Prof Matthias Heil

Professor of Applied Mathematics at the School of Mathematics at the University of Manchester.

[People] [Research] [Publications] [Teaching] [Admin] [Seminars] [Editorial responsbilities/Refereeing] [Family] [Contact]


Here are some people I work with: ...and here are some former PhD students/postdocs:

Research Interests: Computational Mechanics and Biomechanics.

[Check out this recent article in the EPSRC Newsline magazine.]

Scientific Computing:
oomph-lib -- the object-oriented multi-physics finite-element library

Fluid-structure interaction problems

(Flow in collapsible tubes).

Airway reopening:

The propagation of an air finger into a collapsed elastic tube.

Airway closure:
Static liquid bridges in elastic tubes.

Airway closure:
Surface-tension-driven instabilities in elastic tubes.

Computational fluid dynamics.

Finite element methods.

Large displacement shell theory.

Experimental methods.


Crane, L.J. & Heil, M. (1995) The Summation of Slowly Convergent Double Series and Application to Fluid-Mechanics. Zeitschrift fur angewandte Mathematik und Mechanik 75 , 403-407. (abstract)

Heil, M. & Pedley, T.J. (1995) Large Axisymmetric Deformations of a Cylindrical Shell Conveying a Viscous Flow. Journal of Fluids and Structures 9, 237-256. (abstract)

Heil, M. (1996) The Stability of Cylindrical Shells Conveying Viscous Flow. Journal of Fluids and Structures 10, 173-196. (abstract) (pdf) (ps)

Heil, M. & Pedley, T.J. (1996) Large Post-Buckling Deformations of Cylindrical Shells Conveying Viscous Flow. Journal of Fluids and Structures 10 , 565-599. (abstract) (pdf) (ps)

Heil, M. (1997), Stokes Flow in Collapsible Tubes -- Computation and Experiment. Journal of Fluid Mechanics 353, 285-312. (abstract)(ps preprint)

Heil, M. (1998) Stokes Flow in an Elastic Tube -- A Large-Displacement Fluid-Structure Interaction Problem. The International Journal for Numerical Methods in Fluids 28, 243-265. (abstract)(pdf preprint)

Heil, M. (1999) Minimal Liquid Bridges in Non-Axisymmetrically Buckled Elastic Tubes. Journal of Fluid Mechanics 380, 309-337. (abstract) (ps preprint)

Heil, M. (1999) Airway Closure: Liquid Bridges in Strongly Buckled Elastic Tubes. ASME Journal of Biomechanical Engineering 121, 487-493. (ps preprint) (abstract)

Heil, M. (2000) Finite Reynolds number effects in the propagation of an air finger into a liquid-filled flexible-walled channel. Journal of Fluid Mechanics 424, 21-44.(ps preprint) (abstract)

Heil, M. (2001) The Bretherton problem in elastic-walled channels: Finite Reynolds number effects. In: IUTAM Symposium on Free Surface Flows, Editors: A. C. King & Y.D. Shikmurzaev. Kluwer, Dordrecht, Netherlands. pp. 113-120. (ps preprint) (abstract)

Heil, M. (2001) Finite Reynolds number effects in the Bretherton problem. Physics of Fluids 13, 2517-2521. (ps preprint) (abstract)

Heil, M. & Hazel, A. L. (2001) PROSPER: LATEX -based computer presentations. SIAM News 34(10), 19-20. (pdf reprint). (html version of this article with all files).

