__Invited Speakers__

**June Barrow-Green**(Open University) -*Stokes' Mathematical Education*On the 22 January 1841 Joseph Romilly, Registrary of the University of Cambridge, wrote in his diary "Stokes of Pembroke S.W. & a very good one". Romilly was remarking on the fact that Stokes had not only triumphed in in the gruelling Mathematical Tripos examination, but that he had done so in a year when the papers were notoriously difficult. It was a notable achievement but it was a prize hard won after several years of preparation, and not only years spent at Cambridge. When Stokes arrived at Pembroke he had spent the previous two years at Bristol College, a school which prided itself on its success in preparing students for Oxford and Cambridge. In my talk I shall follow Stokes on his path to the senior wranglership, tracing his mathematical steps from his schooldays in Ireland to the end of his final year of undergraduate study.

**Michael Berry**(University of Bristol) -*Asymptotics and optics - Stokes and the rainbow*

**John Brady**(Caltech, Pasadena) -*Stokesian Dynamics*What do corn starch, swimming spermatozoa, DNA and self-assembling nanoparticles have in common? They are all (or can be modeled as) ‘particles’ dispersed in a continuum suspending fluid where hydrodynamic interactions compete with thermal (Brownian) and interparticle forces to set structure and determine properties. These systems are ‘soft’ as compared to molecular systems largely because their number density is much less and their time scales much longer than atomic or molecular systems. In this talk I will trace the pioneering work of Stokes and Einstein in the development of the mechanical framework for modeling these diverse systems and the essential features that any hydrodynamic modeling must incorporate in order to capture the correct behavior. Computing the hydrodynamics in an accurate and efficient manner is the challenge, which led to the development of the Stokesian Dynamics method. I will illustrate past successes and current efforts with examples drawn from the diffusion and rheology of colloids to the ‘swimming’ of microorganisms and catalytic nanomotors.

**John Bush**(MIT, Cambridge, USA) -*Hydrodynamic quantum analogs*Yves Couder and coworkers in Paris discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be exclusive to the microscopic, quantum realm. These walking droplets propel themselves by virtue of a resonant interaction with their own wavefield, and so represent the first macroscopic realization of a pilot-wave system of the form proposed for microscopic quantum dynamics by Louis de Broglie in the 1920s. New experimental and theoretical results allow us to rationalize the emergence of quantum-like behavior in this hydrodynamic pilot-wave system in a number of settings, and explore its potential and limitations as a quantum analog.

**Michael Cates**(DAMTP, Cambridge) -*Hard Sphere Suspensions Beyond Stokes: Friction and Shear Thickening*Hard spheres with a no slip boundary condition, immersed in a fluid that strictly obeys the Stokes equation, never come into contact. Therefore, the nature of any solid-solid frictional forces that might arise on contact should have no consequences for an idealized, Stokesian, dense suspension of hard spheres. In real dense suspensions, however, interparticle contacts do arise, and a consensus has recently emerged whereby friction is not merely significant, but a dominant factor, in determining their rheology. This holds specifically in relation to shear thickening, especially when this is discontinuous. I will summarise the consensus view, explore some of its implications for spatiotemporal flow instabilities, and point to areas where our understanding still falls far short of a predictive theory for dense suspension rheology.

**Olivier Darrigol**(CNRS, Paris) -__Stokes' discovery of fluorescence__

Stokes's works in optics were largely responsible for the prestigious positions he held in the British academia. They covered a wide range of topics and exemplified diverse methodologies. Most important, most beautiful, and most appreciated was his discovery of fluorescence, for at least two reasons: it broke the Newtonian dogma of the immutability of colors, and it opened a fruitful new kind of spectral analysis. We will discuss the sources, contents, style, and impact of Stokes's breakthrough.

**Anne De Wit**(ULB, Brussels) -*Flow control of the yield of chemical fronts*In presence of advection, the properties of A+B-> C reaction fronts may in some cases depend on the flow properties. In particular, it can be shown that, in radial Stokes flows, the amount of product obtained in the course of time can be tuned by varying the flow rate. We compare here the efficiency of rectilinear and radial flow geometries in terms of the amount of product C generated in the course of time. We show that, even though the production of C per unit area of the contact interface between the two reactant solutions is larger in the rectilinear case, the fact that the length of the contact zone increases with the radius in the radial advection case allows in fine to produce more product C for a same injected volume of reactant. These results pave the way to a geometrical flow control of the properties of chemical fronts.

**Nigel Goldenfeld**(University of Illinois) -*How Sir George Gabriel Stokes nearly discovered the renormalization group***Raymond Goldstein**(DAMTP, Cambridge) -*Biological fluid dynamics***Stuart Mathieson**(Queen's University Belfast) -*Stokes as a Religious Man of Science*

**Oreste Piro**(UIB, Palma de Mallorca) - Neutral particles, a review

**Paul Ranford**(UCL, London) -*Stokes as Secretary of the Royal Society 1854-1885 - his influence on science and scientists*The assistance of George Gabriel Stokes was regularly acknowledged by his scientific peers – Lord Rayleigh noted that the RS's publications "abound with grateful recognitions of help thus rendered, and in many cases his suggestions or comments form not the least valuable part of the memoirs that appear under the name of others" . Reviews of the available literature assert the general truth of Rayleigh’s statement. Some in Stokes’ wide network of correspondents regarded him as obsessively diligent, but extremely few regarded him with any sense of disapproval. Evidence will show that Stokes was the principal gatekeeper of publication in the RS’s journals during his time as Secretary. His power to publish or edit for publication – or even to suppress publication – was deftly wielded, impressively minimising (over a very long period) consequential discontent and political ramifications in the RS. Apart from Stokes’ influence on his close friend William Thomson (Lord Kelvin), his correspondence with a stellar cast of C19th scientists – from William Whewell in 1847 to Ernest Rutherford in 1898 – covers a vast range of scientific endeavour in which Stokes appeared as a constant central contributing (and sometimes commanding) character. This talk will summarise the significant (and so far mostly unrecognised) impact of Stokes on C19th science.

