Craters form as the lander's exhaust interacts with the planetary surfaces. Understanding this phenomenon is imperative to ensuring safe landings. We investigate the crater morphology, where a turbulent air jet impinges on granular surfaces. To reveal the fundamental aspect of this phenomenon, systematic experiments are performed with various air-jet velocities, nozzle positions and grain properties. The resultant crater morphology is characterized by an aspect ratio. We find a universal scaling law in which the aspect ratio is scaled by a dimensionless variable consisting of the air velocity at the nozzle, the speed of sound in air, the nozzle diameter, the nozzle-tip distance from the surface, the grain diameter, the density of the grains and the density of air. The obtained scaling reveals the cross-over of the length scales governing the crater aspect ratio, providing a useful guideline for ensuring safe landings. Moreover, we report a novel drop-shaped sub-surface cratering phenomenon.

The motion of rigid particles in complex fluids is ubiquitous in natural and industrial processes. The most popular toy model for understanding the physics of such systems is the settling of a solid sphere in a viscoelastic fluid. There is general agreement that an elastic wake develops downstream of the sphere, causing the breakage of fore-and-aft symmetry, while the flow remains axisymmetric, independent of fluid viscoelasticity and flow conditions. Using a continuum mechanics model, we reveal that axisymmetry holds only for weak viscoelastic flows. Beyond a critical value of the settling velocity, steady, non-axisymmetric disturbances develop peripherally of the rear pole of the sphere, giving rise to a four-lobed fingering instability. The transition from axisymmetric to non-axisymmetric flow fields is characterized by a regular bifurcation and depends solely on the interplay between shear and extensional properties of the viscoelastic fluid under different flow regimes. At higher settling velocities, each lobe tip is split into two new lobes, resembling fractal fingering in interfacial flows. For the first time, we capture an elastic fingering instability under steady-state conditions, and provide the missing information for understanding and predicting such instabilities in the response of viscoelastic fluids and soft media.

Shear-thinning viscosity is a non-Newtonian behaviour that active particles often encounter in biological fluids such as blood and mucus. The fundamental question of how this ubiquitous non-Newtonian rheology affects the propulsion of active particles has attracted substantial interest. In particular, spherical Janus particles driven by self-diffusiophoresis, a major physico-chemical propulsion mechanism of synthetic active particles, were shown to always swim slower in a shear-thinning fluid than in a Newtonian fluid. In this work, we move beyond the spherical limit to examine the effect of particle eccentricity on self-diffusiophoretic propulsion in a shear-thinning fluid. We use a combination of asymptotic analysis and numerical simulations to show that shear-thinning rheology can enhance self-diffusiophoretic propulsion of a spheroidal particle, in stark contrast to previous findings for the spherical case. A systematic characterization of the dependence of the propulsion speed on the particle's active surface coverage has also uncovered an intriguing feature associated with the propulsion speeds of a pair of complementarily coated particles not previously reported. Symmetry arguments are presented to elucidate how this new feature emerges as a combined effect of anisotropy of the spheroidal geometry and nonlinearity in fluid rheology.

The motion of glaciers over their bedrock or drops of fluid along a solid surface can become unstable when these substrates are lubricated. Previous studies modelled such systems as coupled gravity currents (GCs), consisting of one fluid that lubricates the flow of another fluid, and having two propagating fronts. When both fluids are Newtonian and discharged at constant flux, global similarity solutions were found. However, when the top fluid is strain-rate softening, experiments have shown that each fluid front evolved with a different exponent. Here, we explore theoretically and numerically such lubricated GCs consisting of axisymmetric spreading of a power-law fluid on top of a Newtonian fluid, where each fluid volume grows in time like $t^{\alpha }$. We find that the structure imposed by the non-Newtonian flow precludes general self-similarity, unlike purely Newtonian GCs. Consequently, we identify outstripping solutions in which the inner fluid front outstrips the outer fluid front. Despite the absence of a general global similarity solution, we find similarity solutions in several asymptotic limits. These include the purely Newtonian limit for any $\alpha$, the case of $\alpha =5$ for a general power-law fluid, asymptotic limits in the viscosity ratio, and in the vicinity of the fluid fronts. Many of our theoretical predictions are found to be consistent with recent laboratory experiments. Discrepancies suggest the presence of hydrofracturing or wall slip near the fronts, and potentially, a progressive significance of extensional stresses as front outstripping is approached.

