# The Brooke Benjamin Lecture in Fluid Dynamics

**The Seventh Brooke Benjamin Lecture - Wednesday 27 November 2013**

** ****Professor John Toland, Isaac Newton Institute, University of Cambridge**

**The fascination of what's difficult:
Mathematical aspects of classical water wave theory from the past 20 years**

Brooke Benjamin believed that mathematical proofs and data from carefully designed and executed experiments were two pillars upon which scientific progress rests. He made distinguished contributions to both.

Experimental observations about steady water waves have famously challenged mathematicians since Stokes and Scott-Russell in the 19th century and modern methods of global analysis are inadequate to answer the simplest of questions raised by careful numerical experiments in the 20th century.

This lecture concerns mathematical advances that have emerged since Brooke's untimely death in 1995 and elucidates important challenges that remain to the present day.

## About Thomas Brooke Benjamin (1929-1995)

Professor Thomas Brooke Benjamin, born 15 April 1929, developed an aptitude for science at an early age through his avid interest in constructing radio sets. This continued throughout his childhood until his admission to Liverpool University to study for a BEng in 1947.

After graduating from Liverpool with First Class Honours in 1950, Benjamin was granted a Rotary foundation fellowship to study electronics at Yale University, where he gained an MEng in 1952. Following a period of study in the United States, he entered King's College, Cambridge in 1952 where he devised ingenious and accurate new experiments to study cavitation in fluid flow. In 1955, he was elected a Fellow of King's College, and in 1958 was appointed Assistant Director of research in both the engineering department and in the department of applied mathematics and theoretical physics (DAMTP). It was here that Benjamin developed further research into fluid dynamics by converting the departmental basement into a laboratory dedicated to research solely in this field. Whilst at Cambridge, Benjamin made several key contributions to the study of fluid dynamics, most notably in the area of instability in varying kinds of fluid motion. Benjamin also prompted further study of the French school of nonlinear analysis and its unquestionable influence upon certain fluid instabilities.

In 1970, Benjamin was appointed to a professorship at the University of Essex, where he set up the Fluid Mechanics Research Institute. The Institute was incredibly successful with over ninety research reports published during Benjamin's tenure.

After a successful period at Essex, Benjamin was appointed to the Sedleian chair of natural philosophy at Oxford, with a fellowship at Queen's College in 1978, which he held until his death in 1995. He was also appointed as adjunct professor at Pennsylvania State University which he made frequent visits to.

Throughout his career Benjamin was awarded a vast array of honours including the Lewis F. Moody award from the American Society of Civil Engineers in 1966, election to the Royal Society of London (1966), the William Hopkins prize of the Cambridge Philosophical Society in 1969 and in 1993 he was elected a foreign member of the French Academy of Sciences. He received honorary degrees from the universities of Liverpool, Bath, and Brunel.

Perhaps the most eloquent tribute to Brooke Benjamin is given by M. S. Longuet-Higgins in the Oxford Dictionary of National Biography:

Benjamin, known to his friends as Brooke, was tall, modest, and always polite. Among his colleagues he inspired great admiration, affection, and loyalty. With a quiet sense of humour, he was inwardly sensitive and imaginative? He was justifiably proud that despite taking a leading part in many activities he made no known enemies.

Benjamin died on the 16 August 1995 and is survived by his wife, Natalia, their daughter, Victoria, and the three children of his first marriage.

# Past Lectures

**Sixth Lecture - ****17 October 2012**

**Professor Yves Couder, Laboratoire Matière et Systèmes Complexes, Université Paris Diderot**

**A fluid dynamical wave-particle duality**

Wave-particle duality is a quantum behaviour usually assumed to have no possible counterpart in classical physics. We revisited this question when we found that a droplet bouncing on a vibrated bath could become self-propelled by its coupling to the surface waves it excites. A dynamical wave-particle association is thus formed. Through several experiments we addressed the same general question. How can a localized and discrete droplet have a common dynamics with a continuous and spatially extended wave? Surprisingly several quantum-like behaviors emerge; a form of uncertainty and a form of quantization are observed. I will show that both properties are related to the "path memory" contained in the wave field. The relation of this experiment with the pilot-wave models proposed by de Broglie and by Bohm for quantum mechanics will be discussed.

