Andro Mikelić

Sunčica Čanić

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 330 (2002) 661–666Effective equations describing the flow of a viscousincompressible fluid through a long elastic tubeSunčica Čanić a, Andro Mikeli ć ba Department of Mathematics, University of Houston, Houston, TX 77204-3476, USAb UFR mathématiques, Université Claude Bernard Lyon 1, bât. 101, 43, bd. du 11 novembre 1918,69622 Villeurbanne cedex, FranceReceived and accepted 1 July 2002Note presented by Roland Glowinski.Abstract We study the flow of a viscous incompressible fluid through a long and narrow elastic tubewhose walls are modeled by the Navier equations for a curved, linearly elastic membrane.The flow is governed by a given small time dependent pressure drop between the inletand the outlet boundary, giving rise to creeping flow modeled by the Stokes equations. Byemploying asymptotic analysis in thin, elastic, domains we obtain the reduced equationswhich correspond to a Biot type viscoelastic equation for the effective pressure and theeffective displacement. The approximation is rigorously justified by obtaining the errorestimates for the velocity, pressure and displacement. Applications of the model probleminclude blood flow in small arteries. We recover the well-known Law of Laplace and providea new, improved model when shear modulus of the vessel wall is not negligible. To cite thisarticle: S. Čanić, A. Mikelić, C. R. Mecanique 330 (2002) 661–666. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SASfluid mechanics / creeping blood flow / viscoelasticityEquations efficaces décrivant l’écoulement d’un fluide visqueuxincompressible dans un tuyau long et élastiqueRésumé Nous considérons l’écoulement d’un fluide incompressible visqueux à travers un tuyaulong et de faible épaisseur, ayant la paroi latérale obéissant aux équations de Navier pourune membrane courbe élastique linéaire. L’écoulement est régi par une petite chute depression entre l’entrée et la sortie du tuyau et on a un écoulement lent décrit par leséquations de Stokes. En utilisant la méthode des développements asymptotiques, avecla réduction dimensionnelle dans la partie mince, nous obtenons les équations limites.Elles correspondent aux équations viscoélastiques de Biot pour la pression efficace etles déplacements efficaces. L’approximation est justifiée rigoureusement en obtenantune estimation d’erreur pour la vitesse, la pression et les déplacements « dilatés ». Desapplications incluent l’écoulement sanguin dans des arterioles. Nous retrouvons la bienconnue Loi de Laplace et donnons un nouveau modèle amélioré lorsque le module decisaillement de la paroi n’est pas négligeable. Pour citer cet article : S. Čanić, A. Mikelić,C. R. Mecanique 330 (2002) 661–666. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SASmécanique des fluides / écoulement sanguin lent / viscoélasticitéE-mail addresses:canic@math.uh.edu (S. Čanić); andro@lan.univ-lyon1.fr (A. Mikelić). 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservésS1631-0721(02)01517-6 /FLA 661S.