EJH @ 75: Detailed Programme

In addition to his fundamental contributions to areas of fluid mechanics including polymer rheology, capillary driven flows and microhydrodynamics, John Hinch also worked on many industrial problems including two major collaborative projects on advancing industrial inkjet printing technology with a consortium of inkjet companies on which we collaborated for 10 years. One key finding of this project was that across a certain range of molecular weights and concentrations polymeric additives can reduce or eliminate satellite-drop formation and thereby improve inkjet performance, whereas prior to this project polymer additives were considered always to be detrimental to inkjet applications. More recently we have examined the effects of surfactants that are often added to aqueous ink formulations. The rapid expansion of the free surface during jetting process produces local areas of the surface depleted of surfactants leading to surface tension gradients. The resulting Marangoni stresses act to delay and can even prevent the break-off of the main drop from the ligament. The presence of surfactants also reduce the mobility of the surface of the droplet, modifying the internal flow within the droplet and enhancing the viscous dissipation.

John Hinch is well known for his many contributions to non-Newtonian fluids, polymer rheology and microhydrodynamics. However it is not always remembered that he contributed to various other areas as well, including porous media flows. Some years after working with John (on ink jet printing and also on polymers) I became interested in modelling another non-Newtonian fluid, namely foam. By accident, the models upon which I had been working turned out to be relevant to engineering applications of foam flowing in porous media (including foam in 3D inkjet-printed porous media). This talk will outline some of the strange behaviours that foam bubbles might or might not exhibit when flowing along channels within a porous medium. In particular foam structures consisting of a moving train of bubbles might or might not be predicted to break up depending upon the size of the bubbles (relative to the channel size) and also the rate at which flow is driven (relative to a characteristic relaxation rate).

The failure of 2D numerical cohesive granular steps collapsing under gravity is simulated for a large range of cohesion. Focussing on the cumulative displacement of the grains, we establish a sensible criterion for capturing the failure characteristics. We are able to locate the failure in time and to identify the different stages of the destabilisation. Defining a narrow displacement interval, a well-defined linear shear band comes out, revealing the failure and its orientation. Solving the equilibrium of the failing block, with gravity, friction and cohesion as sole ingredients, we are able to make predictions for the dependance between failure angle, friction, and cohesion, thereby disclosing two distinct frictional behaviours: while constant friction is acceptable at small cohesion, no solutions are admitted for constant friction at larger cohesion. Implementing a friction decreasing with cohesion in the analytical prediction allows for matching the simulation points for the range of cohesion simulated. This agreement suggests the existence of a cohesion-induced weakening mechanism at large cohesion.

I will discuss several problems in low-Reynolds-number hydrodynamics where my Research group was able to make contributions, but where the guideposts were established by John Hinch. In particular, I will mention uses of the Reciprocal Theorem, particle-wall interactions, flows involving elastic filaments, and the flow of polymer solutions. I will try to provide some examples of new ideas while highlighting John’s style of linking mathematics and physics to yield new insights.

We study the oscillatory flow of suspensions of non-Brownian isodense particles (volume fraction 20%-40%) in a Hele-Shaw cell at a low Reynolds number (< 0.2) and a period of a few seconds. The velocity profile across the gap in a plane parallel to the flow is determined by tracking individually tagged particles. At early times, the particle velocity is parallel to the mean flow with a blunted profile due to a larger concentration of particles in the gap center, consistent with shear induced migration models. At longer times, there appears a secondary particle velocity component transverse to the mean flow and the cell walls and periodic both with time and spatially along the mean flow: no influence of inertia on the instability has been observed in the range of Reynolds numbers used in the experiments. A comparison of the variations of the characteristic times for the onset of the instability and of shear induced particle diffusion with the gap thickness suggests that they are closely related. Joint work with A.A. Garcia, Y.L. Roht, I. Ippolito, G. Gauthier, D. Salin, G. Drazer.

Hard spheres are the quintessential engineering model for particulate systems of all types because it is the simplest model and captures the important features of many systems. An artifact is the reversibility of pair trajectories under creeping flow conditions because contact between smooth spheres in a flowing suspension is impossible. Particle roughness has been invoked as a model to explain contact between particles and the broken symmetry of particle interactions that leads to cross-stream particle transport in flowing suspensions, as investigated by Da Cunha and Hinch (1996). Here, we consider the effect of particle permeability on lubrication interactions of particles. The leaking of fluid from the lubrication gap into the particles limits the build up of pressure in the near-contact region allowing contact between permeable particles in a suspension. Particle permeability also enhances the coupling between rotation and translation, providing access to non-singular rolling motions that are inaccessible to hard spheres at contact. An analogy between rough and permeable particles is established in terms of an effective roughness for permeability particles. Following Da Cunha and Hinch (1996), a pair-wise description for cross-flow particle transport in dilute suspensions is derived involving a diffusive flux and a drift velocity driven, respectively, by gradients of particle concentration and shear rate, and stationary particle distributions are predicted for Poiseuille flow. Away from regions where the shear rate vanishes, a scale-free power-law particle distribution is obtained. A boundary layer with width set by particle size is predicted in regions where the shear rate vanishes.

