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ISRN Applied Mathematics
Volume 2013 (2013), Article ID 750185, 14 pages
Global Analysis of a Blood Flow Model with Artificial Boundaries
School of Mathematical Sciences, North-West University, Mafikeng 2735, South Africa
Received 3 June 2013; Accepted 4 August 2013
Academic Editors: S. He and K. Karamanos
Copyright © 2013 S. C. Oukouomi Noutchie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A theoretical model for blood flow in ramifying arteries was introduced and studied numerically (Quarteroni and Veneziani, 2003). A special experimental condition was considered on the artificial boundaries. In this paper, the aim is to analyze the well-posedness of this model, with the focus on the stilted boundary conditions. We use Brouwer’s fixed point theorem to show the existence of a solution to the stationary problem. For the evolutionary version, we use some energy estimates and Galerkin’s method to prove global existence, uniqueness, and stability of a weak solution.
Evolution systems with artificial boundaries are difficult to analyze as a result of the complex dynamics at the boundaries and very few papers have attempted to capture these processes from an analytical point of view. This paper explores this question regarding the flow of blood in a portion of a large artery and addresses the analysis of the Navier-Stokes problem, having provided boundary conditions which can be considered as a generalization of the mean pressure drop problem investigated in [1–3], as they arise in bioengineering applications. Our purpose is to consider both the stationary case and the nonstationary case. In this regard, we will prove the existence of a weak solution for the stationary case based on Brouwer’s fixed point theorem and afterwards, we will establish a well-posedness analysis for the nonstationary case based on a suitable energy estimate that we are going to derive as well as a well-known compactness argument.
1.1. Basic Notations
In this subsection, we summarize some notations that will occur throughout the paper. Vectors and tensors are denoted by bold-face letters : location of fluid particle, : velocity field of the flow, : pressure, : the pressure gradient, : identity tensor, : symmetric Cauchy stress tensor, : dynamic viscosity, : fluid mass density, : the set of real numbers, : the absolute value of and correspondingly, the norm of, : a bounded domain in, : the boundary of, : the gradient of, : the divergence of, : the Laplacian of.
The spaces,,, andand their vector-valued analogues,,, and are defined as usual, the superscript indicating continuous derivatives to a certain order and the subscript zero indicating functions with compact support.
The space , the Hölder space , and the Sobolev space and their vector-valued analogues , , and are also defined as usual. In particular, .
For functions depending on space and time, for a given space of space-dependent functions, we define (for some ) with norm .
When considering functions which depend only on time, we define the space endowed with the norm
In this subsection, we recall some assumptions used in [2, 4], and we are going to make use of them throughout our analysis. In what follows, is a bounded domain of with boundary sufficiently smooth. consists of the artery wall denoted by and some artificial sections. The velocity is required to be zero on the artery wall. To account for homogeneous Dirichlet boundary conditions on the artery wall, we define
Note that Poincaré’s inequality holds for [2, 4]. The artificial sections consist of the upstream section on the side of the heart and the downstream sections on the side of the peripheral vessels. Rather than giving serious thought to the artificial sections boundary conditions, in seeking a variational formulation, the test space is left free on these portions of the boundary. Accordingly, we introduce as the test space. To prove an existence theorem for a Navier-Stokes problem, either steady or nonsteady, it is convenient to construct the solution as a limit of Galerkin approximations in terms of the eigenfunctions of the corresponding steady Stokes problem. This use of the Stokes eigenfunctions originated with Prodi and was further developed by Heywood . To define the corresponding Stokes operator, we introduce as the completion of with respect to the norm of . Then for every , there exists exactly one satisfying Moreover, for each , there exist at most one such that (6) holds. In this way, a one-to-one correspondence can be defined between elements of and functions belonging to an allowing suitable subspace of that we denote by . The Stokes operator is defined setting so that (6) is satisfied.
The inverse operator is completely continuous and self-adjoint. Therefore it possesses a sequence of eigenfunctions , which are complete and orthogonal in bothand.
In what follows, () will denote generic constants, not necessarily the same at different places. The inequalities are satisfied for every, provided thatis a bounded domain (see  page 178).
Theorem 1 (Brouwer’s fixed point theorem). Letbe a finite dimensional linear space endowed with a norm. Assume is continuous, wheredenotes the closed-unit ball in. Thenhas a fixed point; that is, there exists a pointwith
Theorem 2 (Rellich-Kondrachov compactness theorem). Assumeis a bounded open subset of, with aboundary. Suppose. Then for each, where.
