In this paper, we study a two-dimensional (2D) bidirectional fluid–solid coupling problem in blood vessels and prove the existence of weak solutions to this kind of problem. When studying the fluid motion of arterial blood flow in arterial vascular tissue, where the density of the vessel wall is roughly equal to that of the blood, the coupling between the blood and the vessel wall is highly nonlinear. In this case, the fluid space where the arterial blood flow is located is no longer fixed in the fluid–structure interaction (FSI) model, and its boundary position is determined by the position of the elastic vessel wall. The contact force exerted by the elastic vascular structure on the blood flow also affects the movement of the blood flow, while the position of the blood vessel is determined by the contact force exerted by the blood flow on the blood vessel; that is, two-way coupling occurs between the vascular diameter and the blood flow space.

MSC2020 Classification: 35D30, 35M31

1. Introduction

When studying biomechanics, the issue of fluid–structure interaction (FSI) is an unavoidable topic, such as the swimming of sperm cells, plasma–red blood cell interactions, and interactions between blood flow and cardiovascular tissue. Due to the strong nonlinear and multiphysics nature of these problems, a comprehensive study of these problems remains a challenge.

Various studies [1–4] have addressed the existence of solutions for the Navier–Stokes (N–S) equation with second-type boundary conditions (BCs) in a given domain. And other studies [5] use artificial intelligence approach nanofluid with energy chemical reaction effects, and Rasool et al. [6] studied entropy generation with nonlinearly stretching surface in nanofluid flow.

It is customary to use Euler coordinates to describe the motion of fluids. Let the velocity u and pressure p be unknown quantities, treated as functions of both space and time. Considering blood to be a Newtonian fluid, the following equations can be used to describe the motion of the blood flow. We determine (u, p) as flow velocity and pressure defined on Ω_f(t) as a tube region (Figure 1), such that [7]

(1)

(2)

where μ represents the viscosity and ρ_f represents the density of blood (which is assumed to be constant in this paper). In addition, f represents the external force due to the external load, which is related to the elasticity of the underlying vessel wall. Equation (1) ensures the conservation of momentum, while Equation (2) represents the incompressibility condition. Let the initial conditions (i.e., the initial blood flow BCs) for these equations be the following:

(3)

The boundary outside the blood flow and the vessel interface Γ_f, where Γ_f = ∂Ω_f(t)∖Σ(t), can be altered to consider different conditions. For example, a homogeneous initial condition can be imposed on Γ_f, as follows:

Details are in the caption following the image — **Figure 1**
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Blood flow and vessel region.

(4)

Consider a thick blood vessel whose displacement d is decomposed into two components, namely, transverse u^s and radial u^r components. For convenience, we assume that u^s and u^r satisfy two linear decoupling equations, corresponding to Euler–Bernoulli beam equations. Furthermore [8], we set the length of the vessel wall as L, where the displacement of the vessel wall

in the range (0, L) satisfies the following elastic balance equation when 0 < t < T:

(5)

(6)

where ρ_s represents the density of the blood vessel (which is considered to be constant), e represents the thickness of the blood vessel wall, γ represents the normal constant, α_i, i ∈ {1, 2} represents material property constants, and β_i, i ∈ {1, 2} represents additional viscous damping constants. There are two types of loads acting on the vascular structure: loads T_f from the blood fluid (defined below) and the given external surface loads g.

Considering the insufficient conditions to obtain energy estimates, the typical way to address this problem is to improve the initial problem and the boundary value conditions, as follows:

(7)

(8)

These equations must satisfy the following initial conditions:

as well as the BCs. It is possible to assume that the ends are fixed, thus allowing for the application of homogeneous first BCs. These types of conditions are simplifications commonly used in the case of hemodynamic modeling. For example, we may consider the following:

(9)

(10)

Next, we wish to determine the coupling condition between the blood flow and vessel structure. First, it is assumed that the blood flow has friction with the blood vessel boundary, such that the blood flow velocity and the blood vessel velocity are equal at the boundary Γ_f. Considering the fact that blood velocities are described in Eulerian coordinates and that structure velocities are described in Lagrangian coordinates,

(11)

This equation can be rewritten as follows:

This principle yields equal normal components of the blood flow and vessel stress tensors, when considering the coupling of blood flow with the thick vessel structures. The blood vessels here are very thin, and so, the effects of the blood on the blood vessels appear as surface loads on the right-hand side of the blood vessel equations. This external force, T_f, can be defined variationally as follows:

(12)

For each and n, denote an outer cell perpendicular to the deformed wall Σ(t) with , where T_f represents the surface load exerted by the blood on the blood vessel.

