Air Bubble Collapse in Non-Newtonian Medium with an Application in Biology

Document Type : Research Article


1 University of Tehran

2 Shahid Beheshti University

3 University of Tehran*


  An unsteady compressible multiphase flow solver is developed and used to simulate shock-bubble interaction in a non-Newtonian fluid. A five-equation multiphase model that accounts for capillary and viscous effects is employed and discretized by finite volume methodology. Harten-Lax-Van Leer-contact Riemann solver is invoked to compute the convective fluxes and tangent of hyperbola for interface capturing interface sharpening scheme is applied to reduce the excessive diffusion at the interface. Multiple benchmark problems such as air-helium shock tube, shock cavity interaction, Rayleigh-Taylor instability and underwater explosion are probed to evaluate the performance and accuracy of this method. The results obtained compare well with the available experimental and numerical data. The developed solver is then used to study shock-interface interaction in both Newtonian and non-Newtonian mediums. Non-Newtonian liquid is resembling the blood modeled by Carreau-Yasuda constitutive equation. The obtained results show an expedition of bubble-collapse with a higher jet tip velocity in non-Newtonian medium compared to that in the Newtonian surrounding liquid. Moreover, a third phase adjacent to the bubble collapse is considered and the penetration depth of the re-entrant jet in the neighboring phase is studied as a measure of tissue injury. Our results show that by increasing post shock pressure, the re-entrant jet velocity and thus the penetration depth increases. Furthermore, increasing the adjacent phase viscosity results into less penetration depth in the tissue..


