Hele-Shaw model for melting/freezing with two dendrits
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Hele-Shaw model for melting/freezing with two dendrits
Sergei Rogosin a , Tatsyana Vaitekhovich b
Mechanical-Mathematical Faculty, Belarusian State University, Minsk, Belarus
a
rogosin@bsu.by, b vaiteakhovich@mail.ru
Keywords: melting/freezing, several dendrits, Hele-Shaw moving problem, Cauchy-Kovalevsky prob-
lem, Riemann-Hilbert problem for a doubly connected domain, Schwarz Alternation Method
Abstract. Melting/freezing process with two dendrits (or freeze “pipes”) is modelled by the
complex Hele-Shaw moving boundary value problem in a doubly connected domain. The later is
equivalently reduced to a couple of problems, namely, to the linear Riemann-Hilbert boundary
value problem in a doubly connected domain and to evolution problem, which can be written
in a form of an abstract Cauchy-Kovalevsky problem. The later is studied on the base of
Nirenberg-Nishida theorem, and for the former a generalization of the Schwarz Alternation
Method is proposed. By using composition of these two approaches we get the local in time
solvability of this couple of problems in appropriate Banach space setting.
Introduction
We are discussing the mathematical model for artificial melting/freezing in porous media. There
exists a big interest to the description of such processes in practice (see, e.g., [1], [2], [3]). There
are several theoretical approaches for the formulation and study of the corresponding model.
We have to mention here the paper [4] as a good reference on subject. One of these approaches
deals with the modelling of the melting/freezing process by the Hele-Shaw moving boundary
value problem. The study of complex-analytic Hele-Shaw model going back to [5] and to [6]
(cf. also the recent monograph [7]). This approach was applied to melting/freezing process in
[8]–[10].
We consider here the mathematical model for melting/freezing process presenting in the
form of the Hele-Shaw moving boundary problem. Following to [11] and [12] we rewrite the
corresponding Hele-Shaw problem as a couple of two problems, namely, the time-independent
Riemann-Hilbert problem for a doubly connected domain and an evolution problem reformu-
lated as an abstract Cauchy-Kovalevsky problem. In contrast to the case of simply connected
domain in our situation there is no explicit formula for the Riemann-Hilbert problem (see, e.g.,
[13]–[15], [16], cf., also [7]).
In order to overcome this difficulty we propose a generalization of the Schwarz Alternation
Method based on the ideas of [17]. The convergence of this method is shown. It gives together
with Nirenberg-Nishida theorem (see [18], [19]) for Cauchy-Kovalevsky problem (applied for
the discussed model in [20]) an existence of a local in time solution to the Hele-Shaw problem
for a doubly connected domain.
Mathematical Model
In the description of the physical situation as well as in formulation of physically consistent
model we follow the article [10], namely, we suppose that in a big enough box of saturated porousmedia two parallel cylindric freezing pipes of different base are situated. At certain moment
the frozen zone (ice) occupies two cylinders growing at freezing (or, wise-versa, unfrozen zone
is growing at melting). The question is to describe the dynamics of the boundary ice-water.
Since the uniformity of freezing with respect to the height one can reformulate the problem in
terms of the Hele-Shaw model. In the case of doubly connected domains the later consists in
the description of the dynamics of the boundary of two “spots” of fluid between two closely
related plates (caused by certain mechanisms). Without loss of generality we can suppose that
these plates are unbounded which means that the porous media box is unbounded too.
In complex-analytic model for Hele-Shaw flow (cf. [5]) we parametrize the unfrozen region
by a conformal mapping of the canonical domain (e.g. exterior of two discs) onto this region.
