In this study, we analyse ‘magneto-Stokes’ flow, a fundamental magnetohydrodynamic (MHD) flow that shares the cylindrical-annular geometry of the Taylor–Couette cell but uses applied electromagnetic forces to circulate a free-surface layer of electrolyte at low Reynolds numbers. The first complete, analytical solution for time-dependent magneto-Stokes flow is presented and validated with coupled laboratory and numerical experiments. Three regimes are distinguished (shallow-layer, transitional and deep-layer flow regimes), and their influence on the efficiency of microscale mixing is clarified. The solution in the shallow-layer limit belongs to a newly identified class of MHD potential flows, and thus induces mixing without the aid of axial vorticity. We show that these shallow-layer magneto-Stokes flows can still augment mixing in distinct Taylor dispersion and advection-dominated mixing regimes. The existence of enhanced mixing across all three distinguished flow regimes is predicted by asymptotic scaling laws and supported by three-dimensional numerical simulations. Mixing enhancement is initiated with the least electromagnetic forcing in channels with order-unity depth-to-gap-width ratios. If the strength of the electromagnetic forcing is not a constraint, then shallow-layer flows can still yield the shortest mixing times in the advection-dominated limit. Our robust description of momentum evolution and mixing of passive tracers makes the annular magneto-Stokes system fit for use as an MHD reference flow.

Lozano-Duran et al. (J. Fluid Mech., vol. 914, 2021, p. A8) have recently identified the ability of streamwise-averaged turbulent streak fields ${\mathcal {U}}(y,z,t)\hat {\boldsymbol {x}}$ in minimal channels to produce short-term transient growth as the key linear mechanism needed to sustain turbulence at $Re_{\tau }=180$. Here, in an attempt to extend this result to larger domains and higher $Re_{\tau }$, we model this streak transient growth as a two-stage linear process by first selecting the dominant streak structure expected to emerge over the eddy turnover time on the turbulent mean profile $U(y)\hat {\boldsymbol {x}}$, and then examining the secondary growth on this (frozen) streak field ${\mathcal {U}}(y,z)\hat {\boldsymbol {x}}$. Choosing the mean streak amplitude and eddy turnover time consistent with simulations captures the growth thresholds found by Lozano-Duran et al. (2021) for sustained turbulence. In a larger domain at $Re_{\tau }=180$, the most energetic near-wall streaks observed in simulations are close to the predicted optimal streaks. This most energetic streak spacing, approaches the optimal streak at $Re_{\tau }=550$ where the secondary growth possible on each also comes together. A key prediction from the model is that the threshold transient growth required to sustain turbulence decreases with increasing $Re_{\tau }$. More fundamentally, the work of Lozano-Duran et al. (2021) and our results suggest a subtle but significant revision of Malkus's (J. Fluid Mech., vol. 1, 1956, pp. 521–539) classic hypothesis concerning realisable turbulent mean profiles. The key property for a realisable turbulent mean profile could be the ability to generate sufficient short-term transient growth rather than dependence on its (long-term) linear stability characteristics, which was Malkus's original idea.

We explore the drawing of a shear-thinning or shear-thickening thread with an axisymmetric hole that evolves due to axial drawing, inertia and surface tension effects. The stress is assumed to be proportional to the shear rate raised to the $n$th power. The presence of non-Newtonian rheology and surface tension forces acting on the hole introduces radial pressure gradients that make the derivation of long-wavelength equations significantly more challenging than either a Newtonian thread with a hole or shear-thinning and shear-thickening threads without a hole. In the case of weak surface tension, we determine the steady-state profiles. Our results show that for negligible inertia the hole size at the exit becomes smaller as $n$ is decreased (i.e. strong shear-thinning effects) above a critical draw ratio, but surprisingly the opposite is true below this critical draw ratio. We determine an accurate estimate of the critical draw ratio and also discuss how inertia affects this process. We further show that the dynamics of hole closure is dominated by a different limit, and we determine the asymptotic forms of the hole closure process for shear-thinning and shear-thickening fluids with inertia. A linear instability analysis is conducted to predict the onset of draw resonance. We show that increased shear thinning, surface tension and inlet hole size all act to destabilise the flow. We also show that increasing shear-thinning effects reduce the critical Reynolds number required for unconditional stability. Our study provides valuable insights into the drawing process and its dependence on the physical effects.