Heil, M. & White, J.P. (2002) Airway Closure: Surface-tension-driven non-axisymmetric instabilities of liquid-lined elastic rings. Journal of Fluid Mechanics 462, 79-109. (ps preprint)(abstract)

Hazel, A. L. & Heil, M. (2002) The steady propagation of a semi-infinite bubble into a tube of elliptical or rectangular cross-section. Journal of Fluid Mechanics 470, 91-114. (pdf preprint) (abstract)

Heil, M. & Jensen, O.E. (2003) Flows in deformable tubes and channels -- Theoretical models and biological applications. Chapter 2 of: Flow in Collapsible Tubes and Past Other Highly Compliant Boundaries. Eds: T.J. Pedley and P.W. Carpenter. pp. 15-50. Kluwer, Dordrecht, Netherlands. (pdf preprint)(abstract)

Hazel, A. L. & Heil, M. (2003) Three-dimensional airway reopening: The steady propagation of a semi-infinite bubble into a buckled elastic tube. Journal of Fluid Mechanics 478 47-70. (pdf preprint) (abstract)

Jensen, O.E. & Heil, M. (2003) High-frequency self-excited oscillations in a collapsible-channel flow. Journal of Fluid Mechanics 481 235-268. (pdf preprint) (abstract)

Hazel, A. L. & Heil, M. (2003) Steady finite-Reynolds-number flows in three-dimensional collapsible tubes. Journal of Fluid Mechanics 486 79-103. (pdf preprint)(abstract)

Heil, M. & Hazel, A. L. (2003) Mass transfer from a finite strip near an oscillating stagnation point --- implications for atherogenesis. Journal of Engineering Mathematics. 47 315-334. (abstract)(pdf preprint)

Heil, M. (2004) An efficient solver for the fully-coupled solution of large-displacement fluid-structure interaction problems. Computer Methods in Applied Mechanics and Engineering. 193 1-23. (abstract) (pdf preprint)

Watton, P. N., Hill, N. A. & Heil, M. (2004) A mathematical model for the growth of the abdominal aortic aneurysm. Biomechanics and Modeling in Mechanobiology 3 98-113. (abstract) (pdf reprint)

White, J.P. & Heil, M. (2005) Three-dimensional instabilities of liquid-lined elastic tubes -- a lubrication theory model. Physics of Fluids 17 (abstract) (pdf preprint)

Hazel, A. L. & Heil, M. (2005) Surface-tension-induced buckling of liquid-lined elastic tubes -- a model for pulmonary airway closure. Proceedings of the Royal Society A 461, 1847-1868.(abstract) (pdf preprint)

Heil, M. & Waters, S.L. (2006) Transverse flows in rapidly oscillating, elastic cylindrical shells. Journal of Fluid Mechanics 547, 185-214. (abstract) (pdf preprint)

Baroud, C.N., Tsikata, S. & Heil, M. (2006) The propagation of low-viscosity fingers into fluid-filled, branching networks. Journal of Fluid Mechanics 546, 285-294. (abstract) (pdf preprint)

Hazel, A. L. & Heil, M. (2006) Finite-Reynolds-Number Effects in Steady, Three-Dimensional Airway Reopening. ASME Journal of Biomechanical Engineering 128, 573-578. (abstract) (pdf preprint)

Heil, M. & Hazel, A. L. (2006) oomph-lib -- An Object-Oriented Multi-Physics Finite-Element Library. In: Fluid-Structure Interaction, Editors: M. Schafer und H.-J. Bungartz. Springer (Lecture Notes on Computational Science and Engineering), pp. 19--49. (abstract) (pdf preprint)

Heil, M. & Waters, S.L. (2008) How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. Journal of Fluid Mechanics 601, 199-227. (abstract) (pdf preprint)

Hazel, A.L. & Heil, M. (2008) The influence of gravity on the steady propagation of a semi-infinite bubble into a flexible channel. Physics of Fluids 20, 092109. (abstract) (pdf preprint)

Heil, M., Hazel, A.L. & Boyle, J. (2008) Solvers for large-displacement fluid-structure interaction problems: Segregated vs. monolithic approaches. Computational Mechanics 43, 91-101. (abstract) (pdf preprint)

Heil, M., Hazel, A.L. & Smith, J.A. (2008) The Mechanics of Airway Closure Respiratory Physiology & Neurobiology 163, 214-221. (abstract) (pdf preprint)

