Updated on 2022/09/25


Faculty of Science and Engineering, Global Center for Science and Engineering
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I am an applied mathematician undertaking research into nonlinear partial differential equations, employing a combination of analytical and numerical techniques in their study. I am particularly interested in studying free boundary problems arising from thin film flows, including investigations of rupture phenomena and the effects of surface topography and driving forces upon the motion of the liquid. Such problems often yield evolution equations for the interfacial film thickness in the form of high-order degenerate parabolic equations combined with additional convective effects. In certain cases, the solutions to these equations may contain discontinuities; the selection criterion and stability of these ‘shocks’ is also both of mathematical and physical interest.

Concurrent Post

  • Faculty of Political Science and Economics   School of Political Science and Economics

Research Institute

  • 2020

    理工学術院総合研究所   兼任研究員


  • 博士


Research Areas

  • Applied mathematics and statistics


  • Pressure-dipole solutions of the thin-film equation

    Mark Bowen, T. P. Witelski

    European Journal of Applied Mathematics    2018  [Refereed]


  • On self-similar thermal rupture of thin liquid sheets

    M. Bowen, B. S. Tilley

    PHYSICS OF FLUIDS   25 ( 10 ) 102105  2013.10  [Refereed]

     View Summary

    We consider the dynamics of a symmetrically heated thin incompressible viscous fluid sheet. We take surface tension to be temperature dependent and consequently the streamwise momentum equation includes the effects of thermocapillarity, inertia, viscous stresses, and capillarity. Energy transport to the surrounding environment is also included. We use a long-wave analysis to derive a single nondimensional system which, with appropriate choices of Reynolds number, recovers two previously studied cases. In both cases, we find conditions under which sufficiently large-amplitude initial temperature profiles induce film rupture in finite time, notably without the inclusion of disjoining pressures from van der Waals effects. When the Reynolds number is large, the similarity solution is governed by a balance of inertia and capillarity near the rupture location, analogous to the isothermal case. When the Reynolds number is small, the thermocapillary transients induce the same similarity solution over intermediate times that is found for the drainage of lamellae in foams. For O(1) Reynolds numbers, the dynamics are governed initially by the large Reynolds number evolution, and then a transition over several orders of magnitude in the sheet thickness needs to take place before the small Reynolds number similarity solution is observed. (C) 2013 AIP Publishing LLC.


  • Dynamics of a viscous thread on a non-planar substrate

    Mark Bowen, John R. King

    Journal of Engineering Mathematics   80 ( 1 ) 39 - 62  2013  [Refereed]

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    We derive a general reduced model for the flow of a slender thread of viscous fluid on a grooved substrate. Specific choices of the substrate topography allow further analytic progress to be made, and we subsequently focus on a convection-diffusion equation governing the evolution of viscous liquid in a wedge geometry. The model equation that arises also appears in the context of foam drainage, and we take the opportunity to review and compare the results from both applications. After summarising the constant mass results, we introduce a time-dependent fluid influx at one end of the wedge. The analytical results are supported by numerical computations. © 2012 Springer Science+Business Media B.V.


  • Thermally induced van der Waals rupture of thin viscous fluid sheets

    Mark Bowen, B. S. Tilley

    PHYSICS OF FLUIDS   24 ( 3 )  2012.03  [Refereed]

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    We consider the dynamics of a thin symmetric fluid sheet subject to an initial temperature profile, where inertia, viscous stresses, disjoining pressures, capillarity, and thermocapillarity are important. We apply a long-wave analysis in the limit where deviations from the mean sheet velocity are small, but thermocapillary stresses and heat transfer from the sheet to the environment are significant and find a coupled system of partial differential equations that describe the sheet thickness, the mean sheet velocity, and the mean sheet temperature. From a linear stability analysis, we find that a stable thermal mode couples the velocity to the interfacial dynamics. This coupling can be utilized to delay the onset of rupture or to promote an earlier rupture event. In particular, rupture can be induced thermally even in cases when the heat transfer to the surrounding environment is significant, provided that the initial phase shift between the initial velocity and temperature disturbances is close to phi = pi/2. These effects suggest a strategy that uses phase modulation in the initial temperature perturbation related to the initial velocity perturbation that assigns priority of the rupture events at particular sites over several spatial periods. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3693700]


  • The linear limit of the dipole problem for the thin film equation

    Mark Bowen, Thomas P. Witelski

    SIAM JOURNAL ON APPLIED MATHEMATICS   66 ( 5 ) 1727 - 1748  2006  [Refereed]

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    We investigate self-similar solutions of the dipole problem for the one-dimensional thin film equation on the half-line {x >= 0}. We study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem. The similarity solutions are generally of the second kind, given by the solution of a nonlinear eigenvalue problem, although there are some notable cases where first-kind solutions also arise. We examine the conserved quantities connected to these first-kind solutions. Difficulties associated with the lack of a maximum principle and the non-self-adjointness of the fundamental linear problem are also considered. Seeking similarity solutions that include sign changes yields a surprisingly rich set of ( coexisting) stable solutions for the intermediate asymptotics of this problem. Our results include analysis of limiting cases and comparisons with numerical computations.