The asymmetric instability in two streamwise orthogonal planes for three-dimensional flow-induced vibration (FIV) of an elastically mounted cube at a moderate Reynolds number of 300 is numerically investigated in this paper. The full-order computational fluid dynamics method, data-driven stability analysis via the eigensystem realization algorithm and the selective frequency damping method and total dynamic mode decomposition (TDMD) are applied here to explore this problem. Due to the unsteady non-axisymmetric wakefield formed for flow passing a stationary cube, the FIV response was found to exhibit separate structural stability and oscillations (including lock-in and galloping behaviour) in the two different streamwise orthogonal planes while the body is released. The initial kinetic energy accompanying the release of the cube could destabilize the above-mentioned structural stability. The observed FIV asymmetric instability is verified by the root trajectory of the structural mode obtained via data-driven stability analysis. The stability of the structural modes dominates regardless of whether the structural response oscillates significantly in various (reduced) velocity ranges. Further TDMD analysis on the wake structure, accompanied by the time–frequency spectrum of time-history structural displacements, suggested that the present FIV unit with galloping behaviour is dominated by the combination of the shifted base-flow mode, structure modes and several harmonics of the wake mode.

We propose a data-driven methodology to learn a low-dimensional manifold of controlled flows. The starting point is resolving snapshot flow data for a representative ensemble of actuations. Key enablers for the actuation manifold are isometric mapping as encoder, and a combination of a neural network and a $k$-nearest-neighbour interpolation as decoder. This methodology is tested for the fluidic pinball, a cluster of three parallel cylinders perpendicular to the oncoming uniform flow. The centres of these cylinders are the vertices of an equilateral triangle pointing upstream. The flow is manipulated by constant rotation of the cylinders, i.e. described by three actuation parameters. The Reynolds number based on a cylinder diameter is chosen to be $30$. The unforced flow yields statistically symmetric periodic shedding represented by a one-dimensional limit cycle. The proposed methodology yields a five-dimensional manifold describing a wide range of dynamics with small representation error. Interestingly, the manifold coordinates automatically unveil physically meaningful parameters. Two of them describe the downstream periodic vortex shedding. The other three describe the near-field actuation, i.e. the strength of boat-tailing, the Magnus effect and forward stagnation point. The manifold is shown to be a key enabler for control-oriented flow estimation.

This study uses high-fidelity simulations (direct numerical simulation or large-eddy simulation) and experimental datasets to analyse the effect of non-equilibrium streamwise mean pressure gradients (adverse or favourable), including attached and separated flows, on the statistics of boundary-layer wall-pressure fluctuations. The datasets collected span a wide range of Reynolds numbers ($Re_\theta$ from 300 to 23 400) and pressure gradients (Clauser parameter from $-0.5$ to 200). The datasets are used to identify an optimal set of variables to scale the wall-pressure spectrum: edge velocity, boundary layer thickness and the peak magnitude of Reynolds shear stress. Using the present datasets, existing semi-empirical models of the wall-pressure spectrum are shown unable to capture effects of strong, non-equilibrium adverse pressure gradients, due to inappropriate scaling of the wall pressure using the wall shear stress, calibration with limited types of flows and dependency on model parameters based on the friction velocity, which reduces to zero at the detachment point. To address these shortcomings, a generalized wall-pressure spectral model is developed with parameters that characterize the extent of the logarithmic layer and the strength of the wake. Derived from the local mean streamwise velocity profile, these two parameters inherently carry the effect of the Reynolds number, as well as those of the non-equilibrium pressure gradient and its history. Comparison with existing models shows that the proposed model behaves well and is more accurate in strong-pressure-gradient flows and in separated-flow regions.