As a result of that, we have
Theorem 3 (trace inequality for solenoidal functions). Letbe a solenoidal function defined on. Then the following inequality holds:
Theorem 4 (weak compactness). Let be a reflexive Banach space and suppose that the sequence is bounded. Then there exists a subsequence and such that
In other words, bounded sequences in a reflexive Banach space are weakly precompact. In particular, a bounded sequence in a Hilbert space contains a weakly convergent subsequence.
2. Formulation of the Problem
Letbe the artery portion where we aim at providing a detailed flow analysis. For each and at any time , we denote by and the blood velocity and pressure, respectively. In larger vessels, it is reasonable to assume that blood has a constant viscosity, because the vessel diameters are large compared with the individual cell diameters and because shear rates are high enough for viscosity to be independent of them. Hence, in these vessels the non-Newtonian behavior becomes insignificant and blood can be considered to be a Newtonian fluid ( and references therein). In what follows, we assume that the vessels are large enough; blood density and blood viscosity are assumed to be constant. Under these assumptions, blood flow can be described by the Navier-Stokes equations where . The equations are expressions of balance of linear momentum and incompressibility. We are neglecting the presence of any external forces. Then (18) is obtained by substitution of the divergence-free constraint into the expression for the stress in (17) as follows
In fact, For the sake of simplicity, we normalizeto 1.
2.1. Initial and Boundary Conditions
The initial condition requires the specification of the flow velocity at the initial time; for example, for ; where the given initial velocity fieldis divergence-free. The system (18) has to be provided with boundary conditions. In this respect, we split the boundaryinto different parts. In the present work, we are assuming that the wall is rigid so that no-slip boundary condition holds. The other parts ofare the artificial boundaries which bound the computational domain. For the sake of clarity, we distinguish the upstream section on the side of the heart denoted byand the downstream sections on the side of the peripheral vessels denoted byand. As introduced in , the following boundary conditions are provided: whereis a suitable nonnegative constant,is the prescribed mean pressure on each artificial section, andrepresents the outward normal unit vector on every part of the vessel boundary. These conditions can be considered as a generalization of the mean pressure drop problem investigated in  in the sense that when (), we recover the usual Neumann or natural conditions associated with (18). The physical justification for the case in whichis provided in ( Figure 4.1).
The mathematical formulation of the problem is therefore described by the system of differential equations:
Next we define the following bilinear and trilinear forms: such that such that Following  we set Furthermore, if we set and consider the sequenceof eigenfunctions of, the following estimates hold for every : wheredepends on the trace inequality (15) and wheredepends on the trace inequality (15) and Poincaré’s inequality .
3. The Stationary Problem
Our purpose is to establish the existence of a weak solution to the stationary version of problem (23). We are going to consider the inner product within order to formulate the weak formulation.
3.1. Weak Formulation
Assume thatis a solution of the boundary value problem described above andis a smooth solenoidal vector-valued function defined on. The functionsatisfies the following identities: We notice thatsince.
We calculate to obtain It is clear that (33) does not depend on the pressure; therefore, the weak formulation of the problem reads as follows.
Given a divergence free velocity field and nonnegative constants and (), find such that holds for all , where, defined in (5), is the space of test functions.
3.2. Galerkin Approximations
The inverse operatorof the Stokes operatoris self-adjoint and possesses a sequence of eigenfunctionswhich are orthogonal in. Following  we fix a positive integer. Galerkin approximations are defined as solutions of the finite system of equations () as follows: This is a system of linear equations for constant unknowns (). The identity (37) foris obtained by multiplying (36) through byand summing over: This implies that Together with (4), we make use of (28) to obtain Likewise, from (29), we have Furthermore, making use of Hölder’s inequality (, page 623), one obtains where depends on Rellich-Kondrachov compactness inequality , Poincaré’s inequality , and Sobolev inequality (, page 270). We make use of these inequalities in (38) and we obtain Set . It follows that where , .