Čanić, A. Mikeli ć / C. R. Mecanique 330 (2002) 661–666Version française abrégéeNous considérons l’écoulement quasi-statique axisymétrique d’un fluide Newtonien incompressible àtravers un cylindre droit, à petite épaisseur,ε ={x ∈ R3;x = (r cosϑ, r sinϑ, z), r < εR, 0< z <L} (1)La vitesse du fluide vε et la pression pε satisfont les équations de Stokes dans des coordonnées cylindriques−µ(∂2vεr∂r2+ ∂2vεr∂z2+ 1r∂vεr∂r− vεrr2)+ ∂pε∂r= 0 dans ε × R+ (2)−µ(∂2vεz∂r2+ ∂2vεz∂z2+ 1r∂vεz∂r)+ ∂pε∂z= 0 et ∂vεr∂r+ ∂vεz∂z+ vεrr= 0 dans ε × R+ (3)Nous supposons que la paroi latérale du cylindreε = {r = εR}× (0,L) est élastique et que sa déformationest décrite par les équations de NavierFr = h(ε)E(ε)1 − σ 2(σεR∂sε∂z+ ηεε2R2)− h(ε)G(ε)k(ε)∂2ηε∂z2+ ρwh(ε)∂2ηε∂t2(4)Fz = −h(ε)E(ε)1 − σ 2(∂2sε∂z2+ σεR∂ηε∂z)+ ρwh(ε)∂2sε∂t2(5)Dans (4) et (5), ηε est le déplacement radial et sε le déplacement longitudinale dans les coordonnées deLagrange (voir Fig. 1), h = h(ε) est l’épaisseur de la membrane, ρw sa densité, E = E(ε) le moduled’Young, 0 < σ < 1 le coefficient de Poisson, G= G(ε) le module de cisaillement et k = k(ε) le facteurde correction de Timoshenko (voir [1]). Fr et Fz sont les composantes radiale et longitudinale de la forceexercée par le fluide sur la paroi et données parFr 	er +Fz	ez =(pεI − 2µD(vε))	er sur ε × R+ (6)où D(vε) est le tenseur des vitesses de déformation. La vitesse du fluide vε est liée avec la vitesse de laparoi parvεr =∂ηε∂tet vεz =∂sε∂tsur ε × R+ (7)Nous ajoutons au système (1)–(7) les conditions aux limites (13)–(16).On va obtenir les équations asymptotiques qui décrivent l’écoulement à l’échelle du temps dominanteet aussi les oscillations de la membrane induites par le réponse du matériau élastique. Nous posonst̃ = ωεt avec ωε = ε2/µ. Ce choix conduit à l’égalité d’énergie (17), qui nous permet d’obtenir lesestimations a priori pour la solution. Soient E0 = limε→0 h(ε)E(ε)/ε et G0 = limε→0G(ε)h(ε)k(ε)ε.Nous utilisons l’approche de Ciarlet et al. (cf. [3]) ou on remplace les inconnues par leurs « dilatées »v(ε)(z, r, t) = vε(z, εr, t) et p(ε)(z, r, t) = pε(z, εr, t). ε devient le domaine « non-mince » =1. Lavitesse et la pression « dilatées » satisfont les estimations a priori (19)–(21). Ensuite nous cherchons leséquations efficaces en utilisant les développements asymptotiques suivantesv(ε)(z, r, t)= ε2µ∑i0εivi(z, r, t), p(ε)(z, r, t)= p0(z, t)+∑i1εipi(z, r, t) (8)ηε(z, t)= ε∑i0εiηi(z, t), sε(z, t)=∑i0εisi(z, t) (9)qui nous donnent en Tableau 1 le problème parabolique pour la pression efficace p = p0, et les expressionssuivants pour les autres inconnus662To cite this article: S. Čanić, A. Mikeli ć, C. R. Mecanique 330 (2002) 661–666E0η0R(1 − σ 2) =R2 − σ(−12p+ h(ε)E(ε)Rε18(1 − σ 2)∂2∂z2∫ t0p)(10)∂s0∂z= R2 − σ(−Rσ8∂2∂z2∫ t0p+ 1 − σ2E0p)(11)v0z (z, r, t)=r2 −R24∂p∂z(z, t)+ ∂s∂t(z, t), v0r =Rr∂η∂t+ (R2 − r2)24r∂2p∂z2− R2 − r22∂2s∂z∂t(12)Notons que pour G0 = 0, on obtient la loi (22), liant la pression efficace p et le déplacement radialefficace η. Pour σ = 1/2 (22) est exactement la Loi de Laplace (voir [4]).Nous justifions le problème efficace par une estimation d’erreur. Soit {βbl,πbl} la solution pour leséquation de la couche limite (23) et (24). Alors on aTHÉORÈME 0.1. – Soientγ (ε)= p(ε)− p − πε , w(ε) = µε2v(ε)− v0z 	ez − εv0r 	er + βε, s(ε)= sε − s0 etη(ε)= ηε − εη0. Alors on a∥∥w(ε)∥∥L2(0,T ;W) +∥∥γ (ε)∥∥L2(×(0,T ))  Cε3/2sup0tT{√G0∥∥∥∥∂η(ε)ε∂z (t)∥∥∥∥L2(0,L)+ √E0∥∥∥∥η(ε)ε (t)∥∥∥∥L2(0,L)+∥∥∥∥∂s(ε)∂z (t)∥∥∥∥L2(0,L)} Cε3/21. Statement of the problemWe study the quasi-static axisymmetric flow of a Newtonian incompressible fluid in a thin right cylinderε , given by (1), whose radius is small with respect to its length Fluid velocity vε and pressure pε satisfyStokes equations in Eulerian cylindrical coordinates (2) and (3). We assume that the lateral wall of thecylinder ε = {r = εR} × (0,L) is elastic and that it satisfies