Since not any two doubly connected domain conformally equivalent [21] we need to determine
the parameters of the canonical domain too. Finally, the mathematical problem is formulated as
follows: given a function f0 (z), holomorphic and univalent in a neighbourhood of the “canonical”
domain
G(0) := {z ∈ C : |z − aj | > rj (0); j = 1, 2; |a1 − a2 | > r1 (0) + r2 (0)} ,
two functions Q1 (t), Q2 (t), continuous in a right-sided neighbourhood of t = 0, find a family of
functions f (z, t), holomorphic and univalent in z in a neighbourhood of the domain
G(t) := {z ∈ C : |z − aj | > rj (t); j = 1, 2; |a1 − a2 | > r1 (t) + r2 (t)} ,
and continuously differentiable in a right-sided neighbourhood of t = 0, satisfying the following
conditions µ ¶
1 ∂f ∂f
Re = Qj (t), (1)
z ∂t ∂z
for all z ∈ Γj (t) := {z ∈ C : |z − aj | = rj (t)} ; j = 1, 2; t ∈ [0, T ),
f (z, 0) = f0 (z), z ∈ G(0), (2)
Z ¯ ¯−2 Z ¯ ¯−2
¯ ∂f ¯ dτ ¯ ∂f ¯
¯ (τ, t)¯ Q1 (t) + ¯ (τ, t)¯ Q2 (t) dτ = 0, t ∈ [0, T ). (3)
¯ ∂z ¯ τ − a1 ¯ ∂z ¯ τ − a2
Γ1 (t) Γ2 (t)
Let us consider the structure of the problem (1)–(3). For each fixed (sufficiently small) t ≥ 0
the problem (1) can be understood as the Riemann-Hilbert boundary value problem
³ ´
Re λj (τ, t)ψ(τ, t) = cj (t), τ ∈ Γj (t), j = 1, 2, t ∈ [0, T ), (4)
with respect to an unknown function
∂f
ψ(z, t) := (z, t), z ∈ G(T ), t ∈ [0, T ), (5)
∂t
and with coefficients defined by
1 ∂f
λj (τ, t) := , cj (t) := Qj (t). (6)
z ∂z z=τ ∈Γj (t)
Since we are looking for univalent solution f (z, t) to (1)–(3), then ∂f
∂z
6= 0 in the considered
domains, and thus the index of problem (4) (or winding number of its coefficients) is vanishing
identically in an appropriate interval for time-variable t:
κ := 4Γ1 (t) arg λ1 (τ, t) + 4Γ2 (t) arg λ2 (τ, t) ≡ 0. (7)Therefore we can froze the variable t in our study of solvability of the problem (4) and con-
struction of the approximation procedure (since the domain for the problem are of the same
type and they will give no influence neither on the proof of solvability nor on approximation
procedure). By freezing variable t we omit it in all components of (4)–(6).
Relation (7), in principal (see [22]), guarantees the uniqueness of the solution to (4) (hence
of the solution to (1) with respect to ∂f
∂t
(z, t)). Moreover, an additional condition (3) yields the
single-valuedness of this solution. Two questions remains. There is no explicit formula for the
solution to (4) and we need to describe the procedure of obtaining of such solution. Moreover
we have to show that this solution is in fact univalent in the corresponding domain.
Let us denote by TG (λ, c) ≡ TG (λ1 , λ2 , c1 , c2 ) the operator determining the above said
solution of (4) under notations (5)–(6). If this operator is already constructed then by unfreezing
the variable t the relation (1) can be rewritten in the form
µ ¶
∂f 1 ∂f 1 ∂f
(z, t) = TG , , Q1 (t), Q2 (t) . (8)
∂t z ∂z |z=τ ∈Γ1 (t) z ∂z |z=τ ∈Γ2 (t)
It has together with initial condition (2) the form of an abstract Cauchy-Kovalevsky problem
(see, e.g. [11]) and can be handled by application of the following theorem due to Nirenberg
and Nishida
Theorem 1. (Nirenberg-Nishida) Let {Bs , k · ks }0Such assumption corresponds to expansion of both holes in the Hele-Shaw cell.
Let us fix constants ri,j i, j = 1, 2; rj (0) > r1,j > r2,j > 0, a positive number b > 0 and
introduce parameter
¡ s ¢∈ (0, 1).
