the liquid (saturated) region Ω = Ω(t) and the unsaturated region
C\Ω¯ . The velocity of the liquid is proportional to the gradient of the pressure:
v = −κ∇ P, (1)
where the permeability κ = κ(z, z¯) is a real function, sufficiently regular inΩ , and z = x + iy, z¯ = x − iy are complex
coordinates on the plane. The flow is incompressible and the velocity satisfies the continuity equation
∇ · v = 0. (2)
The free boundary conditions are
P(∂Ω) = 0 (3)
and the normal velocity of the boundary is
vn = n · v for z ∈ ∂Ω , (4)
where n denotes the outward normal to the boundary.
∗ Corresponding author.
E-mail address: ejam@maths.ox.ac.uk (J.R. Ockendon).
0893-9659/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2006.03.005
94 S.D. Howison et al. / Applied Mathematics Letters 20 (2007) 93–97
The flow is driven by a point multipole source of order k + 1 located at z = z1, so that the pressure is singular at
this point. From (1) and (2) it follows that, away from singularities, the pressure satisfies
∇ · (κ∇ P) = 0, z = z1. (5)
When z approaches the singular point z1 the pressure tends to a linear combination of log |z − z1| and its k first
derivatives w.r.t. z and z¯. For instance, in the case of the homogeneous problem
κ(z, z¯) = 1, (6)
P ∼ −q(t)
2
log |z − z1| +
k∑
j=1
(−1) j−1
( j − 1)!
(
q j (t)
(z − z1) j +
q¯ j (t)
(z¯ − z¯1) j
)
+ O(1), as z → z1 (7)
where q(t) equals the time derivative of the area of Ω , while q j (t) is proportional to the time derivative of its j th
harmonic moment
∫
Ω (z − z1) j dxdy.
In the case of the simply connected domains Ω that we deal with, the boundary conditions (3), (4), the elliptic
equation (5) as well as the pressure asymptotics (e.g. (7)), together with the initial data Ω(0), constitute a free-
boundary problem for this flow.
2. Main result
The class of rational solutions to the homogeneous problem (6) has been considered in [3]. Here “rational” means
that Ω(t) is the image of the unit circle under a rational conformal map. In [3] it is shown that the form of simply
connected fluid domains resulting from the injection of fluid through monopole sources located at a finite number of
points into an initially empty medium is determined only by the fluxes injected at these points and is independent of
the history of the sources.
In the homogeneous case, the multipole-driven flows (7) can be considered as limiting cases of the above rational
solutions and the shape of the corresponding domains depends on the flux Q = ∫ t0 q(t ′)dt ′ as well as the “multipole
fluxes”
Qi =
∫ t
0
qi(t ′)dt ′, i = 1, . . . , k.
The above property results in an infinite number of conservation laws. Similar conservation laws hold for an arbitrary
inhomogeneous problem, with κ = κ(z, z¯) that is regular in Ω [1].
The main result of the present paper establishes a correspondence between homogeneous problems and a class of
inhomogeneous ones. We first take a homogeneous medium flow driven by multipoles located at z = z1 that admits
polynomial solutions, by which we mean that the domain is the image of the unit disc |w| < 1 under the polynomial
conformal map
z(w) = z1 + rw +
k∑
j=1
u jw1+ j , (8)
and its boundary is the image of |w| = 1, where the coefficients
r = r(Q, Q1, . . . , Qk, Q¯1, . . . , Q¯k), u j = u j (Q, Q1, . . . , Qk , Q¯1, . . . , Q¯k), j = 1, . . . , k (9)
are completely determined by the multipole fluxes.
Our main result is that the same fluid domain results from a special combination of multipole sources at z = z1
