This paper addresses the pulsating motion of cerebrospinal fluid in the aqueduct of Sylvius, a slender canal connecting the third and fourth ventricles of the brain. Specific attention is given to the relation between the instantaneous values of the flow rate and the interventricular pressure difference, needed in clinical applications to enable indirect evaluations of the latter from direct magnetic resonance measurements of the former. An order of magnitude analysis accounting for the slenderness of the canal is used in simplifying the flow description. The boundary layer approximation is found to be applicable in the slender canal, where the oscillating flow is characterized by stroke lengths comparable to the canal length and periods comparable to the transverse diffusion time. By way of contrast, the flow in the non-slender opening regions connecting the aqueduct with the two ventricles is found to be inviscid and quasi-steady in the first approximation. The resulting simplified description is validated by comparison with results of direct numerical simulations. The model is used to investigate the relation between the interventricular pressure and the stroke length, in parametric ranges of interest in clinical applications.

The flow in a channel having walls with periodic undulations of small amplitude $\epsilon$ in the streamwise direction is considered as a model for wall roughness. It is shown that the undulations act as a catalyst to allow a new instability related to vortex–wave interactions to grow. The roughness couples a wave disturbance with a roll–streak flow and it is shown that channel flows, both wall and pressure gradient driven, are unstable when the Reynolds number exceeds a critical value proportional to ${\epsilon ^{-({3}/{2})} [\vert {\log \epsilon }\vert ]^{-({3}/{4})}}$, the constant of proportionality depending on the wall wavelengths and amplitudes. The roughness is an integral part of the instability mechanism and not simply the seed for an existing flow instability as in receptivity theory. The mechanism involves an interaction of the rolls, streaks and waves very similar to that in vortex–wave interaction theory but now facilitated by the wall roughness. Surprisingly, the subtle interaction between waves, rolls, streaks and the walls can be solved in closed form, and an explicit form for the neutral configuration is found. The theoretical predictions are in good agreement with numerical investigations of similar problems and are applicable to a wide range of shear flows.

We study the longitudinal spreading of a passive tracer by a two-dimensional pressure-driven flow through a composite layer of porous rock which is bounded above and below by impermeable seal rock. We focus on the flow across the interface between two neighbouring zones of the rock. First, we show that, with isotropic permeability, if the interface between the two zones is tilted relative to the boundaries, then this results in a difference in travel times across the formation which in turns leads to a net shear flow. We explore the strength of this shear as a function of (a) the permeability ratio across the interface, and (b) the interface angle. Second, we show that if one zone of the rock is cross-bedded, then with uniform flow, the pressure gradient is directed at an angle to the boundary. As a result, there is a transition zone across the interface, which again leads to a net shear, even if the interface is normal to the boundaries of the layer. We explore the competition between these effects, showing how they may combine constructively to produce a larger shear, or may negate one another, reducing or reversing the sign of the shear, depending on the angle of the interface, the degree of anisotropy and the change in effective downstream permeability across the interface. We discuss some of the implications of this shear for modelling flow in such composite rocks.

In this paper, we consider two-dimensional steady free-surface flows where gravity is ignored, but the effects of surface tension are included. It is found that the existence of an additional solid boundary can allow for previously unseen limiting configurations as the surface tension tends to infinity. The free surface of these new solutions is formed of straight lines, arcs of circles and a point where the flow turns at a wall. These limiting configurations form endpoints of solution branches of capillary free-surface flows. Other endpoints of these branches include the surface tension free (i.e. free streamline) solution, and a solution whose free surface is composed simply of a straight line. The model we explore is flow incoming along a channel of constant width. One of the walls terminates, where the fluid is forcibly separated from the wall and a free boundary is formed. The other wall meets a second straight boundary with interior angle $\beta$. Far downstream the solution approaches a uniform stream. Making use of Cauchy's integral formula, the unknowns are expressed in terms of values on the boundary. The integral equations are then solved numerically. The solution space relative to the parameter values of the model is discussed.

