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Self-excited oscillations in a collapsible channel with applications to retinal venous pulsation. (English) Zbl 1422.76213
Summary: We consider a theoretical model for the flow of Newtonian fluid through a long flexible-walled channel which is formed from four compliant and rigid compartments arranged alternately in series. We drive the flow using a fixed upstream flux and derive a spatially one-dimensional model using a flow profile assumption. The compliant compartments of the channel are assumed subject to a large external pressure, so the system admits a highly collapsed steady state. Using both a global (linear) stability eigensolver and fully nonlinear simulations, we show that these highly collapsed steady states admit a primary global oscillatory instability similar to observations in a single channel. We also show that in some regions of the parameter space the system admits a secondary mode of instability which can interact with the primary mode and lead to significant changes in the structure of the neutral stability curves. Finally, we apply the predictions of this model to the flow of blood through the central retinal vein and examine the conditions required for the onset of self-excited oscillation. We show that the neutral stability curve of the primary mode of instability discussed above agrees well with canine experimental measurements of the onset of retinal venous pulsation, although there is a large discrepancy in the oscillation frequency.

76Z05 Physiological flows
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[1] Armitstead, J. P., Bertram, C. D. and Jensen, O. E., “‘A study of the bifurcation behaviour of a model of flow through a collapsible tube”, Bull. Math. Biol.58 (1996) 611-641; doi:10.1007/BF02459476. · Zbl 0859.92005
[2] Band, L. R., Hall, C. L., Richardson, G., Jensen, O. E., Siggers, J. H. and Foss, A. J. E., “‘Intracellular flow in optic nerve axons: a mechanism for cell death in glaucoma”, Invest. Ophthalmol. Vis. Sci.50 (2009) 3750-3758; doi:10.1167/iovs.08-2396.
[3] Bertram, C. D. and Pedley, T. J., “‘A mathematical model of unsteady collapsible tube behaviour”, J. Biomech.15 (1982) 39-50; doi:10.1016/0021-9290(82)90033-1.
[4] Bertram, C. D., Raymond, C. J. and Pedley, T. J., “‘Mapping of instabilities for flow through collapsed tubes of differing length”, J. Fluids Struct.4 (1990) 125-153; doi:10.1016/0889-9746(90)90058-D.
[5] Bertram, C. D. and Tscherry, J., “‘The onset of flow-rate limitation and flow-induced oscillations in collapsible tubes”, J. Fluids Struct.22 (2006) 1029-1045; doi:10.1016/j.jfluidstructs.2006.07.005.
[6] Cancelli, C. and Pedley, T. J., “‘A separated-flow model for collapsible-tube oscillations”, J. Fluid Mech.157 (1985) 375-404; doi:10.1017/S0022112085002427.
[7] Coccius, E. A., Ueber die Anwendung des Augen-Spiegels: nebst Angabe eines neuen Instrumentes, (1853), I. Müller: I. Müller, Leipzig
[8] Davies, C. and Carpenter, P. W., “‘Instabilities in a plane channel flow between compliant walls”, J. Fluid Mech.352 (1997) 205-243; doi:10.1017/S0022112097007313. · Zbl 0903.76029
[9] Davies, C. and Carpenter, P. W., “‘Numerical simulation of the evolution of Tollmien-Schlichting waves over finite compliant panels”, J. Fluid Mech.335 (1997) 361-392; doi:10.1017/S0022112096004636. · Zbl 0903.76030
[10] Garhofer, G., Werkmeister, R., Dragostinoff, N. and Schmetterer, L., “‘Retinal blood flow in healthy young subjects”, Invest. Ophthalmol. Vis. Sci.53 (2012) 698-703; doi:10.1167/iovs.11-8624.
[11] Golzan, S. M., Graham, S. L., Leaney, J. and Avolio, A., “‘Dynamic association between intraocular pressure and spontaneous pulsations of retinal veins”, Curr. Eye Res.36 (2011) 53-59; doi:10.3109/02713683.2010.530731.
[12] Golzan, S. M., Kim, M. O., Seddighi, A. S., Avolio, A. and Graham, S. L., “‘Non-invasive estimation of cerebrospinal fluid pressure waveforms by means of retinal venous pulsatility and central aortic blood pressure”, Ann. Biomed. Eng.40 (2012) 1940-1948; doi:10.1007/s10439-012-0563-y.
[13] Grotberg, J. B. and Jensen, O. E., “‘Biofluid mechanics in flexible tubes”, Annu. Rev. Fluid Mech.36 (2004) 121-147; doi:10.1146/annurev.fluid.36.050802.121918. · Zbl 1081.76063
[14] Guidoboni, G., Harris, A., Cassani, S., Arciero, J., Siesky, B., Amireskandari, A., Tobe, L., Egan, P., Januleviciene, I. and Park, J., “‘Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance”, Invest. Ophthalmol. Vis. Sci.55 (2014) 4105-4118; doi:10.1167/iovs.13-13611.