into an initially empty inhomogeneous medium with the permeability
κ = 1
(zs + z¯s)2n(zs − z¯s)2l , n > l ≥ 0, (10)
where s, n, l are non-negative integers. This is a rational homogeneous function of x, y and an invariant of the group of
a regular polygon (4s-gon if l > 0 or 2s-gon otherwise). The multiplicity of the sources at z = z1 is (s(n+l)+1)(k+1).
Note that when stating that the domain dynamics are the same, we mean that the boundary evolutions coincide,
while the flow dynamics differ in the homogeneous and inhomogeneous cases.
S.D. Howison et al. / Applied Mathematics Letters 20 (2007) 93–97 95
Obviously the same result holds for the fluid regions evolving from any initially non-empty regular “polynomial”
domains.
The pressure satisfies the elliptic PDE
∇ ·
(
1
(zs + z¯s)2n(zs − z¯s)2l ∇ P
)
= π ˆ˜q[δ(x − x1)δ(y − y1)], (11)
where ˆ˜q is the differential operator
ˆ˜q = q +
k˜∑
i=1
(
q˜i
∂ i
∂zi
+ ¯˜qi
∂ i
∂ z¯i
)
, k˜ = (k + 1)(s(n + l) + 1) − 1.
The k˜ multipole fluxes of the non-homogeneous problem,
Q˜i = (−1)i
∫ t
0
q˜i (t ′)dt ′, (12)
are fixed functions
Q˜i = Q˜i (Q, Q1, . . . , Qk , Q¯1, . . . , Q¯k, z1, z¯1), i = 1, . . . , k˜
of the homogeneous problem multipole fluxes Qi , i = 1, . . . , k (which are proportional to the harmonic moments∫
Ω (z − z1)i dxdy), as well as of the source position z1.
The elliptic equation (11) is a counterpart of the Poisson equation
P = π qˆ [δ(x − x1)δ(y − y1)] , qˆ = q +
k∑
i=1
(
qi
∂ i
∂zi
+ q¯i ∂
i
∂ z¯i
)
for the pressure in the corresponding homogeneous problem.
Before giving a sketch of the proof of the result, we give some examples.
3. Examples
Let us start with the simplest possible example, where the liquid is injected into an initially empty homogeneous
porous medium (6) through a monopole source at z = z1 = x1 + iy1.
By symmetry, the solution is a circular disc of radius r(t), centered at z = z1:
|z − z1| < r(t). (13)
The pressure satisfies
P = −πq(t)δ(x − x1)δ(y − y1),
where the source power and total flux are
q(t) = d(r(t)
2)
dt
, Q = r2
respectively.
According to Section 2, the variable-coefficient problem with permeability that varies as the inverse square of one
Cartesian coordinate (s = 1, n = 1, l = 0 in (10)),
κ = 1
x2
, (14)
admits the same circular solution (13) if the flow is driven by a combination of the same monopole source (of strength
q(t)) and a dipole source. To be precise, one has to add a dipole source located at the point z = z1, of strength
−q Q/2x1 to the monopole source, to preserve the circular shape of the domain in the non-homogeneous case (14).
96 S.D. Howison et al. / Applied Mathematics Letters 20 (2007) 93–97
The equation for the pressure distribution becomes
∇ ·
(
1
x2
∇ P
)
= −π ˆ˜q[δ(x − x1)δ(y − y1)], ˆ˜q = dr
2
dt
(
1 − r
2
2x1
∂
∂x
)
. (15)
It is not difficult to check that the pressure
P = r dr
dt
(
(2x1x + ρ2 + r2) log ρ − r
2x(x − x1)
ρ2
− ρ2 + x(x − x1) − (2x1x + ρ2) log r
)
,
where ρ = |z − z1|, is constant along the boundary. It satisfies (15) and the kinematic condition
dr
dt
= − 1
x2
(
∂ P
∂n
)
ρ=r
holds at the disc boundary. Therefore, the above pressure distribution is indeed the solution of the inhomogeneous
problem under consideration.
Next, consider an example of a homogeneous medium flow, again in an initially empty porous medium, driven by
a combination of monopole and dipole sources, both located at z = z1. The pressure satisfies the following equation
