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 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.

The axisymmetric steady two-phase flow of a differentially heated thermocapillary liquid bridge in air and its linear stability is investigated numerically, taking into account dynamic interfacial deformations in the basic flow. Since most experiments require a high temperature difference to drive the flow into the three-dimensional regime, the temperature dependence of the material properties must be taken into account. Three different models are investigated for a high-Prandtl-number thermocapillary liquid bridge with nominal Prandtl number ${\textit {Pr}}=28.8$: the Oberbeck–Boussinesq (OB) approximation, a linear temperature dependence of all material properties and a full nonlinear temperature dependence of all material properties. For all models, critical Reynolds numbers are computed as functions of the volume of the liquid bridge, its aspect ratio, its dimensional size and as a function of the strength of a forced axial flow in the ambient air. Under most circumstances the OB approximation overpredicts and the linear model underpredicts the critical Reynolds number, compared with the model based on the full temperature dependence of the material properties. Among the main influence factors are the proper selection of the reference temperature and, at larger temperature differences, the temperature dependence of the viscosity of the liquid.

In numerical linear stability investigations, the rates of change of the kinetic and thermal energy of the perturbation flow are often used to identify the dominant mechanisms by which kinetic or thermal energy is exchanged between the basic and the perturbation flow. Extending the conventional energy analysis for a single-phase Boussinesq fluid, the energy budgets of arbitrary infinitesimal perturbations to the basic two-phase liquid–gas flow are derived for an axisymmetric thermocapillary bridge when the material parameters in both phases depend on the temperature. This allows identifying individual transport terms and assessing their contributions to the instability if the basic flow and the critical mode are evaluated at criticality. The full closed-form energy budgets of linear modes have been derived for thermocapillary two-phase flow taking into account the temperature dependence of all thermophysical parameters. The influence of different approximations to the temperature dependence on the linear stability boundary of the axisymmetric flow in thermocapillary liquid bridges is tested regarding their accuracy. The general mechanism of symmetry breaking turns out to be very robust.

The motion of a finite-size particle in the cuboidal lid-driven cavity flow is investigated experimentally for Reynolds numbers $100$ and $200$ for which the flow is steady. These steady three-dimensional flows exhibit chaotic and regular streamlines, where the latter are confined to Kolmogorov–Arnold–Moser (KAM) tori. The interaction between the moving wall and the particle creates global particle attractors. For neutrally buoyant particles, these attractors are periodic or quasi-periodic, strongly attracting and located in or near KAM tori of the flow. As the density mismatch between particle and fluid increases, buoyancy and inertia become important, and the attractors evolve from those for neutrally buoyant particles, changing their shape, position and attraction rates.

The linear stability of the axisymmetric steady thermocapillary flow in a liquid bridge made from 2 cSt silicone oil (Prandtl number 28) is investigated numerically in the framework of the Boussinesq approximation. The flow and temperature fields in the surrounding gas phase (air) are taken into account for a generic cylindrical container hosting the liquid bridge. The flows in the liquid and in the gas are fully coupled across the hydrostatically deformed liquid–gas interface, neglecting dynamic interface deformations. Originating from a common reference case, the linear stability boundary is computed varying the length of the liquid bridge (aspect ratio), its volume and the gravity level, providing accurate critical data. The qualitative dependence of the critical threshold on these parameters is explained in terms of the characteristics of the critical mode. The heat exchange between the ambient gas and the liquid bridge that is fully resolved has an important influence on the critical conditions.

The linear stability of the incompressible flow in an infinitely extended cavity with rectangular cross-section is investigated numerically. The basic flow is driven by a lid which moves tangentially, but at yaw with respect to the edges of the cavity. As a result, the basic flow is a superposition of the classical recirculating two-dimensional lid-driven cavity flow orthogonal to a wall-bounded Couette flow. Critical Reynolds numbers computed by linear stability analysis are found to be significantly smaller than data previously reported in the literature. This finding is confirmed by independent nonlinear three-dimensional simulations. The critical Reynolds number as a function of the yaw angle is discussed for representative aspect ratios. Different instability modes are found. Independent of the yaw angle, the dominant instability mechanism is based on the local lift-up process, i.e. by the amplification of streamwise perturbations by advection of basic flow momentum perpendicular to the sheared basic flow. For small yaw angles, the instability is centrifugal, similar as for the classical lid-driven cavity. As the spanwise component of the lid velocity becomes dominant, the vortex structures of the critical mode become elongated in the direction of the bounded Couette flow with the lift-up process becoming even more important. In this case the instability is made possible by the residual recirculating part of the basic flow providing a feedback mechanism between the streamwise vortices and the streamwise velocity perturbations (streaks) they promote. In the limit when the basic flow approaches bounded Couette flow the critical Reynolds number increases very strongly.

