This work provides new lower bounds on the global (nonlinear) stability limit of pressure-driven two-dimensional plane Poiseuille flow, improving on the energy stability limit $\textit{Re}_E$ originally computed by Orr in 1907. Using a computer, we carefully construct quartic Lyapunov functionals of the velocity perturbations about the laminar profile, which certify the nonlinear stability of the flow to arbitrary perturbations. The formulation combines a decomposition of the velocity into finitely many energy eigenmodes, referred to as a ‘mode set’, and an infinite-dimensional ‘tail’, together with explicit bounds that recast the Lyapunov inequality conditions as semidefinite programs, whose feasibility is tested. Over the streamwise lengths considered, the certified stability limit exceeds the classical energy bound. In particular, at the critical energy-stable streamwise length, where $\textit{Re}_E\approx 87.59$, the flow is found to be globally stable up to $\textit{Re} \approx 106.8$ (representing a $22\,\%$ improvement). Various modestly sized mode sets, capable of capturing sufficient features of the nonlinear dynamics of energy growth and subsequent decay, are proposed and found to be successful in producing improved bounds, with the simplest one involving only five modes.

This study presents a quantitative analysis of the onset of self-induced thermoacoustic instabilities in a flame propagating through a channel open only at the ignition end. Direct numerical simulations are performed for a stoichiometric mixture to examine flame–acoustic interactions in two computational set-ups: (i) a conventional truncated domain that models only the channel, imposing a constant-pressure boundary condition at the open end; and (ii) an extended domain that more closely replicates experimental conditions by resolving both the internal flow and the surrounding atmosphere, where the jet emerging from the channel exit after ignition generates additional acoustic sources. In the extended domain, both pressure and velocity at the channel exit arise naturally from the calculation rather than being prescribed. The extended-domain simulations predict a markedly different flame response, with an earlier onset of thermoacoustic instability and an accelerated transition to a non-symmetric flame front. These features arise from vortices formed in the mixing layer between the pulsating jet and the surrounding atmosphere, which act as additional sources of acoustic perturbations and trigger the premature growth of flame instabilities. Although thermoacoustic generation has traditionally been regarded as the primary mechanism of acoustic energy production, the results demonstrate that aeroacoustic contributions play an equally significant role in amplifying acoustic perturbations. Our findings underscore the critical interplay between acoustics, combustion and fluid mechanics in determining the flame dynamics and emphasise the importance of accurately representing boundary conditions that reproduce the physical interaction with the surrounding environment to achieve reliable predictive simulations.

In this paper we investigate analytically the formation of finite time singularities in the three-dimensional (3-D) incompressible Euler equations under the Gibbon–Fokas–Doering model for vorticity stretching within a bounded cylindrical domain and under axisymmetric conditions. We derive explicit Lagrangian solutions for the vorticity, its stretching rate, fluid pathlines and velocity components by exploiting constants of motion associated with the field dependent infinitesimal symmetries of the system. The central finding is that the existence and nature of a finite time singularity are determined exclusively by the local geometric structure of the initial vortex stretching rate near its global minimum. Whether a singularity forms depends on how flat this profile is at the minimum. Flatter profiles delay the blowup and sufficient flatness can suppress it entirely. For power law behaviour near the minimum, critical thresholds for the exponent are identified which separate regular solutions from those that develop a finite time singularity. These thresholds differ depending on whether the singularity occurs at the centre of the cylinder or on a ring away from the centre, with minima at the centre requiring higher flatness to avoid blowup. This work provides a rigorous analytical framework that elucidates how the local geometric structure of the initial conditions governs the potential for singularity formation in 3-D fluid flows, offering fundamental insights into the interplay between symmetry, initial data and the development of extreme events in idealized turbulence.

Turbulence statistics of a three-dimensional boundary layer over a rotating disk are measured using molecular tagging velocimetry and laser Doppler velocimetry up to a friction Reynolds number of $\simeq$3000. The present new water-based facility results in the achievement of relatively high Reynolds numbers, and we overcome the challenges associated with mimicking an infinite disk scenario using a finite tank. For the first time, the mean momentum balance and mean stress budget of this flow are investigated in the tangential/streamwise direction and are contrasted with those in a zero-pressure-gradient turbulent boundary layer over a flat plate. Furthermore, the mean momentum and stress balance in the radial direction are examined to better understand the cross-flow effects on the structure of the rotating disk turbulent flow. The tangential and radial mean momentum equations both contribute at leading order to the mean flow dynamics. Analytical approximations for the tangential and radial friction velocities are found using the mean equations and well-supported similarity assumptions. Furthermore, an analytical estimation for the variations of the mean wall-normal velocity across the boundary layer is derived, which shows a good agreement with present measurements. Independent of Reynolds number, the log layer in the rotating disk turbulent boundary layer is seen to extend to the boundary layer edge with an ostensibly negligible wake. This is similar to observations in the turbulent sink flow under equilibrium conditions and channel flow. Through quadrant analysis, it is evidenced that the Reynolds-shear-stress-producing events are weakened in the rotating disk boundary layer.