the Navier equations (4) and (5). We denoteby ηε the radial and by sε the longitudinal displacement from the reference state, see Fig. 1, e= h(ε) is themembrane thickness, ρw the wall volumetric mass, E =E(ε) is the Young modulus, 0< σ < 1 the Poissonratio, G=G(ε) is the shear modulus and k = k(ε) is the Timoshenko shear correction factor (see [1]). Theradial and the longitudinal component of the forces exerted by the fluid on the lateral wall, Fr and Fz, andgiven by (6), where D(vε) is the rate of the strain tensor. Fluid velocity vε is coupled with the velocity ofthe wall through (7). We note that the true coupling between the Eulerian quantity vε and the Lagrangianquantities (ηε, sε) is given by vε(x + (ηε, sε), t)= (∂ηε/∂t, ∂sε/∂t)(x, t). After linearization, compatiblewith linear elasticity and creeping flow, we approximate the interface by its initial position and the interfacecondition for the velocity by (7).Initially, the cylinder is filled with fluid at velocity zero, and the entire structure is in an equilibrium atsome reference pressure P0,ηε = sε = ∂ηε∂t= ∂sε∂t= 0 on ε × {0} (13)The following boundary conditions are motivated by the fact that only the pressure field is correctlyobserved at the outlet boundary:Figure 1. Wall displacement.Figure 1. Déformation de la paroi latérale.663S.Čanić, A. Mikeli ć / C. R. Mecanique 330 (2002) 661–666vεr = 0 and pε = 0 on(∂ε ∩ {z= 0}) × R+ (14)vεr = 0 and pε =A(t) on(∂ε ∩ {z= L}) × R+ (15)∂sε∂z= ηε = 0 for z= 0, sε = ηε = 0 for z= L and ∀t ∈ R+ (16)The time-dependent pressure drop A(t) between the inlet and the outlet boundary drives the flow. We areassuming that the pressure drop is small compared to the reference pressure P0 so that the acceleration ofthe fluid is negligible compared to the effects of the fluid viscosity. Blood flow in small arteries satisfiesthese assumptions [2].2. Main resultsWe derive the asymptotic equations that describe: (a) the flow occurring at the leading order time scaleand (b) the oscillations of the membrane caused by a response of the elastic material. Since they occur atdifferent time scales we introduce the scaling t̃ = ωεt . Energy estimate (17) below implies that ωε = ε2/µ.This choice of ωε balances the term involving the time derivative of the velocity and the inertia term.It needs to be compatible with the oscillations of the pressure drop A(t), which is related to the heartpulsation, blood viscosity and the size of the vessel via the so called Womersley number α [1]. The resultsthat follow are all given in terms of the rescaled time t̃ where the wiggle sign is dropped for notationalsimplicity.Energy estimate. –We first state the energy equalityωεh(ε)εRd2 dt{(ωε)2ρw∫ L0(∣∣∣∣∂ηε∂t∣∣∣∣2+∣∣∣∣∂sε∂t∣∣∣∣2)dz+G(ε)k(ε)∫ L0∣∣∣∣∂ηε∂z∣∣∣∣2dz+ E(ε)1 − σ 2(σ∫ L0(ηεεR− ∂sε∂z)2+ (1 − σ)∫ L0(∣∣∣∣ ηεεR∣∣∣∣2+∣∣∣∣∂sε∂z∣∣∣∣2))}+ 2µ∫ε(t)∣∣D(vε(t))∣∣2r dr dz= −∫ εR0A(t)vεz (t, r,L)r dr (17)with vεr = ωε∂ηε/∂t and vεz = ωε∂sε/∂t on ε × (0, T ), which provides the following a priori estimatePROPOSITION 2.1. – Solution(vεr , vεz , ηε, sε) satisfies the following a priori estimate.