The slow motion of a small buoyant sphere near a right dihedral corner made by tangentially sliding walls is investigated. Under creeping-flow conditions the force and torque on the sphere can be decomposed into eleven elementary types of motion involving simple particle translations, particle rotations and wall movements. Force and torque balances are employed to find the velocity and rotation of the particle as functions of its location. Depending on the ratio of the wall velocities and the gravitational settling velocity of the sphere, different dynamical regimes are identified. In particular, a non-trivial line attractor/repeller for the particle motion exists at a location detached from both the walls. The existence, location and stability of the corresponding two-dimensional fixed point are studied depending on the wall velocities and the buoyancy force. The impact of the line attractors/repellers on the motion of small particles in cavities and its relevance for corner cleaning applications are discussed.

The motion of a single spherical particle in a two-sided lid-driven cavity is investigated experimentally. The flow in which the particle moves is created by two facing cavity sidewalls which move with equal velocity in opposite directions. For a long cavity with width-to-height cross-sectional aspect ratio $\varGamma =W/H=1.6$ the flow field at Reynolds number ${Re}=400$ consists of steady spatially periodic three-dimensional convection cells. Nearly neutrally buoyant particles with radius in units of $H$ ranging from $1.1\times 10^{-2}$ to $7.1\times 10^{-2}$ are found to be attracted to periodic or quasi-periodic orbits in close vicinity of Kolmogorov–Arnold–Moser (KAM) tori of the unperturbed flow. Like the KAM tori the attractors of neutrally buoyant particles arise in mirror-symmetric pairs within each convection cell. The particle attractors are created by a dissipative effect in the dynamical system describing the particle motion which arises when the finite-size particle closely passes the moving walls. When the particle density deviates from that of the fluid, inertial attractors arise whose symmetry is broken by buoyancy, and other periodic attractors are created which do not have KAM tori as counterparts.

The steady axisymmetric incompressible flow in a droplet sitting on or hanging from a flat plate is calculated numerically. In the limit of large mean surface tension the liquid–gas interface is spherical which allows the use of boundary-fitted toroidal coordinates. The flow is driven by thermocapillary and buoyant forces induced by a linear variation of the ambient temperature normal to the perfectly conducting wall. We present benchmark-quality results for the streamfunction and temperature fields, varying the contact angle, the thermocapillary Reynolds number, the Prandtl number, the Grashof number and the interfacial heat-transfer coefficient including the latent heat of evaporation. Scaling laws for the strength of the flow are provided for asymptotically large Marangoni numbers.

The structure of the incompressible steady three-dimensional flow in a two-sided anti-symmetrically lid-driven cavity is investigated for an aspect ratio $\unicode[STIX]{x1D6E4}=1.7$ and spanwise-periodic boundary conditions. Flow fields are computed by solving the Navier–Stokes equations with a fully spectral method on $128^{3}$ grid points utilizing second-order asymptotic solutions near the singular corners. The supercritical flow arises in the form of steady rectangular convection cells within which the flow is point symmetric with respect to the cell centre. Global streamline chaos occupying the whole domain is found immediately above the threshold to three-dimensional flow. Beyond a certain Reynolds number the chaotic sea recedes from the interior, giving way to regular islands. The regular Kolmogorov–Arnold–Moser tori grow with increasing Reynolds number before they shrink again to eventually vanish completely. The global chaos at onset is traced back to the existence of one hyperbolic and two elliptic periodic lines in the basic flow. The singular points of the three-dimensional flow which emerge from the periodic lines quickly change such that, for a wide range of supercritical Reynolds number, each periodic convection cell houses a double spiralling-in saddle focus in its centre, a spiralling-out saddle focus on each of the two cell boundaries and two types of saddle limit cycle on the walls. A representative analysis for $\mathit{Re}=500$ shows chaotic streamlines to be due to chaotic tangling of the two-dimensional stable manifold of the central spiralling-in saddle focus and the two-dimensional unstable manifold of the central wall limit cycle. Embedded Kolmogorov–Arnold–Moser tori and the associated closed streamlines are computed for several supercritical Reynolds numbers owing to their importance for particle transport.

