Modeling of Fluid Flow in a Flexible Vessel with Elastic Walls

We exploit a two-dimensional model (Ghosh et al. in Q J Mech Appl Math 71(3):349–367, 2018; Kozlov and Nazarov in Dokl Phys 56(11):560–566, 2011, J Math Sci 207(2):249–269, 2015) describing the elastic behavior of the wall of a flexible blood vessel which takes interaction with surrounding muscle tissue and the 3D fluid flow into account. We study time periodic flows in an infinite cylinder with such intricate boundary conditions. The main result is that solutions of this problem do not depend on the period and they are nothing else but the time independent Poiseuille flow. Similar solutions of the Stokes equations for the rigid wall (the no-slip boundary condition) depend on the period and their profile depends on time.


Introduction
In any book about the human circulatory system, one can read that the elasticity of the composite walls of the arteries and the muscle material surrounding arteria's bed significantly contributes to the transfer of blood pushed by the heart along the arterial tree.In addition, hardening of the walls of blood vessels caused by calcification or other diseases makes it much more difficult to supply blood to peripheral parts of the system.At the same time, the authors could not find an answer to a natural question anywhere in the medical and applied mathematical literature: how is the elasticity support mechanism for the blood supply system?In numerous publications, modeling the circulatory system, computational or pure theoretical, there are no fundamental differences between the steady flows of viscous fluid in pipes with rigid walls and vessels with elastic walls.Moreover, quite often much attention is ungroundedly paid to nonlinear convective terms in the Navier-Stokes equations, although blood as a multi-component viscoelastic fluid should be described by much more involved integro-differential equations.We also note that for technical reasons, none of the primary one-dimensional models of a single artery or the entire arterial tree obtained using dimension reduction procedures includes the terms generated by these nonlinear terms.
In connection with the foregoing, in this paper we consider the linear nonstationary Stokes equations simulating a heartbeat, we study time-periodic solutions in a straight infinite cylinder with an arbitrary cross-section.
In the case of the Dirichlet (no-slip) condition, according to well-known results, there are many periodic solutions (v, p), where the velocity v has only the component directed along the cylinder (z-axis) and the pressure p depends linear on z with coefficients depending only on the time variable t.For elastic walls, there is only one such solution up to a constant factor, proportional to the steady Poiseuille flow which does not depend on the time variable but can be considered as periodic one with any period.This is precisely the difference in behaviour of blood flow between elastic and rigid walls.The former smooths (at removal from the heart) the blood flow entering the aorta, and then into the artery with sharp, clearly defined jerks, this is how the heart works with a heart valve, and the latter reproduce the frequency of the flow throughout the length of the pipe.
Due to the elastic walls of arteries, an increased heart rate only leads to an increase in the speed of blood flow (the flux grows) without changing the structure of the flow as a whole, and only at an ultra-high beat rate, when the wall elasticity is not enough to compensate for the flow pulsation, the body begins to feel heart beating.The fact is that the human arterial system is geometrically arranged in a very complex system, gently conical shape of blood vessels, their curvature and a considerable number of bifurcation nodes.Therefore, the considered model problem of an infinite straight cylinder gives only a basic approximation to the real circulatory system and on some of its elongated fragments, which acquires periodic disturbances found in the wrists, temples, neck and other periphery of the circulatory system.The correctness of such views is also confirmed by a full-scale experiment on watering the garden: a piston pump delivers water with shocks, but with a long soft hose the water jet at the outlet is unchanged, but with a hard short-pulsating one.
A general two-dimensional model describing the elastic behaviour of the wall of a flexible vessel has been presented in the case of a straight vessel in [6], [8], in the case of a curved vessel in [1] and for numerical results see [3], [4].The wall has a laminate structure consisting of several anisotropic elastic layers of varying thickness and is assumed to be much thinner than the radius of the channel which itself is allowed to vary.This two-dimensional model takes the interaction of the wall with surrounding or supporting elastic material of muscles and the fluid flow into account and is obtained via a dimension reduction procedure.We study the time-periodic flow in the straight flexible vessel with elastic walls.In comparison with the Stokes problem for the vessel with rigid walls, we prove that Compared with the classical works [5] and [16] of J.R. Womersley, for an alternative description of the above works see [15], our formulation of problem has much in common.In Wormerley's works, axisymmetric pulsative blood flow in a vessel with circular isotropic elastic wall is found as a perturbation of the steady Poisseulle flow.Apart from inessential generalizations like arbitrary shape of vessel's cross-section and orthotropic wall, the main difference of our paper is in the coefficient K(s) which describe the reaction of the surrounding cell material on deformation of the wall.In other words, the vessel is assumed in [5], [16] to "hang in air" while in our paper it is placed inside the muscular arteria's bed as in human and animal bodies intended to compensate for external and internal influences.An evident experiment shows that a rubber or plastic hose uses to wriggle under pulsative water supply.

