Fractures provide pathways for fluids and solutes through crystalline rocks and low permeability materials, thus playing a key role in many subsurface processes and applications. In small aperture fractures, solute transport is strongly impacted by the coupling of electrical double layers at mineral–fluid interfaces to bulk ion transport. Yet, most models of flow and transport in fractures ignore these effects. Solving such coupled electrohydrodynamics in realistic three-dimensional (3-D) fracture geometries poses computational challenges which have so far limited our understanding of those electro-osmotic effects’ impact. Starting from the Poisson–Nernst–Planck–Navier–Stokes (PNPNS) equations and using a combination of rescaling, asymptotic analysis and the Leibniz rule, we derive a set of nonlinearly coupled conservation equations for the local fluxes of fluid mass, solute mass and electrical charges. Their solution yields the fluid pressure, solute concentration and electrical potential fields. The model is validated by comparing its predictions to the solutions of the PNPNS equations in 3-D rough fractures. Application of the model to realistic rough fracture geometries evidences several phenomena hitherto not reported in the literature, including: (i) a dependence of the permeability and electrical conductivity on the fracture walls’ charge density, (ii) local (sometimes global) flow reversal, and (iii) spatial heterogeneities in the concentration field without any imposed concentration gradient. This new theoretical framework will allow systematically addressing large statistics of fracture geometry realizations of given stochastic parameters, to infer the impact of the geometry and various hydrodynamic and electrical parameters on the coupled transport of fluid and ions in rough fractures.

Violent respiratory events play critical roles in the transmission of respiratory diseases, such as coughing and sneezing, between infectious and susceptible individuals. In this work, large-scale multiphase flow large-eddy simulations have been performed to simulate the coughing jet from a human's mouth carrying pathogenic or virus-laden droplets by using a weakly compressible smoothed particle hydrodynamics method. We explicitly model the cough jet ejected from a human mouth in the form of a mixture of two-phase fluids based on the cough velocity profile of the exhalation flow obtained from experimental data and the statistics of the droplets’ sizes. The coupling and interaction between the two expiratory phases and ambient surrounding air are examined based on the interaction between the gas particles and droplet particles. First, the results reveal that the turbulence of the cough jet determines the dispersion of the virus-laden droplets, i.e. whether they fly up evolving into aerosols or fall down to the ground. Second, the droplet particles have significant effects on the evolution of the cough jet turbulence; for example, they increase the complexity and butterfly effect introduced by the turbulence disturbance. Our results show that the prediction of the spreading distance of droplet particles often goes beyond the social distancing rules recommended by the World Health Organization, which reminds us of the risks of exposure if we do not take any protecting protocol.

A kinetic model called the $\nu$-model is proposed to replace the complicated Boltzmann collision operator in the simulation of rarefied flows of monatomic gas. The model follows the relaxation-time approximation, but the collision frequency (i.e. inverse relaxation time) is a function of the molecular velocity to reflect part of the collision details of the Boltzmann equation, and the target velocity distribution function (VDF) to which the VDF relaxes is close to that used in the Shakhov model. Based on the numerical simulation of strong non-equilibrium shock waves, a half-theoretical and half-empirical collision frequency is designed for different intermolecular potentials: the $\nu$-model shows significantly improved accuracy, and the underlying mechanism is analysed. The $\nu$-model also performs well in canonical rarefied microflows, especially in thermal transpiration, where kinetic models with velocity-independent collision frequency lack the capability to distinguish the influence of intermolecular potentials.

An analytical framework is proposed to explore the structure and kinematics of iso-scalar fields. It is based on a two-point statistical analysis of the phase indicator field which is used to track a given iso-scalar volume. The displacement speed of the iso-surface, i.e. the interface velocity relative to the fluid velocity, is explicitly accounted for, thereby generalizing previous two-point equations dedicated to the phase indicator in two-phase flows. Although this framework applies to many transported quantities, we here focus on passive scalar mixing. Particular attention is paid to the effect of Reynolds (the Taylor based Reynolds number is varied from 88 to 530) and Schmidt numbers (in the range 0.1 to 1), together with the influence of flow and scalar forcing. It is first found that diffusion in the iso-surface tangential direction is predominant, emphasizing the primordial influence of curvature on the displacement speed. Second, the appropriate normalizing scales for the two-point statistics at either large, intermediate and small scales are revealed and appear to be related to the radius of gyration, the surface density and the standard deviation of mean curvature, respectively. Third, the onset of an intermediate ‘scaling range’ for the two-point statistics of the phase indicator at sufficiently large Reynolds numbers is observed. The scaling exponent complies with a fractal dimension of 8/3. A scaling range is also observed for the transfer of iso-scalar fields in scale space whose exponent can be estimated by simple scaling arguments and a recent closure of the Corrsin equation. Fourth, the effects of Reynolds and Schmidt numbers together with flow or scalar forcing on the different terms of the two-point budget are highlighted.

