The growth of perturbations subject to the Crow instability along two vortex rings of equal and opposite circulation undergoing a head-on collision is examined. Unlike the planar case for semi-infinite line vortices, the zero-order geometry of the flow (i.e. the ring radius, core thickness and separation distance) and by extension the growth rates of perturbations vary in time. The governing equations are therefore temporally integrated to characterize the perturbation spectrum. The analysis, which considers the effects of ring curvature and the distribution of vorticity within the vortex cores, explains several key flow features observed in experiments. First, the zero-order motion of the rings is accurately reproduced. Next, the predicted emergent wavenumber, which sets the number of secondary vortex structures emerging after the cores come into contact, agrees with experiments, including the observed increase in the number of secondary structures with increasing Reynolds number. Finally, the analysis predicts an abrupt transition at a critical Reynolds number to a regime dominated by a higher-frequency, faster-growing instability mode that may be consistent with the experimentally observed rapid generation of a turbulent puff following the collision of rings at high Reynolds numbers.

The observed behaviour of passive objects in simple flows can be surprisingly intricate, and is complicated further by object activity. Inspired by the motility of bacterial swimmers, in this two-part study we examine the three-dimensional motion of rigid active particles in shear Stokes flow, focusing on bodies that induce rapid rotation as part of their activity. In Part 1 we develop a multiscale framework to investigate these emergent dynamics and apply it to simple spheroidal objects. In Part 2 (Dalwadi et al., J. Fluid Mech., vol. 979, 2024, A2) we apply our framework to understand the emergent dynamics of more complex shapes; helicoidal objects with chirality. Via a multiple scales asymptotic analysis for nonlinear systems, we systematically derive emergent equations of motion for long-term trajectories that explicitly account for the strong (leading-order) effects of fast spinning. Supported by numerical examples, we constructively link these effective dynamics to the well-known Jeffery's orbits for passive spheroids, deriving an explicit closed-form expression for the effective shape of the active particle, broadening the scope of Jeffery's seminal study to spinning spheroids.

In this two-part study, we investigate the motion of rigid, active objects in shear Stokes flow, focusing on bodies that induce rapid rotation as part of their activity. In Part 2, we derive and analyse governing equations for rapidly spinning complex-shaped particles – general helicoidal objects with chirality. Using the multiscale framework that we develop in Part 1 (Dalwadi et al., J. Fluid Mech., vol. 979, 2024, A1), we systematically derive emergent equations of motion for the angular and translational dynamics of these chiral spinning objects. We show that the emergent dynamics due to rapid rotation can be described by effective generalised Jeffery's equations, which differ from the classic versions via the inclusion of additional terms that account for chirality and other asymmetries. Furthermore, we use our analytic results to characterise and quantify the explicit effect of rotation on the effective hydrodynamic shape of the chiral objects, expanding significantly the scope of Jeffery's seminal study.

The progressive destabilisation of the incompressible flow in a cubical cavity driven by a constant shear stress is investigated numerically. To that end, one of the square faces of the cube is subjected to a constant shear stress parallel to two opposite edges of that face. The three-dimensional steady basic flow loses its mirror symmetry through a supercritical pitchfork bifurcation leading to a pair of steady stable non-symmetric flow states that are antisymmetric to each other. Upon increase of the strength of the driving, these non-symmetric equilibria become unstable via a Hopf bifurcation resulting in two limit cycles. The bifurcations are investigated using classical linear stability analyses as well as nonlinear simulations. Upon a further increase of the driving shear stress, the limit cycles destabilise through bursts triggering a complex interplay between the unstable equilibria. The transition to turbulence resembles the Pomeau–Manneville scenario.

The fundamental resonance (FR) in the nonlinear phase of the boundary-layer transition to turbulence appears when a dominant planar instability mode reaches a finite amplitude and the low-amplitude oblique travelling modes with the same frequency as the dominant mode, together with the stationary streak modes, undergo the strongest amplification among all the Fourier components. This regime may be the most efficient means to trigger the natural transition in hypersonic boundary layers. In this paper, we aim to reveal the intrinsic mechanism of the FR in the weakly nonlinear framework based on the large-Reynolds-number asymptotic technique. It is found that the FR is, in principle, a triad resonance among a dominant planar fundamental mode, a streak mode and an oblique mode. In the major part of the boundary layer, the nonlinear interaction of the fundamental mode and the streak mode seeds the growth of the oblique mode, whereas the interaction of the oblique mode and the fundamental mode drives the roll components (transverse and lateral velocity) of the streak mode, which leads to a stronger amplification of the streamwise component of the streak mode due to the lift-up mechanism. This asymptotic analysis clearly shows that the dimensionless growth rates of the streak and oblique modes are the same order of magnitude as the dimensionless amplitude of the fundamental mode $(\bar {\epsilon }_{10})$, and the amplitude of the streak mode is $O(\bar {\epsilon }_{10}^{-1})$ greater than that of the oblique mode. The main-layer solution of the streamwise velocity, spanwise velocity and temperature of both the streak and the oblique modes become singular as the wall is approached, and so a viscous wall layer appears underneath. The wall layer produces an outflux velocity to the main-layer solution, inclusion of which leads to an improved asymptotic theory whose accuracy is confirmed by comparing with the calculations of the nonlinear parabolised stability equations (NPSEs) at moderate Reynolds numbers and the secondary instability analysis (SIA) at sufficiently high Reynolds numbers.