P = π
(
q(t) + q1(t) ∂
∂z
+ q¯1(t) ∂
∂ z¯
)
[δ(x − x1)δ(y − y1)]. (16)
According to (8) and (9) the boundary of the domain is the limac¸on
z = z1 + rw + u1w2, w = eiθ , 0 < θ ≤ 2π, (17)
where r, u1 are functions of Q =
∫ t
0 q(t
′)dt ′, Q1 = −
∫ t
0 q1(t
′)dt ′, Q¯1 = −
∫ t
0 q¯1(t
′)dt ′, determined by the following
equations:
Q = r2 + 2|u1|2, Q1 = u¯1r. (18)
The counterpart of (16) for the flow in a medium with permeability k = 1/x2 is
∇ ·
(
x−2∇ P
)
= π
(
q +
3∑
i=1
(
q˜i (t)
∂ i
∂zi
+ ¯˜qi (t)
∂ i
∂zi
))
[δ(x − x1)δ(y − y1)].
Now the parameters of the limacon map (17) are functions of
Q = r2 + 2|u1|2, Q˜1 = 14x1 (r
4 + 4u¯1x1r2 + 6|u1|2r2 + 2|u1|4),
Q˜2 = u¯1r
2
48x21
(12x1r2 − u¯1r2 + 24|u1|2x1), Q˜3 = 124x1 u¯
2
1r
4,
(19)
where the multipole fluxes are defined in (12). Eliminating r, u1 from the union of Eqs. (18) and (19), we can express
the fluxes Q˜i , i = 1, 2, 3 as functions of Q, Q1, Q¯1 (as well as x1).
4. Main result, idea of the proof
In this section we give an idea of the proof of our main result. The proof relies on the fact that an arbitrary solution
φ(z, z¯), regular in Ω , of the elliptic PDE
κ−1∇ · (κ∇φ) = 0, z ∈ Ω (20)
satisfies the quadrature identity∫
Ω
φdxdy = π ˆ˜Q[φ](z1, z¯1) (21)
S.D. Howison et al. / Applied Mathematics Letters 20 (2007) 93–97 97
if Ω is a zero-initial condition solution to the inhomogeneous problem with
∇ · (κ∇ P) = π ˆ˜q[δ(x − x1)δ(y − y1)], ˆ˜Q =
∫ t
0
ˆ˜q(t ′)dt ′.
The above property is a generalization of quadrature identities for harmonic functions [4] to the case of variable-
coefficient elliptic PDEs [1]. Therefore, for our purposes, the first chief idea is the correspondence between time-
parametrized quadrature identities and the evolution in time of the fluid domain in the porous medium. In the proof,
we obtain a complete set of solutions of (20) with κ given by (10) and then check that there exist values of the multipole
fluxes, such that the corresponding φ(z, z¯) indeed satisfy (21) in domains (8) that are solutions for the homogeneous
medium problem (these are special cases of algebraic domains, e.g. see [5]).
The second chief idea is to represent solutions to (10) and (20) in the form [2]
φ(z, z¯) = T [ f (z) + g(z¯)] (22)
where T is a differential operator of order s(n + l) with coefficients polynomial in z, z¯, and f, g are analytic and
antianalytic respectively in Ω . T is an intertwining operator that relates the elliptic equation (10), (20) with the
Laplace equation via
T = κ−1∇ · (κ∇) T .
From (8) and (22), the verification of the quadrature identity (21) for a polynomial map reduces to residue calculus,
leading to an algebraic system of equations for the magnitudes of the fluxes (12). This system has a unique solution [2]
and yields our main result, given in Section 2.
Acknowledgements
The authors are grateful to P. Etingof for useful remarks. The work of I.L. has been supported by the European
Community IIF MIF1-CT-2005-007323.
References
[1] V.M. Entov, P.I. Etingof, Bubble contraction in Hele-Shaw cells, Quart. J. Mech. Appl. Math. 44 (1991) 507–535.
[2] I. Loutsenko, http://xxx.lanl.gov/math-ph/0510070.
[3] S. Richardson, Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972)
609–618.
[4] H.S. Shapiro, The Schwartz Function and its Generalization to Higher Dimension, Wiley, New York, 1992.
[5] A. Varchenko, P. Etingof, Why the Boundary of a Round Drop Becomes a Curve of Order Four, in: University Lecture Series, AMS, 1992.


