Three-dimensional flows in a hyperelastic vessel under external pressure
Tóm tắt
We study the collapsible behaviour of a vessel conveying viscous flows subject to external pressure, a scenario that could occur in many physiological applications. The vessel is modelled as a three-dimensional cylindrical tube of nonlinear hyperelastic material. To solve the fully coupled fluid–structure interaction, we have developed a novel approach based on the Arbitrary Lagrangian–Eulerian (ALE) method and the frontal solver. The method of rotating spines is used to enable an automatic mesh adaptation. The numerical code is verified extensively with published results and those obtained using the commercial packages in simpler cases, e.g. ANSYS for the structure with the prescribed flow, and FLUENT for the fluid flow with prescribed structure deformation. We examine three different hyperelastic material models for the tube for the first time in this context and show that at the small strain, all three material models give similar results. However, for the large strain, results differ depending on the material model used. We further study the behaviour of the tube under a mode-3 buckling and reveal its complex flow patterns under various external pressures. To understand these flow patterns, we show how energy dissipation is associated with the boundary layers created at the narrowest collapsed section of the tube, and how the transverse flow forms a virtual sink to feed a strong axial jet. We found that the energy dissipation associated with the recirculation does not coincide with the flow separation zone itself, but overlaps with the streamlines that divide the three recirculation zones. Finally, we examine the bifurcation diagrams for both mode-3 and mode-2 collapses and reveal that multiple solutions exist for a range of the Reynolds number. Our work is a step towards modelling more realistic physiological flows in collapsible arteries and veins.
Tài liệu tham khảo
Barclay W, Thalayasingam S (1986) Self-excited oscillations in thin-walled collapsible tubes. Med Biol Eng Comput 24(5):482–487
Bertram C (1986) Unstable equilibrium behaviour in collapsible tubes. J Biomech 19(1):61–69
Bertram C, Elliott N (2003) Flow-rate limitation in a uniform thin-walled collapsible tube, with comparison to a uniform thick-walled tube and a tube of tapering thickness. J Fluids Struct 17(4):541–559
Bertram C, Godbole S (1997) LDA measurements of velocities in a simulated collapsed tube. J Biomech Eng 119(3):357–363
Bertram C, Pedley T (1982) A mathematical model of unsteady collapsible tube behaviour. J Biomech 15(1):39–50
Bertram C, Tscherry J (2006) The onset of flow-rate limitation and flow-induced oscillations in collapsible tubes. J Fluids Struct 22(8):1029–1045
Cai Z, Fu Y (1999) On the imperfection sensitivity of a coated elastic half-space. Proc Math Phys Eng Sci 455(1989):3285–3309
Cai Z, Luo X (2003) A fluid-beam model for flow in a collapsible channel. J Fluids Struct 17(1):125–146
Cancelli C, Pedley T (1985) A separated-flow model for collapsible-tube oscillations. J Fluid Mech 157:375–404
Conrad WA (1969) Pressure-flow relationships in collapsible tubes. IEEE Trans Biomed Eng BME–16(4):284–295
Gasser TC, Ogden RW, Holzapfel GA (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3(6):15–35
Gaver DP, Halpern D, Jensen OE, Grotberg JB (1996) The steady motion of a semi-infinite bubble through a flexible-walled channel. J Fluid Mech 319:25–65
Giddens D, Zarins C, Glagov S (1993) The role of fluid mechanics in the localization and detection of atherosclerosis. J Biomech Eng 115(4B):588–594
Gobin Y, Counord J, Flaud P, Duffaux J (1994) In vitro study of haemodynamics in a giant saccular aneurysm model: influence of flow dynamics in the parent vessel and effects of coil embolisation. Neuroradiology 36(7):530–536
Hao Y, Cai Z, Roper S, XY L (2016) Stability analysis of collapsible-channel flows using an arnoldi-frontal approach. Int J Appl Mech pp 1650, 073–1–20
Hazel AL, Heil M (2003) Steady finite-Reynolds-number flows in three-dimensional collapsible tubes. J Fluid Mech 486:79–103
Heil M (1997) Stokes flow in collapsible tubes: computation and experiment. J Fluid Mech 353(1):285–312
Heil M (2004) An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems. Comput Methods Appl Mech Eng 193(1):1–23
Heil M, Pedley T (1996) Large post-buckling deformations of cylindrical shells conveying viscous flow. J Fluids Struct 10(6):565–599
Heil M, Waters SL (2008) How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. J Fluid Mech 601(–1):199–227
Hell M (1999) Airway closure: occluding liquid bridges in strongly buckled elastic tubes. Trans Am Soc Mech Eng J Biomech Eng 121:487–493
Hoi Y, Meng H, Woodward SH, Bendok BR, Hanel RA, Guterman LR, Hopkins LN (2004) Effects of arterial geometry on aneurysm growth: three-dimensional computational fluid dynamics study. J Neurosurg 101(4):676–681
Holzapfel GA (2002) Nonlinear solid mechanics: a continuum approach for engineering science. Meccanica 37(4–5):489–490
Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast Phys Sci Solids 61(1–3):1–48
