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Solvability of a moving contact-line problem with interface formation for an incompressible viscous fluid
Boundary Value Problems volume 2022, Article number: 1 (2022)
Abstract
We investigate the free-boundary problem of a steadily advancing meniscus in a circular capillary tube. The problem is described using the “interface formation model,” which was originally introduced with the aim of avoiding the singularities that arise when classical hydrodynamics is applied to problems with a moving contact line. We prove the existence of an axially symmetric solution in weighted Hölder spaces for low meniscus speeds.
1 Introduction
When a liquid moves across a substrate surface, the line along which the free interface intersects with the surface is called the contact line. Free-surface flows with moving contact lines are commonly encountered in our daily lives. However, mathematical descriptions of such flows are beyond the capabilities of classical hydrodynamics. More precisely, if the classical no-slip condition is applied to a moving contact-line problem, a non-integrable stress singularity arises near the contact line (see, e.g., [5]).
The most common approach for avoiding this situation is to replace the no-slip condition with conditions that allow for slipping near the contact line (see, e.g., the review of slip models in Chap. 3 of [25]). This approach removes the above-mentioned stress singularity, but other issues remain unresolved. One such issue concerns the modeling of the dynamic contact angle. In slip models, the contact angle has to be prescribed, and in many cases it is assumed to be a function of the contact-line speed. However, a previous experimental study [2] suggests that the variation in the contact angle depends not only on the contact-line speed, but also on the flow field in the vicinity of the contact line. Another issue concerns the description of the flow kinematics. The experiments reported in [5] imply that a rolling motion arises in an actual flow. However, as reported in [24], two typical slip models (one that prescribes the fluid velocity on a solid surface and another that adopts the Navier slip condition) fail to reproduce the rolling motion.
With the aim of overcoming the above-mentioned shortcomings of conventional models, Shikhmurzaev developed a new “interface formation model” [22]. Physically, the interface is considered to be a thin layer in which the density varies drastically because of the asymmetric action of intermolecular forces from the different phases. The interface formation model simulates this layer as a separate phase with zero thickness characterized by its surface density and surface velocity, where the surface density is related to the surface tension through an equation of state. This treatment of the interface makes it possible to describe the process of interface formation/disappearance occurring at the contact line and the variation in surface tension induced by these processes. The equations governing the relaxation process for the interface are derived by applying irreversible thermodynamics to the surface phase. The resultant equations consist of the balance laws for mass and momentum in the surface phase, which are coupled with the Navier–Stokes equations in the bulk phase by constitutive relations of the fluxes between the surface and bulk phases derived by applying Onsager’s principle.
The main issues of dynamic wetting described above are consistently resolved by the interface formation model on the basis of the mechanism of interface formation and disappearance. This is the primary advantage of this model. The interface formation model naturally describes a flux at the contact line (see [22]), which is observed in the actual rolling motion. This flux is missing in the conventional models. The interface formation model relates the surface tension to the bulk velocity through boundary conditions. Thus, the contact angle is the response of the overall velocity field. In addition, the thermodynamics applied to the liquid–solid interface means that the no-slip condition is replaced with the generalized Navier slip condition. Thus, the interface formation model eliminates the non-integrable stress singularity at the contact line as a physical consequence.
As seen in [25], the interface formation model can describe various flows with moving contact lines, including dynamic wetting, cusp formation, and drop coalescence, in a unified way. This is another advantage of this model. In addition to Shikhmurzaev’s studies (see [25] and the references therein), this model has been examined through applications to various flows using computational and asymptotic methods (see, e.g., [1, 3, 4, 7, 8, 16, 17, 26]).
By contrast, we have relatively few mathematically rigorous results for the interface formation model. Mathematical studies on the solvability of free boundary problems for the Navier–Stokes (or Stokes) equations with moving contact lines have mainly been conducted for models with conventional boundary conditions (see, e.g., [9, 21, 28, 31, 32, 35]). If we restrict ourselves to the interface formation model, the solvability was studied in [11, 12]; however, no situations involving contact lines were investigated in these studies, and to the best of the author’s knowledge, there are no results on the solvability of Shikhmurzaev’s full model describing flows with moving contact lines.
In this paper, we apply the interface formation model to the steadily advancing meniscus of an incompressible viscous fluid in a circular capillary tube, and prove that there exists an axially symmetric solution for low meniscus speeds. As stated in [24] or Sect. 2.5.2 in [25], for a hydrodynamic model describing the moving contact line to be physically valid, the solution of the model must remain finite near the contact line. We construct the above solution in weighted Hölder spaces to clarify the regularity of the solution near the contact line in the classical sense. As will be described in detail in the next section, we need to add one more condition on the contact line to ensure that the interface formation model is well posed. Following Shikhmurzaev’s treatment in [24], a condition whereby the pressure is finite at the contact line is added. This treatment enables us to determine the contact angle as part of the solution without generating any singularity from the motion of the contact line. However, the case in which the contact angle is \(\pi /2\) is an exception (see the next section in this regard). The pressure diverges at the contact line when the contact angle is in the range \((\pi /2,\pi )\), because the contact line is a ridge of the domain. It is shown that the singularity arising in this problem is caused only by the presence of the corner of the domain, and the regularity of the solution near the contact line is determined by the size of the contact angle.
The remainder of this paper is organized as follows. In Sect. 2, we formulate the problem. In Sect. 3, we introduce function spaces and state our main result. In Sect. 4, we give some results for the Stokes equations with the slip boundary condition defined in a sectorial domain. The regularity of the solution to our problem near the contact line is determined from the distribution of eigenvalues of the operator pencil associated with this model problem. In Sect. 5, we derive an estimate of the solution to the linearized problem. The estimate is obtained by combining Schauder’s method with an exponential decay estimate for the Dirichlet integral obtained by means of differential inequalities techniques. In Sect. 6, we solve the nonlinear problem using the contraction mapping principle with the estimates established for the linearized problem. We construct a solution in a neighborhood of the rest state of the fluid. We assume that the static contact angle differs from 0 and π, and we do not consider the situation where the dynamic contact angle reaches 0 and π. In our problem, the location of the interface is not known a priori; hence, for technical convenience, we transform the problem onto a prescribed domain with fixed boundaries. However, as the contact angle is unknown in advance, the regularity of the solution near the contact line cannot be determined directly from the transformed problem. The regularity of the solution is determined after the contact angle has been found by solving the transformed problem. Finally, in Sect. 7, the overall conclusions from this study and future tasks are described.
2 Formulation of the problem
Consider the situation in which a meniscus is moving at a constant speed \(W>0\) in an infinite circular tube \(\Omega =\{ |x'|\equiv \sqrt{x_{1}^{2}+x_{2}^{2}}< 1\}\). We assume that the total flux across an arbitrary cross-section of the part filled with the liquid is prescribed and given by \(\rho |B|W\), where ρ is the density of the liquid and \(|B|\) is the area of the cross-section. In the formulation below, we assume that the flow is steady in the coordinate system moving with the meniscus.
First, we establish the governing equations. Let the region \(\Omega _{h}=\{x_{3}< h(x'), x'\in B\}\) (\(B:|x'|<1\)) be filled with a liquid and \(\Gamma _{h}=\{x_{3}=h(x'), x'\in B\}\) be the free interface of the liquid. The interface is assumed to meet the surface ∂Ω at the contact line \(M\equiv \bar{\Gamma }_{h}\cap \partial \Omega =\{x\in \partial \Omega | x_{3}=0\}\), and \(\Sigma \equiv \{x\in \partial \Omega | x_{3}<0\}\) is the liquid–solid interface.
We assume that the flow over \(\Omega _{h}\) is governed by the steady Navier–Stokes equations
where v is the velocity, p is the pressure, and ν is the kinematic viscosity.
