Name | ## BOWEN, Mark | Official Title | Associate Professor | |

Affiliation | （Global Center for Science and Engineering） | |||

## Profile

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.

## Contact Information

- Grant-in-aids for Scientific Researcher Number
- 30534341

### URL

## Sub-affiliation

### Sub-affiliation

Faculty of Political Science and Economics（School of Political Science and Economics）

### Affiliated Institutes

*理工学術院総合研究所（理工学研究所）*

兼任研究員 2018-

## Paper

*Pressure-dipole solutions of the thin-film equation*

Mark Bowen, T. P. Witelski

European Journal of Applied Mathematics Peer Review Yes 2018-

*On self-similar thermal rupture of thin liquid sheets*

Mark Bowen, B. Tilley

Physics of Fluids Peer Review Yes 25(10) p.1021052013-

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

Mark Bowen, J. R. King

Journal of Engineering Mathematics Peer Review Yes 80(1) p.39 - 622013-

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

Mark Bowen, B. Tilley

Physics of Fluids Peer Review Yes 24(3) p.0321062012-

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

Mark Bowen, T. P. Witelski

SIAM Journal of Applied Mathematics Peer Review Yes 66(5) p.1727 - 17482006-

*Thermocapillary control of rupture in thin viscous fluid sheets*

B. Tilley, Mark Bowen

Journal of Fluid Mechanics Peer Review Yes 541p.399 - 4082005-

*Nonlinear dynamics of two-dimensional undercompressive shocks*

Mark Bowen, J. Sur, A. L. Bertozzi, R. P. Behringer

Physica D: Nonlinear Phenomena Peer Review Yes 209p.36 - 482005-

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

JB van den Berg, Mark Bowen, J R King, M M A El-Sheikh

European Journal of Applied Mathematics Peer Review Yes 15(3) p.329 - 3462004-

*ADI schemes for higher-order nonlinear diffusion equations*

TP Witelski, Mark Bowen

Applied Numerical Mathematics Peer Review Yes 45p.331 - 3512003-

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

J R King, Mark Bowen

European Journal of Applied Mathematics Peer Review Yes 12(3) p.321 - 3562001-

*Asymptotic behaviour of the thin film equation in bounded domains*

Mark Bowen, J R King

European Journal of Applied Mathematics Peer Review Yes 12(2) p.135 - 1572001-

*Intermediate asymptotics of the porous medium equation with sign changes*

J Hulshof, J R King and Mark Bowen

Advances in Differential Equations Peer Review Yes 6(9) p.1115 - 11522001-

*Anomalous exponents and dipole solutions for the thin film equation*

J Hulshof, J R King and Mark Bowen

SIAM Journal on Applied Mathematics Peer Review Yes 62(1) p.149 - 1792001-

*Thin film dynamics: theory and applications*

AL Bertozzi, Mark Bowen

Modern Methods in Scientific Computing and Applications Peer Review Yes p.31 - 792002-

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

Bowen, Mark;Tilley, B. S.

PHYSICS OF FLUIDS 24(3) 2012-2012

ISSN：1070-6631

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

Bowen, Mark;King, John R.

JOURNAL OF ENGINEERING MATHEMATICS 80(1) p.39 - 622013-2013

ISSN：0022-0833

*Methods of Mathematical Modelling: Continuous Systems and Differential Equations*

Witelski, Thomas; Bowen, Mark

Methods of Mathematical Modelling: Continuous Systems and Differential Equations p.1 - 3052015/09-2015/09

Outline：© Springer International Publishing Switzerland 2015. All rights are reserved. This book presents mathematical modelling and the integrated process of formulating sets of equations to describe real-world problems. It describes methods for obtaining solutions of challenging differential equations stemming from problems in areas such as chemical reactions, population dynamics, mechanical systems, and fluid mechanics. Chapters 1 to 4 cover essential topics in ordinary differential equations, transport equations and the calculus of variations that are important for formulating models. Chapters 5 to 11 then develop more advanced techniques including similarity solutions, matched asymptotic expansions, multiple scale analysis, long-wave models, and fast/slow dynamical systems. Methods of Mathematical Modelling will be useful for advanced undergraduate or beginning graduate students in applied mathematics, engineering and other applied sciences.

## Books And Publication

*Methods of Mathematical Modelling*

(Joint authorship)

Springer2015-

## Research Grants & Projects

### Grant-in-aids for Scientific Research Adoption Situation

Research Classification：

*Self-similar behaviour in thin film flow*

2012/-0-2014/-0

Allocation Class：￥910000

## On-campus Research System

### Special Research Project

*Drainage problems for the multidimensional thin film equation*

2018Collaborator：L. Smolka, T. P. Witelski

Research Results Outline：1) Working with Professor T P Witelski (Duke University, USA), we have been1) Working with Professor T P Witelski (Duke University, USA), we have been studying out-diffusion solutions of the so-calle...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.