This work investigates the hydrodynamics of a rigid cylindrical body with an arbitrary cross-section undergoing arbitrary translational oscillations in a viscous incompressible fluid. An asymptotic solution is derived analytically in a high-frequency, small-amplitude approximation, using a generalised curvilinear non-inertial coordinate system. The solution yields analytical expressions for the three components of the hydrodynamic force: the added mass, the Basset force and the viscous drag. These general dependencies are used to analyse the hydrodynamic loads on various oscillating cylinders, including an elliptical cylinder, a symmetric Zhukovsky aerofoil, a rounded polygonal profile (defined as a hypotrochoid) and a rectangular cylinder. For sharp-edged bodies, limitations are discussed. The range of applicability of the developed theory is assessed by comparing its predictions with numerical simulations and experimental data. The appendix provides formulas for calculating the hydrodynamic force coefficients in the Morison form for bodies of different shapes.

The droplet–microfluidic system under acoustic-field actuation represents a classical multiphase flow problem with broad relevance to engineering and biomedical applications. When an acoustic wave interacts with a viscous droplet suspended in another liquid, it induces acoustic streaming both inside and outside the droplet and exerts acoustic radiation stresses that deform its shape. In this work, we present a comprehensive analytical and numerical framework to investigate the coupled dynamics of droplet shape evolution, acoustic fields and acoustic streaming. The study considers droplets stably positioned within a standing wave at either a pressure node or a velocity node, undergoing deformations analogous to the classical Taylor regime. Our analytical model, supported by numerical simulations, reveals that the acoustic field is predominantly governed by low-order modes, up to the quadrupole. Furthermore, the acoustic-streaming pattern is dictated by the density contrast between the droplet and the surrounding fluid, whereas both density and compressibility contrasts control the droplet deformation. We also demonstrate that impedance mismatch is not the fundamental criterion for acoustic scattering. Instead, mismatches in density, or compressibility, or both, serve as the primary mechanisms. In addition, we develop a phase diagram illustrating droplet shape-deformation and streaming-pattern regimes as functions of the material properties (density ratio and compressibility ratio). These results provide deep mechanistic insights for the design and control of efficient droplet–acoustofluidic platforms and lay the groundwork for next-generation ultrasound-driven strategies for precise droplet manipulation in soft-matter and microfluidic applications.

Contrary to accepted turbulence folklore, which holds that no mathematical relation exists between the Navier–Stokes equations (NSEs) and the multifractal model (MFM) of Parisi and Frisch, we develop a theory that reconciles the MFM with Leray’s weak solutions of Navier–Stokes analysis. From a combination of Euler invariant scaling and the NSEs set in a three-dimensional box of side $L$, we also derive the Paladin–Vulpiani scale $\eta_{h,pav}$ which is related to the Reynolds number Re by $L\eta _{h,\textit{pa}v}^{-1} = \textit{Re}^{1/(1+h)}$, and which acts as a mediator between the two theories. This is achieved by considering $L^{2m}$-norms of the velocity gradient to find a correspondence between $m$ and the local scaling exponent $h$ in the multifractal model. The parameter $m$ acts as if it were the sliding focus control on a telescope which allows us to zoom in and out on different structures. The range $1 \leqslant m \leqslant \infty$ is equivalent to $-{{ {2}/{3}}} \leqslant h_{\textit{min}} \leqslant {{{1}/{3}}}$, which lies precisely in the region where Bandak et al. (Phys. Rev. E, 2022, vol. 105, p. 065113; Phys. Rev. Lett., 2024, vol. 132, p. 104002) have suggested that thermal noise makes the NSEs inadequate and generates spontaneous stochasticity. The implications of this are discussed.