In order to rewrite T_f in an efficient manner, v_f is introduced to represent the deformation of the blood domain, mapping

to ∪_t{t} × Ω_f(t). Thus,

maps to

. Let J_f be the Jacobian determinant of v_f and F_f be the deformation gradient. Equation (12) can be written as follows:

(13)

where

represents the normal direction perpendicular to the reference interface

Due to the fact that the blood flow equations are described with Euler coordinates, the blood flow domain is unknown and depends on the blood vessel’s structural displacements, considered as a third coupling condition.

This is evident when setting the second BC. When considering the first BC, the pressure of the blood vessel is determined by adding an additional mass condition, which is included into the equations as the blood flow is incompressible. Thus, the total volume of the blood domain remains constant, as follows:

(14)

This is equivalent to the following:

(15)

The mean pressure can be obtained using the conditional variational method associated with Equation (14) or Equation (15).

With the above conditional, we obtained the existence of weak solutions for bidirectional FSI coupling problem in two-dimensional (2D) arterial blood flow.

2. Variational Formulation

The calculations below are based on the solution (u, p, d) of the coupled blood flow and vessel equations, with the additional assumption that the trial function has sufficient regularity.

Let v be a blood flow trial function satisfying v = 0 on Γ₀ under Robin BCs or v = 0 on Γ₀ under Γ_f under homogeneous BCs.

Let be a vessel structural trial function that satisfies the homogeneous first BC in Equation (9) as .

For the conservation of fluid momentum in Equation (1), both sides are multiplied by v and integrated over the blood flow region Ω_f(t). Likewise, the vessel structural in Equations (5) and (6) are multiplied by v^s and v^r, respectively, and integrated over the blood vessel region (0, L). Then, after the integration by parts and summing the blood flow with the blood vessel’s structural parts, we obtain the following:

It is worth noting that the term associated with the blood flow and the blood vessel interface Γ_f is as follows:

(16)

due to the definition of Γ_f. By choosing a trial function that satisfies the kinematic conditions, we have the following:

(17)

In the case of Robin BCs, we have the following:

Considering the second BCs on Γ_in and Γ_out, we obtain the following:

(18)

3. Energy Estimate

In order to obtain the energy estimate satisfied by (u, p), the blood flow velocity u and the blood vessel structure velocity ∂_td are selected as trial functions for use in the above variational formulas in Equation (17) or Equation (18). These trial functions are acceptable as they meet the kinematic condition in Equation (11). Due to the incompressibility of the blood flow, in the case of a uniform first BC on Γ_f, and as the interface Γ_fΣ(t) moves with a vessel structural velocity equal to the blood flow velocity, we take into consideration the following conservation law:

(19)

Therefore, the emergence of kinetic conditions of the blood flow leads to the following energy conservation condition:

(20)

Remark 1. Consider elastic and hyperelastic thin blood vessel structures. The energy forms are similar, except for the solid energy, which depends on the blood vessel structure of the considered material.

Considering the energy conservation in Equation (20), by Gronwall–Bellman’s lemma, for t ≤ T, we obtain the following:

(21)

The dissipation from the structure is not used here to obtain an energy estimate, which is why there is a time constant and the characteristic is exponential. Therefore, the above energy estimate also holds true for β₁ = β₂ = 0. By exploiting the dissipation of fluids and structures, exponential estimates of their behavior can be obtained. In particular, an estimate of the following form can be obtained:

where C is not time-dependent. Thus, we have that, if

where L²(Ω) is the space of square integrable functions on Ω, H²(Ω) is the Hilbert space of order 2, W^1,p is the Sobolev space with derivative of first order and pth derivative, where p = ∞.

Then we have the following:

In the case of the second BC on Γ_in and Γ_out, Newton’s equation in Equation (19) must be modified, and the kinetic energy flux occurs at the artificial boundary:

Therefore, the conservation of energy is described as follows:

The signs of the last two additional terms are uncertain, and so, energy estimates for the blood flow are not easy to obtain. In two dimensions, energy estimates can be obtained when considering small initial conditions in time; however, they cannot be obtained in three dimensions.