When a viscous liquid bridge between two parallel substrates is stretched by accelerating one substrate, its interface on the plates recedes in the radial direction. In some cases the interface becomes unstable. Such instability leads to the emergence of a network of fingers. In this study, the mechanisms of such fingering are studied experimentally and analysed theoretically. The experimental set-up allows a constant acceleration of a movable substrate at up to 180 m s$^{-2}$. The phenomena are observed using two high-speed video systems. The number of fingers is measured for different liquid viscosities, liquid bridge sizes and wetting conditions. Linear stability analysis of the bridge interface takes into account the inertial, viscous and capillary effects in the liquid flow. The theoretically predicted maximum number of fingers, corresponding to an instability mode with the maximum amplitude, and a threshold for the onset of finger formation are proposed. Both models agree well with the experimental data up to the start of emerging cavitation bubbles.

A study on ${Re} =2000$ and 4000 vortex rings colliding with V-walls with included angles of $\theta =30^{\circ }$ to 120$^{\circ }$ has been conducted. Along the valley plane, higher Reynolds numbers and/or included angles of $\theta \leqslant 60^{\circ }$ lead to secondary/tertiary vortex-ring cores leapfrogging past the primary vortex-ring cores. The boundary layers upstream of the latter separate and the secondary/tertiary vortex-ring cores pair up with these wall-separated vortices to form small daisy-chained vortex dipoles. Along the orthogonal plane, primary vortex-ring cores grow bulbous and incoherent after collisions, especially as the included angle reduces. Secondary and tertiary vortex-ring core formations along this plane also lag those along the valley plane, indicating that they form by propagating from the wall surfaces to the orthogonal plane as the primary vortex ring gradually comes into contact with the entire V-wall. Circulation results show significant variations between the valley and orthogonal plane, and reinforce the notion that the collision behaviour for $\theta \leqslant 60^{\circ }$ is distinctively different from those at larger included angles. Vortex-core trajectories are compared to those for inclined-wall collisions, and secondary vortex-ring cores are found to initiate earlier for the V-walls, postulated to be a result of the opposing circumferential flows caused by the simultaneous collisions of both primary vortex-ring cores with the V-wall surfaces. These circumferential flows produce a bi-helical flow mode (Lim, Exp. Fluids, vol. 7, issue 7, 1989, pp. 453–463) that sees higher vortex compression levels along the orthogonal plane, which limit vortex stretching along the wall surfaces and produce secondary vortex rings earlier. Lastly, vortex structures and behaviour of the present collisions are compared to those associated with flat/inclined walls and round-cylinder-based collisions for a more systematic understanding of their differences.

Resolving the detailed hydrodynamics of a slender body immersed in highly viscous Newtonian fluid has been the subject of extensive research, applicable to a broad range of biological and physical scenarios. In this work, we expand upon classical theories developed over the past fifty years, deriving an algebraically accurate slender-body theory that may be applied to a wide variety of body shapes, ranging from biologically inspired tapering flagella to highly oscillatory body geometries with only weak constraints, most significantly requiring that cross-sections be circular. Inspired by well known analytic results for the flow around a prolate ellipsoid, we pose an ansatz for the velocity field in terms of a regular integral of regularised Stokes-flow singularities with prescribed, spatially varying regularisation parameters. A detailed asymptotic analysis is presented, seeking a uniformly valid expansion of the ansatz integral, accurate at leading algebraic order in the geometry aspect ratio, to enforce no-slip boundary conditions and thus analytically justify the slender-body theory developed in this framework. The regularisation within the ansatz additionally affords significant computational simplicity for the subsequent slender-body theory, with no specialised quadrature or numerical techniques required to evaluate the regular integral. Furthermore, in the special case of slender bodies with a straight centreline in uniform flow, we derive a slender-body theory that is particularly straightforward via use of the analytic solution for a prolate ellipsoid. We evidence the validity of our simple theory with explicit numerical examples for a wide variety of slender bodies, and highlight a potential robustness of our methodology beyond its rigorously justified scope.

Janus particles propel themselves by generating concentration gradients along their active surface. This induces a flow near the surface, known as the diffusio-osmotic slip, which propels the particle even in the absence of externally applied concentration gradients. In this work, we study the influence of viscoelasticity and shear-thinning (described by the second-order fluid and Carreau model, respectively) on the diffusio-osmotic slip on an active surface. Using matched asymptotic expansions, we provide an analytical expression for the modification of slip induced by the non-Newtonian behaviour. The results reveal that the modification in slip velocity, arising from polymer elasticity, is proportional to the second tangential derivative of the concentration field. Using the reciprocal theorem, we estimate the influence of this modification on the swimming velocity of a Janus sphere: (i)for second-order fluid, the contribution is non-negligible and its sign is dependent on the surface coverage of activity and (ii) for Carreau fluid, the contribution is more pronounced and always enhances the swimming velocity. The current study also has implications on the understanding of the transport of complex fluids in diffusio-osmotic pumps.