Muddle, R.L., Boyle, J.W., Mihajlovic, M.D. & Heil, M. (2009) : The Development of an Object-Oriented Parallel Block Preconditioning Framework Parallel scientific computing and optimization: Advances and applications 27, 37-46 (abstract)

Whittaker, R.J., Waters, S.L., Jensen, O.E., Boyle, J. & Heil, M. (2010) The energetics of flow through a rapidly oscillating tube. Part I: General theory. Journal of Fluid Mechanics 648, 83-121 (abstract) (pdf)

Whittaker, R.J., Heil, M., Boyle, J., Jensen, O.E., & Waters, S.L. (2010) The energetics of flow through a rapidly oscillating tube. Part II: Application to an elliptical tube. Journal of Fluid Mechanics 648, 123-153 (abstract) (pdf)

Heil, M., Boyle, J. (2010) Self-excited oscillations in three-dimensional collapsible tubes: Simulating their onset and large-amplitude oscillations. Journal of Fluid Mechanics 652, 405-426 (abstract) (pdf)

Whittaker, R.J., Heil, M., Jensen, O.E., & Waters, S.L. (2010) A rational derivation of a tube law from shell theory. Quarterly Journal of Mechanics and Applied Mathematics (pdf) (abstract)

Whittaker, R.J., Heil, M., Jensen, O.E., & Waters, S.L. (2010) The onset of high-frequency self-excited oscillations in elastic-walled tubes. Proceedings of the Royal Society A 466, 3635-3657. (abstract) (pdf)

Stewart, P.S., Heil, M., Waters, S.L. & Jensen, O.E. (2010) Sloshing and slamming oscillations in collapsible channel flow. Journal of Fluid Mechanics 662, 288-319. (abstract) (pdf) (Supplementary material (movie))

Heil, M., Hazel, A. L. (2011) Fluid-structure interaction in internal physiological flows. Annual Review of Fluid Mechanics 43, 141-162. (abstract). (pdf)

Whittaker, R.J., Heil, M., & Waters, S.L. (2011) The energetics of flow through a rapidly oscillating tube with slowly varying amplitude. Philosophical Transactions of the Royal Society A 369, 2989-3006. (abstract) (pdf)

Hazel, A. L., Heil, M., Waters, S.L. & Oliver, J.M. (2012) On the liquid lining in fluid-conveying curved tubes. Journal of Fluid Mechanics 705, 213-233. DOI: 10.1017/jfm.2011.346

Pihler-Puzovic, D., Illien, P., Heil, M. & Juel, A. (2012) Suppression of Complex Fingerlike Patterns at the Interface between Air and a Viscous Fluid by Elastic Membranes. Phys. Rev. Lett. 108. DOI: 10.1103/PhysRevLett.108.074502

Heil, M., Kharrat, T., Cotterill, P.A. & Abrahams, I.D. (2012) Quasi-resonances in sound-insulating coatings. Journal of Sound and Vibration 331 4774-4784. DOI: 10.1016/

Muddle, R.L., Mihajlovic, M. & Heil, M. (2012) An efficient preconditioner for monolithically-coupled large-displacement fluid-structure interaction problems with pseudo-solid mesh updates. Journal of Computational Physics 231, 7315-7334. DOI: 10.1016/

Pihler-Puzovic, D., Perillat, R., Russell, M., Juel, A. & Heil, M. (2013) Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells. Journal of Fluid Mechanics 731, 162-183 DOI: 10.1017/jfm.2013.375

Pihler-Puzovic, D., Juel, A. & Heil, M. (2014) The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells. Physics of Fluids 26, 022102. DOI: doi:10.1063/1.4864188.

