The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, Joseph & Fosdick (1973, Arch. Rat. Mech. Anal. vol. 49, pp. 321–380) employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. In this work, we investigate the time-dependent interface height through asymptotic analysis with dimensionless variables and equations using the Giesekus model. We begin by neglecting inertia to focus on the interaction between gravity, viscoelasticity and surface tension. In the small-deformation scenario, the governing equations indicate the presence of a boundary layer in time, where the interface rises rapidly over a short time scale before gradually approaching a steady state. By employing a stretched time variable, we derive the transient velocity field and corresponding interface shape on this short time scale, and recover the steady-state shape on a longer time scale. In contrast to the work of Joseph and Fosdick, which used the method of successive approximations to determine the steady shape of the interface, we explicitly derive the interface shape for both steady and transient cases. Subsequently, we reintroduce small but finite inertial effects to investigate their interaction with viscoelasticity, and propose a criterion for determining the conditions under which rod climbing occurs. Through numerical computations, we obtain the transient interface shapes, highlighting the interplay between time-dependent viscoelastic and inertial effects.

A long-standing conceptual debate regarding the identification and independence of first Mack and cross-flow instabilities is clarified over a Mach 5.9 sharp wing at zero angle of attack and varying sweep angles. Their receptivity of the boundary layers to three-dimensional slow acoustic and vorticity waves is investigated using linear stability theory, direct numerical simulation and momentum potential theory (MPT). Linear stability theory demonstrates that the targeted slow mode appears as the oblique first mode at small sweep angles ($0^\circ$ and $15^\circ$) and transitions to the cross-flow mode at larger sweep angles ($30^\circ$ and $45^\circ$). Direct numerical simulation indicates that both the oblique first mode and cross-flow mode share identical receptivity pathways: for slow acoustic waves, the pathway comprises ‘leading-edge damping–enhanced exponential growth–linear growth’ stages. For vorticity waves, it consists of ‘leading-edge damping–non-modal growth–linear growth’ stages. Momentum potential theory decomposes the fluctuation momentum density into vortical, acoustic and thermal components, revealing unified receptivity mechanisms: for slow acoustic waves, the leading-edge damping is caused by strong acoustic components generated through synchronization. The enhanced exponential growth stage is dominated by steadily growing vortical components, with acoustic and thermal components remaining at small amplitudes. For vorticity waves, leading-edge disturbances primarily consist of vortical components, indicating a distinct mechanism from slow acoustic waves. Non-modal stages originate from adjustments among MPT components. Vortical components dominate the linear growth stage for both instabilities. These uniform behaviours between first Mack and cross-flow modes highlight their consistency.

Investigations into the effects of polymers on small-scale statistics and flow patterns were conducted in a turbulent von Kármán swirling (VKS) flow. We employed the tomographic particle image velocimetry technique to obtain full information on three-dimensional velocity data, allowing us to effectively resolve dissipation scales. Under varying Reynolds numbers ($R_\lambda =168{-}235$) and polymer concentrations ($\phi =0{-}25\ {\textrm{ppm}}$), we measured the velocity gradient tensor (VGT) and related quantities. Our findings reveal that the ensemble average and probability density function (PDF) of VGT invariants, which represent turbulent dissipation and enstrophy along with their generation terms, are suppressed as polymer concentration increases. Notably, the joint PDFs of the invariants of VGT, which characterise local flow patterns, exhibited significant changes. Specifically, the third-order invariants, especially the local vortex stretching, are greatly suppressed, and strong events of dissipation and enstrophy coexist in space. The local flow pattern tends to be two-dimensional, where the eigenvalues of the rate-of-strain tensor satisfy a ratio $1:0:-1$, and the vorticity aligns with the intermediate eigenvector of the rate-of-strain tensor, while it is perpendicular to the other two. We find that these statistics observations can be well described by the vortex sheet model. Moreover, we find that these vortex sheet structures align with the symmetry axis of the VKS system, and orient randomly in the horizontal plane. Further investigation, including flow visualisation and conditional statistics on vorticity, confirms the presence of vortex sheet structures in turbulent flows with polymer additions. Our results establish a link between single-point statistics and small-scale flow topology, shedding light on the previously overlooked small-scale structures in polymeric turbulence.