Various articles [2, 9–11] have addressed the existence of solutions to the N–S equations with a second BC in a given domain. As the L² estimate cannot be derived, one strategy to address this problem involves improving the initial problem and BCs, as follows:

where p is substituted by

. As a result, one can obtain an energy estimate that proves the existence of a weak solution in the energy space (the reader is referred to [7, 12, 13], where such BCs are taken into account).

4. Weak Solutions

Let the lateral displacement of the blood vessels satisfy u^s = 0 (i.e., only consider the radial displacement of blood vessels u^s ≠ 0) and consider the complete nonlinear coupling problem. In this context, the blood flow domain is a subregion defined as follows:

where the radial displacement u^r satisfies a wave equation (and, thus, γ = 0 in Equation (6)). In the first part, an additional viscosity term, β₂ > 0, will be considered. In addition, to simplify the discussion, the homogeneous first BC in Equation (3) is imposed by considering the blood flow BC that yields energy estimates. Furthermore, Equations (7) and (8) can also be applied.

The same type of energy estimate as in Equation (21) indicates that we should look for elastic blood vessel displacement u^r in . However, the regularity condition means that the blood flow and blood vessel interface Γ_f are always continuous but not necessarily Lipschitz continuous. Hence, the region is an open region (until R + u^r(t, x) > 0) which may not be Lipschitz continuous. Two major initial issues are to define the function space suitable for the blood flow velocity and to strictly define the velocity of the blood flow on the moving boundary Γ_f.

Remark 2. The beam equation (i.e., γ > 0) gives an estimate of the energy of the elastic blood vessel displacement belonging to . Therefore, the mapping v_f is C¹ as long as R + u^r > 0, and so, the region is Lipschitz continuous.

In this example, we define an open set

and

as well as

Concerning the significance of the traces for the blood flow velocity on the moving boundary of the interface Γ_f, one can easily obtain u(t, x, R + u^r(t, x)). In fact, the elastic blood vessel displacement is only lateral, while the blood flow velocity is in the space H¹ with the following:

This particularity of coupled problems has been widely addressed, such as in [14]. Notably, it can be found that (0, b(x))^T and b ∈ H¹(0, L) belong to

. This provides an improvement in the nondispersion structure of the blood vessel test function. Furthermore, the lateral nature of the motion implies that

is a subregion. The density properties, Korn’s inequality (even though the blood flow region is not Lipschitz continuous), and the possibility of shrinking the blood flow region with regard to vertical movement follow. The considered weak solution is defined as follows (to simplify the discussion, let all of the constants in the equation, except for viscosity, be equal to 1):

(22)

Integrating by parts in both time and space dimensions, a new advection term appears:

. Furthermore, blood flow dissipation only involves the gradient of u, not its other aspects (e.g., symmetry). This simplification comes from the exclusively lateral movement of the elastic blood vessel structure. In fact, using the same velocity on the interface Γ_f and the incompressibility of the blood flow on Σ(t), we can obtain the following:

This feature can be used to simplify weak forms. The volume of the blood flow region must be kept constant during the deformation process; that is, the mass must be conserved. Unlike the case where u^s ≠ 0, the requirement is met by the blood vessel displacement being linear, and u^r should satisfy

. This simplification also derives from the sole lateral movement. When the boundary satisfies the following compatibility conditions:

(23)

we obtain the following existence theorem.

Theorem 1. Assuming the above compatibility conditions in Equation (23) hold, where

as long as the elasticity of the blood vessel does not reach the opposite of the blood flow region, there exists at least one weak solution (u, u^r) of the coupled system. In addition, the solution satisfies the energy estimation.

The first step is to establish an appropriate approximate solution. Simply decoupling the blood flow system from the structure of blood vessel equations can lead to problems, due to the addition of mass effects. Therefore, the blood flow is not decoupled from the structure. For the purpose of resolving the problems associated with the nonsmooth blood flow region and nonlinearity due to advection, the blood flow domain and advection velocity are regularized. To do so, a regularization procedure must be taken into account to match the regularized flow-to-flow velocity to that of the blood vessel structure or to improve the flow–vessel variational formulation in Equation (22), if one wishes to obtain a solution that satisfies the energy estimate. The most common approach has been previously detailed in [8, 9], which relies on blood flow and vessel velocity expressions. However, this requires higher regularity of vessel structural movement than needed here. Thus, we modify the variational formula by pointing out the following: if

on (0, L), then we have (0, b(x))^T = v(x, R + u^r(x)).