We perform analytical and numerical analyses of the propulsion of a rigid body in a viscous fluid subjected to a periodic force with zero average over a period. This general formulation specifically addresses the significant case where propulsion is generated by the oscillation of a mass located in an internal cavity of the body. We provide a rigorous proof of the necessary and sufficient conditions for propulsion at the second order of magnitude of the force. These conditions are implemented and confirmed by numerical tests for bodies without fore-and-aft symmetry, while they are silent for bodies with such symmetry, like round ellipsoids. Consequently, in this case, propulsion can only occur at an order higher than the second. This problem is investigated by numerically integrating the entire set of equations, and the result shows that, in fact, propulsion does occur, thus opening new avenues for further analytical studies.

Fluid mixture models are essential for describing a wide range of physical phenomena, including wave dynamics and spinodal decomposition. However, there is a lack of consensus in the modelling of compressible mixtures, with limited connections between different classes of models. On the one hand, existing compressible two-phase flow models accurately describe wave dynamics, but do not incorporate phase separation mechanisms. On the other hand, phase-field technology in fluid dynamics consists of models incorporating spinodal decomposition; however, a general phase-field theory for compressible mixtures remains largely undeveloped. In this paper we take an initial step toward bridging the gap between compressible two-phase flow models and phase-field models by developing a theory for compressible, isothermal N-phase mixtures. Our theory establishes a system of reduced complexity by formulating N mass balance laws alongside a single momentum balance law, thereby naturally extending the Navier–Stokes Korteweg model to N phases and providing the Navier–Stokes Cahn–Hilliard/Allen–Cahn model for compressible mixtures. Key aspects of the framework include its grounding in continuum mixture theory and its preservation of thermodynamic consistency despite its reduced complexity.

We investigate the interfacial fluid dynamics and heat transfer at nanoscales using an improved diffuse interface approach for liquid–vapour interfaces in non-equilibrium. Conventional Navier–Stokes–Korteweg (NSK) formulations often fail to accurately capture transport phenomena across extremely thin interfaces due to underestimation of interface resistances. In this work, we improve the NSK model by adding a production term in the momentum equation based on higher-order corrections. To enhance interface resistances, viscosity and thermal conductivity are made dependent on the density gradient, increasing resistance only within the interface region. The gradient-based coefficients are determined by fitting to solutions of the Enskog–Vlasov equation for Couette flow of Struchtrup & Frezzotti (2022 J. Fluid Mech., vol. 940, p. A40). Applying these fitted equations to pure heat conduction and planar evaporation problems shows that the model accurately captures interfacial transport, making it a useful tool for studying nanoscale evaporation, thermal management and the droplet dynamics on solid surfaces.

In Navier–Stokes (NS) turbulence, large-scale turbulent flows inevitably determine small-scale flows. Previous studies using data assimilation with the three-dimensional (3-D) NS equations indicate that employing observational data resolved down to a specific length scale, $\ell ^{\rm 3\text{-}D}_{\ast }$, enables the successful reconstruction of small-scale flows. Such a length scale of ‘essential resolution of observation’ for reconstruction $\ell ^{\rm 3\text{-}D}_{\ast }$ is close to the dissipation scale in three-dimensional NS turbulence. Here, we study the equivalent length scale in two-dimensional (2-D) NS turbulence, $\ell ^{\rm 2\text{-}D}_{\ast }$, and compare with the three-dimensional case. Our numerical studies using data assimilation and conditional Lyapunov exponents reveal that, for Kolmogorov flows with Ekman drag, the length scale $\ell ^{\rm 2\text{-}D}_{\ast }$ is actually close to the forcing scale, substantially larger than the dissipation scale. Furthermore, we discuss the origin of the significant relative difference between the length scales, $\ell ^{\rm 2\text{-}D}_{\ast }$ and $\ell ^{\rm 3\text{-}D}_{\ast }$, based on inter-scale interactions, ‘cascades’ and orbital instabilities in turbulence dynamics.