Preliminaries
Let Ω be a bounded, simple connected domain in the plane R 2 with C 1,1 boundary γ = ∂Ω and let us introduce the spatial cylinder We assume that the curve γ is parameterised as where s is the arc length along γ measured counterclockwise from a certain point and ζ = (ζ 1 , ζ 2 ) is a vector C 1,1 function.The length of the countour γ is denoted by |γ| and its curvature by In a neighborhood of γ, we introduce the natural curvilinear orthogonal coordinates system (n, s), where n is the oriented distance to γ (n > 0 outside Ω).
The boundary of the cylinder C is denoted by Γ, i.e.
The flow in the vessel is described by the velocity vector v = (v 1 , v 2 , v 3 ) and by the pressure p which are subject to the non-stationary Stokes equations: Here ν > 0 is the kinematic viscosity related to the dynamic viscosity µ by ν = µ/ρ b , where ρ b > 0 is the density of the fluid.The elastic properties of the 2D boundary are described by the displacement vector u defined on Γ and they are presented in [7], [5] for a straight cylinder and in [1] for a curve-linear cylinder.If we use the curve-linear coordinates (s, z) on Γ and write the vector u in the basis n, τ and z, where n is the outward unit normal vector, τ is the tangent vector to the curve γ and z is the direction of z axis, then the balance equation has the following form: where ρ(s) is the average density of the vessel wall, σ = ρ b /h, h = h(s) > 0 is the thickness of the wall, A T stands for the transpose of a matrix A, where and K(s)u = (k(s)u 1 , 0, 0).Here k(s) is a scalar function, Q is a 3 × 3 symmetric positive definite matrix of homogenized elastic moduli (see [1]) and the displacement vector u is written in the curve-linear coordinates (s, z) in the basis n, τ and z.Furthermore F = (F n , F z , F s ) is the hydrodynamical force given by where v n and v s are the velocity components in the direction of the normal n and the tangent τ , respectively, whereas v z is the longitudinal velocity component.The functions ρ, k and the elements of the matrix Q are bounded measurable functions satisfying The elements of the matrix Q are assumed to be Lipschitz continuous and Qξ, ξ ≥ q 0 |ξ| 2 for all ξ ∈ R 3 with q 0 > 0, where •, • is the cartesian inner product in R 3 .We note that and one can easily see that on Γ.Here ε ss (u), ε zz (u) and ε sz (u) are components of the deformation tensor in the basis {n, τ , z}.In what follows we will write the displacement vector as u = (u 1 , u 2 , u 3 ), where u 1 = u n , u 2 = u s and u 3 = u z .For the velocity v we will use indexes 1, 2 and 3 for the components of v in y 1 , y 2 and z directions respectively.Furthermore the vector functions v and u are connected on the boundary by the relation The problem (3)-( 9) appears when we deal with a flow in a pipe surrounded by a thin layered elastic wall which separates the flow from the muscle tissue.Since we have in mind an application to the blood flow in the blood circulatory system, we are interested in periodic in time solutions.One of goals of this paper is to describe all periodic solutions to the problem (3), (4), (9) which are bounded in R × C ∋ (t, x).
It is reasonable to compare property of solutions to this problem with similar properties of solutions to the Stokes system (3) supplied with the no-slip boundary condition v = 0 on Γ.
Considering the problem (3), (10) we assume that the boundary γ is Lipschitz only.
The following result about the problem (3), (10) is possibly known but we present a concise proof for reader's convenience.
Theorem 2.1.Let the boundary γ be Lipschitz and Λ > 0. There exists δ > 0 such that if (v, p) are Λ-periodic in time functions satisfying ( 3), ( 10) and may admit a certain exponential growth at infinity where p and v 3 are Λ-periodic functions in t which satisfy the problem Thus the dimension of the space of periodic solutions to the problem (3), ( 10) is infinite and they can be parameterised by a periodic function p * .In the case of elastic wall situation is quite different Theorem 2.2.Let the boundary γ be C 1,1 and Λ > 0. Let also (v, p, u) be a Λ-periodic with respect to t solution to the problem ( 3)-( 9) admitting an arbitrary power growth at infifnity where p 0 and p 1 are constants and v * is the Poiseuille profile, i.e.
The boundary displacement vector u = u(s, z) satisfies the equation If the elements Q 21 and Q 31 vanish then the function u is a polynomial of second degree in z: , where α and β are constants.
Thus, in the case of elastic wall all periodic solutions are independent of t and hence are the same for any period.Moreover inside the cylinder the flow takes the Poiseuille form.The above theorems have different requirements on the behavior of solutions with respect to z, compare ( 11) and ( 14).This is because of the following reason.In the case of the Dirichlet boundary condition we can prove a resolvent estimate on the imaginary axis (λ = iω, ω is real) with exponential weights independent on ω.In the case of the elastic boundary condition exponential weights depends on ω.Becuase of that we can not put in (14) the same exponential weight as in (11).
The structure of our paper is the following.In Sect.3 we treat the Stokes system with the no-slip condition on the boundary of cylinder.Since we are dealing with time-periodic solutions the problem can be reduced to a series of time independent problems with a parameter (frequency).The main result there is Theorem 3.1.Using this assertion it is quite straightforward to proof the main theorem 2.1 for the Dirichlet problem.Parameter dependent problems are studied in Sect.3.2-3.4.Theorem 3.1 is proved in Sect.3.5.
Stokes problem in a vessel with elastic walls is considered in Sect.4.We also reduce the time periodic problem to a series of time independent problems depending on a parameter.The main result there is Theorem 4.1.Using this result we prove our main theorem 2.2 for the case of elastic wall in Sect.4.1.The parameter depending problem is studied in Sect.4.2-4.6.The proof of Theorem 4.1 is given in Sect.4.7.In Sect.4.8 we consider the case when the parameter in the elastic wall problem is vanishing.This consideration completes the proof of Theorem 2.2.