While a postdoc with John Hinch in 1982-83, we initiated a study on elastohydrodynamic collisions of wet particles, predicting that they would stick for Stokes numbers below a critical value and bounce but with reduced restitution coefficient for higher Stokes numbers [1]. Motivated by wet-granular and coating flows, my group has recently resumed study in this field. Our analyses use a rotating coordinate system for each particle pair in near contact to accurately resolve the relative motion. Expressions are derived for the squeezing and sliding lubrication forces, as well as the capillary force. These forces are then used in a discrete element model to follow the particle positions and translational and rotational velocities. For an oblique collision of two wetted particles, three possible outcomes are predicted: (1) rapid rebound and separation, (2) long-term agglomeration and rotation, and (3) initial agglomeration and rotation followed by slow separation due to centrifugal forces [2]. These outcomes support prior experimental observations [3]. Oblique, three-particle collisions yield even richer physics, including full agglomeration and rapid and slow separation of all three spheres or of one sphere from a pair, as previously demonstrated for head-on collisions of three spheres in a row [4].

[1] Davis, R. H., J.-M. Serayssol, and E. J. Hinch, “The Elastohydrodynamic Collision of Two Spheres,” J. Fluid Mech. 163, 479-497 (1986).

[2] Davis, R.H. and J.W. Sitison, “Oblique Collisions of Two Wetted Spheres,” Phys. Rev. Fluids 5: 054305 (2020).

[3] Donahue, C.M., W.M. Brewer, R.H. Davis, and C.M. Hrenya, “Agglomeration and De-agglomeration of Rotating Wet Doublets,” J. Fluid Mech. 708, 128-148 (2012).

[4] Davis, R.H., “Simultaneous and Sequential Collisions of Three Wetted Spheres,” J. Fluid Mech. 881, 983-1009 (2019).

A three-pages paper by John on the mechanism of the instability at the interface between two shearing fluids, in 1984, had a profound influence on my approach to fluid mechanics: what do we mean by "explaining" or "understanding" a physical phenomenon? Trying to extend John's arguments to the related problem of a thin layer sheared by a more or less viscous fluid, I got in touch with him. A fruitful collaboration ensued on interfacial instabilities and bedload particle transport. John spent many one-month stays at the Institut de mécanique des fluides de Toulouse, where his invaluable insights and curiosity benefited numerous researchers and students. Beyond Toulouse, French fluid mechanics owes much to John's great scientific penetration and European feelings.

In 1992, John Hinch, Owen Harris & John Rallison explained the mechanism of an interfacial instability recently discovered by Kang-Ping Chen, in which a jump in the first normal stress difference across an interface causes a co-extrusion flow to become unstable to long waves. The essence of the instability is captured in the additional shear stress caused by tilting an interface across which the normal stress had a discontinuity. This insight is very powerful, and can be used to explain instabilities where there isn't strictly an interface at all, just a region over which the material properties vary rather steeply. One such situation is shear-banding fluids, which we will look at in this talk.

As a postdoc with John Hinch and my interactions with other DAMTP faculty at that time, the groundwork of my professional career was created through our interaction on suspension mechanics and non-Newtonian fluid mechanics. The present talk will detail how some of my most recent work stems from ideas and problems that were originally examined during the early years studying with John. I will first discuss the collective settling behavior of a suspension of rigid, non-Brownian spheres in viscoelastic fluids. It is demonstrated, that hydrodynamic interactions between particles in viscoelastic fluids under quiescent conditions can result in concentration heterogeneities and associated large velocity fluctuations during sedimentation. This is actually one of a class of such concentration instabilities that was examined during my postdoc and is understood through a mean field theory. However, upon later examination of the sedimentation of particles that are “swirled” magnetically in a viscoelastic fluid, it is found that they individually sediment faster than without swirl. This leads to the idea of swirling propulsion, which we then examine in the context of freely suspending swimmers in viscoelastic fluids. Ultimately, we demonstrate a class of propulsion for freely suspended swimmers in elastic fluids that is absent for low Reynolds number Newtonian fluids. We believe this has applications for autonomous process rheometry measurements. Work done in collaboration with William L. Murch, Laurel Kroo, Jeremy Binagia and Manu Prakash.

In June 1987, John Hinch gave a series of lectures at the NATO summer institute on Disorder and Mixing held in Cargese (Corsica). In particular, he gave a lecture on the sedimentation of small particles where he discussed the sedimentation speed but also the velocity fluctuations. He provided theoretical estimates for these fluctuations which triggered a lot of debate and much further work. This talk presents the current state of understanding in the Stokes regime but also expands on the small-but-finite inertial regime. Joint work with Laurence Bergougnoux.

Lubrication theory is modified to include the large normal stresses (tension in the streamlines) which occur when the residence time is comparable with the liquid relaxation time (Deborah number O(1)). Numerical calculations find that the pressure drop through the contraction/constriction is less than for a Newtonian viscous liquid with the same zero shear-rate viscosity. Explanations are sought in terms of (i) the tension in the streamlines pulling the flow along, and (ii) the Lagrangian-transient nature of the flow which does not allow time for the elastic stress to build up to its steady equilibrium value.