Moving boundary problems

Abstract

Moving boundary problems are ubiquitous in nature, technology, and engineering. Examples include the human heart and heart valves interacting with blood flow, biodegradable microbeads swimming in water to clean up water pollution, a micro camera in the human intestine used for early colon cancer detection, and the design of next-generation vascular stents to prop open clogged arteries and to prevent heart attacks. These are time-dependent, dynamic processes, which involve the interaction between fluids and various structures. Analysis and numerical simulation of fluid-structure interaction (FSI) problems can provide insight into the “invisible” properties of flows and structures, and can be used to advance design of novel technologies and improve the understanding of many physical and biological phenomena. Mathematical analysis of FSI models is at the core of this understanding. In this paper we give a brief survey of recent progress in the area of mathematical well-posedness for moving boundary problems describing fluid-structure interaction between incompressible, viscous fluids and elastic, viscoelastic, and rigid solids.

1. Introduction

Moving boundary problems are time-dependent problems describing the motion of a quantity such as fluid in a domain that is moving due to, e.g., the domain’s exterior boundary motion or the motion of an immersed structure within the fluid, or both. An immersed structure within the fluid defines an interior fluid domain boundary described by the fluid-structure interface location, which is, in general, time dependent. Moving boundary problems can be classified into two types: The first type is a moving boundary problem in which the domain boundary motion is given a priori. The second is a moving boundary problem in which the motion of the domain boundary is not known a priori, but this is one of the unknowns in the problem. In both cases the flow inside the domain is strongly affected by the motion of the domain boundary, while in the second case the motion of the domain boundary is simultaneously adjusted as it is being impacted by the flow inside the domain. In this case two sets of coupling conditions need to be prescribed to capture this two-way coupling: the kinematic condition describing the coupling of the kinematic quantities, such as velocity and the dynamic coupling condition describing the dynamic balance of forces. Depending on the problem at hand, this two-way coupling can be evaluated along the “current” location of the moving boundary, or along the fixed domain boundary. The coupling along a fixed domain boundary is said to be linear (i.e., it is linearized around the fixed, reference domain configuration), while the coupling along the current location of the moving boundary not known a priori is said to be nonlinear. See Figure 1. Needless to say, the nonlinear coupling between the flow and boundary motion gives rise to an exceedingly complicated nonlinear moving boundary problem, for which the theory of existence and uniqueness of solutions, and their continuous dependence on data, has only recently become the focus of a systematic mathematical study. Thus, it is of major interest to develop a general framework for the study of solutions to moving boundary problems.

In this paper we survey some recent developments and open problems in this area. In particular, we focus on problems arising from the interaction between incompressible, viscous fluids and elastic, viscoelastic, or rigid solids (also referred to as structures). Fluid-structure interaction (FSI) problems are ubiquitous in nature, technology, and engineering: from environmental science, where pollutant concentration is studied in aquifers through poroelastic media, to biomedical engineering and cardiovascular medicine, where, e.g., designs for vascular stents for treatment of coronary artery disease and heart valve replacement are studied.

Interestingly enough, even though the mathematical theory of the motion of bodies in a liquid is one of the oldest and most classical problems in fluid mechanics, mathematicians have only recently become interested in a systematic study of the basic problems related to fluid-structure interaction. One reason for this may be that problems of this type are notoriously difficult to study. In addition to the nonlinearity in the fluid and possibly the structure equations, the nonlinear coupling between the fluid and structure motion may give rise to strong geometric nonlinearities. Mathematical study of the existence of solutions to the coupled problems must account for nonlinearities due to the strong energy exchange between the fluid and (elastic) structure motion in problems with nonlinear coupling, and they must employ novel compactness arguments to deal with the nonlinearities. Due to the fluid domain motion, the compactness results must hold for a family of operators defined on time-dependent function spaces associated with moving domains not known a priori. Additionally, the compactness arguments must account for the fact that the coupled problem involves two sets of equations of different type (parabolic vs. hyperbolic) accounting for the different physics in the problem. Crucial for the existence proofs and the compactness arguments is to make use of the parabolic regularizing effects (by the fluid viscosity) to keep the high frequency oscillations of the (hyperbolic) structure under control.