Dirichlet problem for the Stokes system
The first step in the proof of Theorem 2.1 is the following reduction of the time dependent problem to time independent one.Due to Λ-periodicity of our solution we can represent it in the form where These coefficients satisfy the following time independent problem with the Dirichlet boundary condition and with ω = 2πk/Λ and F = (F 1 , F 2 , F 3 ) = 0 (for further analysis it is convenient to have an arbitrary F).
Theorem 3.1.There exist a positive number β * depending only on Ω and ν such that for β ∈ (0, β * ) the only solution to problem ( 20), ( 21) with F = 0 which may admit a certain exponential growth where p 0 and p 1 are constants and v satisfies i.e. the flux does not vanish for this solution.
We postpone the proof of the above theorem to Sect.3.5 and in the next section we present the proof of Theorem 2.1

Proof of Theorem 2.1
By ( 11) Applying Theorem 3.1 and assuming δ < β * , we get that where p 0k and p 1k are constants.This implies that v 1 = v 2 = 0, v 3 depends only on y, t and p = p 0 (t)z + p 1 (t), which proves the required assertion.

System for coefficients (20), (21)
To describe the main solvability result for the problem (20), ( 21), let us introduce some function spaces.For β ∈ R we denote by L 2 β (C) the space of functions on C with the finite norm .
By W 1,2 β (C) we denote the space of functions in C with the finite norm We will use the same notation for spaces of vector functions.
Proposition 3.1.Let the boundary γ is Lipschitz and ω ∈ R.There exist β * > 0 independent of ω such that the following assertions are valid: where C may depend on β, ν and Ω.Moreover, with certain constants p 0 and p 1 .Here v is solution to ( 24).
Remark 3.2.If γ is C 1,1 then it follows from [14] that the left-hand side in ( 26) can be replaced by

Operator pencil, weak formulation
We will use the spaces of complex valued functions H 1 0 (Ω), L 2 (Ω) and H −1 (Ω) and the corresponding norms are denoted by Let us introduce an operator pencil by ) is a vector function and p is a scalar function in Ω .This pencil is defined for vectors (v, p) such that v = 0 on γ.Clearly S(λ) : is a bounded operator for all λ ∈ C. The following problem is associated with this operator supplied with the Dirichlet condition The corresponding sesquilinear form is given by . This form is well-defined on H 1 0 (Ω) 3 × L 2 (Ω).The weak formulation of (31)-(33) can be written as As it was shown in the proof of Lemma 3.2(ii) [13] the operator S(λ) is isomorphism for λ = iξ, ξ ∈ R \ {0}.Since the operator corresponding to the difference of the forms A for different λ is compact, the operator pencil S(λ) is Fredholm for all λ ∈ C and its spectrum consists of isolated eigenvalues of finite algebraic multiplicity, see [2].

Operator pencil near the imaginary axis ℜλ = 0
Here we consider the right-hand sides in (31)-(33) as follows f ∈ L 2 (Ω) and The next assertion is proved in Lemma 3.2(i) [13], after a straightforward modification.
has a solution w ∈ H 1 0 (Ω) satisfying the estimate and C depends only on Ω.The mapping h → w can be chosen linear.
The proof of the next lemma can be extracted from the proof of Lemma 3.2(ii) [13].
Then the solution to (31)-(33) admits the estimate and where the constant C depends only on ν and Ω.
Proof.To estimate the norm of v by the right-hand side in (36), we take v = v in (34) and obtain This implies the inequality (36).To estimate the norm of p, we choose ∇ iξ w = p.Then the relation (34) becomes Since by Lemma 3.1 we arrive at (37).
Now we can describe properties of the pencil S in a neighborhood of the imaginary axis ℜλ = 0. Lemma 3.4.There exist β * > 0 such that the following assertion are valid: (i) the only eigenvalue of S in the strip |ℜλ| < β * is zero; and its norm is estimated as follows: where the constant C may depend on β, ν and Ω.
Proof.First, we observe that Thus the first form is a small perturbation of the second one.Now using Lemmas 3.2 and 3.3 for small β we arrive at the existence of β * , which satisfies (i).Moreover, the estimates (36) and (37) are true for λ = β + iξ for a fixed β ∈ (−β * , 0) ∪ (0, β * ) and arbitrary real ξ.With this the constants in ( 36) and (37) may depend now on β, ν and Ω only.To prove (ii) in Proposition 3.1, we observe that and the relation ( 28) is obtained by applying the residue theorem.Now we turn to Proof of Theorem 3.1.Let (V, P) be a solution to (20), (21) satisfying (22).Our first step is to construct a representation of the solution in the form where V (±) ∈ W 1,2 ±β (C), P (±) ∈ L 2 ±β (C) and they solve the problem (20), (21) with certain F = F (±) ∈ L 2 ±β (C).
By the second equation in (20) and by (21) the flux The vector-function (0, 0, p 0 v3 , p 0 ) with a constant p 0 verifies the homogeneous problem (20), (21) and its flux does not vanish in the case p 0 = 0.So subtracting it with appropriate constant p 0 from (V, P) we can reduce the proof of theorem to the case Ψ = 0.In this way we assume in what follows that this is the case.Let ζ(z) be a smooth cut-off function equal 1 for large positive z and 0 for large negative z and let ζ ′ be its derivative.We choose in (42) where the vector function We construct solution W by solving two-dimensional Stokes problem in Ω depending on the parameter z: and W k = 0 on γ, k = 1, 2. This problem has a solution in H 1 0 (Ω) 2 × L 2 (Ω), which is unique if we require Ω Qdy = 0.If we look on the dependence on the parameter z it is the same as in the right-hand side.So where Similar formulas are valid for (V (−) , P (−) ) with k .By Proposition 3.1(ii) this implies (V (+) , P (+) ) + (V (−) , P (−) ) = (0, 0, p 0 v(y), p 0 z + p 1 ) for certain constants p 0 and p 1 , which furnishes the proof of the assertion.