For \(\Gamma _{h} \), where n is the unit outward normal to \(\Gamma _{h}\), we consider the following equations:
where \({\mathbf {v}}_{1}\) and \(\rho _{1}\) are the velocity and surface density of the liquid–gas interface, respectively; \(\mathbf{T}({\mathbf {v}},p)=2\nu {\mathbf{D}}({\mathbf {v}})-p\mathbf{I}\) is the stress tensor, where
is the strain-rate tensor; H is twice the mean curvature of \(\Gamma _{h}\), which is assumed to be negative where \(\Omega _{h}\) is convex; Π is the projection operator onto the tangent plane of \(\Gamma _{h}\); \(\nabla _{\Gamma }\) is the gradient operator defined on \(\Gamma _{h}\); and \(\rho _{1,e}\), \(\tau _{1}\), χ, γ, and \(\rho _{0}\) are positive constants. In particular, \(\rho _{1,e}\) is the equilibrium surface density.
For Σ, we assume that
where \({\mathbf {v}}_{2}\) and \(\rho _{2}\) are the velocity and surface density of the liquid–solid interface, respectively; \({\mathbf {W}}=(0,0,-W)\); n, Π, and \(\nabla _{\Sigma }\) are defined for Σ in the same way as for \(\Gamma _{h}\); and \(\rho _{2,e}\), \(\tau _{2}\), α, and β are positive constants. In particular, \(\rho _{2,e}\) is the equilibrium surface density.
In regard to the interfaces, (2) and (6) represent the momentum balance, (3) and (7) represent the mass balance, (5) and (9) are equations of state, and (4) and (8) are phenomenological laws between fluxes and thermodynamic forces arising in the entropy production rate of the interface (for details of the derivation, see [22, 23, 25]).
Remark 1
In the present paper, we assume that the right-hand sides of (4)1 and (8)1 vanish. In the original version of the interface formation model, Shikhmurzaev derived these conditions by assuming that the effect of mass exchange on the bulk is negligible (see, e.g., [23]). However, as revealed in [24], these terms play an essential role in describing the flow in the immediate vicinity of the contact lines more accurately. We have not yet succeeded in proving the solvability of the same problem when the right-hand sides of (4)1 and (8)1 do not vanish. The main difficulty lies in the derivation of an inequality corresponding to (54), which is essential in proving the solvability of the linear problem.
On M, we assume the conservation of mass, i.e.,
where \(\mathbf{e}_{1}\) and \(\mathbf{e}_{2}\) are unit normal vectors to M that are tangential to \(\Gamma _{h}\) and Σ, respectively. We also assume the following force balance:
where \(\theta _{d}\) is the contact angle determined by \(\cos \theta _{d}=\mathbf{e}_{1}\cdot {\mathbf{e}}_{2}\), and \(\sigma _{sg}\) is the solid–gas surface tension, assumed to be a nonnegative constant.
Now, let us formulate our problem. After eliminating the surface velocity \({\mathbf {v}}_{1}\) with the aid of (4)2 and \({\mathbf {v}}_{2}\) with the aid of (8)2, the above equations can be written as follows:
and
Further, we assume that the liquid–solid interface reaches its equilibrium state and that the flow is fully developed in the far field, and we impose the following condition:
Here, V is the Poiseuille flow given by the pair (\({ \mathbf {V}}, P\)), where
This Poiseuille flow was derived in [34].
As seen in (13)3 and (14)2, the surface densities \(\rho _{1}\) and \(\rho _{2}\) satisfy second-order elliptic partial differential equations. Therefore, from a mathematical point of view, we need two boundary conditions on M to determine the surface densities using these equations. However, between conditions (15) and (16) on M, we can use only (15) for that purpose, because condition (16) is used to determine the contact angle. To use (16) as a boundary condition to determine the surface densities, we need to prescribe the contact angle, but doing so would eliminate the main advantage of this model. This “lack of a condition” was first pointed out in [1].
When the contact line is moving, the term βW arises in the generalized Navier condition (6). Shikhmurzaev [24] adds condition (50) (see [24]) to remove a pressure singularity caused by the above term and constructs an asymptotic solution without a pressure singularity. We follow the above treatment and assume his condition (50) in the following equivalent form:
We can also find this condition from the right-hand side of conditions (13)1,4 and (14)1,3 by setting \(a_{1}=\gamma \nabla _{\Gamma }\rho _{1}\cdot {\mathbf{e}}_{1}\), \(b_{1}=-(\frac{\gamma }{2}\nabla \rho _{2}+\beta \Pi ({\mathbf {v}}-{ \mathbf{W}}))\cdot {\mathbf{e}}_{2}\), \(a_{2}=b_{2}=0\) and applying condition (26) to these terms (in this case the problem domain is not a sector, so the term \(\frac{\partial a_{2}}{\partial \tau }\) is replaced by \(\frac{\partial a_{2}}{\partial \tau }-{\mathbf {v}}\cdot \frac{\partial {\mathbf{n}}}{\partial \tau }\)).
Remark 2
Another approach for adding a condition at the contact line can be found in [1], where a boundary condition is added by accounting for the entropy produced at the contact line. Shikhmurzaev disagrees with this modeling approach from a physical point of view (see Sect. 4.3.5.3 in [25]).
Condition (20) ensures that the compatibility condition corresponding to (34) is satisfied for problem (91), which makes it possible to construct a solution without a singularity at the contact line when \(\theta _{d}<\pi /2\). However, when \(\theta _{d}>\pi /2\), the pressure becomes singular because the model problem (23) has singular eigensolutions. When \(\theta _{d}=\pi /2\), in addition to (20), the condition that
which comes from condition (30), has to be imposed for the pressure to be regular at the contact line. However, if we further assume condition (21), our problem becomes overdetermined. In addition, it is impossible to impose a condition for a specific contact angle in advance for a problem where the contact angle is not known. For these reasons, we cannot construct a regular solution when \(\theta _{d}=\pi /2\).
We hereafter denote the problem consisting of equations (12)–(20) for unknown functions \(({\mathbf {v}}, p, \rho _{1}, \rho _{2}, h)\) as problem \((\mathcal{P})\).
3 Main result
To state our main result, we first introduce the function spaces used throughout this paper. Let D be a domain in \({\mathbb{R}}^{n} \), l be a nonnegative integer, and \(\alpha \in (0,1) \). We use \(C^{l+\alpha }(D) \) to denote the space of functions \(f(x),x\in D\) with the norms
where
Below, we introduce weighted Hölder spaces. Let \(s\in {\mathbb{R}}\) and \(M\subset \partial D\) be a closed set. Using , we denote the function space with the norm
where
\(K(x)=\{y\in D | |y-x|<\frac{1}{2}\rho (x)\}\), and \(\rho (x)=dist(x,M)\).
Let \(s(\notin {\mathbb{Z}})\in (0,l+\alpha ]\). Let \(C^{l+\alpha }_{s}(D,M)\) be the function space with the norm
For \(s<0\), we assume that .
Further, we introduce a weighted Hölder space defined on the infinite semi-cylinder \(\Omega _{h}\). Let \(\mu >0\), \(\Omega _{t}=\{x\in \Omega _{h} | x_{3}<-t\}\), and \(\Omega _{0,t}=\{x\in \Omega _{h} | x_{3}> -t\}\), where \(t>0\). Let \(C^{l+\alpha ,\mu }_{s}(\Omega _{h},M)\) be the function space with the norm
We also introduce the space \(C^{l+\alpha ,\mu }_{s}(\Sigma ,M)\) of functions defined on the lateral surface Σ, which is defined in the same manner as \(C^{l+\alpha ,\mu }_{s}(\Omega _{h},M)\), except that \(\Omega _{h}\) is replaced by Σ.
Finally, let \(W^{l}_{2}(D)\) be the function space with the norm
The following is our main result.
