### Fluid Mechanics

The nonlinear dynamics of fluid flow is key to phenomena in fields as diverse as astrophysics, biology, engineering, physics and the geosciences. Research at the UKAAM focuses on practical fluids problems in many of these applications, but also explores fundamental theory of fluid mechanics itself. Specific directions of research include the instabilities encountered in shear flows and vortices, the dynamics of complex fluids, flow problems in industrial processes and the environment, and glacier mechanics. We are always interested to hear from potential graduate students and postdoctoral fellows. We recruit from many different backgrounds, including mathematics, physics, engineering and the geosciences. We often supervise undergraduate thesis projects and take on summer research undergraduate students.

Research facilities of the UKAAM Fluid Mechanics Group include the Laboratory for Complex and Non-Newtonian Fluid Flow (Fluids Lab), which is operated by the UKAAM faculty in the Mathematics Department. The Lab contains space, tools and equipment for experiments, including several rheometers and other equipment for studying fundamental fluid properties. Recent experiments include skipping and sloshing (the dynamics of skimming stones and reservoirs with movable dams), as well as the pinch-off of pendant drops and liquid bridges of complex fluids.

Complex fluids have microscopic structure that influences the macroscopic flow behaviour. For example, suspended polymers can unravel and intertwine as fluid flows, endowing the material with an effective elasticity; such viscoelastic fluids climb rods rotating in them and extend into strong, fine filaments. Other fluids have networks of interacting particles that build a microstructure capable of holding the fluid up against gravity and other stress; such viscoplastic fluids include mud, hair gel and tomato ketchup.

### Core Faculty

The Fluid Mechanics Group is composed of several core UKAAM faculty who are actively involved in the UKAAM activities and supervise UKAAM students or postdoctoral fellows. Prospective students interested in research in Fluids Mechanics in the UKAAM are encouraged to contact one or more of the coe faculty as potential supervisors and let them know of their interests. The Group’s research areas incude complex fluids, atmospheric and ocean dynamics, geophysics, and engineering fluids.

Neil Balmforth | Neil is a Professor in the Department of Mathematics and in Earth and Ocean Sciences. His research interests include geophysical and astrophysical fluid dynamics and complex fluid flow. He has co-organised and directed a variety of programmes in Geophysical Fluid Dynamics, including the summer school at the Woods Hole Oceanographic Institution. Neil was the Director of the UKAAM from 2008 to 2013. |

Gwynn Elfring | Gwynn is an assistant professor in Mechanical Engineering. His research involves using the methods of applied mathematics, typically asymptotic analysis or numerical methods, to solve problems in science and engineering, often in collaboration with or inspired by experimentalists. His current research interests include: Theoretical Fluid Mechanics, Complex Fluids, Cell Biomechanics, Capillary Phenomena, Applied Mathematics. |

Jimmy Feng | Jimmy is a Professor of Mathematics and of Chemical and Biological Engineering. He is interested in the dynamics and applications of complex fluids, such as polymers, liquid crystals, colloids, emulsions, foams and various biological fluids. Jimmy’s work is highly interdisciplinary, spanning over applied mathematics, soft-matter physics, chemical engineering and biomedical engineering. |

Ian Frigaard | Ian is a Professor of Mathematics and Mechanical Engineering. His research focusses on the mechanics of non-Newtonian fluids, particularly viscoplastic fluids, and in understanding industrial processes that exploit the non-Newtonian fluid properties. Examples of practical applications include oilfield cementing, well control, transport in pipelines, spray forming, etc. Ian’s research combines mathematical, experimental and computational approaches. |

Greg Lawrence | Greg is a Professor of Civil Engineering. His main research area is environmental fluid mechanics, with the primary focus on the impact the fluid flow has on inland and coastal waters. He is also interested in hydraulics, hydrodynamic stability and mixing, physical limnology, and water quality management. |

Mark Martinez | Mark is a Professor of Chemical and Biological Engineering, interested in multiphase flows and computational fluid dynamics with applications to industrial problems. His main research focus is on investigating the papermaking fibre suspensions, which often exhibit complex behaviour not seen in ordinary fluids such as water. Mark actively collaborates with UBC researchers in Mechanical Engineering, Mathematics and in TRIUMF. |

Christian Schoof | Christian is a Professor in the Department of Earth and Ocean Sciences. He is mainly interested in glaciology and in ice-sheet dynamics, which he analyses using various mathematical tools, including PDEs, free boundary problems, applied complex analysis, nonlinear dynamics, perturbation methods, etc. Christian also conducts field work on the site in St. Elias Mountains, collaborating with Gwenn Flowers from Simon Fraser University. |

Anthony Wachs | Anthony is an Associate Professor of Mathematics and of Chemical and Biological Engineering. His research focusses on multiphase flows, non-Newtonian fluid mechanics and computational methods to solve fluid mechanics PDE problems on large supercomputers. His primary interest is on the modelling and parallel computing of particle-laden flows with heat and mass transfer. Examples of application include sediment transport in rivers, fluidized bed in biomass gasification and blood flow in the human body. Anthony’s group develops in-house parallel codes both on fixed and adaptive grids and is a big user of Compute Canada and UBC computing resources. |

### Recommended Courses

Research in fluid mechanics prompted the development of many classical and modern techniques of Applied Mathematics. Matched asymptotic expansions and spectral methods for partial differential equations were both developed with fluid problems in mind, and the theory of solitons and the inverse scattering transform has its roots in the study of water waves. The broad implications for researchers in fluids are that a solid grounding in the tools of applied mathematics are highly recommended, if not essential.

**Preliminary and Foundational Courses**

MATH 400: Partial Differential Equations

MATH 401: Green Functions and Variational Methods

MATH 450/550: Perturbation Methods

MATH 521: Numerical Analysis of PDEs

MATH 552: Dynamical Systems Theory

CHBE 557: Fluid Mechanics

MATH 607E: Numerical Methods for Differential Equations

**Fluids Courses**

EOSC 512: Geophysical Fluid Dynamics

MATH 519: Hydrodynamic Stability

MATH 557: Linear and Nonlinear Waves

MATH 606: Mathematical Modelling of Complex Fluids

**Further Options**

MECH 510: Computational Methods in Transport Phenomena

MATH 551: Asymptotic Analysis for PDEs

MATH 554: Symmetries and Differential Equations

MATH 556: Industrial Mathematical Modelling