**Jayne Ringrose**(Pembroke College, Cambridge) -*Stokes in the context of Pembroke College*The connection of George Gabriel Stokes with Pembroke College stretches from his admission as an undergraduate in 1837 until his death as Master, in 1903. Effectively his college career began in the last few years of the "Long eighteenth century" and ended in the early years of the twentieth. This was a period of constant change and reform for the Nation, the University, and inevitably, for the College. Stokes was caught up in these changes: they affected him and he them. The College over which he presided briefly as Master had undergone extraordinary change and expansion, so as to be almost unrecognisable. This paper seeks to record those changes especially as they affected Stokes's Life and academic career.

**James Robinson**(Warwick) - The Navier-Stokes regularity problemThe question of whether or not the three-dimensional Navier-Stokes equations have smooth solutions that exist for all positive times is one of the Clay Foundation's Seven Millennium Prize Problems. I will give an (I hope accessible) overview of what is currently known in terms of rigorous results for this model, and in particular highlight the link between "regularity" (the smoothness of solutions) and uniqueness.

**Idan Tuval**(IMEDAE, Palma de Mallorca) -*Limits and variations of Stokes' law for living matter*One of the most universally renowned results of Stokes prolific scientific achievements is undoubtedly his derivation of the viscous drag force (and terminal velocity) of a small sinking sphere at low Re. Here we explore the limits and variations of Stokes' law for the case in which the sinking object is not a passive particle but rather a living cell with a plethora of active responses to environmental conditions. We discuss how these physiological responses have a direct effect on the physical properties of the organism that include changes in metabolic activity affecting cell buoyancy. As such, they represent dynamical constraints that modify the interaction between the cell and its fluid environment, ultimately determining its sinking fate. By focusing on sinking diatoms, one of the most abundant groups of unicellular organisms in our oceans and responsible for the production of ~ 20% of our planet's oxygen, we will further link our results to population dynamics, sexual reproduction and large scale biogeochemical cycles.

**Sylvie Vergniolle**(IPGP, Paris) -*The profound impact of Stokes’ work in modern physical volcanology***Andy Woods**(BP Institute, Cambridge) -*Stokes law and turbulent particle-laden flows*

__Contributed talks__

**Snezhana Abarzi**__(Western Australia) -__

*Interface dynamics: New mechanisms of stabilization and destabilization and structure of flow fields*Interfacial mixing and transport are nonequilibrium processes coupling kinetic to macroscopic scales. They occur in fluids, plasmas, and materials over celestial events to atoms. Grasping their fundamentals can advance a broad range of disciplines in science, mathematics, and engineering. This work focuses on the long-standing classic problem of stability of a phase boundary - a fluid interface that has a mass flow across it. We briefly review the recent advances and challenges in theoretical and experimental studies, develop the general theoretical framework directly linking the microscopic interfacial transport to the macroscopic flow fields, discover the new mechanisms of interface stabilization and destabilization that have not been discussed before for both inertial and accelerated dynamics, and chart perspectives for future research.

**Michael Bestehorn**(Brandenburg University of Technology) -*Reduced models in fluid dynamics*In hydrodynamics, instabilities and subsequent non-equilibrium pattern formation is often encountered in spatially extended systems. If one of the space dimensions is distinguished for physical or geometrical reasons, dimension-reduced models may be derived successfully by integrating out that dimension. The nowadays called ’Shallow Water Equations’ were historically the first dimensionreduced set derived by A. Saint-Venant already in 1871, allowing for the theoretical and later on numerical linear and nonlinear computations of long surface gravity waves on rivers, lakes or oceans if viscosity can be neglected. The Shallow Water equations are derived for an inviscid and irrotational liquid, basically governed by the (Hamiltonian) Euler equations. In real fluids described by the Navier-Stokes equations, viscosity plays a crucial role for boundary layers, even for laminar and slow flows. In such systems, another way of dimension reduction is obtained by enslaving many temporally fast modes damped by viscosity by only the few slow modes normally present in the neighborhood of instabilities. The celebrated Swift-Hohenberg equation (J. Swift and P. Hohenberg 1977) and its extensions derived in the 80ies and 90ies of the last century explain qualitatively well the linear and nonlinear behavior of several pattern forming convectively unstable systems. In thin liquid films, normally viscosity is the dominating mechanism and inertia can be neglected (zero Reynolds number limit). The fluid then obeys the (linear) Stokes equations. Fluid velocity is enslaved by the film profile and a single 2D equation for the film thickness turns out, as first demonstrated by A. Vrij in 1966. However, there are cases even in thin films where the Reynolds number can be large (falling films, mechanically vibrated substrates) and inertia must be kept. This leads to extended thin-film equations where the dynamics of the fluid flow occurs as additional degree of freedom. In the second part of this contribution, the situation of parametrically excited films by mechanical vibrations will be studied in more detail. Linear stability as well as nonlinear results of a reduced model will be presented and compared to findings of the full set of hydrodynamic basic equations. Nonlinear results of the model show interesting new patterns like confined pulses or quasi-periodic geometries.