A class of exactly solvable free-boundary inhomogeneous porous medium flows 

Oxford University Research Archive (ORA)

A class of exactly solvable free-boundary inhomogeneous porous medium flows

Oxford University Research Archive

We describe a class of inhomogeneous two-dimensional porous medium flows, driven by a finite number of multipole sources; the free boundary dynamics can be parametrized by polynomial conformal maps. © 2006 Elsevier Ltd. All rights reserved

Howison, Samuel

Loutsenko, I

Ockendon, John

A class of exactly solvable free-boundary inhomogeneous porous medium flows.

Howison, SD

Ockendon, JR

Applied Mathematics Letters

Mathematical Institute Eprints Archive

A Class of Exactly Solvable Free-Boundary
Inhomogeneous Porous Medium Flows
S.D.Howison, I.Loutsenko, J.R.Ockendon
OCIAM, Mathematical Institute, University of Oxford, 24-29 St. Gilles, OX1
3LB, UK
Abstract
We describe a class of inhomogeneous two-dimensional porous medium
flows, driven by a finite number of multipole sources; the free bound-
ary dynamics can be parametrized by polynomial conformal maps.
1 Inhomogeneous Porous-Medium Flows
A class of two-phase porous medium flows in two dimensions involves the dy-
namics of the boundary ∂Ω(t) in the (x, y) plane separating two disjoint, open
regions, the liquid (saturated) region Ω = Ω(t) and the unsaturated region C\Ω¯.
The velocity of the liquid is proportional to the gradient of the pressure
v = −κ∇P, (1)
where the permeability κ = κ(z, z¯) is a real function, sufficiently regular in Ω,
and z = x+ iy, z¯ = x− iy are complex coordinates on the plane. The flow is
incompressible and the velocity satisfies the continuity equation
∇ · v = 0. (2)
The free boundary conditions are
P (∂Ω) = 0 (3)
and the normal velocity of the boundary is
vn = n · v, for z ∈ ∂Ω, (4)
where n denotes the outward normal to the boundary.
The flow is driven by a point multipole source of order k+1 located at z = z1,
so that the pressure is singular at this point. From (1), (2), (7) it follows that,
away from singularities, the pressure satisfies
∇ · (κ∇P ) = 0, z 6= z1. (5)
1
When z approaches the singular point z1 the pressure tends to a linear combi-
nation of log |z − z1| and its k first derivatives w.r.t. z and z¯. For instance, in
the case of the homogeneous problem
κ(z, z¯) = 1, (6)
P ∼ −q(t)
2
log |z−z1|+
k∑
j=1
(−1)j−1
(j − 1)!
(
qj(t)
(z − z1)j +
q¯j(t)
(z¯ − z¯1)j
)
+O(1), as z → z1
(7)
where q(t) equals the time derivative of the area of Ω, while qj(t) is proportional
to the time derivative of its jth harmonic moment
∫
Ω
(z − z1)jdxdy.
In the case of simply-connected domains Ω that we deal with, the boundary
conditions (3), (4), the elliptic equation (5) as well as the pressure asymptotics
(e.g. (7)), together with the initial data Ω(0), constitute a free-boundary prob-
lem for this flow.
2 Main Result
The class of rational solutions to the homogeneous problem (6) has been con-
sidered in [3]. Here “rational” means that Ω(t) is the image of the unit circle
under a rational conformal map. In [3] it is shown that the form of simply-
connected fluid domains, resulting from the injection of fluid through monopole
sources located at a finite number of points into an initially empty medium, is
determined only by the fluxes injected at these points and is independent of the
history of the sources.
In the homogeneous case, the multipole-driven flows (7) can be considered as
limiting cases of the above rational solutions and the shape of the corresponding
domains depends on the flux Q =
∫ t
0
q(t′)dt′ as well as the “multipole fluxes”
Qi =
∫ t
0
qi(t′)dt′, i = 1, ..., k.
The above property results in an infinite number of conservation laws. Simi-
lar conservation laws hold for an arbitrary inhomogeneous problem, with κ =
κ(z, z¯) that is regular in Ω [1].
