Dispersion is a common phenomenon in miscible displacement flows. In the primary cementing process displacement takes place in a narrow eccentric annulus. Both turbulent Taylor dispersion and laminar advective dispersion occur, depending on flow regime. Since dispersion can cause mixing and contamination close to the displacement front, it is essential to understand and quantify. The usual modelling approach is a form of Hele-Shaw model in which quantities are averaged across the narrow annular gap: a so-called two-dimensional narrow gap (2DGA) model. Zhang & Frigaard (J. Fluid Mech., vol. 947, 2022, A732), introduced a dispersive two-dimensional gap-averaged (D2DGA) model for displacement of two Newtonian fluids, by modifying the earlier 2DGA model. This brings a significant improvement in revealing physical phenomena observed experimentally and in three-dimensional computations, but is limited to Newtonian fluids. In this study we adapt the D2DGA model approach for two Herschel–Bulkley fluids. We first obtain weak velocity solutions using the augmented Lagrangian method, while keeping the same two-layer flow assumption as the Newtonian D2DGA model. These solutions are then used to define closure relationships that are needed to compute the dispersive two-dimensional flows. Results reveal that the modified version of the D2DGA model can now predict expected frontal behaviours for two Herschel–Bulkley fluids, revealing dispersion, frontal shock, spike and static wall layer solutions. We then explore the displacement behaviour in more detail by investigating the impact of rheological properties and buoyancy on the mobility of fluids in a planar frontal displacement flow and their vulnerability to fingering-type instabilities. As the underlying flows are dispersive, our analysis reveals three distinct behaviours: (i) stable, (ii) partial penetration of the dispersing front, and (iii) unstable regimes. We explore these regimes and how they are affected by the two fluid rheologies.

We numerically investigate the cellular detonation dynamics in ethylene/oxygen/ozone/nitrogen mixtures considering detailed chemical kinetics. The aim is to elucidate emergent detonation structures and reveal the transition mechanism from single- to double-cellular structures. Ozone is used to induce two-stage reactions within the mixture. Through systematic initiation strength analysis, we demonstrate two distinct propagation regimes: (i) under strong initiation, a stable double-cellular detonation is established; (ii) weak initiation triggers a multi-stage evolutionary process, beginning with a low-speed single-cellular detonation in the initiation zone. During the initial weak stage, the detonation propagates at a quasi-steady velocity with uniform cellular patterning. The subsequent transition phase features spontaneous acceleration accompanied by structural bifurcation into double cells, ultimately stabilising in a normal stage with sustained double-cellular structures. Further analysis reveals that the weak-stage dynamics is governed exclusively by first-stage chemical reactions, resulting in a single-cellular structure propagating at a velocity much lower than the Chapman–Jouguet speed. In contrast, the double-cellular structure observed at the normal stage results from the two-stage exothermic reactions. Thermodynamic perturbations arising from cellular instability and fluid dynamic instability are identified as critical drivers for the transition from single- to double-cellular detonation. Besides, conditions for the formation of double-cellular detonation are explored, and two qualitative requirements are summarised: the reactions of the two stages must proceed as independently as possible, and both heat releases from the two stages must be high enough to sustain the triple-shock configurations.

Microswimming cells and robots exhibit diverse behaviours due to both their swimming and their environment. One key environmental feature is the presence of a background flow. While the influences of select flows, particularly steady shear flows, have been extensively investigated, these only represent special cases. Here, we examine inertialess swimmers in more general flows, specifically general linear planar flows that may possess rapid oscillations, and impose weak symmetry constraints on the swimmer (ensuring planarity, for instance). We focus on swimmers that are inefficient, in that the time scales of their movement are well separated from those associated with their motility-driving deformation. Exploiting this separation of scales in a multiple-time-scale analysis, we find that the behaviour of the swimmer is dictated by two effective parameter groupings, excluding mathematically precise edge cases. These systematically derived parameters measure balances between angular velocity and the rate of strain of the background flow. Remarkably, one parameter governs the orientational dynamics, whilst the other completely captures translational motion. Further, we find that the long-time translational dynamics is solely determined by properties of the flow, independent of the details of the swimmer. This illustrates the limited extent to which, and how, microswimmers may control their behaviours in planar linear flows.