*Investigation of multi-dimensional thin-film equations*

2015Collaborator：Thomas Witelski

Research Results Outline：Working with an international collaborator, I have investigated the dynamicWorking with an international collaborator, I have investigated the dynamics of a thin liquid film draining off of one edge ...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.

*Dynamics of constrained thin films and jets*

2017Collaborator：T. P. Witelski

Research Results Outline：Working with Professor T. P. Witekski (Duke University, USA), we mathematicWorking with Professor T. P. Witekski (Duke University, USA), we mathematically investigated the dynamics of a thin liquid f...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

*Thermo-capillary control of viscous sheets and jets*

2014Collaborator：Burt Tilley

Research Results Outline：An understanding of the dynamics of liquid jets is important in many applicAn understanding of the dynamics of liquid jets is important in many applications such as in inkjet printing, for example. ...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.

*Dynamics of multi-dimensional thin-film equations*

2016Collaborator：T. P. Witelski

Research Results Outline：We have considered self-similar sign-changing solutions to the thin-film eqWe have considered self-similar sign-changing solutions to the thin-film equation on a semi-infinite domain with zero-pressu...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.

## Lecture Course

Course Title | School | Year | Term |
---|---|---|---|

Calculus A | School of Fundamental Science and Engineering | 2019 | fall semester |

Calculus A | School of Creative Science and Engineering | 2019 | fall semester |

Calculus A | School of Advanced Science and Engineering | 2019 | fall semester |

Calculus B | School of Fundamental Science and Engineering | 2019 | spring semester |

Calculus B | School of Creative Science and Engineering | 2019 | spring semester |

Calculus B | School of Advanced Science and Engineering | 2019 | spring semester |

Ordinary Differential Equations (2) | School of Fundamental Science and Engineering | 2019 | fall semester |

Ordinary Differential Equations (2) | School of Creative Science and Engineering | 2019 | fall semester |

Ordinary Differential Equations (2) | School of Advanced Science and Engineering | 2019 | fall semester |

Partial Differential Equations | School of Fundamental Science and Engineering | 2019 | spring semester |

Partial Differential Equations | School of Creative Science and Engineering | 2019 | spring semester |

Partial Differential Equations | School of Advanced Science and Engineering | 2019 | spring semester |

Research Project B | School of Fundamental Science and Engineering | 2019 | spring semester |

Research Project B [S Grade] | School of Fundamental Science and Engineering | 2019 | spring semester |

Research Project C | School of Fundamental Science and Engineering | 2019 | fall semester |

Research Project C [S Grade] | School of Fundamental Science and Engineering | 2019 | fall semester |

Research Project A | School of Fundamental Science and Engineering | 2019 | fall semester |

Research Project D | School of Fundamental Science and Engineering | 2019 | spring semester |

Calculus A (1) | School of Fundamental Science and Engineering | 2019 | fall quarter |

Calculus A (1) | School of Creative Science and Engineering | 2019 | fall quarter |

Calculus A (1) | School of Advanced Science and Engineering | 2019 | fall quarter |

Calculus B (1) | School of Fundamental Science and Engineering | 2019 | winter quarter |

Calculus B (1) | School of Creative Science and Engineering | 2019 | winter quarter |

Calculus B (1) | School of Advanced Science and Engineering | 2019 | winter quarter |

Calculus C (1) | School of Fundamental Science and Engineering | 2019 | spring quarter |

Calculus C (1) | School of Creative Science and Engineering | 2019 | spring quarter |

Calculus C (1) | School of Advanced Science and Engineering | 2019 | spring quarter |

Master's Thesis (Department of Pure and Applied Mathematics) | Graduate School of Fundamental Science and Engineering | 2019 | full year |

Research on Nonlinear Systems (Prof. Bowen) | Graduate School of Fundamental Science and Engineering | 2019 | full year |

Seminar on Nonlinear Systems A (Prof. Bowen) | Graduate School of Fundamental Science and Engineering | 2019 | spring semester |

Seminar on Nonlinear Systems B (Prof. Bowen) | Graduate School of Fundamental Science and Engineering | 2019 | fall semester |

Seminar on Nonlinear Systems C (Prof. Bowen) | Graduate School of Fundamental Science and Engineering | 2019 | spring semester |

Seminar on Nonlinear Systems D (Prof. Bowen) | Graduate School of Fundamental Science and Engineering | 2019 | fall semester |