The main result of the present paper establishes a correspondence between
homogeneous problems and a class of inhomogeneous ones;
We first take a homogeneous medium flow driven by multipoles located at
z = z1 that admits polynomial solutions, by which we mean that the domain is
the image of the unit disc |w| < 1 under the polynomial conformal map
z(w) = z1 + rw +
k∑
j=1
ujw
1+j , (8)
2
and its boundary is the image of |w| = 1, where the coefficients
r = r(Q,Q1, .., Qk, Q¯1, .., Q¯k), uj = uj(Q,Q1, .., Qk, Q¯1, .., Q¯k), j = 1, .., k
(9)
are completely determined by the multipole fluxes.
Our main result is that the same fluid domain results from a special com-
bination of multipole sources at z = z1 into an initially empty inhomogeneous
medium with the permeability
κ =
1
(zs + z¯s)2n(zs − z¯s)2l , n > l ≥ 0, (10)
where s, n, l are non-negative integers. This is a rational homogeneous function
of x, y and an invariant of the group of a regular polygon (4s-gon if l > 0 or 2s-
gon otherwise). The multiplicity of the sources at z = z1 is (s(n+ l)+1)(k+1).
Note that when stating that the domain dynamics are the same, we mean
that the boundary evolutions coincide, while the flow dynamics differ in the
homogeneous and inhomogeneous cases.
Obviously the same result holds for the fluid regions evolving from any ini-
tially non-empty regular “polynomial” domains.
The pressure satisfies the elliptic PDE
∇ ·
(
1
(zs + z¯s)2n(zs − z¯s)2l∇P
)
= piˆ˜q[δ(x− x1)δ(y − y1)], (11)
where ˆ˜q is the differential operator
ˆ˜q = q +
ek∑
i=1
(
q˜i
∂i
∂zi
+ ¯˜qi
∂i
∂z¯i
)
, k˜ = (k + 1)(s(n+ l) + 1)− 1.
The k˜ multipole fluxes of the non-homogeneous problem,
Q˜i = (−1)i
∫ t
0
q˜i(t′)dt′, (12)
are fixed functions
Q˜i = Q˜i(Q,Q1, ..., Qk, Q¯1, ..., Q¯k, z1, z¯1), i = 1..k˜
of the homogeneous problem multipole fluxes Qi, i = 1, .., k (which are pro-
portional to the harmonic moments
∫
Ω
(z − z1)idxdy), as well as of the source
position z1.
The elliptic equation (11) is a counterpart of the Poisson equation
∆P = piqˆ [δ(x− x1)δ(y − y1)] , qˆ = q +
k∑
i=1
(
qi
∂i
∂zi
+ q¯i
∂i
∂z¯i
)
for the pressure in the corresponding homogeneous problem.
Before giving a sketch of the proof of the result, we give some examples.
3
3 Examples
Let us start with the simplest possible example, where the liquid is injected
into an initially empty homogeneous porous medium (6) through the monopole
source at z = z1 = x1 + iy1.
By symmetry, the solution is a circular disc of radius r(t), centered at z = z1
|z − z1| < r(t). (13)
The pressure satisfies
∆P = −piq(t)δ(x− x1)δ(y − y1),
where the source power and total flux are
q(t) =
d(r(t)2)
dt
, Q = r2
respectively.
According to Section 2, the variable-coefficient problem with permeability
that varies as the inverse square of one Cartesian coordinate (s = 1, n = 1, l = 0
in (10))
κ =
1
x2
(14)
admits the same circular solution (13) if the flow is driven by a combination
of the same monopole source (of strength q(t)) and a dipole source. To be
precise, one has to add a dipole source located at the point z = z1, of strength
−qQ/2x1 to the monopole source, to preserve the circular shape of the domain
in the non-homogeneous case (14). The equation for the pressure distribution
becomes
∇ ·
(
1
x2
∇P
)
= −piˆ˜q[δ(x− x1)δ(y − y1)], ˆ˜q = dr
2
dt
(
1− r
2
2x1
∂
∂x
)
. (15)
It is not difficult to check that the pressure
P = r
dr
dt
(
(2x1x+ ρ2 + r2) log ρ− r
2x(x− x1)
ρ2
− ρ2 + x(x− x1)− (2x1x+ ρ2) log(r)
)
,
where ρ = |z − z1|, is constant along the boundary. It satisfies (15) and the
kinematic condition
dr
dt
= − 1
x2
(
∂P
∂n
)
ρ=r
holds at the disc boundary. Therefore, the above pressure distribution is indeed
the solution of the inhomogeneous problem under consideration.
4
Next, consider an example of a homogeneous medium flow, again in an
initially empty porous medium, driven by a combination of monopole and dipole