Intuitively, crest speeds of water waves are assumed to match their phase speeds. However, this is generally not the case for natural waves within unsteady wave groups. This motivates our study, which presents new insights into the generic behaviour of crest speeds of linear to highly nonlinear unsteady waves. While our major focus is on gravity waves where a generic crest slowdown occurs cyclically, results for capillary-dominated waves are also discussed, for which crests cyclically speed up. This curious phenomenon arises when the theoretical constraint of steadiness is relaxed, allowing waves to change their form, or shape. In particular, a kinematic analysis of both simulated and observed open-ocean gravity waves reveals a forward-to-backward leaning cycle for each individual crest within a wave group. This is clearly manifest during the focusing of dominant wave groups essentially due to the dispersive nature of waves. It occurs routinely for focusing linear (vanishingly small steepness) wave groups, and it is enhanced as the wave spectrum broadens. It is found to be relatively insensitive to the degree of phase coherence and focusing of wave groups. The nonlinear nature of waves limits the crest slowdown. This reduces when gravity waves become less dispersive, either as they steepen or as they propagate over finite water depths. This is demonstrated by numerical simulations of the unsteady evolution of two- and three-dimensional dispersive gravity wave packets in both deep and intermediate water depths, and by open-ocean space–time measurements.

In this paper, using multiple-scale analysis, we derive a generalized mathematical model for amplitude evolution, and for calculating the energy exchange in resonant and near-resonant global triads consisting of weakly nonlinear internal gravity wavepackets in weakly non-uniform density stratifications in an unbounded domain in the presence of viscous and rotational effects. Such triad interactions are one of the mechanisms by which high-wavenumber internal waves lead to ocean turbulence and mixing via parametric subharmonic instability. Non-uniform stratification introduces detuning – mismatch in the vertical wavenumber triad condition, which may strongly affect the energy transfer process. We investigate in detail how factors like wavepacket width, group speeds, nonlinear coupling coefficients, detuning and viscosity affect energy transfer in weakly varying stratification. We also investigate the effect of detuning on energy transfer in varying stratification for different daughter wave combinations of a fixed parent wave. We find limitations of the well-known ‘pump-wave approximation’ and derive a non-dimensional number, which can be evaluated from initial conditions, that can predict the maximum energy transferred from the parent wave during the later stages. Two additional non-dimensional numbers, based on various factors affecting energy transfer between near-resonant wavepackets, have also been defined. Moreover, we identify the optimal background stratification in a medium of varying stratification for the parent wave to form a triad with no detuning so that the energy transfer is maximum.

A two-dimensional air sheet in a surrounding liquid contracts under surface tension. We investigate numerically and analytically this contraction dynamics for a range of Ohnesorge numbers $Oh$. In a similar way as for liquid films, three contraction regimes can be identified based on the $Oh$: vortex shedding, smooth contraction and viscous regime. For $Oh\leqslant 0.02$, the rim can even pinch-off due to the rim deformations caused by the vortex shedding. In contrast with a liquid film that continuously accelerates towards the Taylor–Culick velocity when the surrounding fluid can be neglected, the air film contraction velocity first rises to a maximum value $U_{max}$ before decreasing due to the drag of the external fluid on the moving rim. This $U_{max}$ follows a capillary-inertial scaling at low $Oh$ and continuously shifts to a capillary-viscous scaling with increasing $Oh$. We demonstrate that the decreasing contraction velocity scales as $t^{-0.15}$, which is faster than the scaling $t^{-0.2}$ derived under the assumption of a constant drag coefficient. The transition between the capillary-inertial and capillary-viscous regimes can be characterised by the local time evolving Ohnesorge number $Oh_{\unicode[STIX]{x1D6FF}}$ based on the thickness of the rim. The oscillations of the rim appear at a critical local Weber number $We_{\unicode[STIX]{x1D6FF}}$. Then they follow a well-defined oscillation frequency with a characteristic Strouhal number. Beyond a local Reynolds number larger than 200, the oscillations become more irregular with more complex vortex sheddings, eventually leading to the pinch-off of the rim.