**Fifth Lecture - 16 November 2011**

**Professor Vladimir Zakharov, Department of Mathematics, University of Arizona**

**Theory of Wind-Driven Sea**

The self-consistent analytic theory of the wind-driven sea can be developed due to the presence of small parameter, the ratio of atmospheric and water densities. Because of low value of this parameter the sea is "weakly nonlinear" and the average steepness of sea surface is also relatively small. Nevertheless, the weakly nonlinear four-wave resonant interaction is the dominating process in the energy balance. The wind-driven sea can be described statistically in terms of the Hasselmann kinetic equation. This equation has a rich family of Kolmogorov-type solutions perfectly describing "rear faces" of wave spectra right behind the spectral peak. More short waves are described by steeper Phillips spectrum formed by an ensemble of microbreakings. From the practical view-point the most important question is the spatial and temporal evolution of spectral peaks governed by self-similar solutions of the Hasselmann equation. This analytic theory is supported by numerous experimental data and computer simulations.

### Fourth Lecture - 19 May 2010

**Tom Mullin, Manchester Centre for Nonlinear Dynamics, University of Manchester**

The Enigma of the Transition to Turbulence in a Pipe

The Enigma of the Transition to Turbulence in a Pipe

The puzzle of why fluid motion along a pipe is observed to become
turbulent as the flow rate is increased remains the outstanding
challenge of hydrodynamic stability theory, despite more than a century
of research. The issue is both of deep scientific and engineering
interest since most pipe flows are turbulent in practice, even at
modest flow rates. All theoretical work indicates that the flow is
linearly stable ie infinitesimal disturbances decay as they propagate
along the pipe and the flow will remain laminar. Finite amplitude
perturbations are responsible for triggering turbulence and these
become more important as the non-dimensional flow rate, the Reynolds
number *Re*, increases. It is now established that there are several scalings with *Re*
of the amplitude of the disturbance required to cause transition. Each
of these provides insights into the origins of the turbulent motion and
links are beginning to be made with recent discoveries of new solutions
to the Navier Stokes equations.

### Third Lecture - 12th June 2009

#### Vladimir Sverak, School of Mathematics, University of Minnesota, Minneapolis

Mathematical Aspects of Navier-Stokes equations

The flow of fluids is usually modelled by the Navier-Stokes equations. The mathematical theory of these equations remains incomplete, and the problem of reliably calculating their solutions is often out of reach of present-day computers. This lecture will explain the source of the mathematical difficulties, review some classical results, and mention some recent advances. The focus will be on the incompressible case, for example describing the flow of water.

### Second Lecture - 22nd May 2008

#### Howard Stone, Harvard University

#### Manipulating thin-film flows: From patterned substrates to evaporating systems

We describe two variants of thin film flows, one involving wetting and the other involving evaporation. First, we describe the spreading of mostly wetting liquid droplets on surfaces decorated with assemblies of micron-size cylindrical posts arranged in regular arrays. We obtain a variety of deterministic final shapes of the spreading droplets, including octagons, squares, hexagons and circles. Dynamic considerations provide a "shape" diagram and suggest rules for control.

It is then shown how these ideas can be used to explore (and control) splashing and to create polygonal hydraulic jumps.

Second, we consider evaporation of volatile liquid drops. Using experiments and theory we show how the sense of the internal circulation depends on the ratio of the liquid and substrate conductivities. The internal motions control the deposition patterns and so may impact various printing processes. These ideas are then applied to colloid deposition in porous media.

### First Lecture - 17 May 2007

#### Jerry Bona, University of Illinois, Chicago

Water Wave Theory and some applications

Both recent theoretical results in surface water wave theory and some of its recent applications to problems of practical import including the discussion of tsunami propagation, rogue waves and near shore zone processes.