**Silvana Cardoso**(Cambridge) -__On the spreading of bubble plumes__

We consider the flow of turbulent bubble plumes and how it is affected by the water density stratification, the slip speed of the bubbles and their dissolution. Different regimes of behaviour leading to single or multiple spreading levels in the environment are identified. We apply our findings to understand the dynamics of releases of methane and carbon dioxide in the ocean, related to the BP accident in the Gulf of Mexico, and carbon-dioxide storage beneath the North Sea.

____

**Julyan Cartwright**(Granada) -*Chaotic advection: from 19th to 21st century dynamics*Thirty years ago we found that fluids don’t have to be stirred fast to mix them. Even without that complex behaviour of a fast-moving fluid that we call turbulence, with just slow movement -with what is termed Stokes flow- it is possible to have good mixing of liquids or gases. This concept, called chaotic advection, is at the intersection of two fields, fluid dynamics and nonlinear dynamics, and encompasses a range of applications with length scales ranging from at least micrometres to hundreds of kilometres, including systems as diverse as mixing and thermal processing of viscous fluids, microfluidics, biological flows, and oceanographic and atmospheric flows. Repeated stretching and folding of bits of fluid leads to sensitivity to the initial conditions, so that a very small cause can lead to a very large effect. Smale developed the mathematics of the horseshoe map that squishes by stretching and folding in 1967, and Lorenz presented the “butterfly effect” with the question "does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" in 1972. Although chaotic advection, the application of this phenomenon to fluids, was first conceived by Aref in 1982, and as such is only in its third decade, the two fields whose common boundary it springs from are older. Stokes, of course, had a hand in the development of fluid dynamics in the nineteenth century, and as a result the Navier-Stokes equations bear his name. The understanding of nonlinear dynamics emerged little by little in the twentieth century from other nineteenth-century developments by Stokes’ younger contemporaries including Poincaré, who studied the three-body problem in the 1880s, but also his Cambridge colleague Maxwell, who anticipated twentieth century ideas of chaos in the 1870s. Irregular solutions of high-dimensional systems -turbulence- and irregular solutions of low-dimensional systems -chaos- both originated in the 19th century physics of Stokes’ time.

**Colm Caulfield**(DAMTP, Cambridge) -*On the (quasi)-steady motion of incompressible fluids*In his very first paper published in 1842, George Gabriel Stokes considered various steady flows in incompressible fluids.

However, he appreciated the key concept that there was no guarantee that such steady flows would be realised, noting “there may even be no stable steady mode of motion possible, in which case the fluid would continue perpetually eddying”. In the ensuing decades, both flow instabilities and “perpetually eddying” turbulent flow have been areas of exceptionally active research. A particularly important class of such potentially unstable and turbulent flows are stratified shear flows, where the background velocity and density distribution vary over some characteristic length scales. Such flows are ubiquitous in the atmosphere and the ocean, and at least at sufficiently high Reynolds number, they are commonly believed to play a key role in the transition to and maintenance of turbulence, and hence to be central to irreversible (density) mixing. Parameterizations of such irreversible mixing within larger scale models of the ocean in particular is a major area of uncertainty, not least because there is a wide range of highly scattered and apparently inconsistent experimental and observational data. In this talk I consider two idealized flows, one continuously forced and one freely evolving. Both flows clearly exhibit emergent critical behaviour, with the “perpetually eddying” motion overlaying some underlying and apparently robust quasi-steady structure associated with marginal stability of an appropriately defined background flow. This behaviour suggests that consideration of criticality, marginal stability, and indeed “steady motion of incompressible fluids” is still a fruitful approach for the resolution of the apparent inconsistencies in data, as well as for the construction of robust and useful parameterisations of irreversible mixing by sheared stratified turbulence.

**Andrea Ferrari**(Cambridge) -*Stokes and anti-Stokes*

**Gerard Fuller**(Stanford) -*The shape evolution of miscible sessile and pendant drops*The spreading of liquids is a classical problem in interfacial fluid mechanics and, historically, the examination has been limited to immiscible systems. We report on experimental studies and observations of the shape evolution and power law dynamics of both sessile drops and pendant drops in miscible environments. We have complemented these experimental studies with a theoretical scaling analysis as well as numerical simulation. As time evolves, diffusion across the miscible liquid-liquid boundary proceeds due to the chemical potential difference between the two initially distinct, homogeneous phases. Diffusion, in turn, imparts a time-dependence to the properties of the liquids in the diffusive region – notably the density, viscosity, and interfacial tension – that influence the shape evolution. As expected, droplets in miscible environments primarily respond to gravitational forces. The presence of diffusion sets up a fluid flow of free convection in the case of a pendant drop in a miscible environment; in the case of a sessile drop in a miscible environment, free convection occurs in tandem with spreading along the solid substrate. A series of different liquid pairs and volumes of droplets have been studied. Solving the convection-diffusion and Stokes equations numerically and in tandem has been used to simulate these systems, which quantitatively match observations of the experiments. This presentation will encompass the sessile and pendant drop studies, spanning the experimental, theoretical, and numerical analyses.