The linear stability of the axisymmetric steady flow in a thermocapillary liquid bridge made from 2-cSt silicone oil $({Pr}=28)$ is investigated numerically. The liquid bridge is heated either from above or below and exposed to an axial air flow which is confined to a concentric tube surrounding the bridge. At the annular inlet, the air flow is fully developed and has the same temperature as the adjacent support rod. Using an extended Oberbeck–Boussinesq approximation in which the density of both fluids depends linearly on the temperature in all equations, critical thermocapillary Reynolds numbers are obtained depending on the strength of the imposed axial air flow. The critical conditions are sensitive with respect to the direction of a weak air flow, because the air flow changes the plateau value of the interfacial temperature midway between the hot and cold ends. For stronger air flow the critical thermocapillary Reynolds number almost saturates at moderate values. Throughout, the instability arises as a hydrothermal wave with the gas phase being passive. The dynamic interface deformations for axisymmetric flow caused by the thermocapillarity flow in the liquid and by the stresses from the air flow are considered separately. Apart from turning points of the critical curve, the impact of dynamic surface deformations on the critical thermocapillary Reynolds number is moderate.

Capillary folding is the process of folding planar objects into three-dimensional (3-D) structures using capillary force. It has many important industrial applications, e.g. the fabrication of microelectromechanical systems. In this work, we propose a 3-D model for the capillary folding of thin elastic sheets with pinned contact lines. The energy of the system consists of interfacial energies between the different phases and the elastic energy of the sheet. The latter is described by the nonlinear Koiter's model, which can accommodate large deformations of the sheet. From the energy, we derive the governing equations for the static system using a variational approach. We then develop a numerical method to find equilibrium solutions via a relaxation dynamics. At each time step, we evolve the sheet by using the subdivision element method, and update the droplet surface by minimizing a squared area functional using linear finite elements. Qualitatively, numerical solutions for the equilibrium configurations of the sheet–droplet system agree well with those observed in experiments. Furthermore, we identify the critical bendabilities and present bifurcation diagrams for the folding of a triangular sheet. The results exhibit rich and fully 3-D behaviours that were not captured by previous two-dimensional models. Our results provide new insights into the nonlinear process of capillary folding, and may contribute to the advancement of microfabrication techniques.

This study reveals the competitive evolutionary process of the main driving factors in the early, middle and late stages of sandstorms, as shear turbulence becomes dominant and is then suppressed by enhanced thermal stability, based on quadrant analysis of the sand-laden turbulent wind field acquired from field observations over the entire sandstorm process. Moreover, the self-organized state of multiscale structures in the energy-containing region of the sand-laden turbulence is found to change significantly as the sandstorm develops. The logarithmic scaling law that governs the cumulative turbulent kinetic energy for the non-stationary flow in the early and late stages of the sandstorm is different from the existing theoretical formula. The corresponding rate of increase in the cumulative kinetic energy with increasing scale is much higher in these stages than in the middle stage of the sandstorm with steady flow. The change in self-organized state of turbulence is responsible for the flow acceleration and the thermal superimposed effect, rather than the addition of sand particles.

When a liquid jet plunges into a pool, it can generate a bubble-laden jet flow underneath the surface. This common and simple phenomenon is investigated experimentally for circular jets to illustrate and quantify the role played by the net gas/liquid void fraction on the maximum bubble penetration depth. It is first shown that an increase in either the impact diameter or the jet fall height to diameter ratio at constant impact momentum leads to a reduction in the bubble cloud size. By measuring systematically the local void fraction using optical probes in the biphasic jet, it is then demonstrated that this effect is a direct consequence of the increase in air content within the cloud. A simple momentum balance model, including only inertia and the buoyancy force, is shown to predict the bubble cloud depth without any fitting parameters. Finally, a Froude number based on the bubble terminal velocity, the cloud depth and also the net void fraction is introduced to propose a simple criterion for the threshold between the inertia-dominated and buoyancy-dominated regimes.

Analytical expressions for the velocity field and the effective slip length of pressure-driven Stokes flow through slippery pipes and annuli with rotationally symmetrical longitudinal slits are derived. Specifically, the developed models incorporate a finite local slip length and constant shear stress along the slits, and thus go beyond the assumption of perfect slip employed commonly for superhydrophobic surfaces. Thereby, they provide the possibility to assess the influence of both the viscosity of the air or other fluid that is modelled to fill the slits as well as the influence of the micro-geometry of these slits. First, expressions for tubes and annular pipes with superhydrophobic or slippery walls are provided. Second, these solutions are combined to a tube-within-a circular-pipe scenario, where one fluid domain provides a slip to the other. This scenario is interesting as an application to achieve stable fluid–fluid interfaces. With respect to modelling, it illustrates the specification of the local slip length depending on a linked flow field. The comparison of the analytically calculated solutions with numerical simulations shows excellent agreement. The results of this paper thus represent an important instrument for the design and optimization of slippage along surfaces in circular geometries.

We study the capillary rise of viscous liquids into sharp corners formed by two surfaces whose geometry is described by power laws $h_i(x) = c_i x^n$, $i = 1,2$, where $c_2 > c_1$ for $n \geq 1$. Prior investigations of capillary rise in sharp corners have shown that the meniscus altitude increases with time as $t^{1/3}$, a result that is universal, i.e. applies to all corner geometries. The universality of the phenomenon of capillary rise in sharp corners is revisited in this work through the analysis of a partial differential equation for the evolution of a liquid column rising into power-law-shaped corners, which is derived using lubrication theory. Despite the lack of geometric similarity of the liquid column cross-section for $n>1$, there exist a scaling and a similarity transformation that are independent of $c_i$ and $n$, which gives rise to the universal $t^{1/3}$ power law for capillary rise. However, the prefactor, which corresponds to the tip altitude of the self-similar solution, is a function of $n$, and it is shown to be bounded and monotonically decreasing as $n\to \infty$. Accordingly, the profile of the interface radius as a function of altitude is also independent of $c_i$ and exhibits slight variations with $n$. Theoretical results are compared against experimental measurements of the time evolution of the tip altitude and of profiles of the interface radius as a function of altitude.