The advection term can then be transformed as follows:

where ^∗ represents the value after regularization. Then, we write the modified variational formula, using

to represent the regularized area, as follows:

(24)

for the trial function satisfying v(t, x, R + u^r∗(t, x)) = (0, b(t, x))^T and div v = 0, where x belongs to (0, L).

The hydrokinetic energy can then be recovered by using (v = u, b = ∂_tu^r) as a trial function, as

The approximate problem in Equation (24) has at least one weak solution that depends on a fixed point: Consider a given blood flow region movement of δ and a given advection blood flow velocity v, regularize these values to δ^∗ and v^∗, respectively. Considering

we can solve the following linearized approximation problem:

where v is a trial function satisfying v(t, x, R + δ^∗(t, x)) = (0, b(t, x))^T, div v = 0, and x belongs to (0, L). It is worth noting that the trial function is no longer dependent on the solution, the problem is linear, and the blood flow region movement is smooth; however, the blood flow and vessel equations are still coupled. It should also be pointed out that (∂_tu, ∂_tu^r) is not an acceptable trial function; however, the blood flow acceleration can be easily modified to obtain a blood flow trial function that is nondispersive and equal to the acceleration of the structure on the interface Γ_f.

In addition, the bounded of (∂_tu, ∂_ttu^r) also requires a regularization process.

To determine whether there exists a solution to the approximate problem, it is necessary that a limit exists when the normalization factor tends to 0. The energy estimate leads to an approximate solution of the blood flow velocity uniformly bounded in in , which is the vessel distance of movement.

More specifically, consistent convergence of the interface Γ_f displacement sequence in C⁰ is obtained, which enables passing to the limit in the blood flow domain sequence. However, these kinds of bounds are not sufficient to obtain the ideal strong convergence, and further restrictions are required to obtain the

norm of compactness for the velocity. In nonlinear advection terms, this compactness in

requires transitions to the limit, such as

However, the energy estimates obtained above cannot be used, as they depend on the regularization parameters. This is an essential phase in proving the existence of weak solutions. One can use Aubin’s lemma to prove the required compactness; however, it is challenging to use this theorem when defining a divergence-free function over a time-dependent domain of movement. See [10, 15, 16] for an incompressible N–S equation regarding the moving domain or [17] for the existence of weak solutions to flow–solid boundary problems.

One method which can be used to acquire compactness involves studying the following norms: and . In fact, the following lemma regarding compactness and convergence will be used, which describes the compact set of L^p(0, T; X), where X is a Banach space.

Lemma 1. Let X be a Banach space, 1 ⩽ q < ∞, and F↪L^q(0, T; X) be a sequence in X. Then, F is a relatively compact set of L^q(0, T; X) if and only if:

(a)
When ∀0 < t₁ < t₂ < T, is relatively compact in X.
(b)
as h tends to 0, f in F uniformly converges.

We apply Lemma 1 to the function

, using the regularization parameters q = 2 and X = L²(B) × L²(0, L). It is worth noting that a set B, which contains all blood flow domains

for any t ∈ (0, T), is introduced here, expanding the blood flow velocity:

Considering the energy estimate, condition (a) of Lemma 1 is evidently satisfied, and only the second condition remains to be verified. Given any h > 0, this means we must confirm that g⁻(t, ⋅) = g(t − h, ⋅) and g⁺(t, ⋅) = g(t + h, ⋅) ). Condition (b) is obtained as a consequence of the following lemma:

Lemma 2. Let T > 0. If , then, for sufficiently small h > 0, we have

and

When t > 0,

expands to u^r (and, so, 0 expands to ∂_tu^r), and when t > 0,

expands to 0, where ρ(t) represents the characteristic of

. These estimates are consistent with the regularization parameter.

To demonstrate this conclusion, we may choose one of the variational formulas as Equation (24), as well as trial functions of the form and . However, given that we are addressing mobile domains, these trial functions must be slightly modified to obtain acceptable trial functions.