**Edward Hinton**(Cambridge) -__Free-surface Stokes flow past cylinders__

Attempts to divert lava flows away from buildings by constructing barriers have had limited success. Motivated by this problem, we study the gravitationally-driven Stokes flow of viscous fluid as it migrates down a slope and interacts with a cylinder that is oriented with its axis perpendicular to the plane and is sufficiently tall so that it is not overtopped.

A lubrication model is applied to derive the steady governing equation for the flow depth, which is integrated numerically. Asymptotic analysis is used to interpret the perturbation to the flow depth due to a circular cylinder in the regimes of both relatively deep oncoming flow and relatively shallow oncoming flows. For relatively shallow flows, there is a `pond’ of nearly stationary fluid upstream of the cylinder and a `dry’ region in which there is no fluid downstream of the cylinder. We derive asymptotic expressions for the maximum flow depth. The maximum flow depth occurs at the upstream stagnation point on the circular cylinder. We use our asymptotic predictions to obtain a simple empirical expression for the maximum flow depth that is accurate for all cylinder sizes.

Laboratory experiments corroborate the theory and confirm the occurrence of both `pond’ and `dry’ regions. The computations and asymptotic analysis for relatively shallow oncoming flow are extended to cylinders with square cross-sections, which leads to a different scaling for the maximum flow depth. This demonstrates the sensitivity of the pond depth to the shape of the obstruction.

**Herbert Huppert**(DAMTP, Cambridge) -*Stokes flow in rapidly rotating systems*

**Serafim Kalliadasis**(Imperial) -*The resolution of the moving contact line*The moving contact line problem occurs when modelling one fluid replacing another as it moves along a solid surface, a situation widespread throughout industry and nature. Classically, the no-slip boundary condition at the solid substrate, a zero-thickness interface between the fluids, and motion at the three-phase contact line are incompatible - leading to the well-known shear-stress singularity. At the heart of the problem is its multiscale nature: a nanoscale region close to the solid boundary where the continuum hypothesis breaks down, must be resolved before phenomenological macroscale parameters such as contact line friction and slip, often adopted to alleviate the singularity [1], can be obtained.

Here we will review recent progress made by our group to rigorously analyse the moving contact line problem and related physics from the nano- to macroscopic lengthscales. Specifically, to capture nanoscale properties very close to the contact line and to establish a link to the macroscale behaviour, we employ elements from the statistical mechanics of classical fluids, namely density-functional theory (DFT) [2,3]. We formulate a new and general dynamic DFT (DDFT) [4]that carefully and systematically accounts for the fundamental elements of any classical fluid and soft matter system, a crucial step towards the accurate and predictive modelling of physically relevant systems. In a certain limit, our DDFT reduces to a non-local Navier-Stokes-like equation [5]: an inherently multiscale model, bridging the micro- to the macroscale, and retaining the relevant fundamental microscopic information (fluid temperature, fluid-fluid and wall-fluid interactions) at the macroscopic level. Work analysing the contact line in both equilibrium and dynamics will be presented [6,7]. The new model allows us to benchmark existing phenomenological models and reproduce some of their key ingredients. But its multiscale nature also allows us to unravel the underlying physics of moving contact lines, not possible with any of the previous approaches, and indeed show that the physics is much more intricate than the previous models suggest. For instance, a key property captured by our theory is the fluid layering at the wall-fluid interface, amplified as the contact angle decreases. But also the existence of compressive interfacial regions on the vapor side of the vapor-liquid interface and a large shear region close to the wall in which effective slip can be generated. We demonstrate that the stratified fluid structure in the vicinity of the wall has a large effect on the compression and shearing properties of the fluid and determines the width of the shear region on the wall. We also scrutinize the effect of stratification on contact line friction and the dependence of the latter on the imposed temperature of the fluid and motion orientation [8].

- D.N. Sibley, A. Nold and S. Kalliadasis 2015 "The asymptotics of the moving contact line: cracking an old nut," J. Fluid Mech. 764, 445-462.
- P. Yatsyshin, N. Savva and S. Kalliadasis 2015 "Wetting of prototypical one- and two-dimensional systems: Thermodynamics and density functional theory," J. Chem. Phys. 142, Art. No. 034708.
- P. Yatsyshin, A.O. Parry and S. Kalliadasis 2016 "Complete prewetting," J. Phys.: Condens. Matter 28, Art. No. 275001.
- B.D. Goddard, A. Nold, N. Savva, G.A. Pavliotis and S. Kalliadasis 2012 "General dynamical density functional theory for classical fluids," Phys. Rev. Lett. 109, Art. No. 120603.
- B.D. Goddard, A. Nold, N. Savva, P. Yatsyshin and S. Kalliadasis 2013 "Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: derivation and numerical experiments," J. Phys.: Condens. Matter 25, Art. No. 035101.
- A. Nold, D.N. Sibley, B.D. Goddard and S. Kalliadasis 2014 "Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory," Phys. Fluids 26, Art. No. 072001.
- A. Nold, D.N. Sibley, B.D. Goddard and S. Kalliadasis 2015 "Nanoscale fluid structure of liquid-solid-vapor contact lines for a wide range of contact angles," Math. Model. Nat. Phenom. 10, 111-125.
- A. Nold, PhD Thesis, Imperial College London (2016).