For a perturbed trefoil vortex knot evolving under the Navier–Stokes equations, a sequence of $\nu$-independent times $t_m$ are identified that correspond to a set of scaled, volume-integrated vorticity moments $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}$, with this hierarchy $t_\infty \leqslant \ldots \leqslant t_m\ldots t_1=t_x\approx 40$ and $\mathcal{O}_{\textit{Vm}}=(\int _{V\ell }|\omega |^{2m}\,{\rm d}V)^{1/2m}$. For the volume-integrated enstrophy $Z(t)$, convergence of $\sqrt {\nu }Z(t)=\bigl (\nu ^{1/4}\mathcal{O}_{\textit{V}\text{1}}(t)\bigr )^2$ at $t_x=t_1$ marks the end of reconnection scaling. Physically, reconnection follows from the formation of a double vortex sheet, then a knot, which splits into spirals. Meanwhile $Z$ accelerates, leading to approximate finite-time $\nu$-independent convergence of the energy dissipation rate $\epsilon (t)=\nu Z(t)$ at $t_\epsilon \sim 2t_x$. This is sustained over a finite temporal span of at least $\Delta T_\epsilon \searrow 0.5 t_\epsilon$, giving Reynolds number independent finite-time, temporally integrated dissipation, $\Delta E_\epsilon =\int _{\Delta T_\epsilon }\epsilon \,{\rm d}t$, and thus satisfies one definition for a dissipation anomaly, with enstrophy spectra that are consistent with transient $k^{1/3}$ Lundgren-like inertial scaling over some of the $\Delta T_\epsilon$ time. A critical factor in achieving these temporal convergences is how the computational domain $V_\ell =(2\ell \pi )^3$ is increased as $\ell \sim \nu ^{-1/4}$, for $\ell =2$ to 6, then to $\ell =12$, as $\nu$ decreases. Appendix A shows compatibility with established $(2\pi )^3$ mathematics where $\nu \equiv 0$ Euler solutions bound small $\nu$ Navier–Stokes solutions. Two spans of $\nu$ are considered. Over the first factor of 25 decrease in $\nu$, most of the $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}(t)$ converge to their respective $t_m$. For the next factor of 5 decrease (125 total) in $\nu$, with increased $\ell$ to $\ell =12$, there is initially only convergence of $\nu ^{1/4}\varOmega _{V\infty }(t)$ to $t_\infty$, without convergence for $9\gt m\gt 1$. Nonetheless, there is later $\sqrt {\nu }Z(t)$ convergence at $t_1=t_x$ and $\epsilon (t)=\nu Z$ over $t\sim t_\epsilon \approx 2t_x$.

For the first time, an analytical solution has been derived for Stokes flow through a conical diffuser under the condition of partial slip. Recurrent relations are obtained that allow determination of the velocity, pressure and stream function for a certain slip length λ. The solution is analysed in the first order of decomposition with respect to a small dimensionless parameter ${\lambda }/{r}$. It is shown that the sliding of the liquid over the surface of the cone leads to a vorticity of the flow. At zero slip length, we obtain the well-known solution to the problem of a diffuser with a no-slip boundary condition corresponding to strictly radial streamlines. To solve that problem, we use an alternative form of the general solution of the linearised, stationary, axisymmetric Navier–Stokes equations for an incompressible fluid in spherical coordinates. A previously published solution to this problem, dating back to the paper by Sampson (1891 Phil. Trans. R. Soc. A, vol. 182, pp. 449–518), is given in terms of a stream function that leads to formulae that are difficult to apply in practice. By contrast, the new general solution is derived in the vector potential representation and is simpler to apply.

This study presents a modified intermediate long wave (mILW) equation derived from the Navier–Stokes equations via multi-scale analysis and perturbation expansion, aimed at describing internal solitary waves (ISWs) in finite-depth, stratified oceans. Compared to the classical ILW model, the proposed mILW equation incorporates cubic nonlinearities and captures the dynamical behaviour of large-amplitude ISWs more accurately. The equation reduces to the modified Korteweg–de Vries equation and modified Benjamin–Ono equations in the shallow- and deep-water limits, respectively, thus providing a unified framework across varying depth regimes. Soliton solutions are constructed analytically using Hirota’s bilinear method, and numerical simulations investigate wave–wave interactions, including rogue waves and Mach reflection. Furthermore, a smooth tanh-type density profile is adopted to reflect realistic stratification. Associated vertical modal structures and vertical velocity fields are analysed, and higher-order statistics (skewness and kurtosis) are introduced to reveal the density dependence of wave asymmetry. The results offer new insights into the nonlinear dynamics of ISWs, with implications for ocean mixing, energy transport and submarine acoustics.

The propagation of linear waves in non-ideal compressible fluids plays a crucial role in numerous physical and engineering applications, particularly in the study of instabilities, aeroacoustics and turbulence modelling. This work investigates linear waves in viscous and heat-conducting non-ideal compressible fluids, modelled by the Navier–Stokes–Fourier equations and a fully arbitrary equation of state (EOS). The linearised governing equations are derived to analyse the dispersion relations when the EOS differs from that of an ideal gas. Special attention is given to the influence of non-ideal effects and various dimensionless numbers on wave propagation speed and attenuation. By extending classical results from Kovásznay (1953 J. Aeronaut. Sci. vol. 20, no. 10, pp. 657–674) and Chu (1965 Acta Mech. vol. 1, no. 3, pp. 215–234) obtained under the ideal gas assumption, this study highlights the modifications introduced by arbitrary EOSs to the linear wave dynamics in non-ideal compressible flows. This work paves the path for an improved understanding and modelling of wave propagation, turbulence and linear stability in arbitrary viscous and heat-conducting fluids.