Rupture of liquid sheets of power-law fluids surrounded by a gas is analysed under the competing influences of pressure due to van der Waals attraction, inertia, viscous stress and capillary pressure due to surface tension. Results of a combined theoretical and computational study are presented over the entire range of parameters governing the thinning of a power-law fluid of power-law exponent $0 < n \le 1$ ($n=1$: Newtonian fluid) and Ohnesorge number $0 \le Oh < \infty$, where $Oh \equiv \mu _0/\sqrt {\rho h_0 \sigma }$, and $\mu _0, \rho, h_0$ and $\sigma$ stand for the zero-deformation-rate viscosity, density, the initial sheet half-thickness and surface tension, respectively. The dynamics in the vicinity of the space–time singularity where the sheet ruptures is asymptotically self-similar, and thus the variation with time remaining until rupture $\tau \equiv t_R - t$, where $t_R$ is the time instant $t$ at which the sheet ruptures, of sheet half-thickness, lateral length scale and lateral velocity is determined analytically and confirmed by simulations. For sheets for which inertia is negligible ($Oh^{-1}=0$), two distinct viscous scaling regimes are found, one for $0.58 < n \le 1$ and the other for $n \le 0.58$. The thinning dynamics of inviscid sheets ($Oh = 0$) is identical to that of Newtonian ones. For real fluids for which neither viscosity nor inertia is negligible, it is shown that the aforementioned creeping and inertial flow regimes are transitory and the thinning of power-law sheets exhibits a remarkably richer set of scaling transitions compared with Newtonian sheets.

New insight into the contribution of the microscale vortex evolution to convection heat transfer in porous media is presented in this paper. The objective is to determine how the microscale vortices influence convection heat transfer in turbulent flow inside porous media. The microscale temperature distribution is analysed using flow visualization in two dimensions using streamlines and in three dimensions using the Q-criterion. The pertinent observations are supplemented with a comparison of surface skin friction and heat transfer using: (i) surface skin-friction lines and (ii) the joint probability density function of the pressure and skin-friction coefficients, along with the Nusselt number. The microscale flow phenomena observed are corroborated with the features of the frequency spectra of the drag coefficient and macroscale Nusselt number. The large eddy simulation technique is used in this study to investigate the flow field inside a periodic porous medium. The Reynolds numbers of the flow are 300 and 500. The porous medium consists of solid obstacles in the shape of square and circular cylinders. Two distinct flow regimes are represented by using the porosities of 0.50 and 0.87. The results show that the surface Nusselt number distribution is dependent on whether the micro-vortices are attached to or detached from the surface of the obstacle. The spectra of the macroscale Nusselt number and the pressure drag are similar, signifying a correlation between the dynamics of heat transfer and the microscale turbulent structures. Both vortex shedding and secondary flow instabilities are observed that significantly influence the Nusselt number. The fundamental insight gained in this paper can inform the development of more robust macroscale models of convection heat transfer in turbulent flow in porous media.

Aerothermodynamic characteristics of a sphere such as drag and energy transfer coefficients are calculated for Mach numbers varying from 1 to 10 over a wide range of the gas rarefaction degree spanning the free molecular, transitional and near hydrodynamic flow regimes. The effects of major factors determining the gas flow are studied using the direct simulation Monte Carlo method. To reveal the effect of gas species, the simulations are performed based on ab initio interatomic potentials for helium, neon, argon and krypton, as well as based on the hard sphere model. The impact of the accommodation coefficients is evaluated by applying the Cercignani–Lampis model of gas–surface interaction. The calculations are carried out for several values of the free stream and sphere temperatures. It is found that the effects of gas species on the drag and energy transfer coefficients are approximately 3 % and 6 %, respectively. The accommodation coefficients in the Cercignani–Lampis model strongly affect all aerothermodynamic characteristics. The drag and energy transfer coefficients calculated for different accommodation coefficients vary within 30 % and 200 %, respectively. It is found that the variation of the tangential momentum and normal energy accommodation coefficients can induce an increase of the drag coefficient compared to the case of diffuse gas–surface interaction. In hypersonic flows, the drag coefficient varies within 30 % when the sphere temperature varies from the free stream temperature to the stagnation temperature. The drag and energy transfer coefficients are found to be non-monotonic functions of the free stream temperature.