**Aoife Kearins**(Dublin) -*Where it all began: The early life of Stokes in Sligo, and its lasting impact on his research*George Gabriel Stokes was born and raised in Skreen, County Sligo, a small village located in the very west of Ireland. Stokes grew up beside the ocean and spent a large amount of his childhood on the beach, and later in life cited this upbringing as a reason for his research interests in hydrodynamics. This talk covers Stokes' early days in Ireland, how this inspired his research interests later in life, his journey from Ireland to Cambridge and his lasting relationship with the Irish people, even posthumously.

**Davor Krajnovic**(Leibniz-Institut für Astrophysik Potsdam**)**-*A college friend: John Couch Adams*The opening sentences of John Couch Adams' obituary boldly states that "England has lost the greatest mathematical astronomer she has ever produced, Newton alone excepted". Indeed, Adams was the person who predicted the location of Neptune on sky, solved the problem of the mean motion of the Moon and determined the origin and the orbit of the Leonids meteor shower that puzzled astronomers for almost a thousand years. With his achievements Adams can be compared with his good friend George Stokes. Not only were they born in the same year, but were also both senior wranglers, received the Smith's Prizes and Copley medals, lived, thought and researched at Pembroke College (at least partially overlapping), and shared an appreciation of Newton. On the other hand, Adams' prediction of Neptune's location had absolutely no influence on its discovery in Berlin. His Moon calculations were contemporary with those of other astronomers. Adams' refused a knighthood and an appointment as Astronomer Royal. He was reluctant and slow to publish, but loved to derive the values of logarithms to 263 decimal places. The math and calculations at which he so excelled mark the high point of celestial mechanics, but are essentially not thought nowadays. The differences, but also similarities, between Adams and Stokes could not be more striking. In my contribution I will outline Adams' life, focusing on his Cambridge career and links with Stokes. I will also provide an overview of Adams' role in the discovery of Neptune, the high point of the Newtonian physics.

**John Lister**(DAMTP, Cambridge) -

__Stokes flow and unsteady Stokes flows during drop coalescence__When two viscous drops touch, surface tension acts on the tightly curved cusp-like interface around the fluid bridge between the drops and pulls the bridge wider, with radius $r_b(t)$. For many years it was thought that the early behaviour was governed entirely by Stokes flow, because of the small length scale $r_b$. We show that unsteady Stokes flow (linearised Navier-Stokes) on a larger scale also plays an important role. We present a new asymptotic solution for the initial stage of drop coalescence, including both viscosity and inertia, which involves flows on scales proportional to $r_b^3$, $r_b$, $t^{1/2}$ and the drop radius. Using a variety of classical methods in applied mathematics, we find analytical solutions on each scale and match these together for a complete description of drop coalescence at early times.

**Robert MacKay**(Warwick) -*Use of Stokes' theorem for plasma confinement*Stokes’ theorem, in the general form given by Cartan, says that ∫V dw = ∫¶V w for any k-form w and (k+1)-dimensional subset V with boundary ¶V. This has as special cases Gauss’ divergence theorem, Green’s theorem, and the theorem that Kelvin communicated to Stokes and Stokes set as a prize problem. To me it is a definition of the exterior derivative d on forms.

I will describe how the exterior derivative plays a fundamental role in the design of magnetic confinement devices for plasma. To begin, let Ω be the usual volume 3-form in Euclidean 3-space, which assigns to any ordered triple of vectors the signed volume of the parallelepiped they span. Given a magnetic field B, we can define the magnetic flux 2-form β = iBΩ, which assigns to any ordered pair of vectors the magnetic flux through the parallelogram they span (the interior product iB inserts B in the first argument of any covariant tensor). Div B = 0 is equivalent to dβ=0.

Charged particle motion (q,v)’ = X(q,v) in a magnetic field has a Hamiltonian formulation iXω = dH, where H is the energy m|v|2/2 and ω= -d(Σmvi dqi) - eβ is a symplectic form (non-degenerate 2-form with dω=0) on the 6D state space of position q and velocity v. In a strong field, the magnetic moment μ = m|v^|2/2|B| is an adiabatic invariant and to a good approximation the motion can be reduced to guiding-centre motion on the 4D space of guiding-centre position Q and parallel velocity vp, which has Hamiltonian formulation with H = mvp2/2+μ|B(Q)| and symplectic form ω = -d(mvp b♭)-eβ, where b = B/|B| and b♭is the 1-form which applied to any vector ξ gives b.ξ .

Idealising, we consider confinement to be a question of confining the guiding centres. One way to achieve this is to make the guiding-centre motion integrable, with bounded level sets of the integrals. A 4D autonomous Hamiltonian system as here is integrable if there is a function K on the state space such that K is conserved along X and dH and dK are independent almost everywhere. Defining a vector field u by iuω=dK, we obtain that the commutator [u,X]=0, so u is a symmetry of X. Modulo possibly multivalued functions K, this works in the other direction too (Noether). One symmetry that suffices is axisymmetry. This is the principle of tokamaks, but to make the level sets of H,K fit in the machine, a large toroidal current is required, which is hard to sustain and drives instabilities.