In existence proofs, and in numerical schemes, an additional difficulty is imposed by the incompressibility of the fluid. The main difficulties in existence proofs relate to the construction of divergence-free extensions of fluid velocity to a larger domain containing all the moving domains and to obtaining quantitative estimates of the extensions in terms of the changing geometry. Incompressibility is intimately related to the pressure, and pressure is a major component of the load (i.e., contact force) exerted by the fluid onto the solid. Designing constructive existence proofs and numerical schemes that approximate the load “correctly” is a key ingredient for the stability of constructive solution schemes. In particular, the fluid surrounding the structure affects the structure motion as an extra mass that the structure must displace when moving within a fluid. This has long been known in engineering as the added mass effect. Not accounting for the added mass effect can have negative impact on the stability of partitioned FSI schemes, and this is a well-known problem within FSI problems for which the density of the structure is less than or equal to that of the fluid, i.e., for which the structure is light with respect to the fluid. The added mass is a leading order effect in biofluidic FSI problems, since biological tissues (structures) have density which is approximately the same as that of the surrounding fluid. A failure to account for this effect is associated with the lack of uniform energy estimates in constructive existence proofs for nonlinearly coupled FSI problems.

The question of global-in-time existence of solutions to moving boundary problems is affected by two open problems. One is inherited from the Navier–Stokes equations and the open outstanding question of global existence of strong solutions. The other is related to the so-called no-collision paradox: global weak solution existence results for moving boundary problems are typically obtained until a possible fluid domain degeneracy occurs, such as, e.g., collapse of the cylindrical tube leading to the cross-sectional area of the tube approaching zero. The problems of finite-time contact between elastic bodies in a viscous, incompressible fluid remains an open question—the no-collision paradox. As we shall see below, various questions related to the no-collision paradox are being investigated, including the possibility of finite time contact for classical models, investigation and design of mathematical models that would allow finite time contact, and the type of boundary conditions (no-slip versus slip) for which finite-time contact may occur.

Nonlinearities in the coupled FSI problem also affect the study of uniqueness of solutions. It is not surprising that uniqueness of weak solutions to the coupled FSI problems is still largely an outstanding open problem, since even in the case of classical three-dimensional Navier–Stokes equations, the uniqueness of the Leray–Hopf weak solutions has not been resolved. However, recent advances in this area are significant, and we summarize those results below.

To explain the main challenges in more detail, we present a benchmark problem for FSI involving elastic structures as well as a benchmark problem for FSI involving rigid solids, and we provide a literature review of the recent results.

2. FSI with elastic structures

Although the development of numerical methods for fluid-structure interaction problems started almost 40 years ago (see, e.g., 89414246527374757678868790113114117131 and the references therein), the development of existence theory for FSI problems started less than 20 years ago. We state a benchmark problem in this field and summarize some recent results and open problems.

Description of the main problem

To describe the interaction between a fluid and an elastic (or viscoelastic) structure across a moving interface mathematically, two types of coupling conditions have to be prescribed. This contrasts with classical fluid dynamics problems defined on fixed domains, where only one boundary condition (e.g., the no-slip condition) is sufficient to define the problem. As mentioned earlier, the two sets of coupling conditions describe the following: (1) how the kinematic quantities, such as velocity, are coupled (the kinematic coupling condition, e.g., no-slip); and (2) the elastodynamics of the fluid-structure interface (the dynamic coupling condition). While the precise form of the kinematic and dynamic coupling conditions depends on the particular application at hand, the most common coupling is done via the no-slip kinematic condition (which states that the fluid and structure velocities are continuous across the moving interface) and the dynamic coupling condition (which states that the fluid-structure interface, namely the moving boundary, is driven by the jump in traction, i.e., normal stress, across the interface). For problems in which one expects small interface displacements and small displacement gradients, the coupling conditions may be evaluated at a fixed interface without changing the fluid domain; namely, the fluid and structure may be linearly coupled 10114593. For problems where this may not be a good approximation of reality, the coupling conditions must be evaluated across the moving interface, giving rise to an additional nonlinearity in the problem, which is due to the change of geometry of the moving boundary; namely, the fluid and structure are nonlinearly coupled. In the latter case the fluid domain is a function of time, and additionally, it is not known a priori since it depends on the unknowns in the problem, namely, the displacement of the fluid-structure interface.

The geometric nonlinearity, associated with the fluid domain motion not known a priori, presents one of the major, new difficulties in studying this class of problems mathematically.