In this work, nonlinear variational optimization is used for obtaining minimal seeds for the formation of turbulent bands in channel flow. Using nonlinear optimization together with energy bisection, we have found that the minimal energy threshold for obtaining spatially patterned turbulence scales with $Re^{-8.5}$ for $Re>1000$. The minimal seed, which is different to that found in a much smaller domain, is characterized by a spot-like structure surrounded by a low-amplitude large-scale quadrupolar flow filling the whole domain. This minimal-energy perturbation of the laminar flow has dominant wavelengths close to $4$ in the streamwise direction and $1$ in the spanwise direction, and is characterized by a spatial localization increasing with the Reynolds number. At $Re \lesssim 1200$, the minimal seed evolves in time, creating an isolated oblique band, whereas for $Re\gtrsim 1200$, a quasi-spanwise-symmetric evolution is observed, giving rise to two distinct bands. A similar evolution is found also at low $Re$ for non-minimal optimal perturbations. This highlights two different mechanisms of formation of turbulent bands in channel flow, depending on the Reynolds number and initial energy of the perturbation. The selection of one of these two mechanisms appears to be dependent on the probability of decay of the newly created stripe, which increases with time, but decreases with the Reynolds number.

The existence of Bolgiano–Obukhov (BO) scaling in Rayleigh–Bénard convection (RBC) has long been speculated. However, due to the inhomogeneity and anisotropy of the flow, and the lack of clear scale separation, no conclusive evidence has been found. To avoid these non-ideal factors, we construct an idealized isotropic convection system by introducing an additional horizontal buoyancy field to RBC in a doubly periodic domain. We focus on the two-dimensional case so that its upscale kinetic energy flux can enable a long inertial range for detecting the BO scaling. Through direct numerical simulations of this system, we justify the existence of BO scaling using second- and third-order structure functions, which are in good agreement with our theoretically obtained scaling relations from the Kármán–Howarth–Monin equations. These theoretical and numerical results provide direct support for the conjecture that the existence of the BO scaling in RBC is associated with the inverse kinetic energy cascade. For higher-order structure functions, we found strong intermittent effects in the buoyancy field, but not in the velocity. By comparing the present system with the canonical anisotropic RBC in a periodic domain, the effects of anisotropy on the scaling properties are elucidated.

Investigated first is the linear stability of a Newtonian ferrofluid centred on a rigid wire, surrounded by another ferrofluid with a different magnetic susceptibility. An electric current runs through the wire, generating an azimuthal magnetic field that produces a magnetic stress at the interface of the fluids. Three-dimensional disturbances to the system are considered, and the linearised Navier–Stokes equations are solved analytically in terms of an implicit expression for the growth rate of the disturbance. The growth rate is found numerically for arbitrary Reynolds number, and given explicitly in the inviscid and Stokes regimes. Investigated next is a ferrofluid whose magnetic susceptibility varies radially, centred on a rigid wire, subject to a non-uniform azimuthal field. It is proven that if the gradient of the susceptibility is positive anywhere in the fluid, then the system is linearly unstable. Moreover, it is proven that applying an axial field can stabilise disturbances for both continuous and discontinuous susceptibilities.

We perform direct numerical simulations to study the effect of the gravity centre offset in spherical Rayleigh–Bénard convection. When the gravity centre is shifted towards the south, we find that the shift of the gravity centre has a pronounced influence on the flow structures. At low Rayleigh number $Ra$, a steady-state large-scale meridional circulation induced by the baroclinic imbalance, created by the misalignment of the gravity potentials and isotherms, is formed. At high $Ra$, an energetic jet is created on the northern side of the inner sphere that is directed towards the outer sphere. The large-scale circulation induces a strong co-latitudinal dependence in the local heat flux. Nevertheless, the global heat flux is not affected by the changes in the large-scale flow organization induced by the gravity centre offset.