ωεh(ε)E(ε)4εR(1 + σ){∫ L0∣∣ηε(t)∣∣2 dz+ ε2R2∥∥∥∥∂sε∂z(t)∥∥∥∥2L2}+µ∫ t0∫ε∣∣D(vε)∣∣2r dr dz+ωεG(ε)k(ε)εR∫ L0∣∣∣∣∂ηε(t)∂z∣∣∣∣2dz C(R,L)ε4µ∫ t0{∣∣A(τ)∣∣2 + ∣∣∂τA(τ)∣∣2}dτ (18)A priori solution estimates follow from here. We use the approach introduced by Ciarlet et al. (see,e.g., [3] and the references therein) and consider the scaled velocity v(ε)(z, r, t)= vε(z, εr, t) and pressurep(ε)(z, r, t)= pε(z, εr, t) defined on the scaled, fixed domain =1. Let E0 = limε→0 h(ε)E(ε)/ε andG0 = limε→0G(ε)h(ε)k(ε)ε. Then (18) leads to the following a priori estimates for the scaled unknowns.COROLLARY 2.1. – Scaled solution(v(ε),p(ε), ηε, sε) satisfies the a priori estimates∥∥∥∥v(ε)rr∥∥∥∥L2(×(0,T ))+∥∥∥∥∂v(ε)z∂r + ε∂v(ε)r∂z∥∥∥∥L2(×(0,T ))+∥∥∥∥∂v(ε)r∂r∥∥∥∥L2(×(0,T )) C ε2µ‖A‖V (19)µε∥∥∥∥∂v(ε)z∂z∥∥∥∥L2(×(0,T ))+ ∥∥p(ε)∥∥L2(×(0,T )) +1ε∥∥∥∥ ∂∂r p(ε)∥∥∥∥2L2(0,T ;H−1()) C‖A‖V (20)664Pour citer cet article : S. Čanić, A. Mikeli ć, C. R. Mecanique 330 (2002) 661–6661ε2∥∥ηε(t)∥∥2L2(0,L) +∥∥∥∥∂sε∂z(t)∥∥∥∥2L2(0,L)+ G0E01ε2∥∥∥∥∂ηε(t)∂z∥∥∥∥2L2(0,L) CE0‖A‖2V (21)where‖A‖2V =∫ T0 {|A(τ)|2 + |∂τA(τ)|2} dτ .These a priori estimates are insufficient to control the velocity field correctly. Further calculations imply∥∥∥∥∂v(ε)z∂r∥∥∥∥L2(×(0,T ))+ ε∥∥∥∥∂v(ε)r∂z∥∥∥∥L2(×(0,T )) ε2µ(‖A‖V + ε∥∥∥∥∂sε∂t∥∥∥∥L2((0,L)×(0,T )))We observe that the above estimates also hold for the time derivatives of the unknowns, but with A replacedby ∂tA, assuming the compatibility condition A(0)= 0.Asymptotic expansions. –The above a priori estimates provide the multi-scale asymptotic expansionslisted in (8) and (9).The reduced problem. –By plugging the multi-scale expansions into the scaled Stokes equations and bytaking into account the boundary conditions which are the Navier equations for the membrane, we obtainthe initial-boundary value problem (see Table 1) for the effective pressure p = p0,Table 1.The initial boundary value problem.Tableau 1.Le problème parabolique.Small G0 Non-negligible G0(52− 2σ)∂p∂t= E0R8∂2p∂z2∂∂t{(52− 2σ)p− (1 − σ 2)G0R22E0∂2p∂z2}= ∂2∂z2{E0R8 p−G0R38∂2p∂z2}p(0, t)= p(z,0)= 0; p(L, t)=A(t) p(0, t)= 0; p(L, t)=A(t)p(z,0)= 0 ∂2p∂z2(0, t)= 0; E0R8(1 − σ 2)∂2p∂z2(L, t)= 12dAdtand formulas (10)–(12) for the other effective unknowns. Denote by P the initial value problem for pcoupled with (10)–(12). In the case when the shear stress terms are negligible we find that pressure isdirectly related to the radial displacement viap = E0η(1 − σ/2)R2 =(limε→0h(ε)E(ε)(1 − σ/2)εR)(1 −√A(0)A+ O(ηR))(22)where A denotes the cross-sectional area of the vessel. For σ = 1/2 (incompressible materials) ourcalculation implies s = 0 and Eq. (22) reduces to the Law of Laplace [4] typically used in literature [5,2,1] to model vessel walls via the independent ring model. We note that if the inertia terms have been takeninto account to model the flow by the Navier–Stokes equations, the reduced equation for the pressure (oreffective displacement) would have been hyperbolic. Creeping flow leads to a parabolic problem for theeffective pressure, shown above in P.Convergence result. –As in the study of the Stokes and the Navier–Stokes equations in thin, rigid,cylinders (see [6] and subsequent works by numerous authors), the a priori estimates and uniqueness forproblem P allow us to justify the effective model by obtaining the weak limit as ε→ 0.THEOREM 2.2. – Suppose for simplicity thatG0 > 0 and letW = {ϕ ∈ L2() | ∂ϕ∂r ∈ L2()}. Then thesequence(µ/ε2v(ε)z,p(ε), ηε/ε, sε) converges weakly inL2(0, T ;W ×L2()×H 1(0,L)2) to a uniquesolution of the initial value problemP. Furthermore,µε2vr(ε)⇀ 0 weakly inL2(0, T ;W).665S.