2.1. FSI benchmark problem with no-slip

The simplest example of a moving boundary problem with nonlinear coupling involving a deformable (elastic) structure is a benchmark problem derived from modeling blood flow in a segment of an artery. The fluid domain is a cylinder with an elastic (viscoelastic) lateral boundary. For simplicity, we present the problem in two dimensions, although three-dimensional versions of the problem have been studied in, e.g., 103. In this benchmark problem, we will be assuming that the lateral boundary is thin with small thickness, $h \ll 1$, and with the reference configuration $\Gamma$ corresponding to a straight cylinder of length $L$ and radius $R$:

$$\begin{equation*} \Gamma = \{ (z,R) | z \in (0,L) \}. \end{equation*}$$

In most literature involving FSI with thin structures (except for the recent results in 23106), the lateral boundary is assumed to displace only in the vertical (normal, transverse) direction, rendering longitudinal displacement negligible. By using $\eta$ to denote the vertical component of displacement, the fluid domain $\Omega ^\eta (t)$ can be described by

$$\begin{equation*} \Omega ^\eta (t)=\{(z,r)\in \mathbb{R}^2 | z\in (0,L), r\in (0,R+\eta (t,z)\}, \end{equation*}$$

where $R$ is the radius of the reference cylinder. The reference fluid domain will be denoted by $\Omega = (0,L) \times (0,R)$.

The fluid flow is modeled by the Navier–Stokes equations for an incompressible, viscous fluid, defined on a moving domain $\Omega ^\eta (t)$, which are not known a priori,

$$\begin{equation} \left.\begin{array}{rcl} \rho _f\big ({\partial _t{\mathbf{u}}}+({\mathbf{u}}\cdot \nabla ){\mathbf{u}}\big )&=&\nabla \cdot \boldsymbol{\sigma }\\\nabla \cdot {\mathbf{u}}&=&0 \end{array} \right\}\ \text{in}\; \Omega ^\eta (t),\ t\in (0,T), {\tag{2.1}} \end{equation}$$

where $\boldsymbol{\sigma }$ is the Cauchy stress tensor, $\rho _f$ is the fluid density, and $\mathbf{u}=\mathbf{u}(\boldsymbol{x},t) = (u_z,u_r)$ is the fluid velocity. For Newtonian fluids

$$\begin{equation*} \boldsymbol{\sigma }=-p\mathbf{I}+2\mu \mathbf{D}(\mathbf{u}), \end{equation*}$$

where $\mu$ is the dynamic viscosity coefficient and $\mathbf{D}(\mathbf{u})=\frac{1}{2}(\nabla \mathbf{u}+\nabla ^{\tau }\mathbf{u})$ is the symmetrized gradient of ${\mathbf{u}}$.

In this benchmark problem the flow is driven by the inlet and outlet dynamic pressure data, and the flow is normal to the inlet and outlet boundary $\Gamma _{in}=\{0\}\times (0,R)$ and $\Gamma _{out}=\{L\}\times (0,R)$, respectively,

$$\begin{equation} \left. \begin{array}{rcl} {p+\frac{\rho _f}{2}|u|^2}&=&P_{in/out}(t)\\u_r &=& 0 \\\end{array} \right\} \quad \text{on}\ \Gamma _{in/out}, {\tag{2.2}} \end{equation}$$

where $P_{in/out}\in L^{2}_{loc}(0,\infty )$ are given. At the bottom boundary $\Gamma _b=(0,L)\times \{0\},$ the symmetry boundary conditions are prescribed,

$$\begin{equation} u_r=\partial _r u_z=0,\quad \text{on}\ \Gamma _b. {\tag{2.3}} \end{equation}$$

See Figure 2, left.

Different inlet/outlet boundary conditions have been used in numerical simulations, including in Dirichlet data given in terms of the prescribed fluid velocity and in Neumann data given in terms of the prescribed normal stress. Prescribing the correct boundary conditions, especially at the outflow boundary, is very important in correctly capturing the physiological flow conditions within a subregion of the cardiovascular system that is being modeled 117. From the analysis point of view, it is now known that some of the numerically convenient outlet boundary conditions, such as the Neumann, or do nothing, outlet boundary conditions, may produce instabilities 132 or ruin well-posedness by producing multiple solutions to the steady flow, as was recently shown in 77. In this paper we consider dynamic pressure data 2.2 both at the inlet and at the outlet, which is a boundary condition that is consistent with the energy of the coupled problem.