Monitoring fluid flow and pollutant transport is important in many geophysical, environmental and industrial processes, such as geological $\textrm {CO}_2$ sequestration, waste water disposal, oil and gas recovery and sea water invasion. But it can also be challenging. Recent studies revealed a series of self-similar solutions to describe the interface shape evolution between the injecting and the ambient fluids during fluid injection into a confined porous layer. The present work focuses further on the pressure evolution. In particular, we present self-similar solutions for the pressure evolution at both the early and late times. Two dimensionless parameters are recognised, including the viscosity ratio $M$ and the rescaled buoyancy $G$, and their specific role on the pressure evolution is clarified. Laboratory experiments are also performed to measure the pressure evolution at two specific locations during the propagation of a viscous gravity current within a vertically placed Hele-Shaw cell, with a favourable comparison with the model prediction in the unconfined regime. The obtained pressure solutions are also used to explain the field data of bottom-hole-pressure (BHP) evolution from a geological $\textrm {CO}_2$ sequestration project, considering both fluid injection and shut-in operations. The model and solutions might also be of use to assess reservoir injectivity and develop pressure-based monitoring technologies at well bores.

Spatially evolving turbulent/turbulent interfaces (TTIs) in the absence of mean shear are studied using direct numerical simulation (DNS). To this end, a novel approach was developed, allowing for six different TTIs to be created with a Taylor-based Reynolds number in the range of $146 \lesssim {Re}_{\lambda }\lesssim 296$. The analysis of classical statistics of turbulence intensity, fluctuating vorticity and integral length scale clearly indicates that one of the two distinct turbulent regions bounding the interface tends to dominate the other one. The half-width thickness is found to be dependent on the turbulent properties of each layer, ultimately suggesting that the large-scale quantities dictate the spreading of each turbulent region. Small scale quantities, e.g. the enstrophy, exhibit an universal conditional mean profile when normalised by the local Kolmogorov (velocity and time) scales of motion. In contrast, the large-scale properties of the flow do not modify the enstrophy statistics. Additionally, when taking the difference of fluctuating vorticity levels on each layer ad extremum, profiles typical of turbulent/non-turbulent interfaces (TNTIs) are observed. The budget terms of enstrophy and rate-of-strain magnitude support these findings.

The Stokes boundary layer (SBL) is the oscillating flow above a flat plate. Its laminar flow becomes linearly unstable at a Reynolds number of $\textit{Re} = U_0 \sqrt {T_0/\nu } \approx 2511$, where $U_0$ is the amplitude of the oscillation, $T_0$ is the period of oscillation and $\nu$ is the fluid’s kinematic viscosity, but turbulence is observed subcritically for $\textit{Re} \gtrsim 700$. The state space consists of laminar and turbulent basins of attraction, separated by a saddle point (the ‘edge state’) and its stable manifold (the ‘edge’). This work presents the edge trajectories for the transitional regime of the SBL. Despite linear dynamics disallowing the lift-up mechanism in the laminar SBL, edge trajectories are dominated by coherent structures as in other canonical shear flows: streaks, rolls and waves. Stokes boundary layer structures are inherently periodic, interacting with the oscillating flow in a novel way: streaks form near the plate, migrate upward at a speed $2\sqrt {\pi }$ and dissipate. A streak-roll-wave decomposition reveals a spatiotemporally evolving version of the self-sustaining process (SSP): (i) rolls lift fluid near the plate, generating streaks (via the lift-up mechanism); (ii) streaks can only persist in regions with the same sign of laminar shear as when they were created, defining regions that moves upward at a speed $2 \sqrt {\pi }$; (iii) the sign of streak production reverses at a roll stagnation point, destroying the streak and generating waves; (iv) trapped waves reinforce the rolls via Reynolds stresses; (v) mass conservation reinforces the rolls. This periodic SSP highlights the role of flow oscillations in sustaining transitional structures in the SBL, providing an alternative picture to ‘bypass’ transition, which relies on pre-existing free stream turbulence and spanwise vortices.