Theorem 1
Let \(\lambda _{0}>0\) be the smallest noninteger solution of the equation
When \(\theta _{d}=\pi /2\), we assume that \(\lambda _{0}=1\). Let \(0<\alpha <1\) and s satisfy the conditions \(s\notin {\mathbb{Z}}\), \(0< s\leq 2+\alpha \), and \(0< s<\min (\pi /\theta _{d},\lambda _{0})\). Assume that \(2\rho _{2,e}\alpha -\rho _{1,e}\chi \neq0\) and \(|(\sigma _{sg}-\bar{\sigma }_{2})/\bar{\sigma }_{1}|<\delta _{0}<1\) for a constant \(\delta _{0}\), where \(\bar{\sigma }_{1}=\gamma (\rho _{0}-\rho _{1,e})\), \(\bar{\sigma }_{2}=\gamma (\rho _{0}-\rho _{2,e})\). Then, if \(W>0\) is sufficiently small, for some constant \(\mu >0\), problem (\({ \mathcal{P}}\)) has an axisymmetric solution \(({\mathbf {v}}, p, \rho _{1}, \rho _{2}, h)\) with the following properties:
-
The angular component of v around the \(x_{3}\)-axis vanishes;
-
\(\theta _{d}\neq0,\pi \);
-
The solution belongs to the following classes:
$$\begin{aligned} &{\mathbf {v}}-\zeta {\mathbf {V}} \in C^{2+\alpha ,\mu }_{s}(\Omega _{h},M), \qquad \nabla (p-\zeta P)\in C^{\alpha ,\mu }_{s-2}( \Omega _{h},M),\quad p\in C^{1+ \alpha }_{s-1}(\Omega _{0,1},M), \\ &\rho _{1} \in C^{3+\alpha }_{s+1}(\Gamma _{h},M), \qquad\rho _{2}- \rho _{2,e} \in C^{3+\alpha ,\mu }_{s+1}(\Sigma ,M), \quad h \in C^{3+ \alpha }_{s+1}(B,M), \end{aligned}$$where ζ is the function given in (75).
Remark 3
The condition \(2\rho _{2,e}\alpha -\rho _{1,e}\chi \neq0\) is required for conditions (15) and (20) to be written as two independent conditions (91)9,10.
4 Auxiliary problem in a sectorial domain
Assume that \(0<\theta <\pi \). Let \(d_{\theta }\), \(\gamma _{\theta }\), \(\gamma _{0}\), and M be defined by \(\{x_{2}>0, x_{1}>\frac{x_{2}}{\tan \theta }\}\), \(\{x_{2}>0,x_{2}=x_{1}\tan \theta \}\), \(\{x_{1}>0,x_{2}=0\}\), and \(\bar{\gamma }_{\theta }\cap \bar{\gamma }_{0}\), respectively. When \(\theta =\pi /2\), \(d_{\theta }\) and \(\gamma _{\theta }\) are replaced by \(\{x_{2}>0,x_{1}>0\}\) and \(\{x_{2}>0,x_{1}=0\}\), respectively. We consider the following problem:
where \({\mathbf {\tau }}=(1,0) \text{ on } \gamma _{0}, (\cos \theta ,\sin \theta ) \text{ on } \gamma _{\theta }\), and \(\mathbf{n}=(-\tau _{2},\tau _{1})\).
The following lemma for the above problem will be used in the next section. As we can prove this lemma in a standard manner (see, e.g., [29]), the proof is omitted.
Lemma 1
Let \(\lambda _{0}>0\) be the smallest noninteger solution of the equation
If \(\theta =\pi /2\), assume that \(\lambda _{0}=1\). Assume that \(0<\alpha <1\), \(s\notin \mathbb{Z}\), \(0< s<\lambda _{0}\), and \(s\leq 2+\alpha \). Further, assume that \(\mathbf{f}\in C^{\alpha }_{s-2}(d_{\theta },M)\), \(g\in C^{1+\alpha }_{s-1}(d_{\theta },M)\), \(a_{1}\in C^{1+\alpha }_{s-1}(\gamma _{0},M)\), \(a_{2}\in C^{2+\alpha }_{s}(\gamma _{0},M)\), \(b_{1}\in C^{1+\alpha }_{s-1}(\gamma _{\theta },M)\), and \(b_{2}\in C^{2+\alpha }_{s}(\gamma _{\theta },M)\), and that their supports are compact. Additionally, assume that the condition
is satisfied, and, if \(\lambda _{0}>1\), further assume that the following compatibility condition is satisfied:
Then (23) has a unique solution \(({\mathbf {u}},\nabla q)\in C^{2+\alpha }_{s}(d_{\theta },M)\times C^{ \alpha }_{s-2}(d_{\theta },M)\) that satisfies the estimate
for a constant \(C>0\) that is independent of the data.
Rewrite problem (23) with \(\nu =1\) in the polar coordinates \((r,\phi )\) defined by \(x_{1}=r\cos \phi , x_{2}=r\sin \phi \), then apply the Mellin transform with respect to r:
Thus, we have the following system of ordinary differential equations for \(u_{r}=u_{1}\cos \phi +u_{2}\sin \phi \), \(u_{\phi }=-u_{1}\sin \phi +u_{2}\cos \phi \):
We denote the operator of this boundary value problem by \(\mathcal{U}(\lambda )\). If λ is not a root of (24), this problem has a unique solution. The roots of (24) are called eigenvalues of problem (28). A nontrivial solution corresponding to each eigenvalue is called an eigenvector.
The asymptotic formula (29) in the following lemma describes the singularity of solutions to problem (23). (For the derivation, see, e.g., [10, 15].)
Lemma 2
Let and . Assume that \(s_{1}, s_{2}\ (0< s_{1}< s_{2})\notin \mathbb{Z}\) are not roots of (24). Let , and let \(\lambda _{1},\ldots ,\lambda _{n}\) be the eigenvalues of problem (28) that exist in the range \((s_{1},s_{2})\). Then the solutions and of problem (23) are related by the formula
where \(c_{j}\) are constants depending on the data, and \(({\mathbf {u}}(\lambda _{j}),q(\lambda _{j}))\) is the eigenvector of problem (28) corresponding to eigenvalue \(\lambda _{j}\).
From (29), we can see how q behaves near the corner M. Let \(\lambda _{1}\) in (29) be the smallest positive eigenvalue. When \(0<\theta <\pi /2\), \(\lambda _{1}\) is greater than 1, which indicates that q remains bounded near M. In contrast, when \(\pi /2<\theta <\pi \), q diverges at M.
When \(\theta =\pi /2\), all nonnegative eigenvalues are given by \(\lambda =2k+1\ (k=0,1,2,\ldots )\). These eigenvalues are all double roots, and the corresponding eigenvectors are given by \(({\mathbf {u}}(\lambda ),q(\lambda ))=(\cos 2(k+1)\phi , -\sin 2(k+1) \phi ,0)\) and \((-k\cos 2k\phi , (k+1)\sin 2k \phi , -2(2k+1) \cos 2k\phi )\). If the associated eigenvector \(({\mathbf {v}}(\lambda ), p(\lambda ))\) defined as a solution of the equation
exists for \(\lambda _{j}\), the term \(c_{j}(r^{\lambda _{j}}{\mathbf {u}}(\lambda _{j}),r^{{\lambda _{j}}-1}q( \lambda _{j}))\) in (29) is replaced by
However, elementary calculations show that no associated eigenvectors exist for all the eigenvalues given above. This implies that the solution to problem (23) is regular near the corner M. Indeed, we can obtain the following result in a similar manner to Sect. 6 of [20].
Lemma 3
Assume that \(0<\alpha <1\). Assume that \(\mathbf{f}\in C^{\alpha }(d_{\theta })\), \(g\in C^{1+\alpha }(d_{\theta })\), \(a_{1}\in C^{1+\alpha }(\gamma _{0})\), \(a_{2}\in C^{2+\alpha }(\gamma _{0})\), \(b_{1}\in C^{1+\alpha }(\gamma _{\theta })\), and \(b_{2}\in C^{2+\alpha }(\gamma _{\theta })\), and that their supports are compact. Further, assume that compatibility conditions (25), (26), and
hold.