Under fluid loading, and possibly some external loading, the elastic cylinder deforms. See Figure 2. We denote by

$$\begin{equation} \Gamma ^{\boldsymbol{\eta }}(t)=\{(t,z,R+\eta (t,z)) | z\in (0,L) \} {\tag{2.4}} \end{equation}$$

the location of the deformed cylinder lateral boundary at time $t$. The elastic properties of the cylinder’s lateral wall can be described by an operator ${\mathcal{L}}_e$, so that the elastodynamics problem, in Lagrangian formulation, can be written as

$$\begin{eqnarray} \rho _{s} h \partial _{tt} \eta + {\mathcal{L}}_e \eta = { f}, \quad & \text{on} \ \Gamma ,\ t\in (0,T), {\tag{2.5}} \end{eqnarray}$$

where $\rho _s$ is the structure density, $h$ is the thin structure thickness, and ${f}$ is the vertical component of the external loading (force density) experienced by the elastic structure. The loading $f$ in the coupled problem will come from the jump in the normal stress across the structure, i.e., from the fluid load experienced by the structure (assuming that the external loading is zero). The operator ${\mathcal{L}}_e$ is associated with the elastic energy of the structure, such as the membrane or shell energy (see 102), and it is typically continuous, positive-definite, and coercive on some Hilbert space $\chi$.

The coupling

The fluid flow influences the motion of the structure through traction forces (i.e., by the normal stress exerted onto the structure at $\Gamma ^{\boldsymbol{\eta }}(t)$) while the structure influences the fluid through its inertial and elastic forces due to the structure motion and stretching/recoil. Additionally, the fluid and structure “feel” each other through the continuity of the fluid and structure velocities at the interface. Thus, the kinematic and dynamic coupling conditions are, respectively,

$$\begin{align} {({\partial _t \eta }(t,z),0)} = \boldsymbol{u}(t,z,R+\eta (t,z)), {\tag{2.6}}\\ \rho _s h {\partial ^2_t \eta } + {\mathcal{L}}_e \eta = -J \ (\boldsymbol{\sigma }{\boldsymbol{n}})|_{(t,z,R+\eta (t,z))}\cdot \mathbf{e}_r, {\tag{2.7}} \end{align}$$

where $J=\sqrt {1 + \left(\frac{\partial \eta }{\partial t}\right)^2}$, is the Jacobian of the transformation from Eulerian coordinates to Lagrangian coordinates, and $\mathbf{e}_r$ is the unit vector in the vertical direction. Here, we have assumed that external forcing onto the structure is zero. Generalizations to include external forcing due to the presence of another elastic structure or other types of forcing can be found in 104.

The geometric nonlinearity due to the fluid domain motion, described by the composite function $\boldsymbol{u}(t,z,R+\eta (t,z))$, is generally handled by introducing a family of mappings, parameterized by time, called the arbitrary Lagrangian-Eulerian (ALE) mappings, discussed in section 3. In terms of ALE mappings, the trace of the fluid velocity on $\Gamma ^\eta (t)$ is described by a composite function between the velocity and the ALE mapping.

Equations 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7 define a nonlinear moving-boundary problem for the unknown functions $\mathbf{u}$ and $\eta$. The problem is supplemented with the initial conditions

$$\begin{equation} \mathbf{u}(0,.)=\mathbf{u}_0,\quad \eta (0,.)=\eta _0,\quad \partial _t\eta (0,.)=v_0, {\tag{2.8}} \end{equation}$$

satisfying the compatibility conditions

$$\begin{equation} \begin{array}{c} \mathbf{u}_0(z,R+\eta _0(z))=v_0(z)\mathbf{e}_r,\quad z\in (\Gamma ), \\\eta _0(0)=\eta _0(L)=v_0(0)=v_0(L)=0, \\R+\eta _0(z) > 0,\quad z\in [0,L]. \end{array} {\tag{2.9}} \end{equation}$$