Thus we are searching for other possible symmetries, not requiring toroidal current. This is one of the principles of stellarators. Using the Lie derivative along a vector field u, which on forms has the expression Lu = iud + diu, we express the symmetry condition as Lu|B|=0, Luβ=0, and LuB♭=0. Note that dB♭= iJΩ where J is the current density. We derive a large set of constraints on u and B, and wish to prove either an existence result for a non-axisymmetric B with such a symmetry or that there are none.

This work is sponsored by the Simons Foundation and is joint with Josh Burby and Nikos Kallinikos.

**Scott McCue**(Queensland University of Technology) -*Analysing Taylor-Saffman bubbles using Stokes phenomenon*We consider a selection problem involving a Taylor-Saffman bubble propagating in an unbounded Hele-Shaw cell. There is numerical evidence to suggest that solutions to this problem in an unbounded domain behave in an analogous way to other finger and bubble problems in a Hele-Shaw channel; however, the selection of the ratio of bubble speeds to background velocity appears to follow a very different surface tension scaling to the channel cases. Here we apply Stokes phenomenon and other techniques in exponential asymptotics to solve the selection problem analytically, confirming the numerical results, including the predicted surface tension scaling laws. Further, our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane.

**Sebastien Michelin**(Ecole Polytechnique) -

__Modeling chemo-hydrodynamic interactions in phoretic suspensions__Stokes' equations are linear, yet may lead to complex dynamics, particularly when looking at the collective dynamics of many particles, as in sedimenting suspensions. This is even more the case for active phoretic suspensions where each particle self-propels in response to a physicochemical gradient (e.g. concentration of a solute) generated by the surface activity of each particle. Besides classical hydrodynamic interactions, these particles then also interact through their physicochemical signature.

In this work, we will present a generalisation of the classical method of reflections for the Stokes problem, in order to account for these chemo-hydrodynamic couplings in a generic and systematic framework.

**Christopher Ness**(Cambridge) -*Absorbing state transitions in granular materials close to jamming*We consider a model for driven particulate matter in which absorbing states can be reached both by particle isolation and by particle caging.

The model predicts an absorbing phase diagram in which analogues of Stokesian and elastic reversibility emerge at low and high volume fractions respectively, separated by a diffusive, non-absorbing region. We thus find a single phase boundary that spans the onset of chaos in sheared suspensions to the onset of yielding in jammed packings. This boundary has the properties of a non-equilibrium second order phase transition, leading us to write a Manna-like mean-field description that captures the model predictions. We find that dependent on model parameters, jamming can mark either a direct transition between the 2 absorbing states, or can be buried in the diffusive state.

**Jordi Ortin**(Barcelona) -*Viscoelastic shear waves and vortex rings in oscillatory pipe flow of wormlike micellar solutions*We study the oscillatory pipe flow of aqueous solutions of giant wormlike micelles. Due to the viscoelastic nature of the fluid a periodic forcing gives rise to viscoelastic shear waves, in analogy to Stokes second problem. If the wavelength of these waves is comparable to the radius of the cylindrical container, their mutual interference results in an oscillatory rectilinear shear flow which is increasingly reversing at increasingly higher forcing frequencies [1] and displays resonances [2]. In addition, the reversing nature of the flow accumulates vorticity at axisymmetric cylindrical sheets separating regions of reverse motion. Beyond a critical amplitude of the driving, the cylindrical sheets roll up into axisymmetric vortex rings [3]. Here we present our results of time-resolved 2D-PIV measurements of the velocity field in a meridional plane of a vertical pipe of 5 cm diameter for a CPyCl/NaSal [100:60] mM aqueous solution. Experimental measurements of the rectilinear oscillatory flow, as a function of forcing amplitude and frequency, and the resonance frequency spectrum are compared to theoretical predictions based on the upper-convected Maxwell model of the micellar solution. We study also the mechanism of vortex ring formation, by following the development of incipient vortical structures when the driving amplitude is abruptly increased from below to above the instability threshold at fixed forcing frequency. A single-mode Giesekus equation (the first nonlinear correction to the upper-convected Maxwell model accounting for the shear-thinning behaviour of the solution at large strain rates) shows indeed a divergence of radial normal stresses at the instability threshold, qualitatively reproducing our experimental results in the frequency-amplitude parameter space.

[1] L. Casanellas, J. Ortín. Laminar oscillatory flow of Maxwell and Oldroyd-B fluids: theoretical analysis. J. Non-Newtonian Fluid Mech. 166 (2011) 1315-1326.

[2] L. Casanellas, J. Ortín. Experiments on the laminar oscillatory flow of wormlike micellar solutions. Rheol. Acta 51 (2012) 545-557.