Proof. Owing to Poincaré’s inequality (4), is a norm equivalent to for all ; therefore, (45) defines a closed ball in . To prove the solvability of the finite-dimensional problem (36), we follow Fujita in using Brouwer’s fixed point theorem (see Theorem 1.1 in ), applying it to the continuous mapping defined by the linear problem (): Equation (46) is a system of linear equations. These linear equations are uniquely solvable if lies in the ball defined by (45), because then is the only solution of the corresponding homogeneous equation (, ). In fact, if satisfies (45) and satisfies (46) with , we have Together with Poincaré’s inequality (4), this imply that . To see that the mapping takes the ball defined by (45) into itself, suppose that satisfies (45). Then, similarly to (43), we obtain that and therefore, Thus, (46) defines a continuous mapping of the closed ball into itself. The map has at least one fixed point, and any such fixed point is a solution of (36). is chosen to be any one of these fixed points. Hence, Brouwer’s fixed point has been applied and has given the existence of Galerkin approximations satisfying
3.3. Existence of a Weak Solution
Theorem 6. Assume There exists a weak solution to the stationary version of problem (23).
Proof. By Poincaré’s inequality , the fact that the sequence is bounded implies that the sequence is bounded in . As a result, Theorem 4 yields the existence of a subsequence such that
Next we show that the weak limit is in fact a weak solution. In this respect, we are going to show that
for each .
Fix an integer , (). From (36), we have the identity We recall (54), to find upon passing to weak limits that Since is a basis of , it follows that for each .
Note that, in taking this limit, there is no difficulty with the nonlinear term. In fact, we have that Also, is an eigenfunction of the inverse of the stokes operator and . This implies that . It follows that (see , page 651).
4. The Nonstationary Problem
The mathematical analysis of the Navier-Stokes problem is based on its weak formulation.
4.1. Weak Formulation
Assume that is a solution to problem (23) and is a smooth solenoidal vector-valued function defined on . satisfies the following identities: We calculate to obtain where , , and are defined by (24), (25), and (26), respectively. We notice that (63) does not depend on the pressure . Therefore, the weak formulation of the problem reads as follows.
Given a divergence free velocity field , , and a nonnegative constant for , find such that for all , where , defined in (5), is the space of test functions.
4.2. Galerkin Approximations
We are going to follow the same approach as in . We will construct the weak solution of the initial boundary value problem by first solving for a finite dimensional approximation. We recall that the inverse operator of the Stokes operator is self-adjoint and possesses a sequence of eigenfunctions which are orthogonal in and orthonormal in . Fix a positive integer and write where we intend to select the coefficients (, ) to satisfy
Proof. Assuming that is given by (65), we observe using the fact that is an orthonormal basis of that
where is the derivative of with respect to .
Furthermore, from (24) and the fact that is a bilinear form, we have that where .
From (25) and recalling that is a trilinear form, we have that where .
Moreover, we set It follows that (67) becomes the system of ODEs subject to the initial condition (66). According to standard existence theory for ordinary differential equations, there exists a unique function satisfying (66) and (72) for a.e. . And then defined by (65) solves (67) for a.e. .
4.3. Energy Estimate
Our plan hereafter is to let . Before that, we will need some estimates, uniform in .
Theorem 8. For each approximate solution , the following energy estimates hold:
Proof. We denote by the eigenvalue associated with the eigenfunction . Multiplying (63) by and summing over , one obtains Now, and also Moreover, It follows that We now make use of (28) to obtain Likewise, from (29), we have Furthermore, We make use of these inequalities in (78) to find that Now from (9), which implies that and so, inequality (82) now reads We make use of Cauchy’s inequality with epsilon to estimate the right-hand side of (85). We have Also inequality (10) implies that so that Making use of theseinequalities on (85), we obtain that which leads us to the required energy estimates.
4.4. Local Existence of a Weak Solution
In this subsection, we use Galerkin method to build up a local weak solution of the initial/boundary-value problem. We have already constructed the Galerkin approximations sequence in Section 4.2. Our goal now is to extract from this sequence a subsequence that converges to the weak solution. In this respect, we are going to show that this sequence is bounded and thereafter, we will make use of a compactness result.
Lemma 9. Let be a nonnegative function satisfying the inequality and let be a strictly positive real number; then there exists a time interval where is bounded by a positive constant and depends only on , and .
Proof. We have We set to obtain Integrate over , with : We select in such that ; that is, Then so that Considering that and , we obtain In order to obtain , we must have Hence, we choose such that It follows that is bounded by a positive constant for all such that and that depends only on , , and .
Theorem 10. Let be such that there is a time interval on which a weak solution of (23) exists.