The plane Poiseuille flow of a rarefied gas in a finite length channel, driven by an axial pressure gradient, is analysed numerically to probe (i) the role of ‘dilatation’ ($\varDelta ={\boldsymbol \nabla }\boldsymbol {\cdot }{\boldsymbol u}\neq 0$) on its thermohydrodynamics as well as to clarify (ii) the possible equivalence with its well-studied ‘dilatation-free’ or ‘isochoric’ (${\rm D}\rho /{\rm D}t=0$) counterpart driven by a constant acceleration. Focussing on the mass flow rate ${\mathcal {M}}({Kn})$, which is an invariant quantity for both pressure-driven and acceleration-driven Poiseuille flows, it is shown that while ${\mathcal {M}}\sim \log {{Kn}}$ at ${Kn}\gg 1$ in the acceleration-driven case, the mass flow saturates to a constant value ${\mathcal {M}}\sim {{Kn}}^0$ at ${Kn}\gg 1$ in the pressure-driven case due to the finite length ($L_x<\infty$) of the channel. The latter result agrees with prior theory and recent experiments, and holds irrespective of the magnitude of the axial pressure gradient ($G_p$). The pressure-dilatation cooling ($\varPhi _p=-p\varDelta <0$) is shown to be responsible for the absence of the bimodal shape of the temperature profile in the pressure-driven Poiseuille flow. The dilatation-driven reduction of the shear viscosity and the odd signs of two normal stress differences (${\mathcal {N}}_1$ and ${\mathcal {N}}_2$) in the pressure-driven flow in comparison with those in its acceleration-driven counterpart are explained from the Burnett-order constitutive relations for the stress tensor. While both ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ appear at the Burnett order $O({{Kn}}^2)$ in the acceleration-driven flow, the leading term in ${\mathcal N}_1$ scales as $(\mu/p)\varDelta$ due to the non-zero dilatation in the pressure-driven Poiseuille flow which confirms that the two flows are not equivalent even at the Navier–Stokes–Fourier order $O({{Kn}})$. The heat-flow rate (${{\mathcal {Q}}_q}_x=\int q_x(x,y) \,{{\rm d} y}$) of the tangential heat flux is found to be negative (i.e. directed against the axial pressure gradient), in contrast to its positive asymptotic value (at ${Kn}\gg 1$) in the acceleration-driven flow. Similar to the scale-dependence of the mass flow rate, ${{\mathcal {Q}}_q}_x({{Kn}}, L_x)$ is found to saturate to a constant value at ${Kn}\gg 1$ in finite length channels. The double-well shape of the $q_x(y)$-profile in the near-continuum limit agrees well with predictions from a generalized Fourier law. On the whole, the dilatation-driven signatures (such as the pressure-dilatation work and the ‘normal’ shear-rate differences) are shown to be the progenitor for the observed differences between the two flows with regard to (i) the hydrodynamic fields, (ii) the rheology and (iii) the flow-induced heat transfer.

Bistable states for a sufficiently large amount of liquid can appear in an eccentric capillary due to the eccentricity effect under zero gravity (J. Fluid Mech, vol. 863, 2019, pp. 364–385). A transverse body force, which can lead to rich physical phenomena of a droplet, may lead to multistable states (bistability, tristability and the likes) of a sufficiently large amount of liquid in a capillary. We theoretically investigate this situation in a circular or annular capillary tube under a transverse body force. The results show that there can be tristable (bistable) states in an annular (circular) capillary tube: an occluding configuration and two (one) non-occluding configurations. In the annular tube, for one of the non-occluding configurations, the gas–liquid interface in the middle cross-section of the droplet meets both the inner and outer walls of the tube (bridging configuration); for the other non-occluding configuration, the gas–liquid interface in the middle cross-section of the droplet does not meet the inner wall (non-bridging configuration). The multistability is dependent on the Bond number, the contact angle and the cross-sectional shape. The multistability cannot occur for a zero or very large Bond number. A hydrophilic condition (the contact angle smaller than 90°) contributes to the non-occluding non-bridging configuration, while the hydrophobic condition (the contact angle larger than 90°) contributes to the non-occluding bridging configuration (only for the annular capillary). For the annular capillary with a not-so-large contact angle, increasing the inner-to-outer radius ratio can lead to a larger range of Bond numbers, in which the multistability occurs.

The flow of a nematic liquid crystal in a Hele-Shaw cell with an electrically controlled viscous obstruction is investigated using both a theoretical model and physical experiments. The viscous obstruction is created by temporarily electrically altering the viscosity of the nematic in a region of the cell across which an electric field is applied. The theoretical model is validated experimentally for a circular cylindrical obstruction, demonstrating user-controlled flow manipulation of an anisotropic liquid within a heterogeneous single-phase microfluidic device.

Elastoinertial turbulence (EIT) is a chaotic flow resulting from the interplay between inertia and viscoelasticity in wall-bounded shear flows. Understanding EIT is important because it is thought to set a limit on the effectiveness of turbulent drag reduction in polymer solutions. Here, we analyse simulations of two-dimensional EIT in channel flow using spectral proper orthogonal decomposition (SPOD), discovering a family of travelling wave structures that capture the sheetlike stress fluctuations that characterise EIT. The frequency-dependence of the leading SPOD mode contains distinct peaks and the mode structures corresponding to these peaks exhibit well-defined travelling structures. The structure of the dominant travelling mode exhibits shift–reflect symmetry similar to the viscoelasticity-modified Tollmien–Schlichting (TS) wave, where the velocity fluctuation in the travelling mode is characterised by large-scale regular structures spanning the channel and the polymer stress field is characterised by thin, inclined sheets of high polymer stress localised at the critical layers near the channel walls. The travelling structures corresponding to the higher-frequency modes have a very similar structure, but are nested in a region roughly bounded by the critical layer positions of the next-lower-frequency mode. A simple theory based on the idea that the critical layers of mode $\kappa$ form the ‘walls’ for the structure of mode $\kappa +1$ yields quantitative agreement with the observed wave speeds and critical layer positions, indicating self-similarity between the structures. The physical idea behind this theory is that the sheetlike localised stress fluctuations in the critical layer prevent velocity fluctuations from penetrating them.