Thus, the benchmark nonlinear moving-boundary problem, which exemplifies the main difficulties associated with studying moving boundary problems with nonlinear coupling, can be summarized as follows. Find $\mathbf{u}=(u_z(t,z,r),u_r(t,z,r))$, $p$, and $\eta (t,z)$ such that

$$\begin{equation*} \left.\begin{array}{rcl} \rho _f\big ({\partial _t\mathbf{u}}+(\mathbf{u}\cdot \nabla )\mathbf{u}\big )&=&\nabla \cdot \boldsymbol{\sigma }(\mathbf{u},p) \\\nabla \cdot \mathbf{u}&=&0 \end{array} \right\}\ \text{in}\; \Omega ^\eta (t),\ t\in (0,T), \end{equation*}$$

$$\begin{equation*} \left. \begin{array}{rcl} \mathbf{u}|_{\Gamma ^\eta (t)}&=&{{\partial _t\eta }\mathbf{e}_r} \\\rho _s h \partial ^2_t\eta +{\mathcal{L}}_e \eta &=&-J^\eta \boldsymbol{\sigma }\mathbf{n}|_{\Gamma ^\eta (t)} \cdot \mathbf{e}_r \end{array} \right\}\ \text{on}\; (0,T)\times \Gamma , \end{equation*}$$

$$\begin{equation*} \left. \begin{array}{rcl} u_r&=&0\\{\partial _r u_z}&=&0 \end{array} \right\}\ \text{on}\; (0,T)\times \Gamma _b, \end{equation*}$$

$$\begin{equation*} \left. \begin{array}{rcl} p+\frac{\rho _f}{2}|u|^2&=&P_{in/out}(t)\\u_r&=&0 \end{array} \right\}\ \text{on}\; (0,T)\times \Gamma _{in/out}, \end{equation*}$$

$$\begin{equation*} \eta (t,0)=\partial _z\eta (t,0)=\eta (t,L)=\partial _z\eta (t,L)= 0 \quad \text{on}\; (0,T), \end{equation*}$$

$$\begin{equation*} \left. \begin{array}{rcl} \mathbf{u}(0,.)&=&\mathbf{u}_0\\\eta (0,.)&=&\eta _0\\\partial _t\eta (0,.)&=&v_0 \end{array} \right\}\ \text{at}\; t = 0. \end{equation*}$$

The energy

This benchmark problem satisfies the formal energy inequality

$$\begin{equation} \frac{d}{dt}E(t)+D(t)\leq C(P_{in}(t),P_{out}(t)), {\tag{2.10}} \end{equation}$$

where $E(t)$ denotes the sum of the kinetic energy of the fluid and of the structure, and the elastic energy of the membrane shell,

$$\begin{equation} \begin{array}{rcl} E(t)&=&{\frac{\rho _f}{2}\|\mathbf{u}\|^2_{L^2(\Omega ^\eta (t))}+\frac{\rho _sh}{2}\|\partial _t\eta \|^2_{L^2(\Gamma )}} + \left<{\mathcal{L}}_e \eta ,\eta \right>,\\\end{array} {\tag{2.11}} \end{equation}$$

where $\left<{\mathcal{L}}_e \eta ,\eta \right>$ corresponds to the elastic energy of the structure, which for the cylindrical Koiter shell allowing only radial displacement reads

$$\begin{equation*} \left<{\mathcal{L}}_e \eta ,\eta \right> \coloneq {\frac{1}{2}\big (C_0\|\eta \|^2_{L^2(\Gamma )}+C_1\|\partial _z\eta \|^2_{L^2(\Gamma )}+C_2\|\partial ^2_z\eta \|^2_{L^2(\Gamma )}\big )}. \end{equation*}$$

The term $D(t)$ captures dissipation due to fluid viscosity,

$$\begin{equation} D(t)=\mu \|\mathbf{D}(\mathbf{u})\|^2_{L^2(\Omega ^\eta (t)))}, {\tag{2.12}} \end{equation}$$

and $C(P_{in}(t),P_{out}(t))$ is a constant which depends only on the inlet and outlet pressure data, which are both functions of time.

2.2. FSI benchmark problem with Navier-slip

While the assumption on the continuity of normal velocity components is reasonable for impermeable boundaries,

$$\begin{equation} ( \partial _t\boldsymbol{\eta }- \boldsymbol{u}) \cdot \mathbf{n}= 0 \quad {\text{(nonpenetration condition}),} {\tag{2.13}} \end{equation}$$

the continuity of the tangential velocity component in the no-slip condition is justified only when molecular viscosity is considered 99. Navier contested the no-slip condition for Newtonian fluids 109 when he claimed that the tangential, slip velocity should be proportional to the shear stress. For moving boundary problems this means that the jump in the tangential components of the fluid and solid velocities at the moving boundary is proportional to the shear stress,

$$\begin{equation} ( \partial _t\boldsymbol{\eta }- \boldsymbol{u}) \cdot \boldsymbol{\tau }= \alpha \boldsymbol{\sigma }\mathbf{n}\cdot \boldsymbol{\tau }\quad {\text{(Navier slip)},} {\tag{2.14}} \end{equation}$$

where $\mathbf{n}$ and $\boldsymbol{\tau }$ to denote the unit normal and tangent to the fluid domain boundary, respectively, $\boldsymbol{\sigma }$ is the fluid Cauchy stress tensor, and $\alpha$ is the proportionality constant known as the slip length; $1/\alpha$ has the units of friction.