A liquid–gas interface (LGI) on submerged microstructure surfaces has the potential to achieve large slip velocities, which is significant for underwater applications such as drag reduction. However, surfactants adsorbing on the LGI can cause surface tension gradient against the mainstream, which weakens the flow near the LGI and severely limits drag reduction. The mechanism of the effect of surfactants on two-dimensional flows has already been proposed, while the effect of surfactants on the three-dimensional flow near the LGI is still not clear. In our study, we specifically design an experimental system to directly observe a three-dimensional backflow at the LGI. The formation as well as the behaviour of the backflow are demonstrated to be significantly influenced by the surfactant. Combining experimental measurements, theoretical analyses and numerical simulations, we reveal the underlying mechanism of the backflow, which is a competition between the mainstream and the Marangoni flows generated by the interfacial concentration gradients of surfactant simultaneously in streamwise and spanwise directions, reflecting the three-dimensional feature of the backflow. In addition, a kinematic similarity is obtained to characterize the backflow. The current work provides a model system for investigating the three-dimensional backflow at the LGI with surfactants, which is significant for practical applications such as drag reduction and superhydrophobicity.

Approximate analytical expressions for the eigenfrequencies of freely propagating, divergent, barotropic topographic Rossby waves over a step shelf are derived. The amplitude equation, that incorporates axisymmetric topography while retaining full spherical geometry, is analysed by standard asymptotic methods based on the limited latitudinal extent of the polar basin as the natural small parameter. The magnitude of the planetary potential vorticity field, $\Pi _P$, increases poleward in the deep basin and over the shelf. However, everywhere over the shelf $\Pi _P$ exceeds its deep-basin value. Consequently, the polar basin waveguide supports two families of vorticity waves; here, our concern is restricted to the study of topographic Rossby (shelf) waves. The leading-order eigenfrequencies and cross-basin eigenfunctions of these modes are derived. Moreover, the spherical geometry allows an infinite number of azimuthally propagating modes. We also discuss the corrections to these leading-order eigenfrequencies. It is noted that these corrections can be associated with planetary waves that can propagate in the opposite direction to the shelf waves. For parameter values typical of the Arctic Ocean, planetary wave modes have periods of tens of days, significantly longer than the shelf wave periods of one to five days. We suggest that observations of vorticity waves in the Beaufort Gyre with periods of tens of days reported in the refereed literature could be associated with planetary, rather than topographic, Rossby waves.

We use the Lie symmetry theory to derive new scaling laws for passive scalar dynamics under the influence of a constant mean gradient of the scalar in decaying homogeneous isotropic turbulence. For this purpose, we apply symmetry analysis to the equations for two-point correlation of the scalar and velocity fluctuations. It is shown that, in contrast to the classical self-similarity approach, the general invariant solutions, respectively scaling laws, of the two-point functions are constructed using the symmetry approach, without requiring an a priori set of similarity scales to carry on the analysis. In the context of the current analysis also, scaling laws for one-point quantities of the scalar variance $\overline{\unicode[STIX]{x1D703}^{2}}$, the transverse scalar flux $\overline{u_{2}\unicode[STIX]{x1D703}}$ and the variance of the turbulent velocity fluctuations $\overline{u^{2}}$ are established, which are essentially related to the scaling symmetries. A key step to derive the scaling laws is the symmetry breaking induced by the constant mean scalar gradient. We use the results of a highly resolved direct numerical simulation of Gauding et al. (Comput. Fluids, vol. 180, 2019, pp. 206–217) to verify the scaling laws and the self-similarity of the two-point correlation functions. It is shown that the general symmetry solutions obtained from symmetry results provide a very good similarity to these functions.