[15] Hayreh, S. S., “‘The central artery of the retina. Its role in the blood supply of the optic nerve”, Br. J. Ophthalmol.47 (1963) 651-663; doi:10.1136/bjo.47.11.651.
[16] Hayreh, S. S., “‘Non-invasive measurement of intracranial pressure”, Lancet351 (1998) 524-525; doi:10.1016/S0140-6736(05)78719-5.
[17] Heil, M. and Boyle, J., “‘Self-excited oscillations in three-dimensional collapsible tubes: simulating their onset and large-amplitude oscillations”, J. Fluid Mech.652 (2010) 405-426; doi:10.1017/S0022112010000157. · Zbl 1193.74034
[18] Heil, M. and Hazel, A. L., “‘Fluid-structure interaction in internal physiological flows”, Annu. Rev. Fluid Mech.43 (2011) 141-162; doi:10.1146/annurev-fluid-122109-160703. · Zbl 1299.76319
[19] Jensen, O. E., “‘Instabilities of flow in a collapsed tube”, J. Fluid Mech.220 (1990) 623-659; doi:10.1017/S0022112090003408.
[20] Jensen, O. E. and Heil, M., “‘High-frequency self-excited oscillations in a collapsible-channel flow”, J. Fluid Mech.481 (2003) 235-268; doi:10.1017/S002211200300394X. · Zbl 1049.76015
[21] Jonas, J. B., “‘Retinal venous pulsation and glaucoma”, Ophthalmology112 (2005) 948-949; doi:10.1016/j.ophtha.2004.11.014.
[22] Jonas, J., Paques, M., Monés, J. and Glacet-Bernard, A., “‘Retinal vein occlusions”, in: Macular edema, Volume 47 of Dev. Opthalmol. (eds Coscas, G., Cunha-Vaz, J., Loewenstein, A. and Soubrane, G.), (Karger, Basel, 2010) 111-135; doi:10.1159/000320076.
[23] Knowlton, F. P. and Starling, E. H., “‘The influence of variations in temperature and blood-pressure on the performance of the isolated mammalian heart”, J. Physiol.44(3) (1912) 206-219; doi:10.1113/jphysiol.1912.sp001511.
[24] Kramer, M. O., “‘Boundary layer stabilization by distributed damping”, J. Amer. Soc. Naval Eng.72 (1960) 25-34; doi:10.1111/j.1559-3584.1960.tb02356.x.
[25] Levin, B. E., “‘The clinical significance of spontaneous pulsations of the retinal vein”, Arch. Neurol.35 (1978) 37-40; doi:10.1001/archneur.1978.00500250041009.
[26] Levine, D. N., “‘Spontaneous pulsation of the retinal veins”, Microvas. Res.56 (1998) 154-165; doi:10.1006/mvre.1998.2098.
[27] Luo, X. Y., Cai, Z. X., Li, W. G. and Pedley, T. J., “‘The cascade structure of linear instability in collapsible channel flows”, J. Fluid Mech.600 (2008) 45-76; doi:10.1017/S0022112008000293. · Zbl 1151.76455
[28] Luo, X. Y. and Pedley, T. J., “‘A numerical simulation of unsteady flow in a two-dimensional collapsible channel”, J. Fluid Mech.314 (1996) 191-225; doi:10.1017/S0022112096000286.
[29] Luo, X. Y. and Pedley, T. J., “‘The effects of wall inertia on flow in a two-dimensional collapsible channel”, J. Fluid Mech.363 (1998) 253-280; doi:10.1017/S0022112098001062.
[30] McClurken, M. E., Kececioglu, I., Kamm, R. D. and Shapiro, A. H., “‘Steady, supercritical flow in collapsible tubes. Part 2. Theoretical studies”, J. Fluid Mech.109 (1981) 391-415; doi:10.1017/S0022112081001134.
[31] Moghimi, S., Hosseini, H., Riddle, J., Lee, G. Y., Bitrian, E., Giaconi, J., Caprioli, J. and Nouri-Mahdavi, K., “‘Measurement of optic disc size and rim area with spectral-domain OCT and scanning laser ophthalmoscopy”, Invest. Ophthalmol. Vis. Sci.53 (2012) 4519-4530; doi:10.1167/iovs.11-8362.
[32] Morgan, W. H., Hazelton, M. L., Azar, S. L., House, P. H., Yu, D.-Y., Cringle, S. J. and Balaratnasingam, C., “‘Retinal venous pulsation in glaucoma and glaucoma suspects”, Ophthalmology111 (2004) 1489-1494; doi:10.1016/j.ophtha.2003.12.053.
[33] Morgan, W. H., Hazelton, M. L. and Yu, D.-Y., “‘Retinal venous pulsation: expanding our understanding and use of this enigmatic phenomenon”, Prog. Ret. Eye Res.55 (2016) 82-107; doi:10.1016/j.preteyeres.2016.06.003.