Denote by H G(s) the space of functions, holomorphic in the following circular doubly
connected domain:
G(s) ≡ {z ∈ C : |z − aj | > r1,j − s(r1,j − r2,j ), j = 1, 2} .
Then we define the space
n [ ¡ ¢
B := g = g(z, t) ∈ C [0, b(1 − s)), H(G(s) ) ∩ C 1,α (G(s) ) :
04. Get the solution of the exterior Riemann-Hilbert problem on D− 1 via symmetry with
∗ r12
respect to T1 : z 7→ z1 := a1 + z−a1 (see [17]).
5. Get the solution of the exterior Riemann-Hilbert problem on D1− by applying ω1− . We have
got zero-approximation for the solution to the starting Riemann-Hilbert problem on D1− ∩ D2− .
This approximation satisfy the boundary condition (14) on L1 , but in general, does not satisfy
the boundary condition on L2 . We need to make some correction.
6. Substitute the solution for D− 1 onto the boundary condition on L21 = ω1 ◦ L2 . The left
hand side gives us the first corrector.
7. Subtract the first corrector on L21 from the left hand side of the image of the Riemann-
Hilbert problem under the mapping ω1 . We have got the corrected Riemann-Hilbert problem
on L21 .
−
8. Map D21 = extL21 conformally onto D− 21 = extT2 with the Riemann mapping ω2 :
−
D21 → D− 21 . We have obtained the exterior Riemann-Hilbert problem on D− 21 . (The image L112
of T1 = L11 will be now a certain curve outside T2 ).
9. Reformulate the exterior Riemann-Hilbert problem as an interior one (i.e. for D+ 21 ) and
solve it in terms of the Schwarz operator for D+ 21 .
10. Get the solution of the exterior Riemann-Hilbert problem on D− 21 via symmetry with
∗ r22
respect to T2 : z 7→ z2 := a2 + z−a2 (see [17]).
−
11. Get the solution of the exterior Riemann-Hilbert problem on D21 by applying ω2− . Get
the solution of the exterior Riemann-Hilbert problem on D2 by applying ω1− . The sum of zero-
−
approximation and the above obtained solution for D2− gives us first-order approximation for
the solution to the starting Riemann-Hilbert problem on D1− ∩ D2− . This approximation satisfy
the boundary condition (14) on L2 , but in general, does not satisfy the boundary condition on
L1 .
12. Substitute the solution for D− 21 onto the boundary condition on L112 = ω2 ◦ T1 . The left
hand side gives us the second corrector.
13. Subtract the first corrector on L112 from the left hand side of the image of the Riemann-
Hilbert problem under the mapping ω2 . We have got the corrected Riemann-Hilbert problem
on L112 .
14. Map D− − −1
21 = extT2 conformally onto D21 = extL21 with the mapping ω2 : D21 → D21 .
− −
The image of the L112 will be again the circle T1 . The image of the corrected problem on
L112 gives us the corrected exterior Riemann-Hilbert problem on the circle T1 . We can proceed
further beginning from the point 3.
15. Constructed series of the approximation converge to the solution of the problem (14)
due to boundedness of all applied operators and compactness of the operator consisting of two
successful symmetries (first with respect to T1 , then with respect to T2 or wise-versa) (see [17]).
Discussion and Outlook
The procedure of the approximate solution to the Riemann-Hilbert boundary value problem
together with Nirenberg-Nishida theorem give us possibility to construct a family of conformal
mappings f (z, t) solving the Hele-Shaw moving boundary value problem (1)-(3). It gives the
solution of the starting melting/freezing problem up to the situation when two frozen regions
touch each other (and thus the connectivity of the unfrozen region changes which is a cause of
changing of the type of the corresponding Riemann-Hilbert problem).Acknowledgments
The work is partially supported by the Belarusian Fund for Fundamental Scientific Research
(grants F05-036, F06R-106).
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