Main Subjects

[1] R. Cooter, W. Babidge, K. Mutimer, Ultrasound-assistedlipoplasty, ANZ Journal of Surgery, 71(5) (2001) 309-317.
[2] D. Duscher, Z.N. Maan, A. Luan, M.M. Aitzetmüller,E.A. Brett, D. Atashroo, A.J. Whittam, M.S. Hu, G.G.Walmsley, H.-g. Machens, G.C. Gurtner, M.T. Longaker,D.C. Wan, Ultrasound-assisted liposuction providesa source for functional adipose-derived stromal cells,Cytotherapy, 19(12) (2017) 1491-1500.
[3] M. Brock, I. Ingwersen, W. Roggendorf, Ultrasonicaspiration in neurosurgery, Neurosurg Rev, 7(2-3) (1984)173-177.
[4] T. Sun, Y. Zhang, C. Power, P.M. Alexander, J.T. Sutton,M. Aryal, Closed-loop control of targeted ultrasound drug delivery across the blood – brain / tumor barriers in a ratglioma model, Proceedings of the National Academy ofSciences of the United States of America, 114(48) (2017)E10281-E10290.
[5] Y.-T. Wu, A. Adnan, Effect of Shock-Induced CavitationBubble Collapse on the damage in the SimulatedPerineuronal Net of the Brain, Scientific Reports, 7(1)(2017) 5323.
[6] H.B. Dick, T. Schultz, A Review of Laser-AssistedVersus Traditional Phacoemulsification Cataract Surgery, Ophthalmology and Therapy, 6(1) (2017) 7-18.
[7] A.J. Coleman, J.E. Saunders, L.A. Crum, Acousticcavitation generated by an extracorporeal shockwavelithotripter, Ultrasound Med. Biol, 13(2) (1987) 69-76.
[8] S. Cao, Y. Zhang, Assessing the effect of lithotripter focal width on the fracture potential of stones in shockwavelithotripsy, Journal of the Acoustical Society of America,141(5) (2017) 3718.
[9] M. Shim, M. Park, H.K. Park, The efficacy of performing shockwave lithotripsy before retrograde intrarenalsurgery in the treatment of multiple or large (≥1.5 cm) nephrolithiasis: A propensity score matched analysis, investigative and clinical urology, 58(1) (2017) 27-33.
[10] C.K. Turangan, G.J. Ball, A.R. Jamaluddin, T.G.Leighton, Numerical studies of cavitation erosion on anelastic – plastic material caused by shock-induced bubblecollapse Subject Areas, Proceedings of the Royal SocietyA: Mathematical, Physical and Engineering Sciences,473(2205) (2017) 20170315.
[11] D. Igra, O. Igra, Numerical investigation of theinteraction between a planar shock wave with square andtriangular bubbles containing different gases, Physics ofFluids, 30(5) (2018) 056104.
[12] R.O. Cleveland, M.R. Bailey, N. Fineberg, Design andcharacterization of a research electrohydraulic lithotripter patterned after the Dornier HM3, Rev. Sci. Instrum.,71(6) (2000) 2514-2525.
[13] V. Coralic, T. Colonius, Shock-induced collapse of abubble inside a deformable vessel, Eur J Mech B Fluids,40 (2013) 64-74.
[14] M.R. Bailey, Y.A. Pishchalnikov, Cavitation detectionduring shock-wave lithotripsy, Ultrasound Med. Biol.,31(9) (2005) 1245-1256.
[15] M. Lokhandwalla, B. Sturtevant, Fracture mechanicsmodel of stone comminution in ESWL and implicationsfor tissue damage, Phys. Med. Biol, 45(7) (2000) 1923-1940.
[16] L.A. Crum, Cavitation microjets as a contributorymechanism for renal calculi disintegration in ESWL, J.Urol., 140(6) (1988) 1587-1590.
[17] K.G. Wang, Multiphase Fluid-Solid Coupled Analysisof Shock-Bubble-Stone Interaction in ShockwaveLithotripsy, International Journal for Numerical Methodsin Biomedical Engineering, 33(10) (2017) cnm.2855.
[18] H. Chen, A. Brayman, M.R. Bailey, Blood vesselrupture by cavitation, Urol. Res, 38(4) (2010) 321-326.
[19] H. Chen, W. Kreider, A.A. Brayman, M.R. Bailey, T.J.Matula, Blood vessel deformations on microsecond timescales by ultrasonic cavitation, Physical Review Letters,106(3) (2011) 034301.
[20] C. Weber, M.E. Moran, E.J. Braun, Injury of rat renalvessels following extracorporeal shock wave treatment,J.Urology,, 147(2) (1992) 476-481.
[21] P. Zhang, Y.F. Zhu, S.L. Zhu, Dynamics of bubbleoscillation in constrained media and mechanisms ofvessel rupture in SWL., Ultrasound Med. Biol., 27(1)(2001) 119-134.
[22] Rayleigh, On the pressure developed in a liquid duringthe collapse of a spherical cavity, Phil. Mag., 34(200)(1917) 94-98.
[23] R. Hickling, M.S. Plesset, Collapse and rebound of aspherical bubble in water, Physics of Fluids, 7(1) (1964)7-14.
[24] M. Kornfeld, L. Suvorov, On the destructive action ofcavitation, Journal of Applied Physics, 15(6) (1944) 495-506.
[25] T.B. Benjamin, A.T. Ellis, The Collapse of CavitationBubbles and the Pressures thereby Produced against Solid Boundaries, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 260(1110) (1966) 221-240.
[26] C.D. Ohl, R. Ikink, Shock-Wave-Induced Jetting ofMicron-Size Bubbles, Physical Review Letters, 90(21)(2003) 214502.
[27] C.L. Kling, F.G. Hammitt, A Photographic Study ofSpark-Induced Cavitation Bubble Collapse, J. Basic Eng,94(4) (1972) 825-832.