The paper investigates experimentally the flow past a flat-back, taller than wide Ahmed body having rectangular base aspect ratio $H/W=1.11$ in the context of ground vehicle aerodynamics. Parametric studies at Reynolds number $2.1\times 10^5$ explore the sensitivity of the aerodynamic force and body pressure distribution with respect to varying flow conditions defined from variable ground clearance $C$ (taken at mid-distance from the front and rear axles of the body), pitch angle $\alpha$, and yaw angle $\beta$ (equivalent to a crosswind). Two-dimensional parametric maps in the parametric spaces $(\beta,C)$ and $(\beta,\alpha )$ are obtained for parameter ranges covering real road vehicle driving conditions. The study of the base pressure gradient reveals non-trivial and sharp transitions identified as significant changes of near wake orientation in both parametric spaces. All unsteady transitions correspond to fluctuation crises of the vertical gradient component only. These transitions are summarized in phase diagram representations. Both phase diagrams in $(\beta,C)$ and $(\beta,\alpha )$ parametric spaces can be unified at large yaw in the $(\beta,C_r)$ space, where the rear clearance $C_r$ separating the lower edge of the base from the ground is a simple function of the pitch $\alpha$ and the clearance $C$. The impacts of the wake transitions are clearly identified in the base drag, drag, lift and side force coefficients. The contribution of the steady near wake $z$-instability (Grandemange et al., Phys. Fluids, vol. 25, 2013a, pp. 95–103) is assessed by repeating the sensitivity analysis with a rear cavity. As reported previously, the rear cavity suppresses the steady instability by symmetrizing the wake. A domain for the existence of the instability is finally proposed in the attitudes map $(\beta,\alpha )$ defined from regions where the mean lateral force coefficients are significantly decreased by the presence of the rear cavity. In addition, it is found that the steady instability forces the wake to be less unsteady for all attitudes that do not correspond to unsteady transitions.

Diffusiophoresis refers to the movement of colloidal particles in the presence of a concentration gradient of a solute and enables directed motion of colloidal particles in geometries that are inaccessible, such as dead-end pores, without imposing an external field. Previous experimental reports on dead-end pore geometries show that, even in the absence of mean flow, colloidal particles moving through diffusiophoresis exhibit significant dispersion. Existing models of diffusiophoresis are not able to predict the dispersion and thus the comparison between the experiments and the models is largely qualitative. To address these quantitative differences between the experiments and models, we derive an effective one-dimensional equation, similar to a Taylor dispersion analysis, that accounts for the dispersion created by diffusioosmotic flow from the channel sidewalls. We derive the effective dispersion coefficient and validate our results by comparing them with direct numerical simulations. We also compare our model with experiments and obtain quantitative agreement for a wide range of colloidal particle sizes. Our analysis reveals two important conclusions. First, in the absence of mean flow, dispersion is driven by the flow created by diffusioosmotic wall slip such that spreading can be reduced by decreasing the channel wall diffusioosmotic mobility. Second, the model can explain the spreading of colloids in a dead-end pore for a wide range of particle sizes. We note that, while the analysis presented here focuses on a dead-end pore geometry with no mean flow, our theoretical framework is general and can be adapted to other geometries and other background flows.

Many volcanoes emit a significant portion of the gas they transport to the atmosphere during continual passive degassing rather than during eruptions. To maintain a high gas and thermal flux without erupting magma, the flow field in the volcanic conduit must be approximately balanced with gas-rich, buoyant magma ascending and degassed, heavy magma descending. In vertical conduits, this exchange flow takes the form of core–annular flow, where the gas-rich magma forms a core enclosed by an annulus of degassed magma. The flow dynamics of core–annular flow have been studied extensively in fluid dynamics, but mostly for constant material properties. Our study aims to advance our understanding of how core–annular flow responds to volatile exsolution – a simple, yet ubiquitous disruption in volcanic conduits, which alters both the density and the viscosity of the core fluid. By deriving an evolution equation for the core–annular interface based on a generalized exchange-flow condition using a lubrication approximation, we find that the response of the system to volatile exsolution depends on the conduit flow regime. The same nucleation event can lead to a flow adjustment only in the upper, only in the lower or in both portions of the volcanic conduit. Our results emphasize that the thermodynamic evolution of magma properties and volcanic conduit flow are intricately linked, which may help understand the observed variability of eruptive behaviour at persistently degassing volcanoes.