The benchmark problem defined on the domain shown in Figure 2, incorporating the Navier-slip condition as the kinematic coupling condition, can be summarized as follows: find $(\mathbf{u},p,\boldsymbol{\eta })$ such that the following holds.

The fluid equations:

$$\begin{equation*} \left. \begin{array}{rcl} \rho _F (\partial _t\mathbf{u}+\mathbf{u}\cdot \nabla \mathbf{u})&=&\nabla \cdot \boldsymbol{\sigma }(\mathbf{u},p) \\\nabla \cdot \mathbf{u}&=&0 \end{array} \right\}\ \text{in}\ \Omega ^{\eta }(t),\ t\in (0,T). \end{equation*}$$

The elastic structure (Navier-slip coupling on $(0,L)\times (0,T)$):

$$\begin{eqnarray*} \rho _{S} h \partial _{tt}\boldsymbol{\eta }(t,z) +{\mathcal{L}}_e\boldsymbol{\eta }(t,z) &=& -J(t,z)\boldsymbol{\sigma }\big (\boldsymbol{\varphi }(t,z)\big )\mathbf{n}(t,z), \\ {\partial _t\boldsymbol{\eta }(t,z)\cdot \mathbf{n}(t,z)} &=& \boldsymbol{u}(\boldsymbol{\varphi }(t,z))\cdot \mathbf{n}(t,z), \\ {(\partial _t\boldsymbol{\eta }(t,z)-\mathbf{u}(\boldsymbol{\varphi }(t,z))\cdot \boldsymbol{\tau }(t,z)} &=& \alpha \boldsymbol{\sigma }\big (\boldsymbol{\varphi }(t,z)\big )\mathbf{n}(t,z)\cdot \boldsymbol{\tau }(t,z), \end{eqnarray*}$$

with

$$\begin{equation*} \boldsymbol{\eta }(t,0)=\partial _z\boldsymbol{\eta }(t,0)=\boldsymbol{\eta }(t,L)=\partial _z\boldsymbol{\eta }(t,L)=0,\; t\in (0,T). \end{equation*}$$

Boundary data at the inlet/outlet boundary $\Gamma _{in/out}\times (0,T)$:

$$\begin{equation*} \begin{array}{rl} p+\frac{\rho _F}{2}|\mathbf{u}|^2 =P_{i},\; \mathbf{u}\cdot \boldsymbol{\tau }=0. \end{array} \end{equation*}$$

Boundary data at the bottom, symmetry boundary $\Gamma _b\times (0,T)$:

$$\begin{equation*} \begin{array}{rl} \mathbf{u}\cdot \mathbf{n}=0,\; \partial _\mathbf{n}u_{\tau }=0, \end{array} \end{equation*}$$

with $u_\tau$ denoting the tangental component of velocity $\mathbf{u}$.

Initial conditions:

$$\begin{equation} \mathbf{u}(0,.)=\mathbf{u}_0,\;\boldsymbol{\eta }(0,.)=\boldsymbol{\eta }_0,\; \partial _t\boldsymbol{\eta }(0,.)=\mathbf{v}_0. {\tag{2.15}} \end{equation}$$ The following energy estimate holds:

$$\begin{equation} \begin{split} \frac{1}{2}\frac{d}{dt} &\big (\rho _F\|\mathbf{u}\|^2_{L^2(\Omega ^\eta (t))}+\rho _S h\|\partial _t\boldsymbol{\eta }\|^2_{L^2(\Gamma )}+c\|\boldsymbol{\eta }\|^2_{\chi (\Gamma )}\big )\\ &+\mu \|\mathbf{D}(\mathbf{u})\|^2_{L^2(\Omega ^{\eta }(t))}+\frac{1}{\alpha }\|u_{\tau }-\partial _t\eta _{\tau }\|^2_{L^2(\Gamma ^\eta (t))}\leq \mathbf{C}, \end{split} {\tag{2.16}} \end{equation}$$

where $\mathbf{C}$ depends on the initial and boundary data, and the constant $c$ in front of the $\chi$-norm of $\boldsymbol{\eta }$ is associated with the coercivity of the structure operator ${{\mathcal{L}}}_e$. The reference configuration of the lateral boundary $\Gamma$ is $\Gamma = (0,L) \times \{R\}$.