[3] L. Casanellas, J. Ortín. Vortex ring formation in oscillatory pipe flow of wormlike micellar solutions. J. Rheol. 58 (2014) 149-181

**Tim Pedley**(DAMTP, Cambridge) -

__Towards the rheology of a concentrated array of spherical squirmers__Continuum models of dilute suspensions of swimming micro-organisms are well established and can incorporate external (gravitational) forces and torques as well as the particle stress generated by the swimming activity [1]. In a semi-dilute suspension, for example of steady spherical squirmers, hydrodynamic and steric interactions between cells can be computed in a pairwise manner, and Stokesian Dynamics has been developed for higher concentrations. These computations have been used to compute the stress response to externally applied simple shear for planar or fully three-dimensional suspensions [2], but the results cannot be described as a continuum model. Recently we have examined the stability of a concentrated planar array of identical bottom-heavy squirmers, accounting for cell-cell interactions by the use of lubrication theory [3]. The present work seeks to extend this theory to externally driven, unidirectional shear flows, with a view to representing the macroscopic shear stress and normal stresses as functions of the shear-rate, the orientation of the applied shear to gravity, and to dimensionless parameters of the squirmer properties.

[1] Pedley, T.J. & Kessler, J.O. (1992)

*Ann. Rev. Fluid Mech*. 24:313-358.

[2] Ishikawa, T. & Pedley, T.J. (2007)

*J.Fluid Mech*. 588

**:**399-435 and (2010)

*Phys.Rev.Lett*. 100:088103.

[3] Brumley, D.R. & Pedley, T.J. (2019)

*Phys.Rev.Fluids*4:053102.

**Marco Polin**(Warwick) -

*Light Control of Localised Photo-Bio-Convection*Microorganismal motility is often characterised by complex responses to environmental physico-chemical stimuli. Although the biological basis of these responses is often not well understood, their exploitation already promises novel avenues to directly control the motion of living active matter at both the individual and collective level. Here we leverage the phototactic ability of the model microalga Chlamydomonas reinhardtii to precisely control the timing and position of localised cell photo-accumulation, leading to the controlled development of isolated bioconvective plumes. This novel form of photo-bio-convection allows a precise, fast and reconfigurable control of the spatio-temporal dynamics of the instability and the ensuing global recirculation, which can be activated and stopped in real time. A simple continuum model accounts for the phototactic response of the suspension and demonstrates how the spatio-temporal dynamics of the illumination field can be used as a simple external switch to produce efficient biomixing.

**Sandalo Roldan-Vargas**(Max Planck & Granada) -*Rare events, anomalies, and Brownian motion*In one of his celebrated 1905 papers, Albert Einstein proposed for the first time a statistical interpretation of Robert Brown’s innocent observation based on the corpuscular constitution of matter. On one hand, his theory predicted a long-time diffusive motion of the Brownian particles, being the diffusion coefficient expressed in terms of a ratio between the particle thermal energy and the energy dissipated by Stokes friction. On the other hand, the probability distribution of the particle displacement was predicted to be Gaussian. For more than one hundred years these predictions were systematically validated on real systems and the coexistence between Diffusivity and Gaussianity became a paradigm. However, recent experiments on mesoscopic particle systems have claimed the existence of a time regime where diffusion is not accompanied by a purely Gaussian distribution of displacements. In our work we discuss critically these recent observations as well as the minimal stochastic models proposed to rationalize the new intriguing experimental phenomenology. Using molecular dynamics simulations we further analyze the emergence of the hypothetical diffusive yet non-Gaussian regime in glass- and gel-forming systems. Finally, we establish for these systems a connection between non-Gaussian dynamics and system topology.

**Michael Sandford**(formerly at RAL Space)__-__

*Stokes: tracing his family, and their life in Cambridge*The Stokes ancestors and relatives of George Gabriel Stokes are briefly reviewed. Until recently less has been published about his maternal lines: Haughton, Hughes, Boswell and a possible Whitley line. His daughter's memoir in the 1907 publication by Lamor is the principal source for his boyhood in Ireland and later, including the events leading to his marriage to Mary Robinson and family life at Lensfield Cottage. Over the last 20 years research into papers held by family members has added to this, and in particular the events of the 1899 Jubilee celebrations for his 50 years as Lucasian Professor.

**Osamu Sano**(Tokyo) -*Viscous flow around macroscopic cavities in a granular material in terms of darcylet*Viscous flow through a granular material that has "macroscopic cavities" is studied on the basis of the Stokes equation and the generalized Darcy’s (or Brinkman’s) equation. Here, a "macroscopic cavity" refers to a region with higher permeability in an otherwise homogeneous granular material. We have already elucidated the significant effect of the cavities in an otherwise homogeneous granular material exposed to a uniform flow. We assume that the viscous flow inside the cavity is governed by the Stokes equation, whereas the flow outside of the cavity is governed by the generalized Darcy’s equation. The continuities of the velocity components and stress components at the macroscopic boundary are imposed. We found that the viscous flow concentrates to the cavity region, so that the volume flux and the velocity at the center amount to 2 times and 3 times, respectively, for the cylindrical cavity, whereas they are respectively 3 times and 6 times for the spherical cavity, compared with those without the cavity1−6) . Concentration of the flow enhances the stresses at the cavity boundary. If the magnitude of the flow exceeds certain criteria, which depends on the viscous stress and the friction between particles, constituent particles on the upstream-side boundary are carried away to the other side of the cavity, Note that the granular boundary is hard for normal stress toward the granular material because of the excluded volume effect, whereas it is vulnerable to negative pressure, so that the constituent particles at the upstream cavity boundary are more easily removed and are carried downward with fluid flow. This process effectively results in the migration of the cavity toward upstream direction 7,8). The flow field around the cavity looks similar to the one due to the translation of the solid body, but the flow direction is opposite to the latter case, which we term here a "Darcylet". In this paper, we firstly propose a new thee-dimensional singularity "Darcylet", which characterizes suction force in the granular material and acts like a point force with negative direction in contrast to a solid sphere in a clear fluid. Secondly, interaction of cavities is examined, which can explain the increase of volume flow into the cavity and higher local enhancement of stresses that leads to collapse of cavities, micro-scale waterway formation in that material, etc. Flow concentration is further enhanced in the presence of many macroscopic cavities, depending on the configuration of cavities with respect to the flow filed 9−11). This implies the reduction of critical magnitude of the far upstream flow that causes the collapse of cavities. If many such cavities are connected in a global scale, a passage of underground water with fast stream, or a sheet of fluid layer is generated. The former can be a cause of an extraordinarily fast transport of contaminated material (regarded as convective diffusion in contrast to macroscopic molecular diffusion), whereas the latter can be a cause of landslides or collapse of the river banks and cliffs.