Floating particles deform the liquid–gas interface, which may lead to capillary repulsion or attraction and aggregation of nearby particles (e.g. the Cheerios effect). Previous studies employed the superposition of capillary multipoles to model interfacial deformation for circular or ellipsoidal particles. However, the induced interfacial deformation depends on the shape of the particle and becomes more complex as the geometric complexity of the particle increases. This study presents a generalised solution for the liquid–gas interface near complex anisotropic particles using the domain perturbations approach. This method enables a closed-form solution for interfacial deformation near particles with an anisotropic shape, as well as the varying height of the pinned liquid–gas contact line. We verified the model via experiments performed with fixed particles held at the water level with shapes such as a circle, hexagon and square, which have either flat or sinusoidal pinned contact lines. Although in this study we concentrate on the equilibrium configuration of the liquid–gas interface in the vicinity of particles placed at fixed positions, our methodology paves the way to explore the interactions among multiple floating anisotropic particles and, thus, the role of particle geometry in self-assembly processes of floating particles.

Directional freezing of brine is widely found in numerous environmental and industrial settings. Despite extensive studies, the microscopic evolution of ice-brine structures remains unclear. By combining in situ micro-computed tomography visualisation and theoretical analyses, we reveal new details inside the porous ice structure and its evolution towards a cleaner ice layer. We identify three distinct stages characterised by different brine exclusion rates during solidification: a rapid initial stage possibly lasting seconds from nucleation to local equilibrium without long-range heat or mass transfer; a second stage where the system reaches global thermal equilibrium, involving brine expulsion by volume expansion and convection associated with gravity; and a final prolonged stage dominated by diffusion. Comparison between analytical solutions and the migration rates of microstructural features such as brine stripes, columns and pockets extracted from photographic images confirms these understandings. Morphologically, we capture the formation of random striped patterns together with brine columns during downward freezing and brine skirts during upward freezing, all of which gradually transform into vertically aligned polygonal patterns. The volume fraction of brine pockets in porous ice near the cold end reduces to less than 10 % after 22 h in most experiments. The residual brine pockets, however, are not rejected out of the porous ice as fast as predicted by diffusion and remain persistent. Our findings provide new insights into the brine freezing dynamics, with implications ranging from sea ice formation to freeze desalination and general solidification of binary melts.

This study applies the scaling patch approach to investigate the influence of pressure gradients on the mean-momentum balance in turbulent boundary layers (TBLs). Under strong pressure gradients, the force balance in the outer region is dominated by advective and pressure forces, with gradients of Reynolds stresses playing a minimal role. To retain the relevance of Reynolds stress gradients within the scaling patch framework, we propose a redistribution of the component $U_e \textrm {d}U_e/\textrm {d}x$ from the advective term to the pressure-gradient term. Here, $U_e$ is the mean streamwise velocity at the boundary layer edge. This reformulation enhances the outer-scaling framework of Wei & Knopp (2023 J. Fluid Mech. 958, 1–21), ensuring consistency across a wide range of pressure gradients, including those involving flow separation. Remarkably, the new outer-scaled gradient of Reynolds shear stress in TBLs under a pressure gradient closely resembles that observed in zero-pressure-gradient TBLs. In the inner region, the impact of pressure gradient is well captured by the Stratford–Mellor parameter $\beta _{\textit{in}}$. For weak pressure gradients ($|\beta _{\textit{in}}| \ll 0.07$), traditional inner scaling remains valid. However, for stronger pressure gradients $|\beta _{\textit{in}}| \gtrsim 0.07$, the near-wall dynamics is governed by a balance between pressure gradient and viscous force, as described by Stratford (1959 J. Fluid Mech. 5, 1–16) and Mellor (1966 J. Fluid Mech. 24, 255–274). In this sub-layer, viscosity and the imposed wall pressure gradient dictate the relevant velocity and length scales. Moreover, when $|\beta _{\textit{in}}| \gtrsim 0.7$ and the wall pressure $P_{w\textit{all}}$ gradient $\textrm { d}P_{w\textit{all}}/\textrm {d}x \gt 0$, a distinct sub-layer emerges outside the pressure–viscous balance region, characterised by a dominant balance between the imposed pressure gradient and the gradient of the Reynolds shear stress. In this region, the Reynolds shear stress increases linearly with distance from the wall. These findings provide new insights into the structure of TBLs under pressure gradients and establish a refined framework for modelling their dynamics.