The no-slip condition is reasonable for a great variety of problems for which the slip-length $\alpha$ is indeed very close to zero. However, in many cases of practical significance, no-slip is not adequate. Examples include flows over hydrophobic surfaces or surfaces treated with a no-stick coating (see Figure 3), flows over rough surfaces (such as those of, e.g., grooved vascular tissue scaffolds), and problems involving contact of smooth solids immersed in a viscous, incompressible fluid. More precisely, for flows over rough (rigid and fixed) surfaces, it has been shown that the Navier-slip boundary condition is the appropriate effective boundary condition 99100. Instead of using the no-slip condition at the small groove scale, the effective Navier-slip boundary condition is applied at the corresponding groove-free, smooth boundary 99100. Regarding contact of smooth bodies immersed in a viscous, incompressible fluid, recent studies have shown that contact is not possible if the no-slip boundary condition is considered 8182122. A resolution to this no-collision paradox is to employ a different boundary condition, such as the Navier-slip boundary condition, which allows contact between smooth rigid bodies 112. Problems of this type arise, e.g., in modeling an elastic heart valve closure where different kinds of ad hoc gap conditions with the no-slip boundary condition have been used to get around this difficulty.

3. The arbitrary Lagrangian-Eulerian (ALE) mappings

To deal with the problems associated with the motion of the fluid domain, different approaches have been taken. One approach is to consider the entire moving boundary problem written in Lagrangian coordinates, as was done in 293091. This is possible to do when the fluid domain is contained in a closed container and no fluid escapes the fluid domain, which is not the case with the benchmark problem considered above. Another approach is to map the problems from the moving domain onto a fixed reference domain using a family of mappings, known as the ALE mappings.

The ALE mappings are a family of (diffeomorphic) mappings, parameterized by $t$, such that

$$\begin{equation} {\mathcal{A}}^{\eta }(t):\Omega \rightarrow \Omega ^\eta (t). {\tag{3.1}} \end{equation}$$

ALE mappings have been extensively used in numerical simulations of moving boundary problems; see, e.g., 4142131. Recently, they have proven to be useful in mathematical analysis as well 23102103. In numerics, one of the reasons for the introduction of ALE mappings is the calculation of the discretized time derivative $\partial _t \mathbf{u}$ since a finite difference approximation of the time derivative (e.g., $(\mathbf{u}^{n+1} - \mathbf{u}^n)/\Delta t$) contains the functions $\mathbf{u}^{n+1}$ and $\mathbf{u}^n$ which are defined on two different domains, one corresponding to the time $t^{n+1} = (n+1)\Delta t$, and the other to $t^n= n\Delta t$. A way to calculate the time derivative of the fluid velocity is then to map the fluid velocities at times $t^{n+1}$ and $t^n$ onto a fixed domain $\Omega$ via the ALE mappings corresponding to $t^{n+1}$ and $t^n$, evaluate the time derivative there, and then map everything back to the physical domain $\Omega ^n \coloneq \Omega ^{\eta ^n}(t^n)$ to solve the fluid equations on the current domain $\Omega ^n$. This introduces an extra advection term in the Navier–Stokes equations, describing the contribution due to the fluid domain motion, so that the Navier–Stokes equations in ALE form become

$$\begin{equation} \left. \begin{array}{rcl} \rho _f\big ({\partial _t\mathbf{u}}+((\mathbf{u}-\mathbf{w}^{\eta ^n})\cdot \nabla )\mathbf{u}\big )&=&\nabla \cdot \sigma \\\nabla \cdot \mathbf{u}&=&0 \end{array} \right\}\ \text{in}\; (t^n,t^{n+1})\times \Omega ^n, {\tag{3.2}} \end{equation}$$

where $\mathbf{w}^{\eta ^n} = \partial _t {\mathcal{A}}^{\eta ^n}(t^n)$ describes the fluid domain velocity, defined by the time derivative of the ALE mapping. In numerical solvers, the ALE mapping is often defined by the harmonic extension of the boundary data onto the fluid domain, i.e., as a solution to the elliptic problem,