We provide a rigorous analysis of the self-similar solution of the temporal turbulent boundary layer, recently proposed by Biau (2023 Comput. Fluids 254, 105795), in which a body force is used to maintain a statistically steady turbulent boundary layer with periodic boundary conditions in the streamwise direction. We derive explicit expressions for the forcing amplitudes which can maintain such flows, and identify those which can hold either the displacement thickness or the momentum thickness equal to unity. This opens the door to the first main result of the paper, which is to prove upper bounds on skin friction for the temporal turbulent boundary layer. We use the Constantin–Doering–Hopf bounding method to show, rigorously, that the skin-friction coefficient for periodic turbulent boundary layer flows is bounded above by a uniform constant which decreases asymptotically with Reynolds number. This asymptotic behaviour is within a logarithmic correction of well-known empirical scaling laws for skin friction. This gives the first evidence, applicable at asymptotically high Reynolds numbers, to suggest that Biau’s self-similar solution of the temporal turbulent boundary layer exhibits statistical similarities with canonical, spatially evolving, boundary layers. Furthermore, we show how the identified forcing formula implies an alternative, and simpler, numerical implementation of periodic boundary layer flows. We give a detailed numerical study of this scheme presenting direct numerical simulations up to a momentum Reynolds number of $\textit{Re}_\theta = 2000$ and implicit large-eddy simulations up to $\textit{Re}_\theta = 8300$, and show that these results compare well with data from canonical spatially evolving boundary layers at equivalent Reynolds numbers.

Equilibrium, travelling-wave and periodic-orbit solutions of the Navier–Stokes equations provide a promising avenue for investigating the structure, dynamics and statistics of transitional flows. Many such invariant solutions have been computed for wall-bounded shear flows, including plane Couette, plane Poiseuille and pipe flow. However, the organisation of invariant solutions is not well understood. In this paper we focus on the role of symmetries in the organisation and computation of invariant solutions of plane Poiseuille flow. We show that enforcing symmetries while computing invariant solutions increases the efficiency of the numerical methods, and that redundancies between search spaces can be eliminated by consideration of equivalence relations between symmetry subgroups. We determine all symmetry subgroups of plane Poiseuille flow in a doubly periodic domain up to translations by half the periodic lengths and classify the subgroups into equivalence classes, each of which represents a physically distinct set of symmetries and an associated set of physically distinct invariant solutions. We calculate fifteen new travelling waves of plane Poiseuille flow in seven distinct symmetry groups and discuss their relevance to the dynamics of transitional turbulence. We present a few examples of subgroups with fractional shifts other than half the periodic lengths and one travelling-wave solution whose symmetry involves shifts by one third of the periodic lengths. We conclude with a discussion and some open questions about the role of symmetry in the behaviour of shear flows.

This work experimentally explores the alignment of the vorticity vector and the strain-rate tensor eigenvectors at locations of extreme upscale and downscale energy transfer. We show that the turbulent von Kármán flow displays vorticity–strain alignment behaviour across a large range of Reynolds numbers, which is very similar to previous studies on homogeneous, isotropic turbulence. We observe that this behaviour is amplified for the largest downscale energy transfer events, which tend to be associated with sheet-like geometries. These events are also shown to have characteristics previously associated with high flow field nonlinearity and singularities. In contrast, the largest upscale energy transfer events display very different structures which showcase a strong preference for vortex compression. Notably, in both cases we find that these trends are strengthened as the probed scales approach the Kolmogorov scale. We then show further evidence for the argument that strain self-amplification is the most salient feature in characterising the cascade direction. Finally, we identify possible invariant behaviour for the largest energy transfer events, even at scales near the Kolmogorov scale.

Over the past few decades, numerous N-phase incompressible diffuse-interface flow models with non-matching densities have been proposed. Despite aiming to describe the same physics, these models are generally distinct, and an overarching modelling framework is absent. This paper provides a unified framework for N-phase incompressible Navier–Stokes Cahn–Hilliard Allen–Cahn mixture models with a single momentum equation. The framework emerges naturally from continuum mixture theory, exhibits an energy-dissipative structure, and is invariant to the choice of fundamental variables. This opens the door to exploring connections between existing N-phase models and facilitates the computation of N-phase flow models rooted in continuum mixture theory.