Shepherd, D., Miles, J., Heil, M., Mihajlovic, M. (2014) Discretisation induced stiffness in micromagnetic simulations. IEEE Trans. Magn., 50(11) 7201304. DOI: 10.1109/TMAG.2014.2325494

Evatt, G.W., Abrahams, I.D., Heil, M., Mayer, C., Kingslake, J., Mitchell, S.L., Fowler, A.C. & Clark, C.D. (2015) Glacial melt under a debris layer: the DADDI model. Journal of Glaciology 61, 825-836. DOI: 10.3189/2015JoG14J235. (pdf)

Cimpeanu, R., Martinsson, A. & Heil, M. (2015) A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation. Journal of Computational Physics 296 329–347. DOI: doi:10.1016/

Heil, M. & Hazel, A.L. (2015) Flow in flexible/collapsible tubes. In: Fluid-Structure Interactions in Low-Reynolds-Number Flows. Eds: Duprat, C. & Stone, H.A. Royal Society of Chemistry, RSC Publishing. pp. 280-311. See details on the publisher's webpage.

Pihler-Puzovic, D., Juel, A., Peng, G., Lister, J. & Heil, M. (2015) Displacement flows under elastic membranes. Part 1: Experiments and direct numerical simulations. Journal of Fluid Mechanics 784 487- 511. DOI: doi:10.1017/jfm.2015.590. (pdf)

Peng, G., Pihler-Puzovic, D., Juel, A., Heil, M. & Lister, J. (2015) Displacement flows under elastic membranes. Part 2: Analysis of interfacial effects. Journal of Fluid Mechanics 784 512- 547. DOI: doi:10.1017/jfm.2015.589. (pdf)

Pestana, J., Muddle, R., Heil, M., Tisseur, F. & Mihajlovic M. (2016) Efficient block preconditioning for a C1 finite element discretisation of the Dirichlet biharmonic problem. SIAM Journal on Scientific Computing 38(1), A325-A345. DOI: 10.1137/15M1014887 . (pdf)

Heil, M. & Bertram, C. (2016). A poroelastic fluid-structure interaction model of syringomyelia. Journal of Fluid Mechanics, 809, 360-389. DOI:

Bertram, C. & Heil, M. (2017). A Poroelastic Fluid/Structure-Interaction Model of Cerebrospinal Fluid Dynamics in the Cord with Syringomyelia and Adjacent Subarachnoid-Space Stenosis. Journal of Biomechanical Engineering, 139(1), 1-10. DOI: 10.1115/1.4034657

Heil, M., Rosso, J., Hazel, A.L., Brons, M. (2017). Topological fluid mechanics of the formation of the Karman-vortex street. Journal of Fluid Mechanics 812 199-221. DOI: (Open Access).

Cisonni, J., Lucey, A.D., Elliott, S.J & Heil, M. (2017) The stability of a flexible cantilever in viscous channel flow Journal of Sound and Vibration 369 186-202. DOI: 10.1016/j.jsv.2017.02.045

Walters, M.C., Heil, M., Whittaker, R.J. (2017) The Effect of Wall Inertia on High-Frequency Instabilities of Flow Through an Elastic-Walled Tube The Quarterly Journal of Mechanics and Applied Mathematics, hbx024. Direct electronic access to article. DOI:

Evatt, G., Mayer, C., Mallinson, A., Abrahams, I., Heil, M., & Nicholson, L. (2017). The secret life of ice sails. Journal of Glaciology, 1-14. doi:10.1017/jog.2017.72

Juel, A., Pihler-Puzovic, D. & Heil, M. (2018) Instabilities in blistering. Annual Review of Fluid Mechanics 50, 691-714. DOI:10.1146/annurev-fluid-122316-045106.