Then (23) has a unique solution \(({\mathbf {u}},\nabla q)\in C^{2+\alpha }(d_{\theta })\times C^{\alpha }(d_{\theta })\) that satisfies the estimate
for a constant \(C>0\) that is independent of the data.
5 Linear problem
In this section, we consider the following problem:
where ν, γ, \(\kappa _{1}\), \(\kappa _{2}\), β, \(\chi _{1}\), and \(\chi _{2}\) are positive constants, and Ω is defined for a given function \(h \in C^{3+\alpha }_{s+1}(B)\) in the same manner as \(\Omega _{h}\) was defined in Sect. 2. Similarly, Γ, Σ, M, n, \(\mathbf{e}_{1}\), \(\mathbf{e}_{2}\), \(\nabla _{\Gamma }\), and \(\nabla _{\Sigma }\) are defined in the same way as in Sect. 2. We assume that the domain Ω and the data f, g, a, \(a_{3}\), b, \(b_{3}\), \(h_{1}\), \(h_{2}\), \(m_{1}\), and \(m_{2}\) do not depend on the rotation angle ϕ around the \(x_{3}\)-axis, and the ϕ-components of f, a, and b vanish in the cylindrical coordinates \((r,\phi ,z)\) defined by \((x_{1},x_{2},x_{3})=(r\cos \phi ,r\sin \phi ,z)\). We denote the contact angle between Γ and Σ as θ and assume that \(\theta \neq0, \pi \).
The following is the main result in this section.
Theorem 2
Let \(0<\alpha <1\) and s be a constant satisfying the conditions \(s\leq 2+\alpha \), \(s\notin {\mathbb{Z}}\), and \(0< s<\min (\lambda _{0},\pi /\theta )\), where \(\lambda _{0}\) is the constant given in Lemma 1. Assume that \(\mathbf{f}\in C^{\alpha ,\mu }_{s-2}(\Omega ,M)\), \({g}\in C^{1+\alpha ,\mu }_{s-1}(\Omega ,M)\), \(\mathbf{a}\in C^{1+\alpha }_{s-1}(\Gamma ,M)\), \(a_{3}\in C^{2+\alpha }_{s}(\Gamma ,M)\), \(\mathbf{b}\in C^{1+\alpha ,\mu }_{s-1}(\Sigma ,M)\), \(b_{3}\in C^{2+\alpha ,\mu }_{s}(\Sigma ,M)\), \(h_{1}\in C^{1+\alpha }_{s-1}(\Gamma ,M)\), \(h_{2}\in C^{1+\alpha ,\mu }_{s-1}(\Sigma ,M)\), and \(m_{1}, m_{2}\in C^{s}(M)\). In addition, the following condition holds:
For the case \(\theta <\frac{\pi }{2}\), we also assume the compatibility condition at the contact line M:
with
where τ is given by \((\frac{1}{\sqrt{1+h'(r)^{2}}},0,\frac{h'(r)}{\sqrt{1+h'(r)^{2}}})\) in the cylindrical coordinates introduced above. Then there exists a constant \(\mu >0\) such that problem (32) has a unique axisymmetric solution with the following properties: the ϕ-component of v vanishes, \({\mathbf {v}}\in C^{2+\alpha ,\mu }_{s}(\Omega ,M)\), \(\nabla p\in C^{\alpha ,\mu }_{s-2}(\Omega ,M)\), \(\rho _{1}\in C^{3+\alpha }_{s+1}(\Gamma ,M)\), and \(\rho _{2}\in C^{3+\alpha ,\mu }_{s+1}(\Sigma ,M)\), and these functions satisfy the estimate
for some constant \(C>0\) that is independent of the data.
We begin with the following lemma.
Lemma 4
Under the assumption stated in Theorem 2for g, \(a_{3}\), and \(b_{3}\), there exists a vector field \({\mathbf {w}}\in C^{2+\alpha ,\mu }_{s}(\Omega ,M)\) satisfying
and the estimate
for a constant \(C>0\) that is independent of the data.
Proof
We construct the function in the form \({\mathbf {w}}=\nabla \Phi \), where Φ is a solution of
We first construct a function \(\Phi '\) satisfying conditions (38)2,3 and the estimate
Then we introduce the new unknown \(\Phi -\Phi '\) to reduce problem (38) to one in which \(a_{3}=b_{3}=0\). In the following argument, we again denote \(\Phi -\Phi '\) and \(g-\Delta \Phi '\) by Φ and g, respectively.
Now, let us set
For \(u\in D^{1}_{2}(\Omega )\), we set
and let \(\dot{D}^{1}_{2}(\Omega )\) be the space of all equivalence classes \([u]\). As shown in [6], the space \(\dot{D}^{1}_{2}(\Omega )\) is a Hilbert space with the scalar product
We prove the weak solvability of problem (38) with \(a_{3}=b_{3}=0\) in class \(\dot{D}^{1}_{2}(\Omega )\) for arbitrary \(g \in L_{2}(\Omega )\). Let \(\zeta _{k}\) (\(k\in \mathbb{N}\)) be the cut-off function such that
Set \(g_{k}=\zeta _{k} g\) and let \(\Phi _{k}\) be the corresponding solution of the problem with \(g=g_{k}\). We multiply the equation \(\Delta \Phi _{k} =g_{k}\) by \(\phi \in \dot{D}^{1}_{2}(\Omega )\) and integrate both sides to obtain
The Riesz representation theorem then implies that there exists a unique solution \(\Phi _{k} \in \dot{D}^{1}_{2}(\Omega )\) of (39), and we obtain the estimate
Thus, by taking the limit \(k\to \infty \) in (39), we have a solution \(\Phi \in \dot{D}^{1}_{2}(\Omega )\) satisfying the estimate
For problem (38) with \(a_{3}=b_{3}=0\), we can obtain a decay estimate that is similar to (56). Based on this estimate, we can obtain estimate (37) using a method of localization similar to that used to obtain estimates (71) and (73). Because of the axial symmetry of the data and the domain, the solution constructed above is axially symmetric. Therefore, we can perform the above localization procedure for a two-dimensional problem using estimate (41) given below. In Lemma 5, \(d_{\theta }\), \(\gamma _{\theta }\), \(\gamma _{0}\), and M are defined in the manner specified in Sect. 4. As the result is well known, the proof is omitted.
Lemma 5
Assume that \(0<\alpha <1\), \(s\notin {\mathbb{Z}}\), \(0< s<\pi /\theta \), and \(s\leq 2+\alpha \). Further, assume that \(g\in C^{1+\alpha }_{s-1}(d_{\theta },M)\), \(a_{3}\in C^{2+\alpha }_{s}(\gamma _{\theta },M)\), and \(b_{3}\in C^{2+\alpha }_{s}(\gamma _{0},M)\), and that their supports are compact. Additionally, assume that the condition
is satisfied. Then problem (38) with \(\Omega =d_{\theta }\), \(\Gamma =\gamma _{\theta }\), and \(\Sigma =\gamma _{0}\) has a unique solution \(\nabla \Phi \in C^{2+\alpha }_{s}(d_{\theta },M)\) that satisfies the estimate
for a constant \(C>0\) that is independent of the data.
Thus, we have proved Lemma 4. □
Lemma 6
Under the assumptions stated in Theorem 2for \(h_{1}\), \(h_{2}\), \(m_{1}\), and \(m_{2}\), there exists a unique solution \((s_{1},s_{2})\) to the problem
satisfying the estimate
for a constant \(C>0\) that is independent of the data.