Proof. Considering (101), the energy estimates (73) now read:
Since , (102) implies that
we see that inequality (103) has the following form:
where and are some constants depending on the data.
Note that for each integer , the function verifies the requirements of Lemma 9 with since the initial condition is the same for all functions , . Consequently, the sequence is bounded by a real number which depends only on the initial data. We make use of Poincaré’s inequality (4) to find that for each integer , Therefore, the sequence is bounded.
Furthermore, the space being a Hilbert space, Theorem 4 can be applied to the sequence . In fact, the sequence is bounded in . Consequently, there exists a subsequence such that Next, we show that the weak limit is in fact a weak solution. In this respect, we are going to show first that and, thereafter, (1)Fix an integer and choose a function having the form where are given functions and is the basis of . We choose , multiply (67) by , sum , and then integrate with respect to to find that We recall (107) to find upon passing to weak limits (see remark 3.1 and ) that Equality (112) then holds for all functions , as functions of the form (110) are dense in this space . Hence, in particular, for each and a.e. .(2)In order to prove that for every , we first note from (112) that for each with . Similarly, from (111) we deduce that We once again employ (107) to find that since . As is arbitrary, comparing (114) and (116), we conclude that for each . Therefore, is a local weak solution of (23).
4.5. Global Existence of a Weak Solution for Small Data
In this section, we make use of Galerkin’s method to establish the global existence of a weak solution. We are going to show that under certain circumstances, the weak solution is defined at any time . In this regard, we are going to consider the energy estimate and the following lemma.
Lemma 11. Let be a nonnegative, absolutely continuous function satisfying inequality and let a real number be such that , if , and , then is bounded by for all .
Proof. This result is proven by contradiction . Suppose that there exists a such that , and define we have and . (1)We first show that . Set and choose any ; we have and so does not belong to the set . It follows that , and this is true for each ; hence On the other hand, for each natural number , we have . We make use of the fact that to see that there exists such that where is not necessarily unique. We choose one value that may take and we denote it by . This defines a real sequence . For each natural number , since , we have that . The fact that implies that the sequence converges to . We make use of the continuity of to see that the sequence Since for each , we have , and then implies that It follows that and ; therefore, (2)Next we show that . Suppose that ; then there exists a nonnegative natural number such that decreases on the interval . Also we have that this implies that But from part (1), we have and ; because , it follows that Hence, and , which is impossible. Therefore, Finally that we show that and this contradicts the result obtained in (2). From (117), we have Therefore, for all .
Theorem 12. Assume that the initial and boundary data are sufficiently small, precisely Then, for all , there exists a weak solution to problem (23), and it satisfies the inequality
Proof. We make use of the estimates (10), (73), and (101) in order to obtain
The energy estimate (132) takes the form
We make use of Lemma 11 to find that is bounded by for all and the bound does not depend on . Poincaré’s inequality (4) implies that
Let be any positive real number,
Thus, for each nonnegative real number , the sequence is bounded. We make use of Lemma 9 to extract a subsequence that converges to an element of , and we use the same steps as we did for the local existence to show that is a weak solution of problem (23).
Furthermore, the fact that the sequence is bounded by implies that the subsequence is also bounded by .
Therefore, since is the limit of this subsequence, it follows that
5. Uniqueness of the Solution
Theorem 13. There exists a time interval where the solution of problem (23) is unique.
Proof. Assume that there exist two solutions and associated with the same data. Set . Consider (63) with . By substraction, we obtain
We recall that is a bilinear form, is a trilinear form, and is a linear form described by (24), (25), and (26), respectively. We evaluate every term of (142):
It follows that
We now make use of some classical estimates to evaluate the rights hand side of (145) as follow:
This implies that
It follows that
We now make use of these estimates in (145) to find that
Since , there exists such that with . On the interval , we have
is a real function depending on .
The uniqueness theorem follows from Gronwall’s inequality , as .
6. Stability of the Solution
In this section, we are going to prove that there exists a time interval of continuous dependence on the data. We start with the boundary conditions and afterward we move on to the initial condition.
6.1. Boundary Conditions Stability
Theorem 14. There exists a time interval where the solution of problem (23) depends continuously on the prescribed data as the real functions () are varied.
Proof. Denote by the solution associated with the data () and, correspondingly, by the solution associated with ().
Set We make use of (63) for and with . By substraction, we obtain where This gives us that Also, It follows that Since , there exists