Stokes flow in a vessel with elastic walls
This section is devoted to the proof of Theorem 2.2.As in the case of the Dirichlet problem considered in Sect.3 we represent solutions to the problem (3)-( 9) in the form (18) (for the velocity v and the pressure p) and for the displacements u.The coefficients in (18) are given by (19) and in (45) by The above introduced coefficients satisfy the time independent problem where F = 0, G = 0 and ω = 2πk/Λ (for forthcoming analysis it is convenient to have arbitrary right-hand sides in this problem).
Theorem 4.1.Let ω ∈ R and ω = 0. Then there exists β > 0 depending on ω such that the only solution to the homogeneous (F = 0 and G = 0) problem ( 46)-( 48) subject to We postpone the proof of the formulated theorem to Sect.4.7 and in the next section we give the proof of Theorem 2.2

Proof of Theorem 2.2
By (49), Applying Theorem 4.1 we get V k = 0, P k = 0 and U k = 0 for k = 0. Now using Theorem 3.1 and consideration in forthcoming Sect.4.8 for ω = 0 we arrive at the required assertion.

System for coefficients (46)-(48)
To formulate the main solvability result for the system (46)-( 48), we need the following function spaces where C may depend on ω, β, ν and Ω.Moreover, ) is the solution from (i) for k = 1, 2 respectively, then they coincide.3), ( 4), (9) It is convenient to rewrite the Stokes system (20) in the form

Transformations of the problem (
where and δ i,j is the Kronecker delta.Moreover, relations ( 4) and ( 9) become and Here F = e −iωt F .Next step is the application of the Fourier transform.We set V(x) = e λz v(y), P (x) = e λz p(y) and U(x) = e λz u(y).
As the result we obtain the system where The equations ( 52) and (53) take the form and Here Φ(s) = (Φ n , Φ z , Φ s ) and