Proof
The proof is similar to that of Lemma 4. The existence of a weak solution of problem (42) is easily shown. In addition, for the problem
where \(\chi \geq 0\) is a constant, we can obtain the following estimate in the same manner as in the proof of Theorem 7.1 in [30] under the assumption that the supports of the data are compact:
Using this estimate, in addition to a decay estimate for \(s_{1}, s_{2}\) similar to that in (56), we achieve the desired result in a similar manner to that used to obtain estimates (71) and (73). □
With the aid of Lemmas 4 and 6, problem (32) reduces to the problem with \(g=a_{3}=b_{3}=h_{1}=h_{2}=m_{1}=m_{2}=0\). Therefore, we hereafter consider (32) under this assumption.
Let us prove the weak solvability of the problem. After multiplying (32)1 by \({\mathbf {v}}'\), (32)5 by \(\rho _{1}'\), and (32)8 by \(\rho _{2}'\), we use integration by parts to obtain
Let \(J(\Omega )\) be the function space defined by
We now prove the following inequality. The main difficulty is in the estimation of \(\|{\mathbf {v}}\|_{2,\Omega }\). A similar inequality in an infinite strip domain is derived in [18]. Because the domain is three-dimensional in the present case, more complicated arguments are necessary.
Lemma 7
For arbitrary \({\mathbf {v}}\in J(\Omega )\), the following inequality holds:
Proof
For any \({\mathbf {v}}\in J(\Omega )\), with the aid of Korn’s inequality, we have
As Γ is compact, using a similar argument as in the proof of Lemma 4 in [33] implies that there exists a constant \(C(\epsilon )\) satisfying the inequality
for arbitrary \(\epsilon >0\). Combining these inequalities, we have
Now, from the conditions \(\nabla \cdot {\mathbf {v}}=0\) and \({\mathbf {v}}\cdot {\mathbf{n}}|_{\Sigma }=0\), \(\int _{B} v_{3}(x',x_{3})\,dx'=0\) holds for arbitrary \(x_{3}<0\). From this, we have
By integrating the identity with \(n=2\),
on B under the condition that \({\mathbf {v}}\cdot {\mathbf{n}}|_{\partial B}=0\), we obtain
where \({\mathbf {v}}'=(v_{1},v_{2})\). We integrate (48) and (50) with respect to \(x_{3}\) to obtain
Next, we integrate (49) with \(n=3\) in the domain \(\Omega _{0,1}\) and find that
and we note the estimate
Estimate (46) is obtained by combining inequalities (47) and (53). □
We now introduce the Lax–Milgram theorem.
Theorem 3
(Lax–Milgram [14])
Let H be a real Hilbert space with the norm \(\|\cdot \|\) and \(B: H\times H\to \mathbb{R}\) be a bilinear mapping. If there exist constants \(\alpha , \beta >0\) such that
Then, for every linear functional \(f: H\to \mathbb{R}\), there exists a unique element \(u\in H\) such that \(B[u,v]=f(v)\) for all \(v\in H\).
Lemma 7 indicates that the bilinear form defined by the left-hand side of (45), which we denote as \(B[({\mathbf {v}},\rho _{1},\rho _{2}),({\mathbf {v}}',\rho _{1}',\rho _{2}')] \), satisfies the condition
Thus, from Theorem 3, we have the following.
Theorem 4
For arbitrary \((\mathbf{f}, \mathbf{a}, \mathbf{b}) \in L_{2}(\Omega )\times L_{2}(\Gamma ) \times L_{2}(\Sigma )\), there exists a unique solution \(({\mathbf {v}},\rho _{1},\rho _{2})\in J(\Omega )\times W^{1}_{2}( \Gamma )\times W^{1}_{2}(\Sigma )\) satisfying (45) for all \(({\mathbf {v}}',\rho '_{1},\rho '_{2})\in J(\Omega )\times W^{1}_{2}( \Gamma )\times W^{1}_{2}(\Sigma )\) and the estimate
for a constant \(C>0\) that is independent of the data.
For the solution \(({\mathbf {v}},\rho _{1},\rho _{2})\) obtained above, the pressure p is determined from the following equation defined for arbitrary \({\mathbf {\eta }} \in H(\Omega )\equiv \{ \mathbf{f}\in W^{1}_{2}( \Omega ) | \mathbf{f}\cdot {\mathbf{n}}|_{\Gamma \cup \Sigma }=0 \} \) with a compact support:
If we take the vector η such that
where \(\Omega '\subset \Omega \) is a bounded domain (for the construction of η, see, e.g., [13]), we have the estimate
Here, p is normalized by the condition \(\int _{\Omega '}p\,dx=0\).
The axial symmetry of the solution constructed above readily follows from the symmetry of the data and the domain. In addition, when \(f_{\phi }=a_{\phi }=b_{\phi }=0\), if we take \(v_{\phi }{\mathbf{e}}_{\phi }\) (\({ \mathbf{e}}_{\phi }=(-x_{2}/|x'|,x_{1}/|x'|,0)\)) as \({\mathbf {v}}'\) in (45) and take 0 as \(\rho _{1}'\) and \(\rho _{2}'\), we have the equality
which indicates that \(v_{\phi }=0\).
The following decay estimate is obtained in a similar manner to the proofs of Theorem 5.3 in [27] and Lemma 1 in [19].
Lemma 8
Let \(\Omega (\lambda ,t)=\{x\in \Omega | \lambda -t < x_{3}<\lambda +t \}\) and \(\Sigma (\lambda ,t)=\{x\in \Sigma | \lambda -t< x_{3}<\lambda +t \}\ (\lambda <-4)\). Assume that f, a, and b possess the same regularity assumed in Theorem 2. Then the solutions v and \(\rho _{2}\) of (45) satisfy the inequality
for constants \(\mu >0\) and \(C>0\) that are independent of the data.
Proof
We use integration by parts in the domain \(\Omega (\lambda ,t)\) in a manner similar to that for (45) to obtain
where \(\mathbf{e}_{3}=(0,0,1)\) and \(\omega (t)=\{x_{3}=t, x'\in B\}\).
We integrate (57) with respect to t over the interval \((\eta -1,\eta ) (1<\eta )\) to obtain
where
The terms on the right-hand side are estimated as follows:
and
where \(\epsilon >0\) is an arbitrary constant.
Thus, from (58), we have
Multiplying this inequality by \(e^{-\eta /c_{1}}\) and integrating over the interval \((2,-\lambda /2)\) with respect to η gives
Now, we note the following inequalities:
and
to obtain the desired estimate (56) for \(0<\mu <1/2C_{1}\). □
Let us proceed to the estimation of the higher-order derivatives of the solution through Schauder’s method. We introduce a covering \(\{U_{i}\}_{i\in \mathbb{N}}\) of Ω̄, where \(\{U_{i}\}\) is a family of balls with the following properties:
(i) if \(U_{i}\cap M \neq \phi \), then \(\xi _{i}\), the center of \(U_{i}\), is on M; if \(U_{i}\cap M = \phi \) and \(U_{i}\cap \partial \Omega \neq \phi \), then \(\xi _{i}\) is on Γ or Σ;
(ii) the radius of \(U_{i}\) (\(U_{i}\cap \partial \Omega \neq \phi \)) is sufficiently small that \(U_{i}\) divides the surface ∂Ω into two connected parts; and
(iii) there exists an integer \(N_{0}\) such that the intersection of \(N_{0}+1\) arbitrary different balls is empty.
We further introduce a partition of unity \(\{\zeta _{i}\}\) corresponding to the covering \(\{U_{i}\}\).
For \(U_{i}\) with \(\xi _{i} \notin M\), the following estimates are obtained (for the derivation, see [11]):
where \(D_{i}\) and \(D'_{i}\) denote the domains \(D\cap U_{i}\) and \(D\cap U'_{i}\), respectively, for \(D=\Omega , \Gamma , \Sigma \), and \(U'_{i}\) denotes a subset of \(U_{i}\) on which \(\zeta _{i}=1\) holds. To estimate the solution defined on \(U_{i}\) with \(\xi _{i} \in M\), we use the axial symmetry of the solution and rewrite problem (32) as the following two-dimensional problem in cylindrical coordinates:
where
and \(f_{r}\) and \(f_{z}\) denote the r-component and z-component, respectively, of the term f.
