Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-07-01T14:41:54.402Z Has data issue: false hasContentIssue false
  • English
  • Français

Forced motion of a cylinder within a liquid-filled elastic tube – a model of minimally invasive medical procedures

Published online by Cambridge University Press:  29 October 2019

Amit Vurgaft ,
Shai B. Elbaz  and
Amir D. Gat Open the ORCID record for Amir D. Gat [Opens in a new window]
Amit Vurgaft*
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
Shai B. Elbaz
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
Amir D. Gat
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
*
†Email address for correspondence: amirgat@technion.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

This work analyses the viscous flow and elastic deformation created by the forced axial motion of a rigid cylinder within an elastic liquid-filled tube. The examined configuration is relevant to various minimally invasive medical procedures in which slender devices are inserted into fluid-filled biological vessels, such as vascular interventions, interventional radiology, endoscopies and laparoscopies. By applying the lubrication approximation, thin shell elastic model, as well as scaling analysis and regular and singular asymptotic schemes, the problem is examined for small and large deformation limits (relative to the gap between the cylinder and the tube). At the limit of large deformations, forced insertion of the cylinder is shown to involve three distinct regimes and time scales: (i) initial shear dominant regime, (ii) intermediate regime of dominant fluidic pressure and a propagating viscous-peeling front, (iii) late-time quasi-steady flow regime of the fully peeled tube. A uniform solution for all regimes is presented for a suddenly applied constant force, showing initial deceleration and then acceleration of the inserted cylinder. For the case of forced extraction of the cylinder from the tube, the negative gauge pressure reduces the gap between the cylinder and the tube, increasing viscous resistance or creating friction due to contact of the tube and cylinder. Matched asymptotic schemes are used to calculate the dynamics of the near-contact and contact limits. We find that the cylinder exits the tube in a finite time for sufficiently small or large forces. However, for an intermediate range of forces, the radial contact creates a steady locking of the cylinder inside the tube.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Low-Reynolds-number flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 881 , 25 December 2019 , pp. 1048 - 1072
Check for updates
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abreu, D., Levant, M., Steinberg, V. & Seifert, U. 2014 Fluid vesicles in flow. Adv. Colloid Interface Sci. 208, 129–141.10.1016/j.cis.2014.02.004Google Scholar
Barenblatt, G. I. 1952 On some unsteady motions of a liquid and gas in a porous medium. Prikl. Mat. Mekh. 16 (1), 67–78.Google Scholar
Camalet, S. & Jülicher, F. 2000 Generic aspects of axonemal beating. New J. Phys. 2, 24.10.1088/1367-2630/2/1/324Google Scholar
Chew, D. J., Buffington, T., Kendall, M. S., Osborn, S. D. & Woodsworth, B. E. 1996 Urethroscopy, cystoscopy, and biopsy of the feline lower urinary tract. Veterinary Clinics: Small Animal Practice 26 (3), 441–462.Google Scholar
Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J. & Knuth, D. E. 1996 On the Lambert-W function. Adv. Comput. Maths 5, 329–359.10.1007/BF02124750Google Scholar
Davis, T. 2015 No-fluoroscopy crossing of chronic total occlusions using ocelot optical coherence tomography guided catheter. Vascular Disease Management 12 (12), E230–E241.Google Scholar
Dunn, M. E. & Weisse, C. 2015 Thrombectomy and thrombolysis: the interventional radiology approach. In Veterinary Image-Guided Interventions, p. 464. Wiley-Blackwell.Google Scholar
Duprat, C. & Stone, H. A. 2015 Fluid-Structure Interactions in Low-Reynolds-Number Flows. Royal Society of Chemistry.10.1039/9781782628491Google Scholar
Elbaz, S. B. & Gat, A. D. 2016 Axial creeping flow in the gap between a rigid cylinder and a concentric elastic tube. J. Fluid Mech. 806, 580–602.10.1017/jfm.2016.587Google Scholar
