# A Special Class of Univalent Functions in Hele-Shaw Flow Problems

Authors: Paula Curt; Denisa Fericean
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection. Applying methods from the theory of univalent functions, we prove the invariance
in time ofΦ-likeness property (a geometric property which includes starlikeness
and spiral-likeness) for two basic cases: the inner problem and the
outer problem. We study both zero and nonzero surface tension models.
Certain particular cases are also presented.


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## Body

## 1. Introduction

The time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection was studied by many authors. By using methods of univalent functions theory, they proved that certain geometric properties (such as starlikeness and directional convexity) are preserved in time [1–6]. In this paper, we continue their study by proving the invariance in time of another geometric property: Φ-likeness.In the first section of the paper, we review the main results that are needed later. We start by presenting the basic notions regarding the bounded case.In this case, we study the flow of a viscous fluid in a planar Hele-Shaw cell under injection through a source (of constant strengthQ, Q<0 in case of injection) which is situated at the origin. Suppose that at the initial moment the domain Ω(0) occupied by the fluid is simply connected and is bounded by an analytic and smooth curve Γ(0)=∂Ω(0). By using the well-known Riemann mapping theorem, the domain Ω(t) (occupied by the fluid at the moment t) can be described by a univalent function f(ζ,t) of the unit disk U={ζ:|ζ|<1} onto Ω(t) normalized by f(0,t)=0, f′(0,t)>0. We denote Γ(t)=∂Ω(t). The function f(ζ,0)=f0(ζ) produces a parametrization of Γ0={f0(eiθ),θ∈[0,2π)}, and the moving boundary is parameterized by Γ(t)={f(eiθ,t),θ∈[0,2π)}.The equation satisfied by the free boundary was first derived by Galin [7] and Polubarinova-Kochina [8, 9]:(1.1)Re[ḟ(ζ,t)ζf′(ζ,t)¯]=-Q2π,ζ=eiθ
(in the previous equality we have used the notations: f′=∂f/∂ζ, ḟ=∂f/∂t).A classical solution in the interval[0,T) is a function f(ζ,t), t∈[0,T), that is univalent on U¯ and C1 with respect to t in [0,T). It is known that, starting with an analytic and smooth boundary Γ(0), the classical solution exists and is unique locally in time (see [10]; see e.g., [11, Chapter 1]). Note that T is called the blow-up time.In the case of the problem of injection(Q<0) of the fluid into a bounded simply connected with small surface tension γ>0, the Polubarinova-Galin equation [6] is of the form:(1.2)Re[ḟ(ζ,t)ζf′(ζ,t)¯]=-Q2π+γH[i∂k∂θ(eiθ,t)](θ),ζ=eiθ,
where k(eiθ,t)=(1/|f′(eiθ,t)|)Re(1+(eiθf′′(eiθ,t)/f′(eiθ,t))), θ∈[0,2π), and the Hilbert transform in (1.2) is given by (1/π)P.V.θ∫02πΦ(eiθ′)/(1-ei(θ-θ′))dθ′=H[Φ](θ).We mention the following technical results:(∂k/∂θ)(eiθ,t)=-Im(e2iθSf(eiθ,t))/|f′(eiθ,t)| where Sf is the Schwarzian derivative given by Sf(ζ)=(f′′(ζ,t)/f′(ζ,t))′-(1/2)(f′′(ζ,t)/f′(ζ,t))2 and(1.3)∂∂θ[i∂k∂θ(eiθ,t)](θ)=-∂∂θ[iIm⁡(e2iθSf(eiθ))|f′(eiθ)|]=H[iA](θ),
where
(1.4)A(ζ)=(1|f′(ζ)|)[Re(2ζ2Sf(ζ)+ζ[(f′′(ζ)f′(ζ))′-f′′(ζ)f′(ζ)(f′′(ζ)f′(ζ))′])+Im⁡ζf′′(ζ)f′(ζ)Im⁡ζ2Sf(ζ)(2ζ2Sf(ζ)+ζ[(f′′(ζ)f′(ζ))′-f′′(ζ)f′(ζ)(f′′(ζ)f′(ζ))′])].The case of unbounded domain with bounded complement can be viewed as the dynamics of a contracting bubble in a Hele-Shaw cell since the fluid occupies a neighbourhood of infinity and injection (of constant strengthQ<0) is supposed to take place at infinity. Again, we denote by Ω(t) the domain occupied by the fluid at the moment t, Γ(t)=∂Ω(t). By using the Riemann mapping theorem, the domain Ω(t) can be described by a univalent function F(ζ,t) from the exterior of the unit disk U-={ζ∣|ζ|>1} onto Ω(t), F(ζ,t)=aζ+a0+a1/ζ+⋯, a>0.The equation satisfied by the free boundary is [4, 6](1.5)Re[Ḟ(ζ,t)ζF′(ζ,t)¯]=Q2π,ζ=eiθ
for the zero tension surface model and(1.6)Re[Ḟ(ζ,t)ζF′(ζ,t)¯]=Q2π-γH[i∂k∂t(eiθ,t)](θ),ζ=eiθ
for the small surface tension model.