Instability and rupture dynamics of a liquid nano-thread, subjected to external hydrodynamic perturbations, are captured by a stochastic lubrication equation (SLE) incorporating thermal fluctuations via Gaussian white noise. Linear instability analysis of the SLE is conducted to derive the spectra and distribution functions of thermal capillary waves influenced by external perturbations and thermal fluctuations. The SLE is also solved numerically using a second-order finite difference method with a correlated noise model. Both theoretical and numerical solutions, validated through molecular dynamics, indicate that surface tension forces due to specific external perturbations overcome the random effects of thermal fluctuations, determining both the thermal capillary waves and the evolution of perturbation growth. The results also show two distinct regimes: (i) the hydrodynamic regime, where external perturbations dominate, leading to uniform ruptures, and (ii) the thermal-fluctuation regime, where external perturbations are surpassed by thermal fluctuations, resulting in non-uniform ruptures. The transition between these regimes, modelled by a criterion developed from linear instability theory, exhibits a strong dependence on the amplitudes and wavenumbers of the external perturbations.

The recent work of Siegelman & Young (Proc. Natl Acad. Sci. USA, vol. 120, issue 44, 2023, e2308018120) revealed two extreme states reached by the evolution of unforced and weakly damped two-dimensional turbulence above random rough topography, separated by a critical kinetic energy $E_\#$. The low- and high-energy solutions correspond to topographically locked and roaming vortices, surrounded by non-uniform and homogeneous background potential vorticity (PV), respectively. However, we found that these phenomena are restricted to the particular intermediate length scale where the energy was initially injected into the system. Through simulations initialized by injecting the energy at larger and smaller length scales, we found that the long-term state of topographic turbulence is also dependent on the initial length scale and thus the initial enstrophy. If the initial length scale is comparable to the domain size, the long-term flow field resembles the minimum-enstrophy state proposed by Bretherton & Haidvogel (J. Fluid Mech., vol. 78, issue 1, 1976, pp. 129–154), with very few topographically locked vortices; the long-term enstrophy is quite close to the minimum value, especially when the energy is no larger than $E_\#$. As the initial length scale becomes smaller, more vortices nucleate and become more mobile; the long-term enstrophy increasingly deviates from the minimum value. Simultaneously, the background PV tends to homogenization, even if the energy is below $E_\#$. These results complement the phenomenology of topographic turbulence documented by Siegelman & Young, by showing that the minimum-enstrophy and background PV homogenization states can be adequately approached by large- and small-scale initial fields, respectively, with relatively arbitrary energy.

Cavity evolution in granular media is crucial in explosion-driven gas–particle flows, particularly in many engineering applications. In this study, a theoretical model was first proposed to describe the cavity evolution in granular media by extending the classical Rayleigh–Plesset model. A closed equation set comprising the radius, pressure and gas leak-off velocity equations was built by considering the gas expansion and non-Darcy gas-penetration effects. Both centrally symmetric and non-centrally symmetric cases of gas injection into granular media were investigated. Especially for modelling the non-symmetric scenario, the radius and gas leak-off velocity equations were proposed in each radial direction with angle $\theta$, and then the pressure equation was built up based on the integral gas leak-off along the cavity outline. Through non-dimensionalizing the theoretical equations, four key dimensionless numbers $\varPi_1,\ \varPi_4$ were obtained to characterize the response time of cavity expansion and the intensity of non-Darcy effects for both cases. This allowed us to determine a scaling law of effective cavity radius $R_{eff}^*=\sqrt {2\varPi _2/(7{\rm \pi} )}t^{*1/2}$ and the critical time $t_{cr}^*=\sqrt {12.5/\varPi _1}$ for two-dimensional cavity evolution. Additionally, the necessity of incorporating non-Darcy effects was ascertained under conditions of $\varPi _4>400$. The findings demonstrate that the proposed theoretical equations effectively predict the cavity evolution results under various operational conditions ($0.7<\varPi _1<7\times 10^2, 3<\varPi _4<1.1\times 10^3$), as validated by refined Euler–Lagrange numerical simulations.