We rewrite the above problem in the coordinates \((y_{1},y_{2})\), which are related to the original coordinates by \(\binom{y_{1}}{y_{2}}=\mathcal{R}\binom{r-1}{z}\), where \(\mathcal{R}\) is the matrix of rotation \(\theta -(3/2)\pi \) around the origin. In the coordinates \(\{y\}\), Σ is given by the line \(y_{2}=y_{1}\tan \theta \) (or \(y_{1}=0\) when \(\theta =\pi /2\)), and we represent Γ as the curve \(y_{2}=g(y_{1})\).
We now introduce a mapping \(y=\Phi (\eta )=(\eta _{1},\eta _{2})+(\Phi _{1}(\eta ),\Phi _{2}( \eta ))\), where \(\Phi _{1}\) and \(\Phi _{2}\) are constructed to satisfy the following conditions:
in which g̃ is an extension of g on the half-line \({\mathbb{R}}_{+}\) in the class \(C^{3+\alpha }_{s+1}({\mathbb{R}}_{+},0)\).
Then, the functions \({\mathbf {u}}=(u_{1},u_{2})=\hat{\zeta }\hat{\mathbf {v}}\equiv (\zeta { \mathbf {v}})\circ \Phi \), \(q=\hat{\zeta }\hat{p}\equiv (\zeta p)\circ \Phi \), \(r_{1}=\hat{\zeta }\hat{\rho }_{1}\equiv (\zeta \rho _{1})\circ \Phi |_{ \gamma _{0}}\), and \(r_{2}=\hat{\zeta }\hat{\rho }_{2}\equiv (\zeta \rho _{2})\circ \Phi |_{ \gamma _{\theta }}\) (here the same symbols v, p, \(\rho _{1}\), and \(\rho _{2}\) are used to denote the unknowns defined in the coordinates \(\{y\}\)), where
for a constant \(\delta >0\) satisfy the following equations in the sectorial domain \(d_{\theta }\):
where
When \(\theta =\pi /2\), equations (67)3 and (68)2 are replaced by \(-\kappa _{2}\frac{d^{2}r_{2}}{d\eta _{2}^{2}}+\chi _{2}r_{2}=- \frac{1}{2}\hat{\zeta } \hat{\nabla }_{\Sigma }\cdot \hat{\mathbf {v}}\) and \(-\kappa _{2}\frac{dr_{2}}{d\eta _{2}}=-\frac{1}{2}\hat{\zeta } \hat{\mathbf {v}}\cdot \hat{\mathbf{e}}_{2}\), respectively.
Here, \(\hat{\nabla }=({}^{t}{\mathcal{A}}^{-1})\nabla \), where \(\mathcal{A}\) is the Jacobian matrix of the mapping Φ and tA denotes the transpose of matrix A; \(\hat{\mathbf{D}}(\hat{\mathbf {v}})\) denotes the tensor with elements \(\frac{1}{2}\sum_{k=1}^{3} (a_{ik}\partial _{\eta _{k}} \hat{v}_{j}+a_{jk}\partial _{\eta _{k}} \hat{v}_{i} )\ (i,j=1,2) \), where \(a_{ij}\) is the \((i,j)\)th element of \({}^{t}{\mathcal{A}}^{-1}\); \(\hat{\nabla }_{\Gamma }\) and \(\hat{\nabla }_{\Sigma }\) are defined by \(\hat{\nabla }-(\hat{\mathbf{n}}\cdot \hat{\nabla })\hat{\mathbf{n}}\); \(\hat{\frac{d}{d\eta _{1}}}=\hat{\partial }_{\eta _{1}}+ \frac{d\hat{\tilde{g}}}{d\eta _{1}}\hat{\partial }_{\eta _{2}}\) (\(\binom{\hat{\partial }_{\eta _{1}}}{\hat{\partial }_{\eta _{2}}}= \hat{\nabla }\)) on \(\gamma _{0}\); \(\hat{\frac{d}{d\eta _{1}}}=\hat{\partial }_{\eta _{1}}+\tan \theta \hat{\partial }_{\eta _{2}}\) on \(\gamma _{\theta }\); \(\mathcal{G}_{1}= \frac{1}{(1-y_{1}\sin \theta +y_{2}\cos \theta )\sqrt{1+({\tilde{g}'})^{2}}} \circ \Phi \); and \(\mathcal{G}_{2}= \frac{1-y_{1}\sin \theta +y_{2}\cos \theta }{\sqrt{1+({\tilde{g}'})^{2}}} \circ \Phi \).
By applying estimate (27) to the system consisting of (65), (66)1,2, and (67)1,2 and applying estimate (44) to the system consisting of (66)3, \((\text{67})^{3}\), and (68), we have
Let \(\{U_{i}\}_{i\in I}\) be a finite covering of \(\bar{\Omega }_{0,4}\) which is chosen from the set \(\{U_{i}\}_{i\in \mathbb{N}}\). If we assume that each element of \(\{U_{i}\}_{i\in I_{1}(\subset I)}\) satisfies \(U_{i}\cap M=\phi \), then the norms \(|\cdot |^{(l+\alpha )}_{s,\Omega _{i},M}\) and \(|\cdot |^{(l+\alpha )}_{\Omega _{i}}\) are equivalent for \(i\in I_{1}\), since any point in \(\Omega _{i}\ (i\in I_{1})\) is away from M by a positive distance. Thus, by summing estimates (60)–(62) and (70) for all \(i\in I\) and then using (55), we can obtain the estimate
In a similar manner, we take estimates (60) and (62) and use (56) in the domain \(\Omega (\lambda ,1-\delta )\) to obtain the estimate
where \(0<\delta <1\) is a constant. Multiplying both sides of (72) by \(e^{-\mu \lambda }\) and taking the supremum for \(\lambda <-4\), and then noting that the norms \(|e^{-\mu x_{3}}{\mathbf {v}}|^{(2+\alpha )} _{\Omega _{3+\delta }}\) and \(\sup_{\lambda <-4}e^{-\mu \lambda }|{\mathbf {v}}|^{(2+\alpha )}_{ \Omega (\lambda ,1-\delta )}\) are equivalent, we have
6 Nonlinear problem
In this section, we prove Theorem 1.
Assume that \(W=0\) and \({\mathbf {v}}={\mathbf {v}}_{1}={\mathbf {v}}_{2}=\mathbf{0}\). Then, from (1)1, (3), and (7), it follows that \(p=\bar{p}=\text{const.}\), \(\rho _{1}=\rho _{1,e}\), and \(\rho _{2}=\rho _{2,e}\).
Thus, the functions
describe the rest state of problem (\(\mathcal{P}\)). Here, h̄ is a solution of the equation
where \(\nabla '=(\partial _{x_{1}},\partial _{x_{2}})\) and ν is the unit outward normal to ∂Ω with the starting point located on M.
It is easy to see that h̄ is defined uniquely by a part of the sphere
with \(\cos \bar{\theta } =(\sigma _{sg}-\bar{\sigma }_{2})/\bar{\sigma }_{1} \). Note that p̄ is uniquely determined as \(\bar{p}=-2(\sigma _{sg}-\bar{\sigma }_{2})\) from the compatibility condition of problem (74).