Heil, M. 1996 The stability of cylindrical shells conveying viscous flow. J. Fluids Struct. 10 (2), 173–196.10.1006/jfls.1996.0012Google Scholar
Heil, M. 1998 Stokes flow in an elastic tube – a large-displacement fluid-structure interaction problem. Intl J. Numer. Meth. Fluids 28 (2), 243–265.10.1002/(SICI)1097-0363(19980815)28:2<243::AID-FLD711>3.0.CO;2-U3.0.CO;2-U>Google Scholar
Hewitt, I. J., Balmforth, N. J. & De Bruyn, J. R. 2015 Elastic-plated gravity currents. Eur. J. Appl. Maths 26 (1), 1–31.10.1017/S0956792514000291Google Scholar
Karahalios, G. T. 1990 Some possible effects of a catheter on the arterial wall. Medical Phys. 17 (5), 922–925.10.1118/1.596448Google Scholar
Kumar, H., Chandel, R. S., Kumar, S. & Kumar, S. 2013 A mathematical model for blood flow through a narrow catheterized artery. Intl J. Theoret. Appl. Sci. 5 (2), 101–108.Google Scholar
Lighthill, M. J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34 (1), 113–143.10.1017/S0022112068001795Google Scholar
Lister, J. R., Peng, G. G. & Neufeld, J. A. 2013 Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett. 111 (15), 154501.10.1103/PhysRevLett.111.154501Google Scholar
Marzo, A., Luo, X. Y. & Bertram, C. D. 2005 Three-dimensional collapse and steady flow in thick-walled flexible tubes. J. Fluids Struct. 20 (6), 817–835.10.1016/j.jfluidstructs.2005.03.008Google Scholar
Nacey, J. & Delahijnt, B. 1993 The evolution and development of the urinary catheter. Australian New Zealand J. Surgery 63 (10), 815–819.10.1111/j.1445-2197.1993.tb00347.xGoogle Scholar
Park, K., Tixier, A., Christensen, A. H., Arnbjerg-Nielsen, S. F., Zwieniecki, M. A. & Jensen, K. H. 2018 Viscous flow in a soft valve. J. Fluid Mech. 836, R3.10.1017/jfm.2017.805Google Scholar
Pellerin, O., Maleux, G., Déan, C., Pernot, S., Golzarian, J. & Sapoval, M. 2014 Microvascular plug: a new embolic material for hepatic arterial skeletonization. Cardiovascular Interventional Radiology 37 (6), 1597–1601.10.1007/s00270-014-0889-yGoogle Scholar
Rogers, J. H. & Laird, J. R. 2007 Overview of new technologies for lower extremity revascularization. Circulation 116 (18), 2072–2085.10.1161/CIRCULATIONAHA.107.715433Google Scholar
Sarkar, A. & Jayaraman, G. 2001 Nonlinear analysis of oscillatory flow in the annulus of an elastic tube: application to catheterized artery. Phys. Fluids 13 (10), 2901–2911.10.1063/1.1389285Google Scholar
Serruys, P. W., Foley, D. P. & De Feyter, P. J. 1993 Quantitative Coronary Angiography in Clinical Practice, vol. 145. Springer Science & Business Media.Google Scholar
Tani, M., Cambau, T., Bico, J. & Reyssat, E. 2017 Motion of a rigid sphere through an elastic tube with a lubrication film. In APS Meeting Abstracts. American Physical Society.Google Scholar
Timoshenko, S. P. & Woinowsky-Krieger, S. 1959 Theory of Plates and Shells. McGraw-Hill.Google Scholar
Tözeren, A., Özkaya, N. & Tözeren, H. 1982 Flow of particles along a deformable tube. J. Biomech. 15 (7), 517–527.10.1016/0021-9290(82)90005-7Google Scholar
Vajravelu, K., Sreenadh, S., Devaki, P. & Prasad, K. 2011 Mathematical model for a herschel-bulkley fluid flow in an elastic tube. Open Phys. 9 (5), 1357–1365.10.2478/s11534-011-0034-3Google Scholar
Vázquez, J. L. 2007 The Porous Medium Equation: Mathematical Theory. Oxford University Press.Google Scholar
Vlahovska, P. M., Young, Y.-N., Danker, G. & Misbah, C. 2011 Dynamics of a non-spherical microcapsule with incompressible interface in shear flow. J. Fluid Mech. 678, 221–247.10.1017/jfm.2011.108Google Scholar
Wiggins, C. H. & Goldstein, R. E. 1998 Flexive and propulsive dynamics of elastica at low Reynolds number. Phys. Rev. Lett. 80 (17), 3879.10.1103/PhysRevLett.80.3879Google Scholar
Zaitsev, V. F. & Polyanin, A. D. 2002 Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press.10.1201/9781420035339Google Scholar
Zel’dovich, Y. B. & Kompaneets, A. S. 1950 Towards a theory of heat conduction with thermal conductivity depending on the temperature. In Collection of Papers Dedicated to 70th Birthday of Academician AF Ioffe, Izd. Akad. Nauk SSSR, Moscow, pp. 61–71. Izd. Akademii Nauk SSSR.Google Scholar