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

The two-dimensional, incompressible flow in a plane sudden expansion is investigated numerically for a systematic variation of the geometry, covering expansion ratios (steps-to-outlet heights) from to . By means of a three-dimensional linear stability analysis global temporal modes are scrutinized. In a symmetric expansion the primary bifurcation is stationary and two-dimensional, breaking the mirror symmetry with respect to the mid-plane. The secondary asymmetric flow experiences a secondary instability to different three-dimensional modes, depending on the expansion ratio. For a moderately asymmetric expansion only one of the two secondary flows (the connected branch) is realized at low Reynolds numbers. Since the perturbed secondary flow does not deviate much from the symmetric secondary flow, both secondary stability boundaries are very close to each other. For very small and very large expansion ratios an asymptotic behaviour is found for suitably scaled critical Reynolds numbers and wavenumbers. Representative instabilities are analysed in detail using an a posteriori energy transfer analysis to reveal the physical nature of the instabilities. Depending on the geometry, pure centrifugal and elliptical amplification processes are identified. We also find that the basic flow can become unstable due to the effects of flow deceleration, streamline convergence and high shear stresses, respectively.

The global, temporal stability of the two-dimensional, incompressible flow over a forward-facing step in a plane channel is investigated numerically. The geometry is varied systematically covering constriction ratios (step-to-inlet height) from 0.23 to 0.965. A three-dimensional linear stability analysis shows that the stability boundary is a smooth continuous function of the constriction ratio. If the critical Reynolds and wavenumbers are scaled appropriately, they approach a linear asymptotic behaviour for large step heights. The critical mode is found to be stationary and confined to the region of separated flow downstream of the step for all constriction ratios. An energy-transfer analysis reveals that the basic flow becomes unstable due to a combined effect involving lift-up and flow deceleration, leading to a critical mode exhibiting steady streaks. Moreover, the receptivity of the flow to initial as well as to structural perturbations is studied by means of an adjoint analysis.

The two-dimensional, incompressible flow over a backward-facing step is considered for a systematic variation of the geometry covering expansion ratios (step to outlet height) from 0.25 to 0.975. A global temporal linear stability analysis shows that the basic flow becomes unstable to different three-dimensional modes depending on the expansion ratio. All critical modes are essentially confined to the region behind the step extending downstream up to the reattachment point of the separated eddy. An energy-transfer analysis is applied to understand the physical nature of the instabilities. If scaled appropriately, the critical Reynolds number approaches a finite asymptotic value for very large step heights. In that case centrifugal forces destabilize the flow with respect to an oscillatory critical mode. For moderately large expansion ratios an elliptical instability mechanism is identified. If the step height is further decreased the critical mode changes from oscillatory to stationary. In addition to the elliptical mechanism, the strong shear in the layer emanating from the sharp corner of the step supports the amplification process of the critical mode. For very small step heights the basic state becomes unstable due to the lift-up mechanism, which feeds back on itself via the recirculating eddy behind the step, resulting in a steady critical mode comprising pronounced slow and fast streaks.

The dynamics of thermocapillary flows in differentially heated cylindrical liquid bridges is investigated numerically using a mixed finite volume/pseudo-spectral method to solve the Navier–Stokes equations in the Boussinesq approximation. For large Prandtl numbers (Pr = 4 and 7) and sufficiently high Reynolds numbers, the axisymmetric basic flow is unstable to three-dimensional hydrothermal waves. Finite-amplitude azimuthally standing waves are found to decay to travelling waves. Close to the critical Reynolds number, the former may persist for long times. Representative results are explained by computing the coefficients in the Ginzburg–Landau equations for the nonlinear evolution of these waves for a specific set of parameters. We investigate the nonlinear phenomena characteristic of standing and pure travelling waves, including azimuthal mean flow and time-dependent convective heat transport. For Pr [Lt ] 1 the first transition from the two-dimensional basic flow to the three-dimensional stationary flow is inertial in nature. Particular attention is paid to the secondary transition leading to oscillatory three-dimensional flow, and this mechanism is likewise independent of Pr. The spatial and temporal structure of the perturbation flow is analysed in detail and an instability mechanism is proposed based on energy balance calculations and the vorticity distribution.