When a rising bubble in a Newtonian liquid reaches the liquid–air interface, it can burst, leading to the formation of capillary waves and a jet on the surface. Here, we numerically study this phenomenon in a yield-stress fluid. We show how viscoplasticity controls the fate of these capillary waves and their interaction at the bottom of the cavity. Unlike Newtonian liquids, the free surface converges to a non-flat final equilibrium shape once the driving stresses inside the pool fall below the yield stress. Details of the dynamics, including flow energy budgets, are discussed. The work culminates in a regime map with four main regimes, all with different characteristic behaviours.

The motion of glaciers over their bedrock or drops of fluid along a solid surface can vary dramatically when these substrates are lubricated. We investigate the coupled flow of a gravity current (GC) of a strain-rate softening fluid that is lubricated by a denser, lower-viscosity Newtonian fluid. We present a set of experiments in which such GCs are discharged axisymmetrically and at constant flux over a flat surface. Using imaging techniques, we follow the evolution of the front and thickness field of each fluid. We find that, unlike purely Newtonian lubricated GCs, the fronts of the non-Newtonian and lubricating Newtonian fluids have a power-law time evolution with different exponents. Each of these exponents is similar to that of a non-lubricated GC of the same fluid. Nevertheless, the fronts of our lubricated GCs evolve faster than those of the corresponding non-lubricated GCs owing to larger intercepts. In addition, in contrast with the monotonically declining thickness of non-lubricated GCs, the thickness of the lubricated, non-Newtonian fluid is nearly uniform, and that of the lubricating fluid is non-monotonic with localised spikes. Despite these complex thickness patterns, lubricated GCs remain axisymmetric as long as the flux of the lubricating fluid is sufficiently smaller than that of the non-Newtonian fluid.

The spreading under surface tension and gravity of a droplet of yield-stress fluid over a thin film of the same material is studied. The droplet converges to a final equilibrium shape once the driving stresses inside the droplet fall below the yield stress. Scaling laws are presented for the final radius and complemented with an asymptotic analysis for shallow droplets. Moreover, numerical simulations using the volume-of-fluid method and a regularized constitutive law, and experiments with an aqueous solution of Carbopol, are presented.

For flows in microchannels, a slip on the walls may be efficient in reducing viscous dissipation. A related issue, addressed in this article, is to decrease the effective viscosity of a dilute monodisperse suspension of spheres in Poiseuille flow by using two parallel slip walls. Extending the approach developed for no-slip walls in Feuillebois et al. (J. Fluid Mech., vol. 800, 2016, pp. 111–139), a formal expression is obtained for the suspension intrinsic viscosity $[\mu ]$ solely in terms of a stresslet component and a quadrupole component exerted on a single freely suspended sphere. In the calculation of $[\mu ]$, the hydrodynamic interactions between a sphere and the slip walls are approximated using either the nearest wall model or the wall-superposition model. Both the stresslet and quadrupole are derived and accurately calculated using bipolar coordinates. Results are presented for $[\mu ]$ in terms of $H/(2a)$ and $\tilde{\lambda}=\lambda /a\leq 1$, where $H$ is the gap between walls, $a$ is the sphere radius and $\lambda$ is the wall slip length using the Navier slip boundary condition. As compared with the no-slip case, the intrinsic viscosity strongly depends on $\tilde{\lambda}$ for given $H/(2a)$, especially for small $H/(2a)$. For example, in the very confined case $H/(2a)=2$ (a lower bound found for practical validity of single-wall models) and for $\tilde{\lambda}=1$, the intrinsic viscosity is three times smaller than for a suspension bounded by no-slip walls and five times smaller than for an unbounded suspension (Einstein, Ann. Phys., vol. 19, 1906, pp. 289–306). We also provide a handy formula fitting our results for $[\mu ]$ in the entire range $2\leq H/(2a)\leq 100$ and $\tilde{\lambda}\leq 1$.