  • Thermocapillary control of rupture in thin viscous fluid sheets

    BS Tilley, M Bowen

    JOURNAL OF FLUID MECHANICS   541   399 - 408  2005.10  [Refereed]

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    We consider the evolution of a thin Viscous fluid sheet subject to thermocapillary effects. Using a lubrication approximation we find, for symmetric interfacial deflections, coupled evolution equations for the interfacial profile, the streamwise component of the fluid velocity and the temperature variation along the surface. Initial temperature profiles change the initial flow field through Marangoni-induced shear stresses. These changes then lead to preferred conditions For rupture prescribed by the initial temperature distribution. We show that the time to rupture may be minimized by varying the phase difference between the initial velocity profile and the initial temperature profile. For sufficiently large temperature differences, the phase difference between the initial velocity and temperature profiles determines the rupture location.

  • Nonlinear dynamics of two-dimensional undercompressive shocks

    M Bowen, J Sur, AL Bertozzi, RP Behringer

    PHYSICA D-NONLINEAR PHENOMENA   209 ( 1-4 ) 36 - 48  2005.09  [Refereed]

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    We consider the problem of a thin film driven by a thermal gradient with an opposing gravitational force. Under appropriate conditions, an advancing film front develops a leading undercompressive shock followed by a trailing compressive shock. Here, we investigate the nonlinear dynamics of these shock structures that describe a surprisingly stable advancing front. We compare two-dimensional simulations with linear stability theory, shock theory, and experimental results. The theory/experiment considers the propagation of information through the undercompressive shock towards the trailing compressive shock. We show that a local perturbation interacting with the undercompressive shock leads to nonlocal effects at the compressive shock. (c) 2005 Elsevier B.V. All rights reserved.


  • The self-similar solution for draining in the thin film equation

    JB Van den Berg, M Bowen, King, JR, MMA El-Sheikh

    EUROPEAN JOURNAL OF APPLIED MATHEMATICS   15 ( 3 ) 329 - 346  2004.06  [Refereed]

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    We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become untenable. Numerical and power-series solutions are also presented.


  • ADI schemes for higher-order nonlinear diffusion equations

    TP Witelski, M Bowen

    APPLIED NUMERICAL MATHEMATICS   45 ( 2-3 ) 331 - 351  2003.05  [Refereed]

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    Alternating Direction Implicit (ADI) schemes are constructed for the solution of two-dimensional higher-order linear and nonlinear diffusion equations, particularly including the fourth-order thin film equation for surface tension driven fluid flows. First and second-order accurate schemes are derived via approximate factorization of the evolution equations. This approach is combined with iterative methods to solve nonlinear problems. Problems in the fluid dynamics of thin films are solved to demonstrate the effectiveness of the ADI schemes. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.


  • Thin film dynamics: theory and applications

    AL Bertozzi, M Bowen


     View Summary

    This paper is based on a series of four lectures, by the first author, on thin films and moving contact lines. Section 1 presents an overview of the moving contact line problem and introduces the lubrication approximation. Section 2 summarizes results for positivity preserving schemes. Section 3 discusses the problem of films driven by thermal gradients with an opposing gravitational force. Such systems yield complex dynamics featuring undercompressive shocks. We conclude in Section 4 with a discussion of dewetting films.

  • Anomalous exponents and dipole solutions for the thin film equation

    M Bowen, J Hulshof, King, JR

    SIAM JOURNAL ON APPLIED MATHEMATICS   62 ( 1 ) 149 - 179  2001.10  [Refereed]

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    We investigate similarity solutions of the thin lm equation. In particular we look at solutions on the half-line x greater than or equal to 0 with compact support and zero contact angle boundary conditions in x = 0. uch dipole solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind. Using a combination of phase space analysis and numerical simulations, we numerically construct trajectories representing these solutions, at the same time obtaining broader insight into the nature of the four-dimensional phase space. Additional asymptotic analysis provides further information concerning the evolution to self-similarity.

  • Moving boundary problems and non-uniqueness for the thin film equation

    King, JR, M Bowen

    EUROPEAN JOURNAL OF APPLIED MATHEMATICS   12 ( 3 ) 321 - 356  2001.06  [Refereed]

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    A variety of mass preserving moving boundary problems for the thin film equation, u(t) = -(u(n)u(xxx))(x), are derived (by formal asymptotics) from a number of regularisations, the case in which the substrate is covered by a very thin pre-wetting film being discussed in most detail. Some of the properties of the solutions selected in this fashion are described, and the full range of possible mass preserving non-negative solutions is outlined.

  • Asymptotic behaviour of the thin film equation in bounded domains

    M Bowen, King, JR

    EUROPEAN JOURNAL OF APPLIED MATHEMATICS   12 ( 2 ) 135 - 157  2001.04  [Refereed]

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    We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.