## 2. The Inner Problem (Bounded Domains)

In this section, we obtain the invariance in time ofΦ-likeness property for the inner problem. Starting with an initial bounded domain Ω(0) which is Φ-like, we prove that at each moment t∈[0,T) the domain Ω(t) is Φ-like (both for zero and nonzero surface tension models).Definition 2.1.
Letf be a holomorphic function on U such that f(0)=0 and f′(0)≠0. Let Φ be a holomorphic function on f(U) such that Φ(0)=0 and ReΦ′(0)>0. We say that f is Φ-like on U (or Φ-like) if
(2.1)Re[zf′(z)Φ(f(z))]>0,for eachz∈U.We remark that aΦ-like function is univalent. In fact, any univalent function is Φ-like for some Φ.Remark 2.2.
(a) The concept ofΦ-likeness was introduced and studied by Brickman in 1973 [12] and generalizes the notions of starlikeness and spiral-likeness. Applications of this notion in the study of univalence may be found in [13].
(b) IfΦ(w)=w in the above definition, then f is starlike.
(c) IfΦ(w)=λw and Reλ>0, then f is spiral-like of type -argλ.We restate that a holomorphic functionf on U such that f(0)=0 and f′(0)≠0 is said to be spiral-like of type α∈(-π/2,π/2) if Re(eiαzf′(z)/f(z))>0, z∈U [13, 14].The following result is a generalization of [1,Theorem 1] to the case of Φ-like functions. The mentioned theorem may be obtained by taking Φ(w)≡w in Theorem 2.3 below.Theorem 2.3.
LetQ<0 and f0 be a function which is Φ-like on U and univalent on U¯. Let f(ζ,t) be the classical solution of the Polubarinova-Galin equation (1.1) with the initial condition f(ζ,0)=f0(ζ). Also let Ω=⋃0≤t<TΩ(t)=⋃0≤t<Tf(U,t), where T is the blow-up time. If Φ is holomorphic on Ω¯ and satisfies the condition
(2.2)ReΦ′(w)>0,∀w∈Ω¯,
then f(ζ,t) is Φ like for t∈[0,T).Proof.
Taking into account that all the functionsf(ζ,t) have analytic univalent extensions to U¯ for each t∈[0,T) and in consequence their derivatives f′(ζ,t) are continuous and do not vanish in U¯, we can replace with “≥” the inequality in the definition (2.1) of a Φ-like function. The equality can be attained only for |ζ|=1.
We suppose by contrary that the conclusion of Theorem2.3 is not true. Then there exist t0≥0 and ζ0=eiθ0 such that arg(ζ0f′(ζ0,t0)/Φ(f(ζ0,t0)))=π/2(or-π/2), which is equivalent to
(2.3)Reζ0f′(ζ0,t0)Φ(f(ζ0,t0))=0,Im⁡ζ0f′(ζ0,t0)Φ(f(ζ0,t0))≠0,
and for each ɛ>0, there are t>t0 and θ∈(θ0-ɛ,θ0+ɛ) such that
(2.4)argeiθf′(eiθ,t)Φ(f(eiθ,t))≥π2(or≤-π2).
Lett0 be the first such point, t0∈[0,T). Without loss of generality, we assume that
(2.5)Im⁡ζ0f′(ζ0,t0)Φ(f(ζ0,t0))>0.
In fact,(1,θ0) is a maximum point for the function g(r,θ)=arg(reiθf′(reiθ,t0)/Φ(f(reiθ,t0))), where r∈[0,1], θ∈[0,2π]. Hence, (∂/∂θ)g(1,θ0)=0 and (∂/∂r)g(1,θ0)≥0 (the stationarity condition at an endpoint of an interval), and in consequence we obtain
(2.6)Re(1+ζ0f′′(ζ0,t0)f′(ζ0,t0)-Φ′(f(ζ0,t0))ζ0f′(ζ0,t0)Φ(f(ζ0,t0)))=0,(2.7)Im⁡(1+ζ0f′′(ζ0,t0)f′(ζ0,t0)-Φ′(f(ζ0,t0))ζ0f′(ζ0,t0)Φ(f(ζ0,t0)))≥0.
By straightforward calculations, we get(2.8)∂∂targζf′(ζ,t)Φ(f(ζ,t))=Im⁡((∂/∂t)f′(ζ,t)f′(ζ,t)-Φ′(f(ζ,t))(∂/∂t)f(ζ,t)Φ(f(ζ,t))).
By differentiating the Polubarinova-Galin equation with respect toθ, we obtain Im((∂/∂t)f′(ζ,t)f′(ζ,t)¯-ζf′(ζ,t)¯(∂/∂t)f(ζ,t)-ζ2f′′(ζ,t)¯(∂/∂t)f(ζ,t))=0 for ζ=eiθ. The previous equality yields the following relation:
(2.9)|f′(ζ,t)|2Im⁡((∂/∂t)f′(ζ,t)f′(ζ,t)-Φ′(f(ζ,t))⋅(∂/∂t)f(ζ,t)Φ(f(ζ,t)))=Im⁡(ζf′(ζ,t)¯ḟ(ζ,t))(1+(ζf′′(ζ,t)f′(ζ,t))¯-Φ′(f(ζ,t))⋅ζf′(ζ,t)Φ(f(ζ,t))).