Amphibious unmanned vehicles promise next-generation water-based missions by eliminating the need for multiple vehicles to traverse water and air separately. Existing research-grade quadrotors can navigate in water and air and cross the water–air boundary, but it remains unclear how their transition is affected by rotor kinematics and geometry. We present here experimental results from isolated small rotors (diameters $\sim 10\,\mathrm{cm}$) dynamically transitioning from water to air. We discovered that rotors experience an abrupt change in frequency, lift and torque before reaching the interface, and the change is linked to the surface depression caused by a free surface vortex. We explored how the surface dynamics are affected by advance ratio, rotor diameter, number of rotor blades and input throttle. Free surface vortices above rotating objects have been studied in the context of unbaffled stirred tanks, but not in the field of small amphibious rotorcraft. We show that existing free surface vortex models can be adapted to explain water-to-air rotor performance. A better understanding of water–air rotor transitions helps to (i) assess the amphibious capability of existing aerial rotors, and (ii) suggest efficient water–air transition strategies for next-generation amphibious vehicles.

The integration of electro-osmotic effect to the underlying flow enhances solute dispersion precision in microfluidic systems, which is crucial for applications such as drug delivery and on-chip fluidic functionalities. We investigate, in this study, the solute dispersion characteristics of couple-stress fluids in a two-dimensional microchannel configuration under the combined effects of electro-osmotic actuation and applied pressure gradients. We consider both homogeneous and heterogeneous reactions in the present analysis. Couple-stress fluids, which account for additional stresses due to the presence of the microstructures in the fluids, offer a more accurate model to describe the rheological behaviour of biofluids. While previous studies have addressed longitudinal Gaussianity and transverse uniformity of solute distribution, we focus uniquely in this endeavour on longitudinal uniformity. Using Mei’s multiscale homogenisation technique, we solve a two-dimensional convection–diffusion model, extending it to third-order approximation to analyse the dispersion coefficient, concentration profiles, and variation rates of concentration within microchannel flow. Results show that forcing and couple-stress parameters enhance the gradients of the longitudinal variation rate, while boundary absorption reduces this variation rate near the walls. The couple-stress parameter exhibits dual behaviour: initially, it enhances solute dispersion, but beyond a certain value of couple-stress parameter $B_{cr}$ (which depends on forcing comparison and the Debye–Hückel parameter), it reduces dispersion. In the absence of pressure, solute distribution remains longitudinally uniform. However, as the pressure gradient increases, concentration levels drop sharply, and the distribution shifts to a parabolic profile, underscoring the significant influence of pressure on flow behaviour in electro-osmotic flow.

In 1910, Cunningham developed a heuristic expression to predict the drag on a slow-moving spherical particle in a gas; a drag that deviates from Stokes’ law when the particle’s size is comparable to the gas’s mean free path. More than a decade later, Millikan proposed a physical argument for correcting Cunningham’s work: the resulting expression is known today as the ‘Cunningham correction factor’. Despite his contribution, Millikan missed a simpler way to correct Cunningham’s expression, one that would have preserved its generality. In this article, this new, simpler form of the Cunningham correction factor is expanded to provide a predictive heuristic for non-spherical particles through the definition of a ‘correction tensor’. Its accuracy is tested against experiments and kinetic theory for the sphere, and solutions to the Boltzmann equation for a range of spheroids and an infinitesimally thin circular disc.

Supersonic turbulent channels subjected to sudden spanwise acceleration at initial friction Reynolds numbers of approximately 500 and different Mach numbers are studied through direct numerical simulations. The response to the spanwise acceleration creates a transient period where the flow exhibits three-dimensionality in the mean statistics. This enables a detailed study of the thermal transport and development of velocity transformations and Reynolds analogies for compressible turbulent flows in swept-like conditions. Extensions of velocity transformations to three-dimensional (3-D) flows demonstrate near-wall self-similarity of the velocity, providing evidence for Morkovin’s hypothesis in non-equilibrium conditions. A similarity solution for the spanwise velocity, valid during the initial transient, is also presented. During the transient, both the thermal fluctuations and turbulent kinetic energy (TKE) decrease, consistent with previous observations in incompressible flows (Lozano-Durán et al. 2020 J. Fluid Mech. 883, A20, Moin et al. 1990 Phys. Fluids A: Fluid Dyn. 2, 1846–1853). For sufficiently strong spanwise acceleration, $Q_{3}$ $(+T',+v')$ and $Q_{1}$ $(-T',-v')$ events become more significant than sweep and ejections across the channel, creating changes in sign in the velocity–temperature covariances. The temporal evolution of the orientation and sizes of the TKE and temperature-carrying structures is quantified through structure identification and spectra. Finally, the generalized Reynolds analogy (Zhang et al. 2012 Phys. Rev. Lett. 109, 054502) is derived for a transient 3-D flow, allowing predictions of the mean temperature from the velocity.