Flows enabled by phoretic mechanisms are of significant interest in several biological and biomedical processes, such as bacterial motion and targeted drug delivery. Here, we develop a homogenisation-based macroscopic boundary condition that describes the effective flow across a diffusio-phoretic microstructured membrane, where the interaction between the membrane walls and the solute particles is modelled via a potential approach. We consider two cases where potential variations occur (i) at the pore scale and (ii) only in the close vicinity of the boundary, allowing for a simplified version of the macroscopic flow description, in the latter case. Chemical interactions at the microscale are rigorously upscaled to macroscopic phoretic solvent velocity and solute flux contributions, and added to the classical permeability and diffusivity properties of the membrane. These properties stem from the solution of Stokes advection–diffusion problems at the microscale, some of them forced by an interaction potential term. Eventually, we show an application of the macroscopic model to develop minimal phoretic pumps, showcasing its suitability for efficient design and optimisation procedures.

Surface quasi-geostrophic (SQG) theory describes the two-dimensional active transport of a scalar field, such as temperature, which – when properly rescaled – shares the same physical dimension of length/time as the advecting velocity field. This duality has motivated analogies with fully developed three-dimensional turbulence. In particular, the Kraichnan – Leith – Batchelor similarity theory predicts a Kolmogorov-type inertial range scaling for both scalar and velocity fields, and the presence of intermittency through multifractal scaling was pointed out by Sukhatme & Pierrehumbert (2002 Chaos 12, 439–450), in unforced settings. In this work, we refine the discussion of these statistical analogies, using numerical simulations with up to $16\,384^2$ collocation points in a steady-state regime dominated by the direct cascade of scalar variance. We show that mixed structure functions, coupling velocity increments with scalar differences, develop well-defined scaling ranges, highlighting the role of anomalous fluxes of all the scalar moments. However, the clean multiscaling properties of SQG transport are blurred when considering velocity and scalar fields separately. In particular, the usual (unmixed) structure functions do no follow any power-law scaling in any range of scales, neither for the velocity nor for the scalar increments. This specific form of the intermittency phenomenon reflects the specific kinematic properties of SQG turbulence, involving the interplay between long-range interactions, structures and geometry. Revealing the multiscaling in single-field statistics requires us to resort to generalised notions of scale invariance, such as extended self-similarity and a specific form of refined self-similarity. Our findings emphasise the fundamental entanglement of scalar and velocity fields in SQG turbulence: they evolve hand in hand and any attempt to isolate them destroys scaling in its usual sense. This perspective sheds new lights on the discrepancies in spectra and structure functions that have been repeatedly observed in SQG numerics for the past 20 years.

This research examines in detail the complex nonlinear forces generated when steep waves interact with vertical cylindrical structures, such as those typically used as offshore wind turbine foundations. These interactions, particularly the nonlinear wave forces associated with the secondary load cycle, present unanswered questions about how they are triggered. Our experimental campaigns underscore the occurrence of the secondary load cycle. We also investigate how the vertical distributions of the scattering force, pressure field and wave field affect the nonlinear wave forces associated with the secondary load cycle phenomena. A phase-based harmonic separation method isolates harmonic components of the scattering force’s vertical distribution, pressure field and wave field. This approach facilitates the clear separation of individual harmonics by controlling the phase of incident waves, which offers new insights into the mechanisms of the secondary load cycle. Our findings highlight the importance of complex nonlinear wave–structure interactions in this context. In certain wave regimes, nonlinear forces are locally larger than the linear forces, highlighting the need to consider the secondary load cycle in structural design. In addition, a novel discovery emerges from our comparative analysis, whereby very high-frequency (over the fifth in harmonic and order) oscillations, strongly correlated to wave steepness, have the potential to play a role in structural fatigue. This new in-depth analysis provides a unique insight regarding the complex interplay between severe waves and typical cylindrical offshore structures, adding to our understanding of the secondary load cycle for applications related to offshore wind turbine foundations.

An arbitrary Lagrangian–Eulerian finite element method and numerical implementation for curved and deforming lipid membranes is presented here. The membrane surface is endowed with a mesh whose in-plane motion need not depend on the in-plane flow of lipids. Instead, in-plane mesh dynamics can be specified arbitrarily. A new class of mesh motions is introduced, where the mesh velocity satisfies the dynamical equations of a user-specified two-dimensional material. A Lagrange multiplier constrains the out-of-plane membrane and mesh velocities to be equal, such that the mesh and material always overlap. An associated numerical inf–sup instability ensues, and is removed by adapting established techniques in the finite element analysis of fluids. In our implementation, the aforementioned Lagrange multiplier is projected onto a discontinuous space of piecewise linear functions. The new mesh motion is compared to established Lagrangian and Eulerian formulations by investigating a pre-eminent numerical benchmark of biological significance: the pulling of a membrane tether from a flat patch and its subsequent lateral translation.