If we substitute (2.6), (2.7), (2.8), and (1.1) in the above expression and replace θ by θ0 and t by t0, we obtain
(2.10)∂∂targ⁡ζf′(ζ,t)Φ(f(ζ,t))|ζ=ζ0,t=t0=Q2π|f′(ζ0,t0)|2[Im⁡(ζ0f′′(ζ0,t0)f′(ζ0,t0)-Φ′(f(ζ0,t0)ζ0f′(ζ0,t0))Φ(f(ζ0,t0)))+2ReΦ′(f(ζ0,t0))Im⁡ζ0f′(ζ0,t0)Φ(f(ζ0,t0))]<0,
due to (2.2), (2.3) and (2.5). Finally, we get (∂/∂t)arg(ζf′(ζ,t)/Φ(f(ζ,t)))|ζ=ζ0,t=t0<0. Therefore, arg(eiθf′(eiθ,t)/Φ(f(eiθ,t)))<π/2, for t>t0 (close to t0) in some neighbourhood of θ0. This contradicts the assumption (2.4) and completes the proof.If in the previous theorem, we takeΦ(w)≡e-iαw, α∈(-π/2,π/2); then we obtain the following corollary.Corollary 2.4.
LetQ<0, and let f0 be a function which is spiral-like of type α∈(-π/2,π/2) on U and univalent on U¯. Then the classical solution of the Polubarinova-Galin equation (1.1) with the initial condition f(ζ,0)=f0(ζ) is spiral-like of type α for t∈[0,T), where T is the blow-up time.The following result is a generalization of [6,Theorem 1] to the case of Φ-like functions. The mentioned theorem may be obtained by taking Φ(w)≡w in Theorem 2.5 below.Theorem 2.5.
LetQ<0 and the surface tension γ be sufficiently small. If f0 is a function which is Φ-like on U and univalent on U¯, then there exists t(γ)≤T such that the classical solution f(ζ,t) of (1.2) with the initial condition f(ζ,0)=f0(ζ) is Φ-like for t∈[0,t(γ)), where T is the blow-up time, Ω=⋃0≤t<t(γ)Ω(t)=⋃0≤t<t(γ)f(U,t), and Φ is a holomorphic function on Ω¯ which satisfies the condition (2.2).Proof.
If we considerf in the closure of U, then the inequality sign in (2.1) can be replaced by “≥” where equality can be attained for |ζ|=1.
Suppose by contrary that the conclusion of Theorem2.5 is not true. Then there exist t0≥0 and ζ0=eiθ0 such that (2.3), (2.4), (2.5), (2.6), and (2.7) are true. At the same time the equality (2.8) is fulfilled. By differentiating the Polubarinova-Galin equation (1.2) with respect to θ, we obtain
(2.11)Im⁡(∂∂tf′(ζ,t)f′(ζ,t)¯-ζf′(ζ,t)¯∂∂tf(ζ,t)-ζ2f′′(ζ,t)¯∂∂tf(ζ,t))=γH[iA](θ),
for ζ=eiθ. The previous equality is equivalent to the following:
(2.12)|f′(ζ,t)|2Im⁡((∂/∂t)f′(ζ,t)f′(ζ,t)-Φ(f(ζ,t))(∂/∂t)f(ζ,t)Φ(f(ζ,t)))|ζ=ζ0=eiθ0,t=t0=(Q2π-γH(i∂k∂θ(eiθ,t)))×[Im⁡(ζf′′f′-Φ′(f)Φ(f)ζf′)+2ReΦ′(f)Im⁡ζf′Φ(f)]|ζ=ζ0,t=t0,θ=θ0-γH[iA](θ)∣ζ=ζ0,t=t0,θ=θ0.
The right-hand side of this equality is strictly negative for smallγ because of (2.5), (2.7), and the fact that ReΦ′(f)>0.
By using (2.8), we obtain that arg(eiθf′(eiθ,t)/Φ(f(eiθ,t)))<π/2 for t>t0 (close to t0) in a neighbourhood of θ0. This is in contradiction with our assumption and ends the proof.Remark 2.6.
LetQ<0, and let f0 be a function that is Φ-like on U and univalent on U¯. If f0 satisfies the condition (2.1) for each ζ∈U¯, then there exist a surface tension γ (which depends on f0) sufficiently small and t(γ)≤T such that the classical solution f(ζ,t) of (1.2) with the initial condition f(ζ,0)=f0(ζ) are Φ-like for t∈[0,t(γ)), where T is the blow-up time.Proof.
The conclusion is immediate from the smoothness of the classical solution of (1.1).IfΦ(w)≡e-iαw in Theorem 2.5, where α∈(-π/2,π/2), then we obtain the following corollary:Corollary 2.7.
LetQ<0 and the surface tension γ be sufficiently small. If f0 is a function which is spiral-like of type α∈(-π/2,π/2) on U and univalent on U¯, then there exists t(γ)≤T such that the classical solution f(ζ,t) of (1.2) with the initial condition f(ζ,0)=f0(ζ) is spiral-like of type α for t∈[0,t(γ)), where T is the blow-up time.