Surfactants – molecules and particles that preferentially adsorb to fluid interfaces – play a ubiquitous role in the fluids of industry, of nature and of life. Since most surfactants cannot be seen directly, their behaviour must be inferred from their impact on observed flows, like the buoyant rise of a bubble, or the thickness of a coating film. In so doing, however, a difficulty arises: physically distinct surfactant processes can affect measurable flows in qualitatively identical ways, raising the spectre of confusion or even misinterpretation. This Perspective describes, in one coherent piece, both the equilibrium properties and dynamic processes of surfactants, to better enable the fluid mechanics community to understand, interpret and design surfactant/fluid systems. Specifically, we treat the equilibrium thermodynamics of surfactants at interfaces, including surface pressure, isotherms of soluble and insoluble surfactants and surface dilatational moduli (Gibbs and Marangoni). We describe surfactant dynamics in fluid systems, including surfactant transport and interfacial stress boundary conditions, the competition between surface diffusion, advection and adsorption/desorption, Marangoni stresses and flows and surface-excess rheology. We discuss paradigmatic problems from fluid mechanics that are impacted by surfactants, including translating drops and bubbles, surfactant adsorption to clean and oscillating interfaces; capillary wave damping, thin-film dynamics, foam drainage and the dynamics of particles and probes at surfactant-laden interfaces. Finally, we discuss the additional richness and complexity that frequently arise in ‘real’ surfactants, including phase transitions, phase coexistence and polycrystalline phases within surfactant monolayers, and their impact on non-Newtonian surface rheology.

We present laboratory experiments that show that fingering patterns can emerge when circular interfaces of strain-rate-softening fluids displace less viscous fluids in extensionally dominated flows. The fingers were separated by regions in which the fluid appeared to be torn apart. Initially, the interface had a large dominant wavenumber, but some of the fingers progressively merged so that the number of fingers gradually declined in time. We find that the transition rate to a lower wavenumber during this cascade is faster the larger is the discharge flux of the displacing fluid. At late times, depending on the discharge flux, the pattern either converged into an integer wavenumber or varied stochastically within a finite range of wavenumbers, implying convergence to a fractional wavenumber. In that stage of the evolution we find that the average wavenumber declines with the discharge flux of the displacing fluid.

The interface of a strain-rate-softening fluid that displaces a low-viscosity fluid in a circular geometry with negligible drag can develop finger-like patterns separated by regions in which the fluid appears to be torn apart. Such patterns were observed and explored experimentally in Part 1 using polymeric solutions. They do not occur when the viscosity of the displacing fluid is constant, or when the displacing fluid has no-slip conditions along its boundaries. We investigate theoretically the formation of tongues at the interface of an axisymmetric initial state. We show that finger-like patterns can emerge when circular interfaces of strain-rate-softening fluids displace low-viscosity fluids between stress-free boundaries. The instability, which is fundamentally different from the classical Saffman–Taylor viscous fingering, is driven by the tension that builds up along the circular front of the propagating fluid. That destabilising tension is a geometrical consequence and is present independently of the nonlinear properties of the fluid. Shear stresses stabilise the growth either along extended circumferential streamlines or through a street of vortices. However, such stabilising processes become weaker, thereby allowing the instability to develop, the more strain-rate-softening the fluid is. The theoretical model that we present predicts the main experimental observations made in Part 1. In particular, the patterns we predict using linear-stability theory are consistent with the strongly nonlinear experimental patterns. Our model depends on a single dimensionless number representing the power-law exponent, which implies that the instability we describe could arise in any extensional flow of strain-rate-softening material, ranging from suspensions that rupture in squeeze experiments to rifts formed in ice shelves.

Laser-induced forward transfer (LIFT) is a nozzle-free printing technology that can be used for two- and three-dimensional printing. In LIFT, a laser pulse creates an impulse inside a thin film of material that results in the formation of a liquid jet. We experimentally study LIFT of viscoplastic materials by visualizing the process of jetting with high-speed imaging. The shape of the jet depends on the laser energy, focal height, surface tension and material rheology. We theoretically identify the characteristic jetting velocity and how it depends on the control parameters, and define non-dimensional groups to classify the regimes of jetting. Based on the results, we propose the optimal conditions for printing with LIFT technology.