The propagation of detonations in a non-uniform mixture exhibits notable distinctions from that in a uniform mixture. This study first delves into the analytical analysis of the one-dimensional shock transmission problem and the two-dimensional shock propagation in a mixture with temperature non-uniformity. Additionally, the research extends to the numerical simulation of the propagation of shocks and detonations, building upon the insights garnered from the analytical analysis. The numerical results indicate that introducing a temperature interface in a non-uniform gas creates a discrete flow field and wavefront, resulting in oblique shocks that connect hot and cold layers. A competitive mechanism between the transverse waves and non-uniformity is responsible for the detonation propagation. The temperature amplitude tends to inhibit the propagation of transverse waves. In contrast, the wavelengths primarily affect the spacing and strength of these transverse waves, especially during the early stages of propagation. In a Zel’Dovich–von Neumann–Döring detonation, the non-uniformities distort the detonation front, creating transverse wave spacings comparable to the wavelength and reducing the front velocity. However, the detonation can recover its Chapman–Jouguet velocity and approach a steady states as intrinsic instabilities come into play. In the steady state, the cell sizes are found to be determined by the temperature amplitude. When the temperature amplitude is sufficiently high, the detonation cells effectively disappear.

We study the stability of plane Poiseuille flow (PPF) and plane Couette flow (PCF) subject to streamwise system rotation using linear stability analysis and direct numerical simulations. The linear stability analysis reveals two asymptotic regimes depending on the non-dimensional rotation rate ($\textit{Ro}$): a low-$\textit{Ro}$ and a high-$\textit{Ro}$ regime. In the low-$\textit{Ro}$ regime, the critical Reynolds number $\textit{Re}_c$ and critical streamwise wavenumber $\alpha _c$ are proportional to $\textit{Ro}$, while the critical spanwise wavenumber $\beta _c$ is constant. In the high-$\textit{Ro}$ regime, as $\textit{Ro} \rightarrow \infty$, we find $\textit{Re}_c = 66.45$ and $\beta _c = 2.459$ for streamwise-rotating PPF, and $\textit{Re}_c = 20.66$ and $\beta _c = 1.558$ for streamwise-rotating PCF, with $\alpha _c\propto 1/Ro$. Our results for streamwise-rotating PPF match previous findings by Masuda et al. (J. Fluid Mech., vol. 603, 2008, pp. 189–206). Interestingly, the critical values of $\beta _c$ and $\textit{Re}_c$ at $\textit{Ro} \rightarrow \infty$ in streamwise-rotating PPF and PCF coincide with the minimum $\textit{Re}_c$ reported by Lezius & Johnston (J. Fluid Mech., vol. 77, 1976, pp. 153–176) and Wall & Nagata (J. Fluid Mech., vol. 564, 2006, pp. 25–55) for spanwise-rotating PPF at $\textit{Ro}=0.3366$ and PCF at $\textit{Ro}=0.5$. We explain this similarity through an analysis of the perturbation equations. Consequently, the linear stability of streamwise-rotating PCF at large $\textit{Ro}$ is closely related to that of spanwise-rotating PCF and Rayleigh–Bénard convection, with $\textit{Re}_c = \sqrt {Ra_c}/2$, where $Ra_c$ is the critical Rayleigh number. To explore the potential for subcritical transitions, direct numerical simulations were performed. At low $\textit{Ro}$, a subcritical transition regime emerges, characterised by large-scale turbulent–laminar patterns in streamwise-rotating PPF and PCF. However, at higher $\textit{Ro}$, subcritical transitions do not occur and the flow relaminarises for $\textit{Re} \lt Re_c$. Furthermore, we identify a narrow $\textit{Ro}$ range where turbulent–laminar patterns develop under supercritical conditions.