## 3. The Outer Problem (Unbounded Domain with Bounded Complement)

In this section, we obtain the invariance in time of the same geometric property (denoted byΦ̃) for the outer problem.Definition 3.1.
LetF be a holomorphic function on U-={ζ∣|ζ|>1} such that F(ζ)=aζ+a0+a-1/ζ+⋯, where a≠0. Let Φ̃ be a holomorphic function on F(U-) such that limζ→∞Φ̃(ζ)=∞ and limζ→∞Φ̃′(ζ)>0. We say that F is Φ̃-like on U- if
(3.1)Re[ζF′(ζ)Φ̃(F(ζ))]>0,ζ∈U-.Remark 3.2.
(a) IfF is a Φ̃-like function on U-, then the function f:U→C given by f(z)=1/F(1/z), z≠0, and f(0)=0, is Φ-like on U, where Φ:f(U)→C, Φ(w)=w2Φ̃(1/w), for allw∈f(U)∖{0} and Φ(0)=0.
(b) Iff is a Φ-like function on U, then the function F:U-→C, F(ζ)=1/f(1/ζ) is Φ̃-like on U-, where Φ̃:F(U-)→C, Φ̃(ω)=ω2Φ(1/ω), for all ω∈F(U-).
(c) AnyΦ̃-like holomorphic function F on U- is univalent on U-.Proof.
This remark can be obtained by straightforward computations and its proof is, therefore, left to the reader.The following result is a generalization of [5,Theorem 3]. The mentioned theorem may be obtained by taking Φ̃(w)≡w in Theorem 3.3 below.Theorem 3.3.
LetF0 be a function which is Φ̃-like on U- and univalent on U-¯. Then the solution F(ζ,t) of the Polubarinova-Galin equation (1.5) with the initial condition F(ζ,0)=F0(ζ) is Φ̃-like for t∈[0,T), where T is the blow-up time, Ω=⋃0≤t<TΩ(t)=⋃0≤t<TF(U-,t), and Φ̃ is a holomorphic function on Ω¯ which satisfies the following conditions:
(3.2)ReΦ̃(ω)ω>0,ReΦ̃′(ω)<2ReΦ̃(ω)ω,∀ω∈Ω¯.Proof.
By considering the functionf(ζ,t)=1/F(1/ζ,t), the Polubarinova-Galin equation (1.5) can be rewritten in terms of f as follows:
(3.3)Reḟ(ζ,t)ζf′(ζ,t)¯=-Q|f(ζ,t)|42π,|ζ|=1.
Due to the previous remark, the function F(ζ,t), ζ∈U-, is Φ̃-like if and only if f(ζ,t), ζ∈U, is Φ-like, where the relationship between Φ̃ and Φ is Φ̃(ω)=ω2Φ(1/ω), for all ω∈F(U-)(orΦ(w)=w2Φ̃(1/w), for all w∈f(U)). Thus, it suffices to prove that the functions f(ζ,t) are Φ-like for t∈[0,T).
Suppose by contrary that the previous statement is not true. Then there existt0≥0 and ζ0=eiθ0 such that (2.3), (2.4), (2.5), (2.6), and (2.7) are true. At the same time, the equality (2.8) is fulfilled. We have to determine the sign of the function (∂/∂t)arg(ζf′(ζ,t)/Φ(f(ζ,t))) at the point (ζ0,t0).
By differentiating (3.3) and using (2.6), (2.8), and (3.3), we get
(3.4)∂∂targζf′(ζ,t)Φ(f(ζ,t))|ζ=ζ0,t=t0=Q|f|42π|f′|2Im⁡[(ζf′′f′-Φ′(f)ζf′Φ(f))+2Im⁡ζf′Φ(f)(ReΦ′(f)+2ReΦ(f)f)]|ζ=ζ0,t=t0.