Horgan C, Saccomandi G (2003) A description of arterial wall mechanics using limiting chain extensibility constitutive models. Biomech Model Mechanobiol 1(4):251–266
Irons B (1970) A frontal solution scheme for finite element analysis. Int J Numer Methods Eng 2:5–32
Jensen O (1990) Instabilities of flow in a collapsed tube. J Fluid Mech 220:623–659
Jensen O (1992) Chaotic oscillations in a simple collapsible-tube model. J Biomech Eng 114(1):55–59
Jensen OE, Heil M (2003) High-frequency self-excited oscillations in a collapsible-channel flow. J Fluid Mech 481:235–268
Keller HB (1977) Numerical solution of bifurcation and nonlinear eigenvalue problems. Appl Bifurc Theory 1(38):359–384
Ku DN, Giddens DP, Zarins CK, Glagov S (1985) Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. Arterioscler Thromb Vasc Biol 5(3):293–302
Liepsch D (2002) An introduction to biofluid mechanics—basic models and applications. J Biomech 35(4):415–435
Liu H, Luo X, Cai Z (2012) Stability and energy budget of pressure-driven collapsible channel flows. J Fluid Mech 705:348–370
Lowe T, Pedley T (1995) Computation of stokes flow in a channel with a collapsible segment. J Fluids Struct 9(8):885–905
Luo X, Pedley T (1996) A numerical simulation of unsteady flow in a two-dimensional collapsible channel. J Fluid Mech 314:191–225
Luo X, Pedley T (2000) Flow limitation and multiple solutions in 2-d collapsible channel flow. J Fluid Mech 420:301–324
Luo X, Cai Z, Li W, Pedley T (2008) The cascade structure of linear instability in collapsible channel flows. J Fluid Mech 600:45–76
Marzo A, Luo X, Bertram C (2005) Three-dimensional collapse and steady flow in thick-walled flexible tubes. J Fluids Struct 20(6):817–835
Moore J, Ethier C (1997) Oxygen mass transfer calculations in large arteries. J Biomech Eng 119(4):469–475
Nerem R (1992) Vascular fluid mechanics, the arterial wall, and atherosclerosis. J Biomech Eng 114(3):274–282
Papaioannou TG, Karatzis EN, Vavuranakis M, Lekakis JP, Stefanadis C (2006) Assessment of vascular wall shear stress and implications for atherosclerotic disease. Int J Cardiol 113(1):12–18
Pedley T, Luo X (1998) Modelling flow and oscillations in collapsible tubes. Theoret Comput Fluid Dyn 10(1):277–294
Pedley T, Pihler-Puzović D (2015) Flow and oscillations in collapsible tubes: physiological applications and low-dimensional models. Sadhana 40(3):891–909
Pedley TJ, Brook BS, Seymour RS (1996) Blood pressure and flow rate in the giraffe jugular vein. Philos Trans R Soc Lond B Biol Sci 351(1342):855–866
Prendergast P, Lally C, Daly S, Reid A, Lee T, Quinn D, Dolan F (2003) Analysis of prolapse in cardiovascular stents: a constitutive equation for vascular tissue and finite-element modelling. J Biomech Eng 125(5):692–699
Rast M (1994) Simultaneous solution of the navier-stokes and elastic membrane equations by a finite element method. Int J Numer Methods Fluids 19(12):1115–1135
Shapiro AH (1977) Steady flow in collapsible tubes. J Biomech Eng 99(3):126–147
Stewart PS (2017) Instabilities in flexible channel flow with large external pressure. J Fluid Mech 825:922–960
Stewart PS, Heil M, Waters SL, Jensen OE (2010) Sloshing and slamming oscillations in a collapsible channel flow. J Fluid Mech 662:288–319
Sun Z, Al Moudi M, Cao Y (2014) CT angiography in the diagnosis of cardiovascular disease: a transformation in cardiovascular CT practice. Quant Imaging Med Surg 4(5):376–396
Truong N, Bertram C (2009) The flow-field downstream of a collapsible tube during oscillation onset. Commun Numer Methods Eng 25(5):405–428
Wang L, Roper SM, Hill NA, Luo X (2016) Propagation of dissection in a residually-stressed artery model. Biomech Model Mechanobiol 16(1):139–149
Wang L, Hill NA, Roger S, Luo X (2017) Modelling peeling- and pressure-driven propagation of arterial dissection. J Eng Math 109(1):227–238
Whittaker RJ (2015) A shear-relaxation boundary layer near the pinned ends of a buckled elastic-walled tube. IMA J Appl Math 80(6):1932–1967
Whittaker RJ, Waters SL, Jensen OE, Boyle J, Heil M (2010a) The energetics of flow through a rapidly oscillating tube. Part 1: general theory. J Fluid Mech 648:83–121
Whittaker RJ, Heil M, Boyle J, Jensen OE, Waters SL (2010b) The energetics of flow through a rapidly oscillating tube. Part 2: application to an elliptical tube. J Fluid Mech 648:123–153
Whittaker RJ, Heil M, Jensen OE, Waters SL (2010c) Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes. Proc R Soc A Math Phys Eng Sci 466(2124):3635
Whittaker RJ, Heil M, Jensen OE, Waters SL (2010d) A rational derivation of a tube law from shell theory. Q J Mech Appl Math 63(4):465
Zhu Y, Luo X, Ogden R (2008) Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes under axial loading and external pressure. Int J Solids Struct 45(11):3410–3429
Zhu Y, Luo X, Ogden RW (2010) Nonlinear axisymmetric deformations of an elastic tube under external pressure. Eur J Mech A Solids 29(2):216–229
Zhu Y, Luo X, Wang H, Ogden R, Berry C (2012) Nonlinear buckling of three-dimensional thick-walled elastic tubes under pressure. Int J Non Linear Mech 48:1–14