Čanić, A. Mikeli ć / C. R. Mecanique 330 (2002) 661–666Error estimates. –Since weak convergence does not provide enough information about the accuracyof the approximation of the true solution by the solution of the reduced problem P, we find the order ofapproximation by calculating the error estimates of the difference between the two solutions. We found thatthe reduced problem gives rise to a boundary layer in the velocity and pressure at the outlet boundary z= L.We construct the boundary layer explicitly by considering the following abstract problem on a semi-infiniterigid-wall cylinder Z− = S × R−, where S = {r < R} × {y3 = 0}:−7βbl + ∇πbl = 0 and divβbl = 0 in Z− (23)βblr = v0r (t, r,L), −2∂βblz∂y3+ πbl = −2∂v0z∂z(t, r,L) on y3 = 0 and βbl = 0 on ∂S × R− (24)The results from [7] give the existence of a unique smooth velocity and pressure fields, exponentiallydecreasing when |y3| → ∞. We now define the boundary layer velocity and the boundary layer pressureto be βε(t, r, z) = εβbl(t, r, z−Lε) and πε(t, r, z) = ε2πbl(t, r, z−Lε), respectively. These functions areexponentially small away from the outlet boundary. Assuming that the solution of problem P is sufficientlysmooth and that the convergence toward G0 and E0 is sufficiently fast, we obtainTHEOREM 2.3. – Let γ (ε) = p(ε) − p − πε , w(ε) = µε2v(ε) − v0z 	ez − εv0r 	er + βε, s(ε) = sε − s0 andη(ε)= ηε − εη0 . Then we have∥∥w(ε)∥∥L2(0,T ;W) +∥∥γ (ε)∥∥L2(×(0,T ))  Cε3/2 (25)sup0tT{√G0∥∥∥∥∂η(ε)ε∂z (t)∥∥∥∥L2(0,L)+ √E0∥∥∥∥η(ε)ε (t)∥∥∥∥L2(0,L)+∥∥∥∥∂s(ε)∂z (t)∥∥∥∥L2(0,L)}Cε3/2 (26)Therefore, we have shown that the error between the solution of the full set of axisymmetric Stokesequations coupled with the Navier equations for the elastic tube wall, and the solution of the reduced prob-lem P, modified by the boundary layer velocity and pressure, is of order ε3/2, where ε is the ratio betweenthe radius and the length of the elastic tube. We point out that in the interior of the domain, away from theoutlet boundary, boundary layer has negligible effect and the accuracy of the approximation of the solutionto the original, non-reduced problem, by the solution of problem P away from the outlet boundary is ε2.Acknowledgements.This research was made possible by the support from the Texas Higher Education Board, ARPMathematics, grant number 004652-0112-2001 and in part by the National Science Foundation, grant DMS 9970310.References[1] A. Quarteroni, M. Tuveri, A. Veneziani, Computational vascular fluid dynamics: problems, models and methods.Survey article, Comput. Visual. Sci. 2 (2000) 163–197.[2] M.S. Olufsen, C.S. Peskin, W.Y. Kim, E.M. Pedersen, A. Nadim, J. Larsen, Numerical simulation and experimentalvalidation of blood flow in arteries with structured-tree outflow conditions, Ann. Biomed. Engrg. 28 (2000) 1281–1299.[3] P.G. Ciarlet, Plates and Junctions in Elastic Multi-Structures, in: RMA, Vol. 14, Masson, Paris, 1990.[4] Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues, Springer, New York, 1993.[5] S. Čanić, Blood flow through compliant vessels after endovascular repair: wall deformations induced by thediscontinuous wall properties, Comput. Visual. Sci. 4 (3) (2002) 147–155.[6] H. Dridi, Comportement asymptotique des équations de Navier–Stokes dans des domaines applatis, Bull. Sci.Math. 106 (1982) 369–385.[7] W. Jäger, A. Mikelić, On the effective equations for a viscous incompressible fluid flow through a filter of finitethickness, Comm. Pure Appl. Math. LI (1998) 1073–1121.666

Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube

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Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube

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