For σ > 1, we define v_σ as follows:

If v is divergence-free, then v_σ is too. Let

The function v belongs to

, and b belongs to

. For t > 0, u^r can be expanded by

. For t > 0,

and ∂_tu^r can be expanded by 0. The function v is nondispersive. Furthermore, as

and

we have

and

Thus, if

, we have the following:

Once the desired bound on the solution sequence is obtained, the limit can be reached when the regularization parameter tends to 0. In fact, by using ε to represent the regularization parameter, there exists a solution with respect to the sequence , which complies with the ensuing convergence condition.

Let T > 0 such that . Then, T is independent, as is bounded in C⁰. Let represent the limit of the subsequence of . Denote any subsequence of as .

As ε tends to zero, the following convergence occurs:

First, note the equation

on the time and space domain (0, T) × (0, L). For the left term, consider the function w_ε, defined for almost all t as

where R is a continuous linear operator from

, such that

is discrete, and where C_β = (0, L) × (0, β). Then,

is independent of ε, and so, this space is controlled. Therefore, in

, the subsequence

weakly converges to

. There exists w₀ = 0 on Γ_f and

, as

uniformly converges to u^r. Therefore,

and

. Finally, w₀(t, x, R + u^r(t, x)) = 0 on (0, L), but

, such that

on (0, T) × (0, L).

Then, moving to the weak form of the limit, the blood flow trial function depends a priori on ε. However, we consider trial functions that do not depend on ε or are acceptable if ε is small enough.

First, examine the function (v⁰, 0), where and divv⁰ = 0. These trial functions satisfy the property that, for every t, there exists a . Therefore, for sufficiently small ε, , as converges uniformly to u^r as ε → 0.

Consider the trial function pair (v¹, b), where b belongs to

, ∫_ωb = 0, and, for almost all t,

Considering the fact that

, (v¹, b) is a set of trial functions for ε.

With these two kinds of trial functions, the limit can easily be obtained in the weak form when ε tends to 0. Thus, there exists a weak solution satisfying the energy estimate on (0, T).

Finally, insofar as , the solution can be extended. First, create an increasing time series , as follows: Begin with a time step T₁ > 0, such that there is a weak solution until T₁, where . It is possible to slightly change T₁, assuming that ∂_tu^r(T₁) ∈ L²(0, L) and .

Next, let i ≥ 1. Presuming a resolution through the time step T_i has been established,

. The blood vessel makes it possible to build an extension of the solution over several time step intervals, starting from T_i. Considering the prior energy estimate, the following conclusions can be drawn for t ≥ T_i:

For C(T_i, t), we have

where

is non-negative and nondecreasing, with respect to its argument, andC(T_i, t)⩽C(0, t). This prior estimate shows that if

then a solution can be built starting from u(T_i), u^r(T_i) and ∂_tu^r(T_i), until time T_i + τ_i (this is equivalent to picking α = m_i/2). Time T_i+1 should be chosen close to T_i + τ_k (at [T_i + τ_k/2, T_i + τ_i]), in order to have ∂_tu^r(T_i+1) ∈ L²(ω) and

Let . If T^∗ < +∞. Then, ; otherwise, as τ_i ≥ m^∗ for all i, . However, T_i+1 − T_i ≥ τ_i/2 and tends to 0, which contradicts itself.

This makes it possible to prove the existence of a weak solution to this coupling system, as long as the blood vessel has not reached the opposite side of the blood flow region.

5. Conclusions

In this paper, we studied a 2D bidirectional FSI problem in blood vessels. The main novelty in this paper is the boundary which satisfies compatibility conditions as Equation (23), with the conditional we have obtained in the existence theorem for bidirectional FSI coupling problem in 2D arterial blood flow.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

This research was funded by Shanghai Engineering Technology Research Center grant number 18072250900.

Open Research

Data Availability Statement

No underlying data was collected or produced in this study.

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The Existence of Weak Solutions for Bidirectional Fluid–Structure Coupling Problem in 2D Arterial Blood Flow

Abstract

1. Introduction

2. Variational Formulation

3. Energy Estimate

4. Weak Solutions

5. Conclusions

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

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References

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The Existence of Weak Solutions for Bidirectional Fluid–Structure Coupling Problem in 2D Arterial Blood Flow

Abstract

1. Introduction

2. Variational Formulation

3. Energy Estimate

4. Weak Solutions

5. Conclusions

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

Figures

References

Related

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