[1] O. Sano, Nagare 2 (1983) 252 [in Japanese]. [2] K. Momii, K. Jinno, T. Ueda, H. Motomura, F. Hirano and T. Honda, J. Groundwater Hydrol. 31 (1989) 13. [3] G.P. Raja Sekhar and O. Sano: J. Phys. Soc. Jpn. 69 (2000) 2479. [4] G. P. Raja Sekhar and O. Sano: Fluid Dyn. Res. 28 (2001) 281. [5] G. P. Raja Sekhar and O. Sano: Phys. Fluids 15 (2003) 554. [6] Y. Kaneko and O. Sano: Fluid Dyn. Res. 32 (2003) 15. [7] Y. Kaneko and O. Sano: Phys. Fluids 17 (2005) 033102. [8] O. Sano and Y. Nagata: Phys. Fluids 18 (2006) 121507. [9] S. Koizumi, Y.Shirahashi and O. Sano: J. Phys. Soc. Jpn. 78 (2009) 084404. [10] O. Sano: Comp. Phys. Comm. 182 (2011) 1870.

**David Smith**(Birmingham) -*NEAREST - An Accessible and Efficient Method for Biological Stokes Flow*Microscale biological flow problems, exemplified by swimming cells, fluid propulsion by cilia, and the dynamics of macromolecular structures, are described by the Stokes flow equations. Typically these problems involve complexshaped moving boundaries, however the linearity of the Stokes flow equations enables both physical insight and efficient numerical approximation, based on (force) weighted sums of fundamental solutions or 'Stokeslet'. Approaches based on this idea enable natural treatment of curved surfaces such as cell bodies and epithelia, and slender appendages such as flagella and cilia. These methods can be technically challenging to implement due to the presence of singularities, and the need to generate connected 'mesh' structures, which has motivated the development of 'regularized Stokeslet' methods which are both nonsingular and can be implemented with a simple point discretization rather than a true mesh. A practical limitation of the regularized Stokeslet method is its relatively high computational cost to achieve desired accuracy. To address this issue while retaining practical 'accessbility' we have developed NEAREST: NEArest neighbours for REgularized STokeslets, a numerical method and open source implementation based on applying nearestneighbour approximation to the boundary integral equation. The technique combines a fine discretisation for the rapidlyvarying kernel, and a coarse discretisation for the slowlyvarying unknown force weight. The technique enables orderofmagnitude improvements in efficiency and accuracy. The method will be demonstrated in its application to analyse sperm motility data and ciliadriven flows in development.

**Joseph Webber**(Cambridge) -*Stokes drift over corals*Stokes drift - the net drifting of fluid undergoing wave motion due to a difference in Lagrangian and Eulerian velocities - is a well-understood and well-researched topic, and has been applied to understand ocean processes like the motion of driftwood and the spreading of pollutants. In this talk, I consider the consequences of a saturated porous bed layer underlying essentially inviscid fluid on the surface of which are propagating gravity waves, and discuss how the damping effect of this porous bed affects the Stokes drift velocities both above and within the porous layer, including the introduction of a novel vertical drift. Knowledge of how such drifts arise is seen to be important in understanding the associated physics and biology of coral reef processes, such as the transport of nutrients and the prevention of coral bleaching.

**Grae Worster**(DAMTP, Cambridge) -

__Fingering of shear-thinning, radially extensional flows__Viscous gravity currents flowing over horizontal, rigid surfaces are dominated by vertical shear stresses and remain axi-symmetric when spreading from a point source. Conversely, a viscous gravity current that flows along the interface between two inviscid or significantly less viscous fluids is dominated by extensional stresses. Experiments show that such currents that spread radially can be unstable to the formation of fingers when the viscous fluid is sufficiently shear-thinning. An idealization of such a flow consists of viscous fluid spreading radially in the narrow gap between parallel plates if there is no stress exerted by the plates (a stress-free or slippy Hele-Shaw cell). The equations describing such a flow are the 2D Stokes Equations, generalised for a shear-thinning rheology. A linear-stability analysis of axi-symmetric, annular flow shows that the instability mechanism is related entirely to geometry and exists even for Newtonian fluids; shear-thinning does not introduce an instability mechanism but rather weakens a stabilising mechanism that operates on short wavelengths. These instabilities are related to the longitudinal fracturing of ice shelves, for example.