  • Intermediate asymptotics of the porous medium equation with sign changes

    J Hulshof, J R King, Mark Bowen

    Advances in Differential Equations   6 ( 9 ) 1115 - 1152  2001  [Refereed]

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Books and Other Publications

  • Methods of Mathematical Modelling: Continuous Systems and Differential Equations

    Witelski, Thomas, Bowen, Mark( Part: Joint author)

    Springer  2015.09

Research Projects

  • Self-similar behaviour in thin film flow

    Project Year :


     View Summary

    Working with international collaborators, we have made substantial progress into identifying the role of special (self-similar) solutions to equations arising in the study of surface tension driven thin-film flow. This has led to a number of publications in international journals with additional papers to be submitted in the very near future. The research has also been presented at several international meetings

Specific Research

  • Dynamics of monolayer thin-film flows


     View Summary

    We have been analytically and computationally investigating the motion of a monolayer thin film (tether) held between two quasi-steady masses.  The monolayer tether is assumed to be stretched by an externally imposed symmetric stagnation point flow and interesting dynamics arise due to the competition between surface tension and convective flow.  In particular, we have been interested in whether the tether breaks in finite time and how such rupture occurs.Mathematical modelling of this phenomena leads to a higher-order non-autonomous nonlinear diffusion-convection equation containing a parameter that takes different values depending on the thickness of the monolayer film.We have analysed the behaviour of the evolution equation for general values of the parameter and have identified that for large enough values of the parameter rupture of the tether does not occur in finite time.We are now writing this research up for publication in the near future.

  • Drainage problems for the multidimensional thin film equation

    2018   L. Smolka, T. P. Witelski

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    1) Working with Professor T P Witelski (Duke University, USA), we have been studying out-diffusion solutions of the so-called thin film equation (a fourth order parabolic partial differential equation) on a finite multi-dimensional domain; this extends our recent previous work on the one-dimensional problem.While considering this problem, we decided first to make a preliminary study of the related problem for the lower (second) order porous medium equation.  In this context, we have constructed analytically self-similar solutions that act as large time attractors for solutions defined on sectorial [quarter, half-plane and three-quarter-plane] domains.  We have confirmed these results using numerical simulations.2) While working on this project, I established a new working relationship with Professor L. Smolka (Bucknell University, USA) looking at how thin films evolve in a periodic domain (corresponding physically to the external surface of a cylinder) under the combined effects  of gravity (drainage) and thermal stresses (leading to a non-convex convective flux of fluid).  The interaction of convective effects and surface tension (fourth order parabolic terms) yields solutions containing non-classical shock dynamics, such as undercompressive-compressive shock pairs and undercompressive shocks-rarefaction fans.  We are currently writing up the results of this research for publication in the near future.

  • Dynamics of constrained thin films and jets

    2017   T. P. Witelski

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    Working with Professor T. P. Witekski (Duke University, USA), we mathematically investigated the dynamics of a thin liquid film draining from the edge of a (constrained) domain.  Such problems frequently arise in industrial coating processes where domains are of finite extent.This work has now been published in EJAM (European Journal of Applied Mathematics):Pressure-dipole solutions of the thin-film equationM. BOWEN  and T. P. WITELSKI  European Journal of Applied Mathematicshttps://doi.org/10.1017/S095679251800013XPublished online: 02 April 2018

  • Dynamics of multi-dimensional thin-film equations

    2016   T. P. Witelski

     View Summary

    We have considered self-similar sign-changing solutions to the thin-film equation on a semi-infinite domain with zero-pressure-type boundary conditions imposed at the fixed boundary.  In particular, we have identified classes of both first- and second-kind compactly supported self-similar solutions and have explained how these solutions interact as parameters vary.  

  • Investigation of multi-dimensional thin-film equations

    2015   Thomas Witelski

     View Summary

    Working with an international collaborator, I have investigated the dynamics of a thin liquid film draining off of one edge of a flat substrate. Such a scenario frequently arises in the natural sciences, engineering and industry. We have also extended these results to the (non-physical) case where the solution (corresponding to film height) can change sign. The analytical results are supported by detailed numerical calculations.

  • Thermo-capillary control of viscous sheets and jets

    2014   Burt Tilley

     View Summary

    An understanding of the dynamics of liquid jets is important in many applications such as in inkjet printing, for example. Working with an international collaborator, we have been analytically and computationally investigating how externally applied temperature gradients can be used to better control (from the point-of-view of application to inkjet printers) both the printing speed and resolution.We have developed advanced computational methods that are capable of following the jet dynamics over many different length and time scales; we are also considering the application of parallelisation to the computations in order to improve the time taken for the simulations to run.The computations support additional analytical results, allowing us to understand, in particular, the approach to rupture of the jets (separation into droplets) and also how to control the initial instability of a uniform jet that leads to rupture.

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