Floating rather stably above the convective mantle, continents exert a thermal blanket effect on the mantle by locally accumulating heat and altering the flow structure, which in turn affect continent motion. This thermal-mechanic feedback has been captured by a simplified model―a thermally insulating plate floating freely over a bottom-heated convecting fluid. Previous studies have revealed that, as plate size increases, plate motion transitions from long-term stagnancy over cold downwelling to short-term stagnancy over hot upwelling, termed stagnant modes (SMs) I and II, respectively, and eventually to a unidirectionally moving mode (UMM) with no stagnancy. During SM I/II, the adjacent hot/cold plumes are observed to move toward/away from the plate. The present study quantifies these plume motions through analysis on a partially insulated loop model established for SM I and SM II, respectively. The model predictions are well verified by results of numerical simulation. Moreover, the dominant frequency of plate speed is shown to increase linearly with plate size, with an abrupt jump as the coupling mode transitions from SM I to SM II. This suggests that the enhanced thermal blanket effect with plate size enables an increasingly fast feedback between the slowdown of a plate and the strengthening of hot upwelling which boosts plate motion. When plate motion is boosted before it stops, the UMM emerges. The predicted critical plate size for the UMM is well validated by simulation results. Geological implications of the results for continent motion, continent–mantle coupling and particularly the migration of plume and subduction sites are discussed.

We report a laboratory study on the scattering, energy dissipation and mean flow induced by internal gravity waves incident upon slopes with varying surface roughness. The experiment was performed in a rectangular box filled with thermally stratified water. The roughness of the slope surface, $\lambda$, defined as the height of a roughness element over its base width, and the off-criticality $\gamma =(\alpha -\beta )/\beta$, with $\alpha$ and $\beta$ being the angles of the incident wave and the slope, are used as two control parameters. The distribution of energy dissipation in the direction normal to the slope is found to be more uniform in the rough surface cases. Counter-intuitively, both the maximum value in the dissipation profile and the total energy dissipation near the slope are reduced by surface roughness under most circumstances. The measured peak width (the full width at half-maximum of the peaks) of the dissipation profile is found to be broadened significantly in the rough surface cases. We also observed that there exists a non-zero optimal off-criticality ($\gamma =0.17$ for the present measurement resolution) for the normalized average dissipation and total dissipation, which may be due to the strongest wave energy near the slope at this $\gamma$. Unlike surface roughness, the off-criticality has a small effect on the distribution of energy dissipation. Moreover, surface roughness is also found to change the structure of the scattering-induced mean flow and enhance its strength. The present study provides new perspectives on how the surface roughness on topographic features influences energy dissipation.

We present high-resolution three-dimensional (3-D) direct numerical simulations of breaking waves solving for the two-phase Navier–Stokes equations. We investigate the role of the Reynolds number (Re, wave inertia relative to viscous effects) and Bond number (Bo, wave scale over the capillary length) on the energy, bubble and droplet statistics of strong plunging breakers. We explore the asymptotic regimes at high Re and Bo, and compare with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to 3-D turbulent flow on a time scale that depends on the turbulent Re up to a limiting value $Re_\lambda \sim 100$, consistent with the mixing transition in other canonical turbulent flows. We characterize the role of capillary effects on the impacting jet and ingested main cavity shape and subsequent fragmentation process, and extend the buoyant-energetic scaling from Deike et al. (J. Fluid Mech., vol. 801, 2016, pp. 91–129) to account for the cavity shape and its scale separation from the Hinze scale, $r_H$. We confirm two regimes in the bubble size distribution, $N(r/r_H)\propto (r/r_H)^{-10/3}$ for $r>r_H$, and $\propto (r/r_H)^{-3/2}$ for $r<r_H$. Bubbles are resolved up to one order of magnitude below $r_H$, and we observe a good collapse of the numerical data compared to laboratory breaking waves (Deane & Stokes, Nature, vol. 418 (6900), 2002, pp. 839–844). We resolve droplet statistics at high Bo in good agreement with recent experiments (Erinin et al., Geophys. Res. Lett., vol. 46 (14), 2019, pp. 8244–8251), with a distribution shape close to $N_d(r_d)\propto r_d^{-2}$. The evolution of the droplet statistics appears controlled by the details of the impact process and subsequent splash-up. We discuss velocity distributions for the droplets, finding ejection velocities up to four times the phase speed of the wave, which are produced during the most intense splashing events of the breaking process.