We develop perturbation theories to describe the flow dynamics of a droplet with a thin layer of insoluble surfactant whose mechanics are described by interfacial viscosity, i.e. a Boussinesq–Scriven constitutive law. The theories quantify droplet deformation in the limit of small capillary number, large viscosity ratio, or large shear Boussinesq number, to a sufficient level of approximation where one can extract information about nonlinear rheology and droplet breakup. In the first part of this manuscript, we quantify the Taylor deformation parameter and inclination angle in shear and extensional flows, developing expressions that resolve discrepancies between current analytical theories and boundary element simulations. Interestingly, the theories we develop appear to accurately describe the inclination angle of a clean droplet over a wider range of viscosity ratios and capillary numbers than previous works. In the second part of the manuscript, we calculate how interfacial viscosity alters the extra stress of a dilute suspension of droplets, in particular the shear stress, normal stress differences, shear thinning and extensional thickening. The normal stresses are intimately related to the lateral migration of droplets in wall-bound shear flow, and we explore the influence of interfacial viscosity on this phenomenon. We conclude by discussing how one can use these theories to describe droplet breakup, and how one can incorporate additional effects into the perturbation theories such as viscoelastic membranes and/or Marangoni flows.

The equations describing classical viscous fluid flow are notoriously challenging to solve, even approximately, when the flow is host to one or many immersed bodies. When an immersed body is slender, the smallness of its aspect ratio can sometimes be used as a basis for a ‘slender-body theory’ describing its interaction with the surrounding environment. If the fluid is complex, however, such theories are generally invalid and efforts to understand the dynamics of immersed bodies are almost entirely numerical in nature. In a valiant effort, Hewitt & Balmforth (J. Fluid Mech., vol. 856, 2018, pp. 870–897) have unearthed a theory to describe the motion of slender bodies in a viscoplastic fluid, ‘fluids’ such as mud or toothpaste which can be coaxed to flow, but only with a sufficiently large amount of forcing. Mathematical theories for some tremendously complicated physical systems may now be within reach.

The margination and adhesion of micro-particles (MPs) have been extensively investigated separately, due to their important applications in the biomedical field. However, the cascade process from margination to adhesion should play an important role in the transport of MPs in blood flow. To the best of our knowledge, this has not been explored in the past. Here we numerically study the margination behaviour of elastic MPs to blood vessel walls under the interplay of their deformability and adhesion to the vessel wall. We use the lattice Boltzmann method and molecular dynamics to solve the fluid dynamics and particle dynamics (including red blood cells (RBCs) and elastic MPs) in blood flow, respectively. Additionally, a stochastic ligand–receptor binding model is employed to capture the adhesion behaviours of elastic MPs on the vessel wall. Margination probability is used to quantify the localization of elastic MPs at the wall. Two dimensionless numbers are considered to govern the whole process: the capillary number $Ca$, denoting the ratio of viscous force of fluid flow to elastic interfacial force of MP, and the adhesion number $Ad$, representing the ratio of adhesion strength to viscous force of fluid flow. We systematically vary them numerically and a margination probability contour is obtained. We find that there exist two optimal regimes favouring high margination probability on the plane $Ca{-}Ad$. The first regime, namely region I, is that with high adhesion strength and moderate particle stiffness; the other one, region II, has moderate adhesion strength and large particle stiffness. We conclude that the existence of optimal regimes is governed by the interplay of particle deformability and adhesion strength. The corresponding underlying mechanism is also discussed in detail. There are three major factors that contribute to the localization of MPs: (i) near-wall hydrodynamic collision between RBCs and MPs; (ii) deformation-induced migration due to the presence of the wall; and (iii) adhesive interaction between MPs and the wall. Mechanisms (i) and (iii) promote margination, while (ii) hampers margination. These three factors perform different roles and compete against each other when MPs are located in different regions of the flow channel, i.e. near-wall region. In optimal region I, adhesion outperforms deformation-induced migration; and in region II, the deformation-induced migration is small compared to the coupling of near-wall hydrodynamic collision and adhesion. The finding of optimal regimes can help the understanding of localization of elastic MPs at the wall under the adhesion effect in blood flow. More importantly, our results suggest that softer MP or stronger adhesion is not always the best choice for the localization of MPs.