The right-hand side of the previous equality is strictly negative in view of (2.5), (2.7), ReΦ′(f)>0, and Re(Φ(f)/f)>0 (the previous inequalities are easy consequences of (3.2)). Therefore, arg(eiθf′(eiθ,t)/Φ(f(eiθ,t)))<π/2, for t>t0 (close to t0) in some neighbourhood of θ0. This contradiction completes the proof.The following result is a generalization of [6,Theorem 3.1]. The mentioned theorem may be obtained by taking Φ̃(w)≡w in Theorem 3.4 below.Theorem 3.4.
LetQ<0, and let the surface tension γ be sufficiently small. If F0 is a function which is Φ̃-like on U- and univalent on U-¯, then there exists t(γ)≤T such that the solution F(ζ,t) of (1.6) with the initial condition F(ζ,0)=F0(ζ) is Φ̃-like for t∈[0,t(γ)), where T is the blow-up time, Ω=⋃0≤t<t(γ)Ω(t)=⋃0≤t<t(γ)F(U-,t), and Φ̃ is a holomorphic function on Ω¯ which satisfies the conditions (3.2).Proof.
We introduce (as in the proof of Theorem3.3) the function f(ζ,t)=1/F(1/ζ,t),ζ∈U, which is Φ-like (Φ(w)=w2Φ̃(1/w)) if and only if F(ζ,t) is Φ̃-like.
The Polubarinova-Galin equation can be written in terms off as
(3.5)Reḟ(ζ,t)ζf′(ζ,t)¯=-|f|4(Q2π-γH[i∂k∂θ](θ)),|ζ|=1.
Suppose by contrary that the conclusion of Theorem 3.4 is not true. Then there exist t0≥0 and ζ0=eiθ0 such that (2.3), (2.4), (2.5), (2.6), and (2.7) are true. At the same time, the equality (2.8) is fulfilled. We differentiate (3.5) with respect to θ. Since the left side is differentiable with respect to θ and the solution of (3.5) exists and is unique, then the right-hand side is differentiable and its derivative is bounded on [0,2π].
If we denoteA(θ,t)=(∂/∂θ)H[i∂k/∂θ](θ); then by using (2.6), (2.8) and (3.5), we obtain
(3.6)∂∂targζf′(ζ,t)Φ(f(ζ,t))|ζ=ζ0=eiθ0,t=t0=|f|4|f′|2(Q2π-γH(i∂k∂θ)(θ))×[Im⁡(ζf′′f′-Φ′(f)ζf′Φ(f))+Im⁡ζf′Φ(f)(2ReΦ′(f)+4ReΦ(f)f)]|ζ=ζ0,t=t0+|f|4|f′|2⋅γA(θ,t)|ζ=ζ0,t=t0.
The right-hand side of the previous equality is strictly negative for smallγ     because of (2.5), (2.7), ReΦ′(f)>0, and Re(Φ(f)/f)>0. Therefore, arg(eiθf′(eiθ,t)/Φ(f(eiθ,t)))<π/2, for t>t0 (close to t0) in some neighbourhood of θ0. This contradiction completes the proof.Remark 3.5.
LetQ<0, and let F0 be a function that is Φ̃-like on U-and univalent on U-̅. If F0 satisfies the condition (3.1) for each ζ∈U-̅, then there exist a surface tension γ  (which depends on F0) sufficiently small and t(γ)≤T such that the classical solution F(ζ,t) of (1.6) with the initial condition F(ζ,0)=F0(ζ) is Φ̃-like for t∈[0,t(γ)), where T is the blow-up time.Proof.
The conclusion is immediate from the smoothness of the classical solution of (1.6)



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