Oftentimes, turbulent flows exhibit a high-frequency turbulent component developing on a strong low-frequency periodic motion. In such cases, the low-frequency motion may strongly influence the spatio-temporal features of the high-frequency component. A typical example of such behaviour is the flow around bluff bodies, for which the high-frequency turbulent component, characterized by Kelvin–Helmholtz structures associated with thin shear layers, depends on the phase of the low-frequency vortex-shedding motion. In this paper, we propose extended versions of spectral proper orthogonal decomposition (SPOD) and of resolvent analysis that respectively extract and reconstruct the high-frequency turbulent fluctuation field as a function of the phase of the low-frequency periodic motion. These approaches are based on a quasi-steady (QS) assumption, which may be justified by the supposedly large separation between the frequencies of the periodic and turbulent components. After discussing their relationship to more classical Floquet-like analyses, the new tools are illustrated on a simple periodically varying linear Ginzburg–Landau model, mimicking the overall characteristics of a turbulent bluff-body flow. In this simple model, we in particular assess the validity of the QS approximation. Then, we consider the case of turbulent flow around a squared-section cylinder at a Reynolds number of $Re=22\,000$, for which we show reasonable agreement between the extracted spatio-temporal fluctuation field and the prediction of QS resolvent analysis at the various phases of the periodic vortex-shedding motion.

Electrolyte solutions play an important role in energy storage devices, whose performance relies heavily on the electrokinetic processes at sub-micron scales. Although fluctuations and stochastic features become more critical at small scales, the long-range Coulomb interactions pose a particular challenge for both theoretical analysis and simulation of fluid systems with fluctuating hydrodynamic and electrostatic interactions. Here, we present a theoretical framework based on the Landau–Lifshitz theory to derive closed-form expressions for fluctuation correlations in electrolyte solutions, indicating significantly different decorrelation processes of ionic concentration fluctuations from hydrodynamic fluctuations, which provides insights for understanding transport phenomena of coupled fluctuating hydrodynamics and electrokinetics. Furthermore, we simulate fluctuating electrokinetic systems using both molecular dynamics (MD) with explicit ions and mesoscopic charged dissipative particle dynamics (cDPD) with semi-implicit ions, from which we identify that the spatial probability density functions of local charge density follow a gamma distribution at sub-nanometre scale (i.e. $0.3\,{\rm nm}$) and converge to a Gaussian distribution above nanometre scales (i.e. $1.55\,{\rm nm}$), indicating the existence of a lower limit of length scale for mesoscale models using Gaussian fluctuations. The temporal correlation functions of both hydrodynamic and electrokinetic fluctuations are computed from all-atom MD and mesoscale cDPD simulations, showing good agreement with the theoretical predictions based on the linearized fluctuating hydrodynamics theory.

The flow of an unbounded Newtonian fluid above a flat solid plate, set into motion suddenly with uniform velocity, is a canonical problem investigated by Stokes (On the effect of the internal friction of fluids on the motion of pendulums, Trans. Camb. Phil. Soc., vol. 9, 1851) and Rayleigh (Lond. Edinb. Dubl. Philos. Mag. J. Sci., vol. 21, issue 126, 1911, pp. 697–711). We tackle the same problem but with the fluid replaced by a yield-stress material of finite height with a free surface; the latter ensures both yielded (fluid-like) and rigid behaviour. The interface between the yielded and rigid regions – the so-called ‘yield surface’ – always emerges from the plate at a rate faster than that for momentum diffusion. The rigid region, adjacent to the free surface, is accelerated by the constant yield stress acting at this yield surface. As the velocity of this region increases following start-up, its acceleration climaxes and subsequently diminishes. In this latter period, the thickness of the rigid region increases owing to relaxation of the velocity gradients and the stress. The yield surface eventually collides with the plate in finite time, after which the entire material moves in concert with the plate as a rigid body. Analytical and numerical results are presented that can form the basis for practical applications. This includes the development of a ‘rheological microscope’ to directly detect and measure the yield stress. The analysis focuses on exploring the flow physics of a Bingham material. This is shown to be similar to that of a general Herschel–Bulkley material with shear-thinning or shear-thickening rheology.