[28] B.H.T. Goh, Y.D.A. Oh, E. Klaseboer, S.W. Ohl,B.C. Khoo, A low-voltage spark-discharge method forgeneration of consistent oscillating bubbles., Review ofScientific Instruments, 84(1) (2013) 014705.
[29] J.A. Cook, A.M. Gleeson, R.M. Roberts, R.L. Rogers,A spark-generated bubble model with semi-empiricalmass transport, J. Acoust. Soc. Am., 101(4) (1997) 1908-1920.
[30] Y. Tomita, A. Shima, Mechanisms of impulsive pressure generation and damage pit formation by bubble collapse,J.Fluid Mech, 169 (1986) 535–564.
[31] T. Kodama, K.A.T. Takayama, Dynamic behavior ofbubbles during extracorporeal shock-wave lithotripsy,Ultrasound in Medicine and Biology, 24(5) (1998) 723-738.
[32] S. Li, A.M. Zhang, R. Han, Y.Q. Liu, Experimental andnumerical study on bubble-sphere interaction near a rigidwall Experimental and numerical study on bubble-sphereinteraction near a rigid wall, Physics of Fluids, 29(9)(2017) 092102.
[33] M.S. Plesset, R.B. Chapman, Collapse of an initiallyspherical vapour cavity in the neighbourhood of a solidboundary, J. Fluid Mech, 47(2) (1971) 283–290.
[34] J.R. Blake, B.B. Taib, G. Doherty, Transient cavitiesnear boundaries. Part 1. Rigid boundary., J. Fluid Mech,170 (1986) 479–497.
[35] E. Klaseboer, C. Turangan, S.W. Fong, T.G. Liu,Simulations of pressure pulse-bubble interaction usingboundary element method, Comput. Methods Appl.Mech. Engrg, 195 (2006) 4287–4302.
[36] S. Popinet, S. Zaleski, Bubble collapse near a solidboundary: a numerical study of the influence of viscosity,Journal of Fluid Mechanics, 464 (2002) 137-163.
[37] G.J.J. Ball, B.P.P. Howell, T.G.G. Leighton, M.J.J.Schofield, Shock-induced collapse of a cylindrical aircavity in water: a Free-Lagrange simulation, ShockWaves, 10(4) (2000) 265-276.
[38] A.R. Jamaluddin, Free-lagrange simulations ofshock-bubble interaction in extracorporeal shock wavelithotripsy, University of Southampton, 2005.
[39] S.k. Hara, Dynamics of nonspherical bubbles surrounded by viscoelastic fluid, Journal of Non -Newtonian FluidMechanics, 14 (1984) 249-264.
[40] C. Kim, Collapse of spherical bubbles in Maxwellfluids, Journal of Non-Newtonian fluid Mechanic, 55(1)(1994) 37-58.
[41] E.A. Brujan, Y. Matsumoto, T. Ikeda, Dynamics ofultrasound-induced cavitation bubbles in non-Newtonianliquids and near a rigid boundary, Physics of Fluids,16(7) (2004) 2402.
[42] M.J. Walters, An Investigation into the Effects ofViscoelasticity on Cavitation Bubble Dynamics withApplications to Biomedicine, school of MathematicsCardiff University, 2015.
[43]  S.J. Lind, T.N. Phillips, The influence ofviscoelasticity on the collapse of cavitation bubbles neara rigid boundary, Theoretical and Computational FluidDynamics, 26(1-4) (2012) 245–277.
[44] C. F.Rowlatt, S. J.Lind, Bubble collapse near a fluid-fluid interface using the spectral element marker particlemethod with applications in bioengineering, International Journal of Multiphase Flow, 90 (2017) 118-143.
[45] A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problems, Journal ofComputational Physics, 202(2) (2005) 664-698.
[46] F.H. Harlow, A.A. Amsden, Fluid dynamics: A LASLmonograph(Mathematical solutions for problems in fluiddynamics).
[47] L. Zhang, C. Yang, Z.S. Mao, Numerical simulation ofa bubble rising in shear-thinning fluids, Journal of Non-Newtonian Fluid Mechanics, 165(11-12) (2010) 555-567.
[48] F. Toro, Riemann solvers and numerical methods forfluid dynamics, a practical introduction,Springer Science& Business Media, 2009.
[49] K.-M. Shyue, An Efficient Shock-Capturing Algorithmfor Compressible Multicomponent Problems, Journal ofComputational Physics, 142(1) (1998) 208-242.
[50] B. van Leer, Towards the ultimate conservativedifference scheme. V. A second-order sequel toGodunov’s method, Journal of Computational Physics,32(1) (1979) 101-136.
[51] K. So, X. Hu, N. Adams, Anti-Diffusion InterfaceSharpening Technique for Two Phase CompressibleFlow Simulations, J. Comput. Phys., 231(11) (2012)4304-4323.
[52] N.K. Bourne, J.E. Field, Shock-induced collapse ofsingle cavities in liquids, Journal of Fluid Mechanics,244 (1992) 225-240.
[53] H. Terashima, G. Tryggvason, A front-tracking/ghost-fluid method for fluid interfaces in compressible flows,Journal of Computational Physics, 228(11) (2009) 4012-4037.
[54] R.R. Nourgaliev, T.N. Dinh, T.G. Theofanous, Adaptive characteristics-based matching for compressiblemultifluid dynamics, Journal of Computational Physics,213(2) (2006) 500-529.
[55] W. Bo, J.w. Grove, A volume of fluid method basedghost fluid method for compressible multi-fluid flows,Computers and Fluids, 90 (2014) 113-122.
[56] S.S. Shibeshi, W.E. Collins, The Rheology of BloodFlow in a Branched Arterial System, Appl Rheol, 15(6)(2005) 398-405.