[34] Morgan, W. H., Yu, D.-Y. and Balaratnasingam, C., “‘The role of cerebrospinal fluid pressure in glaucoma pathophysiology: the dark side of the optic disc”, J. Glaucoma17 (2008) 408-413; doi:10.1097/IJG.0b013e31815c5f7c.
[35] Pedley, T. J., The fluid mechanics of large blood vessels (Cambridge University Press, Cambridge, 1980); doi:10.1017/CBO9780511896996. · Zbl 0449.76100
[36] Pihler-Puzović, D. and Pedley, T. J., “‘Flutter in a quasi-one-dimensional model of a collapsible channel”, Proc. R. Soc. Lond. Ser. A470 (2014) 20140015; doi:10.1098/rspa.2014.0015.
[37] Sen, P. K., Carpenter, P. W., Hedge, S. and Davies, C., “‘A wave driver theory for vortical waves propagating across junctions with application to those between rigid and compliant walls”, J. Fluid Mech.625 (2009) 1-46; doi:10.1017/S0022112008005545. · Zbl 1171.76355
[38] Singh, S. and Dass, R., “‘The central artery of the retina I. Origin and course”, Br. J. Ophthalmol.44 (1960) 193-212; doi:10.1136/bjo.44.4.193.
[39] Singh, S. and Dass, R., “‘The central artery of the retina II. A study of its distribution and anastomoses”, Br. J. Ophthalmol.44 (1960) 280-299; doi:10.1136/bjo.44.5.280.
[40] Stewart, P. S., “‘Instabilities in flexible channel flow with large external pressure”, J. Fluid Mech.825 (2017) 922-960; doi:10.1017/jfm.2017.404. · Zbl 1374.76243
[41] Stewart, P. S., Heil, M., Waters, S. L. and Jensen, O. E., “‘Sloshing and slamming oscillations in collapsible channel flow”, J. Fluid Mech.662 (2010) 288-319; doi:10.1017/S0022112010003277. · Zbl 1205.76077
[42] Stewart, P. S., Jensen, O. E. and Foss, A. J. E., “‘A theoretical model to allow prediction of the CSF pressure from observations of the retinal venous pulse”, Invest. Ophthalmol. Vis. Sci.55 (2014) 6319-6323; doi:10.1167/iovs.14-14331.
[43] Stewart, P. S., Waters, S. L. and Jensen, O. E., “‘Local and global instabilities of flow in a flexible-walled channel”, Eur. J. Mech. B28 (2009) 541-557; doi:10.1016/j.euromechflu.2009.03.002. · Zbl 1167.76329
[44] Walsh, T. J., Garden, J. W. and Gallagher, B., “‘Obliteration of retinal venous pulsations: during elevation of cerebrospinal-fluid pressure”, Amer. J. Ophthalmol.67(6) (1969) 954-956; doi:10.1016/0002-9394(69)90094-4.
[45] Whittaker, R. J., Heil, M., Jensen, O. E. and Waters, S. L., “‘A rational derivation of a tube law from shell theory”, Quart. J. Mech. Appl. Math.63 (2010) 465-496; doi:10.1093/qjmam/hbq020. · Zbl 1256.74017
[46] Williamson, T. H., Lowe, G. D. and Baxter, G. M., “‘Influence of age, systemic blood pressure, smoking, and blood viscosity on orbital blood velocities”, Br. J. Ophthalmol.79 (1995) 17-22; doi:10.1136/bjo.79.1.17.
[47] Wong, T. Y. and Scott, I. U., “‘Retinal-vein occlusion”, New Engl. J. Med.363 (2010) 2135-2144; doi:10.1056/NEJMcp1003934.
[48] Xie, X.et al., “‘Noninvasive intracranial pressure estimation by orbital subarachnoid space measurement: the Beijing intracranial and intraocular pressure (iCOP) study”, Crit. Care17 (2013) R162; doi:10.1186/cc12841.
[49] Xu, F., Billingham, J. and Jensen, O. E., “‘Divergence-driven oscillations in a flexible-channel flow with fixed upstream flux”, J. Fluid Mech.723 (2013) 706-733; doi:10.1017/jfm.2013.97. · Zbl 1287.76253
[50] Xu, F., Billingham, J. and Jensen, O. E., “‘Resonance-driven oscillations in a flexible-channel flow with fixed upstream flux and a long downstream rigid segment”, J. Fluid Mech.746 (2014) 368-404; doi:10.1017/jfm.2014.136. · Zbl 1309.76236
[51] Xu, F. and Jensen, O. E., “‘A low-order model for slamming in a flexible-channel flow”, Quart. J. Mech. Appl. Math.68 (2015) 299-319; doi:10.1093/qjmam/hbv009. · Zbl 1331.74060
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