We seek a solution of the form
where ζ is a smooth cut-off function satisfying
In terms of the new unknowns in the above formulation, conditions (13)2 and (16) can be written as
and
where \(\mathbf{A}(x')\) and \(\mathbf{B}^{(j)}(\xi ')\) are matrices whose \((k,l)\)th components are \(\partial _{\xi _{l}}F_{k}|_{\xi '=\nabla '\bar{h}}\) and \(\partial ^{2}_{\xi _{k}\xi _{l}}F_{j}(\xi ')\), respectively, and \(F_{j}(\xi ')=\xi _{j}/\sqrt{1+|\xi '|^{2}}\) (\(\xi '=(\xi _{1}, \xi _{2})\)).
We now prove the following theorem, which will be applied to the problem consisting of (76) and (77).
Theorem 5
Let \(0<\alpha <1\), \(s\notin {\mathbb{Z}}\), and \(0< s\leq 3+\alpha \). Assume that \(\bar{h}\in C^{3+\alpha }(B)\), \(f\in C^{\alpha +1}_{s-1}(B,M)\), and \(g\in C(M)\), and that h̄, f, and g are axially symmetric with respect to the \(x_{3}\)-axis. Further, assume that the compatibility condition
is satisfied. Then the problem
has an axisymmetric solution satisfying the estimate
for a constant \(C>0\) that is independent of the data. This solution is uniquely determined if \(u(P)=0\) is satisfied for a point \(P\in M\).
Proof
When \(s>1\), the above result is readily obtained with the aid of a standard theory of second-order elliptic partial differential equations. Therefore, we consider only the case \(0< s<1\).
As the higher-order derivatives of the solution are estimated using Schauder’s method, we derive only the following estimate:
In the polar coordinate system, problem (79) is written as
where \(H= \{ 1+(\frac{\partial \bar{h}}{\partial r})^{2} \} ^{3/2} \).
The following function u satisfies the above problem:
From this expression, we have the estimate
The first term on the right-hand side is estimated as follows:
The second term is estimated in the same way.
Next, we derive the estimate
The relation \(u''(r)= (\frac{H(r)}{r} )'\int ^{r}_{0}\xi f(\xi ) \,d\xi +H(r)f(r) \) can be used to obtain
The first term on the right-hand side is estimated as
Now, let us estimate \([u']^{(s)}_{I}\). For this purpose, we use the following two relations:
and
When \(0< r,r+\tau \leq \frac{1}{2}\), from (85), \(|u''(t)|\leq C\langle f \rangle ^{(1+\alpha )}_{s-1,I,1}\) holds. Therefore, from (89), we have the estimate
When \(\frac{1}{2}\leq r,r+\tau <1\), with the aid of (88), we have (90). Finally, when \(0< r\leq \frac{1}{2}< r+\tau <1\), noting the inequality
we have (90).
Thus, we obtain estimate (81). □
The remaining equations are given by
where \(\Omega _{\delta h} =\Omega _{h}\) and \(\Gamma _{\delta h}=\Gamma _{h}\) (\(\Omega _{h}\) and \(\Gamma _{h}\) are defined in Sect. 2); \(\nabla _{\delta h}\) is the gradient operator on \(\Gamma _{\delta h}\), \(\kappa _{1}=\chi \gamma \), \(\chi _{1}=1/(\tau _{1}\rho _{1,e})\), \(\chi _{2}=1/(\tau _{2}\rho _{2,e})\), \(\kappa _{2}=\alpha \gamma \), and
Let \(X_{\epsilon }\) and \(Y_{\epsilon ,\delta h}\) be function spaces defined by
and
where ϵ and K are positive constants.
It is easy to show that the following lemmas hold. Note that an additive constant of q is determined from the compatibility condition of the problem consisting of (76) and (77). Thus, the following estimate holds for q: \(|q|^{(1+\alpha )}_{s-1,\Omega _{0,1},M}\leq C(|{\mathbf {u}}|^{(2+ \alpha ,\mu )}_{s,\Omega _{\delta h},M}+|r_{1}|^{(3+\alpha )}_{s+1, \Gamma _{\delta h},M}+|r_{2}|^{(3+\alpha ,\mu )}_{s+1,\Sigma ,M}+| \delta h|^{(3+\alpha )}_{s+1,B,M}(|\delta h|^{(3+\alpha )}_{s+1,B,M}+|r_{1}|^{(3+ \alpha )}_{s+1,\Gamma _{\delta h},M}))\).
Lemma 9
Assume that \(({\mathbf {u}},\nabla q,r_{1}, r_{2}) \in Y_{\epsilon ,\delta h}\). Then, for \(\delta h \in X_{\epsilon }\), the following inequality holds:
where C is a positive constant that is independent of ϵ and K.
Lemma 10
Assume that \(0< W<1\) and \(\delta h \in X_{\epsilon }\). Then, for \(({\mathbf {u}}, r_{1},r_{2}), ({\mathbf {u}}', r_{1}', r_{2}') \in Y_{ \epsilon ,\delta h}\), the following inequalities hold:
where C is a positive constant that is independent of ϵ and W, and \(C(\epsilon )\) and \(C(\epsilon ,W)\) denote positive constants possessing the properties \(C(\epsilon )\to 0\) as \(\epsilon \to 0\) and \(C (\epsilon ,W )\rightarrow 0\) as \(\epsilon , W\rightarrow 0\), respectively.
With the aid of the estimates in Lemma 10 and Theorem 2, we have the following lemma. The proof is almost the same as that of Lemma 7 in [11], and is thus omitted.
Lemma 11
There exist \(\epsilon >0\) and \(W>0\) such that, for any \(\delta h \in X_{\epsilon }\), there exists a unique solution \(({\mathbf {u}}, \nabla q ,r_{1},r_{2})\in Y_{\epsilon ,\delta h}\) for problem (91).
As the regularity of the solution \(({\mathbf {u}}, \nabla q ,r_{1},r_{2})\) is determined by the size of the contact angle, it depends on δh. To treat the solutions to problem (91) for various \(\delta h \in X_{\epsilon }\) in a common function space, we now assume that the order s satisfies \(s=s'\equiv \min (\bar{\lambda }_{0}, \pi /\bar{\theta })-\delta >0\), where \(\delta >0\) is a constant and \(\bar{\lambda }_{0}\) is the constant defined in the same way as \(\lambda _{0}\) in (24) with \(\theta =\bar{\theta }\).
Now, let us introduce a mapping \(\Phi _{\delta h}\) defined by
where \(\delta h \in X_{\epsilon }\) and \(\chi (y)\) is a smooth function such that
For sufficiently small \(\epsilon >0\), \(\Phi _{\delta h}\) is injective and maps \(\Omega _{\bar{h}}\) onto \(\Omega _{\delta h}\), and the Jacobian matrices \({\mathcal{J}}_{\delta h}\) and \({\mathcal{J}}_{\delta h'}\) of the mappings \(\Phi _{\delta h}\) and \(\Phi _{\delta h'}\) satisfy the estimate
for a positive constant C that is independent of δh and \(\delta h'\).
We write problem (91) simply as
and let \(({\mathbf {u}},q,r_{1},r_{2})\) and \(({\mathbf {u}}',q',r_{1}',r_{2}')\) be solutions to the problems \(L_{\delta h}({\mathbf {u}},q,r_{1},r_{2})=R({\mathbf {u}},r_{1},r_{2}) \) and \(L_{\delta h'}({\mathbf {u}}',q',r'_{1},r'_{2})=R({\mathbf {u}}',r'_{1},r'_{2}) \) for given δh and \(\delta h' \in X_{\epsilon }\).