We propose to explain shear-thinning behaviour observed in most concentrated non-Brownian suspensions by variable friction between particles. Considering the low magnitude of the forces experienced by the particles of suspensions under shear flow, it is first argued that rough particles come into solid contact through one or a few asperities. In such a few-asperity elastic–plastic contact, the friction coefficient is expected not to be constant but to decrease with increasing normal load. Simulations based on the force coupling method and including such a load-dependent friction coefficient are performed for various particle volume fractions. The results of the numerical simulations are compared to viscosity measurements carried out on suspensions of polystyrene particles ($40~\unicode[STIX]{x03BC}\text{m}$ in diameter) dispersed in a Newtonian silicon oil. The agreement is shown to be satisfactory. Furthermore, the comparison between the simulations conducted either with a constant or a load-dependent friction coefficient provides a model for the shear-thinning viscosity. In this model the effective friction coefficient $\unicode[STIX]{x1D707}^{eff}$ is specified by the effective normal contact force which is simply proportional to the bulk shear stress. As the shear stress increases, $\unicode[STIX]{x1D707}^{eff}$ decreases and the jamming volume fraction increases, leading to the reduction of the viscosity. Finally, using this model, we show that it is possible to evaluate the microscopic friction coefficient for each applied shear stress from the rheometric measurements.

The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour–liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide ($\text{CO}_{2}$) at a pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number ($Re=\unicode[STIX]{x1D70C}_{w}^{\ast }u_{r}^{\ast }h^{\ast }/\unicode[STIX]{x1D707}_{w}^{\ast }$), the product of the Prandtl ($Pr=\unicode[STIX]{x1D707}_{w}^{\ast }C_{pw}^{\ast }/\unicode[STIX]{x1D705}_{w}^{\ast }$) and Eckert numbers ($Ec=u_{r}^{\ast 2}/C_{pw}^{\ast }T_{w}^{\ast }$) and the wall temperature that in addition to pressure determine the thermodynamic reference condition. For low Eckert numbers, the flow is essentially isothermal and no difference with the well-known stability behaviour of incompressible flows is observed. However, if the Eckert number increases, the viscous heating causes gradients of thermodynamic and transport properties, and non-ideal gas effects become significant. Three regimes of the laminar base flow can be considered: the subcritical (temperature in the channel is entirely below its pseudo-critical value), transcritical and supercritical temperature regimes. If compared to the linear stability of an ideal gas Poiseuille flow, we show that the base flow is modally more unstable in the subcritical regime, inviscid unstable in the transcritical regime and significantly more stable in the supercritical regime. Following the principle of corresponding states, we expect that qualitatively similar results will be obtained for other fluids at equivalent thermodynamic states.

We study numerically the dynamics of an insoluble surfactant-laden droplet in a simple shear flow taking surface viscosity into account. The rheology of drop surface is modelled via a Boussinesq–Scriven constitutive law with both surface tension and surface viscosity depending strongly on the surface concentration of the surfactant. Our results show that the surface viscosity exhibits non-trivial effects on the surfactant transport on the deforming drop surface. Specifically, both dilatational and shear surface viscosity tend to eliminate the non-uniformity of surfactant concentration over the drop surface. However, their underlying mechanisms are entirely different; that is, the shear surface viscosity inhibits local convection due to its suppression on drop surface motion, while the dilatational surface viscosity inhibits local dilution due to its suppression on local surface dilatation. By comparing with previous studies of droplets with surface viscosity but with no surfactant transport, we find that the coupling between surface viscosity and surfactant transport induces non-negligible deviations in the dynamics of the whole droplet. More particularly, we demonstrate that the dependence of surface viscosity on local surfactant concentration has remarkable influences on the drop deformation. Besides, we analyse the full three-dimensional shape of surfactant-laden droplets in simple shear flow and observe that the drop shape can be approximated as an ellipsoid. More importantly, this ellipsoidal shape can be described by a standard ellipsoidal equation with only one unknown owing to the finding of an unexpected relationship among the drop’s three principal axes. Moreover, this relationship remains the same for both clean and surfactant-laden droplets with or without surface viscosity.

The presence and the microscopic origin of normal stress differences in dense suspensions under simple shear flows are investigated by means of inertialess particle dynamics simulations, taking into account hydrodynamic lubrication and frictional contact forces. The synergic action of hydrodynamic and contact forces between the suspended particles is found to be the origin of negative contributions to the first normal stress difference $N_{1}$, whereas positive values of $N_{1}$ observed at higher volume fractions near jamming are due to effects that cannot be accounted for in the hard-sphere limit. Furthermore, we found that the stress anisotropy induced by the planarity of the simple shear flow vanishes as the volume fraction approaches the jamming point for frictionless particles, while it remains finite for the case of frictional particles.