Bergemann, N., Juel, A. & Heil, M. (2018) Viscous drop spreading on a layer of the same fluid: from sinking, wedging and spreading to their long-time evolution. Journal of Fluid Mechanics 843, 1-28. DOI:

Pihler-Puzovic, D., Peng, G., Lister, J. R., Heil, M. & Juel, A. (2018) Viscous fingering in a radial elastic-walled Hele-Shaw cell. Journal of Fluid Mechanics 849, 163-191. DOI:

Bergemann, N., Heil, M., Smith, B. & Juel, A. (2018) From elastic deformation to flow in tempered chocolate. Journal of Rheology 62, 1187. DOI: 10.1122/1.5038253

Pearce, S.P., Heil, M., Jensen, O.E., Jones, G.W., Prokop, A. (2018) Curvature-Sensitive Kinesin Binding Can Explain Microtubule Ring Formation and Reveals Chaotic Dynamics in a Mathematical Model. Bulletin of Mathematical Biology 80, 2003-3222. 10.1007/s11538-018-0505-4

Nielsen, A.R., Heil, M., Andersen, M. & Brons, M. (2019) Bifurcation theory for vortices with application to boundary layer eruption. Journal of Fluid Mechanics 865, 831-849. DOI:

Shepherd, D., Miles, J., Heil, M. & Mihajlovic, M. (2019) An adaptive step implicit midpoint rule for the time integration of Newton's linearisations of non-linear problems with applications in micromagnetics. Journal of Scientific Computing 80, 1058-1082. DOI: 10.1007/s10915-019-00965-8

Vaquero-Stainer, C., Heil, M., Juel, A. & Pihler-Puzovic, D. (2019) Self-similar and disordered front propagation in a radial Hele-Shaw channel with time-varying cell depth. Physical Review Fluids 4, 064002. DOI: 10.1103/PhysRevFluids.4.064002.

Haner, E., Heil, M. & Juel, A. (2020) Deformation and sorting of capsules in a T-junction. Journal of Fluid Mechanics 885, A4. DOI: 10.1017/jfm.2019.979.

Xu, D., Heil,M., Seeboeck, T. & Avila, M. (2020) Resonances in Pulsatile Channel Flow with an Elastic Wall Phys. Rev. Lett. 125, 254501. DOI: 10.1103/PhysRevLett.125.254501

Heil, M. (2021). Think Before You Compute: A Prelude to Computational Fluid Dynamics (Cambridge Texts in Applied Mathematics). By E.J. Hinch. Cambridge University Press, 2020. ISBN: 978-1-108-85529-7. Journal of Fluid Mechanics, 910.

Nogueira Fontana, J.V., Juel, A., Bergemann, N., Heil, M. & Hazel, A. (2021) Modelling finger propagation in elasto-rigid channels. Journal of Fluid Mechanics 916 A27. doi:10.1017/jfm.2021.219

Matharu, P., Hazel, A., & Heil, M. (2021). Spatio-temporal symmetry breaking in the flow past an oscillating cylinder. Journal of Fluid Mechanics, 918, A42. doi:10.1017/jfm.2021.358. See also the accompanying "Focus on Fluids" article: Hourigan, K. (2021). Exotic wakes of an oscillating circular cylinder: How singles pair up. Journal of Fluid Mechanics, 922, F1. doi:10.1017/jfm.2021.492

Garg, A., Bergemann, N., Smith, B., Heil, M., Juel, A. (2021) Fluidisation of yield stress fluids under vibration Journal of Non-Newtonian Fluid Mechanics 294, 104595,


Current courses

Previous courses

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Currently, my main administrative job is to act as the School's Postgraduate Director. In the past I used to be responsible for the following administrative jobs:

Applied & NA Seminars

I used to be in charge of organising the Applied & NA Seminars in our department. Now Rich Hewitt has taken over.

Reviewing/Editorial responsibilities

I am an Associate Editor for the

I am a member of the EPSRC College and regularly act as a referee for the following journals


Meet the family!

[very old pictures]

How to contact me:

Snail Mail: School of Mathematics
University of Manchester
Oxford Road
Manchester M13 9PL

Phone: +44 (0)161 275 5808
Fax: +44 (0)161 275 5819
WWW: (this page)

Page last modified: September 7, 2018

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