We rewrite these solutions as equations defined in \(\Omega _{\bar{h}}\) with the aid of the mappings \(\Phi _{\delta h}\) and \(\Phi _{\delta h'}\), and then subtract one from the other to obtain the following equations:
In the above equations, f̂ and f̃ denote the functions \(f\circ \Phi _{\delta h}\) and \(f\circ \Phi _{\delta h'}\), respectively; \(\bar{\mathbf{e}}_{1}\) and \(\nabla _{\bar{h}}\) are defined for the surface \(y_{3}=\bar{h}(y')\) in the same way as \(\mathbf{e}_{1}\) and \(\nabla _{\delta h}\), respectively, are defined; Π̂ is the operator defined by \(\hat{\Pi }{\mathbf{f}}=\mathbf{f}-(\hat{\mathbf{n}}\cdot {\mathbf{f}})\hat{\mathbf{n}}\); ∇̂ and \(\hat{\nabla }_{\delta h}\) are the operators defined by \(\hat{\nabla }= (^{t}{\mathcal{J}}_{\delta h} )^{-1}\nabla \) and \(\hat{\nabla }_{\delta h}=\hat{\nabla }-(\hat{\mathbf{n}}\cdot \hat{\nabla }) \hat{\mathbf{n}}\); \(\hat{\mathbf{D}}(\hat{\mathbf {u}})\) denotes a tensor with elements \((1/2)\sum_{k=1}^{3}(A_{jk}\partial _{y_{k}}\hat{u}_{i}+A_{ik} \partial _{y_{k}} \hat{u}_{j})\) (\(i,j=1,2,3\)), where \(A_{ij}\) is the (i,j)th element of \((^{t}{\mathcal{J}}_{\delta h})^{-1}\); and the operators Π̃, ∇̃, and \(\tilde{\nabla }_{\delta h'}\) and the tensor \(\tilde{\mathbf{D}}(\widetilde{\mathbf {u}'})\), which arise as a result of the change of variables \(x=\Phi _{\delta h'}(y)\), are defined in the same way.
For the terms on the right-hand sides of (102)–(111), we obtain the estimates given in the following lemma.
Lemma 12
For δh, \(\delta h' \in X_{\epsilon }\), \(({\mathbf {u}},\nabla q,r_{1},r_{2})\in Y_{\epsilon ,\delta h}\), and \(({\mathbf {u}}',\nabla q',r_{1}',r_{2}')\in Y_{\epsilon ,\delta h'}\), the following inequalities hold:
where \(C(\epsilon )\) and \(C (\epsilon ,W )\) denote positive constants possessing the same property as the corresponding expressions in Lemma 10.
With the aid of the above estimates, applying Theorem 2 to the problem given in (102)–(111) yields the following estimate for sufficiently small ϵ and W:
We now turn to the problem given by (76)–(77). For notational simplicity, we represent the problem as \(\mathcal{L}(\delta h)=\mathcal{R}({\mathbf {u}},q,r_{1},r_{2},\delta h)\).
Let the mapping F map each \(\delta h \in X_{\epsilon }\) to the solution \(\widetilde{\delta h}\) of the problem
where \(({\mathbf {u}},q,r_{1},r_{2})\in Y_{\epsilon ,\delta h}\) is a solution to problem (91) defined for δh given above.
Then, with the aid of Theorem 5 and estimate (93), we have
Thus, if we choose \(C_{1}\) as K and choose ϵ such that \((K+K^{2})\epsilon <1/2\), we obtain \(|\widetilde{\delta h}|^{(3+\alpha )}_{s'+1,B,M}\leq (3/2)K\epsilon \). This implies that F maps \(X_{\epsilon }\) into itself.
Next, we consider the problem
where \(({\mathbf {u}}',q',r_{1}',r_{2}')\in Y_{\epsilon ,\delta h'}\) is the solution to problem (91) with \(\delta h=\delta h'\).
Then, with the aid of estimate (120), we have
Thus, if we choose ϵ and W such that \(C (\epsilon ,W )<1\), the estimate indicates that F is a contraction in \(X_{\epsilon }\). Thus, by the contraction mapping principle, we have proved the existence of a solution to our problem.
Finally, we prove that the solution constructed above belongs to the classes stated in Theorem 1. For that purpose, it is convenient to rewrite problem (91) in cylindrical coordinates:
Here, \(\Omega , \Gamma , \mathbf{n}\), τ, and \(\mathbf {e}_{1}\) are defined in the same way as in (64) for \(h=\bar{h}+\delta h\).
After localizing the above problem near the contact point M in the same manner as used to obtain (65)–(68), we have the following equations for the functions \({\mathbf {w}}=(w_{1},w_{2})=\hat{\zeta }\hat{\mathbf {u}}\), \(\pi =\hat{\zeta }\hat{q}\), \(s_{1}=\hat{\zeta }\hat{r}_{1}\), and \(s_{2}=\hat{\zeta }\hat{r}_{2}\):
Here, \(H_{i}\) (\(i=1,\ldots ,11\)) are the terms involving the highest-order derivatives of u, q, \(r_{1}\), and \(r_{2}\), while \(L_{i}\) (\(i=1,\ldots ,11\)) comprises the remaining terms. For example, \(H_{1}\) and \(L_{1}\) are specifically expressed as follows:
where \(u_{1}=-(u_{r}-1)\sin \theta -u_{z}\cos \theta \), \(F_{1}=-(F_{r}-1)\sin \theta -F_{z}\cos \theta \), \(a_{ij}\) denotes the \((i,j)\)th component of \((^{t}{\mathcal{A}})^{-1}\), in which \(\mathcal{A}\) is the Jacobian matrix of the mapping Φ, and \(\delta _{ij}\) denotes the Kronecker delta.
We note that \(\delta _{ij}\delta _{ik}-a_{ij}a_{ik}\) (the coefficients of the highest-order derivatives of u) vanish at the origin and that \(\delta _{ij}\delta _{ik}-a_{ij}a_{ik} \in C^{2+\alpha }_{s'}(d_{ \theta _{d}},M)\), from which we obtain
With the aid of this inequality, it is easily found that ; therefore, \(H_{1}+L_{1} \in C^{\alpha }_{2s'-2}(d_{\theta _{d}},M)\). In a similar manner (we also use the fact that to estimate \(H_{5}(\hat{\mathbf {u}})=\hat{\zeta }\hat{\mathbf {u}}\cdot ({\mathbf {n}}- \hat{\mathbf {n}})\)), we see that the terms \(H_{i}+L_{i}\) \((i=2, \ldots ,11)\) are smoother by the order of \(s'\) than the highest-order derivatives of the solutions involved in each \(H_{i}\). Therefore, the solution \(({\mathbf {w}},\nabla \pi , s_{1},s_{2})\) becomes smoother by the same order:
As a result, we have
Then, from equations (76)–(77), we have \(\delta \in C^{3+\alpha }_{2s'+1}(B,M)\). By repeating this procedure, we reach the desired result.
7 Conclusion
A free boundary problem describing a steadily advancing meniscus in an infinite circular tube has been investigated. The problem was formulated using the interface formation model, and it was proved that an axisymmetric solution exists in weighted Hölder spaces when the velocity of the meniscus is low. The problem was closed by adding a condition stating that the pressure must be finite at the contact line. Our analysis shows that this treatment determines the contact angle as part of the solution and removes the singularity caused by the motion of the contact line (with the exception of the case in which the contact angle is \(\pi /2\)). It was also shown that the singularity of the solution is caused by the presence of a corner of the domain, and the regularity of the solution near the contact line is determined by the size of the contact angle. For a technical reason, this paper neglected the term representing the mass exchange between the bulk and the interface arising in the normal component of the velocity on the interface. Shikhmurzaev [24] has suggested that the corner of the domain causes no singularity if this term is not neglected—a rigorous verification of this point will be considered in future work.
Availability of data and materials
Not applicable.
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We thank the anonymous reviewers for their valuable comments to improve the quality of the manuscript. We also thank Stuart Jenkinson, PhD, from Edanz (https://jp.edanz.com/ac) for editing a draft of the manuscript.
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Kusaka, Y. Solvability of a moving contact-line problem with interface formation for an incompressible viscous fluid. Bound Value Probl 2022, 1 (2022). https://doi.org/10.1186/s13661-021-01582-x
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DOI: https://doi.org/10.1186/s13661-021-01582-x