Weak formulation and function spaces
Let us introduce an energy integral and put where •, • is the euclidian inner product in C 3 .Since the matrix Q is positive definite where ξ ∈ R and c 1 is a positive constant independent of ξ.Another useful inequality is the following or by using Korn's inequality where c 3 does not depend on q.
To define a weak solution, we introduce the vector function spaces: and We supply the space Z with the inner product Since ω = 0, the norm ||u 1 ; H 1/2 (γ)|| is estimated by the norm of v in the space H 1 (γ) therefore we do not need a term with u 1 and û1 in (63), indeed.Let also v, p, u; v, p, û We introduce also a sesqui-linear form corresponding to the formulation (54), ( 55), (57), (58): Clearly, this form is bounded in Z × L 2 (Ω).For F ∈ Z * , H ∈ L 2 (Ω) and h ∈ L 2 (Ω) the weak formulation reads as the integral identity which has to be valid for all (v, p, û) for all (v, û) ∈ Z.
It will be useful to introduce the operator pencil in the space Z × L 2 (Ω) depending on the parameter λ ∈ C by Lemma 4.2.Let ω ∈ R and ω = 0. Then the operator pencil C ∋ λ → Θ(λ) possesses the following properties: (i) Θ(λ) is a Fredholm operator for all λ ∈ C and its spectrum consists of isolated eigenvalues of finite algebraic multiplicity.The line ℜλ = 0 is free of the eigenvalues of Θ.
(ii) Let λ = iξ, ξ ∈ R. Then there exists a positive constant ρ(|ω|) which may depend on |ω| such that the solution of problem ( 65 where The constant c here may depend on ω but it is independent of ξ.
Proof.Let λ = iξ and (i) Consider the integral identity We want to apply the Lax-Milgram lemma to find solution (v, u) ∈ X .First, we note that and where the constant c does not depend on ω and ξ.
We use the representation where τ ∈ (0, 1].Let us estimate the last term in (76).We have Using above inequalities together with (62) for a small ǫ, we arrive at the estimate where C ω is a positive constant which may depend on ω and |ξ| is chosen to be sufficiently large with respect to |ω| + 1.On the basis of one can continue the estimation in (78) as follows: with possibly another constant C ω .Application of the Lax-Milgram lemma gives existence of a unique solution in X and the following estimate for this solution Ω (|ξ| 2 |v| 2 + |∇ y v| 2 )dy (80) with a constant C which may depend on ω and ξ.It remains to estimate the function p.We chose the test function (v, û) in the following way: iω û = v on ∂Ω, v3 = 0 and v′ ∈ H 1 (Ω) solves the problem where h ∈ L 2 (Ω).According to Lemma 4.1 the mapping h → v′ can be chosen linear and satisfying the estimate (69).The pressure p must satisfy the relation One can verify using (80) that the right-hand side of (81) is a linear bounded functional with respect to h ∈ L 2 (Ω) and therefore there exists p ∈ L 2 (Ω) solving (81) and estimated by the corresponding norm of F .Thus the operator pencil (67) is isomorphism for large |ξ|.