sources, both located at z = z1. The pressure satisfies the following equation
∆P = pi
(
q(t) + q1(t)
∂
∂z
+ q¯1(t)
∂
∂z¯
)
[δ(x− x1)δ(y − y1)]. (16)
According to (8), (9) the boundary of the domain is the limacon
z = z1 + rw + u1w2, w = eiθ, 0 < θ ≤ 2pi, (17)
where r, u1 are functions ofQ =
∫ t
0
q(t′)dt′, Q1 = −
∫ t
0
q1(t′)dt′, Q¯1 = −
∫ t
0
q¯1(t′)dt′,
determined by the following equations:
Q = r2 + 2|u1|2, Q1 = u¯1r. (18)
The counterpart of (16) for the flow in a medium with permeability k = 1/x2 is
∇ · (x−2∇P ) = pi(q + 3∑
i=1
(
q˜i(t)
∂i
∂zi
+ ¯˜qi(t)
∂i
∂zi
))
[δ(x− x1)δ(y − y1)],
Now the parameters of the limacon map (17) are functions of
Q = r2 + 2|u1|2, Q˜1 = 14x1 (r4 + 4u¯1x1r2 + 6|u1|2r2 + 2|u1|4),
Q˜2 = u¯1r
2
48x21
(12x1r2 − u¯1r2 + 24|u1|2x1), Q˜3 = 124x1 u¯21r4
(19)
where the multipole fluxes are defined in (12). Eliminating r, u1 from the union
of equations (18) and (19), we can express the fluxes Q˜i, i = 1, 2, 3 as functions
of Q,Q1, Q¯1 (as well as x1).
4 Main result, idea of the proof
In this section we give an idea of the proof of our main result. The proof relies
on the fact that an arbitrary solutions φ(z, z¯), regular in Ω, of the elliptic PDE
κ−1 · (∇κ∇φ) = 0, z ∈ Ω (20)
satisfies the quadrature identity∫
Ω
φdxdy = pi ˆ˜Q[φ](z1, z¯1) (21)
if Ω is a zero-initial condition solution to the inhomogeneous problem with
∇ · (κ∇P ) = piˆ˜q[δ(x− x1)δ(y − y1)].
5
The above property is a generalization of quadrature identities for harmonic
functions [4] to the case of variable-coefficient elliptic PDEs [1]. Therefore, for
our purposes, the first chief idea is the correspondence between time-parametrized
quadrature identities and the evolution in time of the fluid domain in the porous
medium. In the proof, we obtain a complete set of solutions of (20) with κ given
by (10) and then check that there exist values of multipole fluxes, such that cor-
responding φ(z, z¯) indeed satisfy (21) in domains (8) that are solutions for the
homogeneous medium problem (these are special cases of algebraic domains,
e.g. see [5]).
The second chief idea is to represent solutions to (10), (20) in the form [2]
φ(z, z¯) = T [f(z) + g(z¯)] (22)
where T is a differential operator of order s(n+l) with coefficients polynomial in
z, z¯, and f, g are analytic and antianalytic respectively in Ω. T is an intertwining
operator that relates the elliptic equation (10), (20) with the Laplace equation
via
T∆ = κ−1∇ · (κ∇)T
From (8), (22), the verification of the quadrature identity (21) for a polynomial
map reduces to residue calculus, leading to an algebraic system of equations for
the magnitudes of the fluxes (12) injected. This system has a unique solution
[2] and yields our main result, given in Section 2.
Acknowledgements
The authors are grateful to P. Etingof for useful remarks. The work of I.L.
has been supported by the European Community IIF MIF1-CT-2005-007323.
References
[1] Entov, V. M.; Etingof, P. I. Bubble contraction in Hele-Shaw cells. Quart.
J. Mech. Appl. Math. 44 (1991), 507–535
[2] Loutsenko I., http://xxx.lanl.gov/math-ph/0510070
[3] Richardson S. Hele Shaw flows with a free boundary produced by the injec-
tion of fluid into a narrow channel. J.Fluid Mech., 56, (1972), pp.609-618
[4] Shapiro H.S., The Schwartz function and its generalization to higher dimen-
sion, Wiley, new York, (1992)
[5] Varchenko A., Etingof P., Why the boundary of a round drop becomes a
curve of order four, University Lecture Series, AMS, (1992)
6


Bubble contraction in Hele-Shaw cells.

Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel.

The Schwartz function and its generalization to higher dimension,

Why the boundary of a round drop becomes a curve of order four,

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A class of exactly solvable free-boundary inhomogeneous porous medium flows

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