Since the operator Θ(λ 1 ) −Θ(λ 2 ) is compact we obtain that the spectrum of the operator pencil Θ consists of isolated eigenvalues of finite algebraic multiplicities, see [2].
To obtain the estimate for p, we proceed as in (i).
is valid, where N is given by ( 72).The positive constant C here may depend on β, ω, ν and Ω.
Proof.Let λ = β + iξ.It is straightforward to verify that is a small operator for small β.Therefore the estimate (83) for large |ξ| follows from (71).From Lemma 4.2(i) it follows that this can be extended on ξ ∈ R if β is chosen sufficiently small.Thus we arrive at both assertions of the lemma.To conclude with (ii) we observe that the same proposition provides and the assertion (ii) in Proposition 4.1 is obtained by applying the residue theorem.
Proof of Theorem 4.1.Let (V, U, P) be a solution to (46)-(48) satisfying (49).Our first step is to construct a representation of the solution in the form where ±β (Γ).Let ζ be the same cut-off function as in the proof of Theorem 3.1.We choose in (84) where the vector function ( Ṽ, Ũ, P) solves the problem (64) for λ = 0 and with F = 0, H = 0 and h = V 3 (y, z) i.e.
In this problem the variable z is considered as a parameter.In order to apply Lemma 4.2(i) we reduce the above formulation to the case h = 0. Applying for this purpose Lemma 4.1 we find a function V(y) = (V 1 , V 2 , V 3 )(y) solving (86) and satisfying (69) or (70) where iωU = V.The function (V, U) ∈ Z and the above formulation is reduced to the case h = 0 but with some nonzero F .Applying to the new problem Lemma 4.2(i) we find solution satisfying First we consider the case p 0 = p 1 = 0. Namely, we want to solve the homogeneous equation where D 0 and D 1 are defined by (5).First, we are looking for solution independent of z.Then it must satisfy where c 2 and c 3 are constants.
Next let U be a linear function in z, i.e.U(s, z) = zu 0 (s) + u 1 (s).Then ) Since u 0 = (0, α, β) T , α and β are constant, equation (92) takes the form and it is solvable since the term containing u 0 is orthogonal to constant vectors (0, a 1 , a 2 ).Thus there exists linear in z solutions.Let us find these solutions.We have where Now system (93) takes the form and where Using (95) we can write the compatibility condition for (96) as This is equivalent to the following three equations This implies Solving the system we get .
We write the equations (102) as These equations has periodic solutions if The last relation means that at each point (y 1 , y 2 ) on γ the corresponding vector is orthogonal to the normal to this curve at the same point, what is impossible.

3. 5
Proof of Proposition 3.1 and Theorem 3.1 Proof.The assertion (i) in Proposition 3.1 is obtained from Lemma 3.4(ii) by using the inverse Fourier transform together with Parseval's identity.

4. 7
Proof of Proposition 4.1 and Theorem 4.1 Proof.The assertion (i) in Proposition 4.1 is obtained from Proposition 4.2(ii) by using the inverse Fourier transform together with Parseval's identity.