Direct numerical simulations (DNS) of spatially evolving turbulent planar jets of viscoelastic fluids described by the FENE-P model, such as those consisting of a Newtonian fluid solvent carrying long chain polymer molecules, are carried out in order to develop a theory for the far field of turbulent jets of viscoelastic fluids. New evolution relations for the jet shear-layer thickness $\unicode[STIX]{x1D6FF}(x)$, centreline velocity $U_{c}(x)$ and maximum polymer stresses $\unicode[STIX]{x1D70E}_{c}^{[p]}(x)$ are derived and validated by the new DNS data, yielding $\unicode[STIX]{x1D6FF}(x)\sim x$, $U_{c}(x)\sim x^{-1/2}$, and $\unicode[STIX]{x1D70E}_{c}^{[p]}(x)\sim x^{-5/2}$, respectively, where $x$ is the coordinate in the streamwise direction. It is shown that, compared with a classical (Newtonian) turbulent jet, the effect of the polymers is to reduce the spreading rate, centreline velocity decay, Reynolds stresses and viscous dissipation rate. The self-preserving character of the flow is analysed and it is shown that profiles of mean velocity, Reynolds stresses and polymer stresses are self-similar provided the proper scales are used in the normalisation of these quantities. A fundamental difference from the Newtonian jet in this regard is the necessity for two, instead of only one, different velocity and length scales to properly characterise the evolution of the turbulent flow. These extra velocity and length scales are directly related to a time scale associated with the characteristic fading memory property of viscoelastic fluids.

The weakest possible waves in nature are detectable by improving sensitive measurements, but the attainable imaging resolution of low-frequency waves is still challenging, especially in aeroacoustic experiments. In this work, we show how extended-resolution imaging of low-frequency wave sources can be achieved by incorporating acoustic analogy into tomography. First, an equivalent source of sound, which is dependent on the low-frequency target sound field, is produced due to the nonlinear coupling and interaction with an external high-frequency incident plane wave. Next, the low-frequency sources are reconstructed based on the induced sound waves recorded at the receivers. The induced sound waves are of high frequency to enable the extended-resolution imaging. The physical processes involved are theoretically explained based on the insightful acoustic analogy theory and the Born approximation. The numerical and experimental demonstration cases, with representative but straightforward configurations, show that the proposed method can identify the isolated target sources (at low frequencies) with a separation distance smaller than one-tenth to one-thirtieth of the wavelength, yielding much better resolution than the conventional acoustic imaging approaches. The results suggest that the proposed method will be a promising candidate to investigate the properties of an acoustic source within small regions, and, therefore, likely to be used in the study of the associated fluid physics.

For flows in microchannels, a slip on the walls may be efficient in reducing viscous dissipation. A related issue, addressed in this article, is to decrease the effective viscosity of a dilute monodisperse suspension of spheres in Poiseuille flow by using two parallel slip walls. Extending the approach developed for no-slip walls in Feuillebois et al. (J. Fluid Mech., vol. 800, 2016, pp. 111–139), a formal expression is obtained for the suspension intrinsic viscosity $[\mu ]$ solely in terms of a stresslet component and a quadrupole component exerted on a single freely suspended sphere. In the calculation of $[\mu ]$, the hydrodynamic interactions between a sphere and the slip walls are approximated using either the nearest wall model or the wall-superposition model. Both the stresslet and quadrupole are derived and accurately calculated using bipolar coordinates. Results are presented for $[\mu ]$ in terms of $H/(2a)$ and $\tilde{\lambda}=\lambda /a\leq 1$, where $H$ is the gap between walls, $a$ is the sphere radius and $\lambda$ is the wall slip length using the Navier slip boundary condition. As compared with the no-slip case, the intrinsic viscosity strongly depends on $\tilde{\lambda}$ for given $H/(2a)$, especially for small $H/(2a)$. For example, in the very confined case $H/(2a)=2$ (a lower bound found for practical validity of single-wall models) and for $\tilde{\lambda}=1$, the intrinsic viscosity is three times smaller than for a suspension bounded by no-slip walls and five times smaller than for an unbounded suspension (Einstein, Ann. Phys., vol. 19, 1906, pp. 289–306). We also provide a handy formula fitting our results for $[\mu ]$ in the entire range $2\leq H/(2a)\leq 100$ and $\tilde{\lambda}\leq 1$.

We consider the dynamics of a gravity-driven flow coating a vertical fibre rotating about its axis. This flow exhibits rich dynamics including the formation of bead-like structures and different types of steady or oscillatory travelling waves driven by a Rayleigh–Plateau mechanism modified by the presence of gravity and rotation. Linear stability shows that the axisymmetric mode dominates the instability when the rotation is slow, which allows us to derive a two-dimensional model equation under the long-wave assumption. The spatio-temporal dynamics and nonlinear wave solutions are then investigated by the model equation. The spatio-temporal stability analysis showed that the absolute instability is enhanced by the rotation. Steady travelling-wave states and relative periodic states are observed in the numerical simulations of the model equation, which show that the rotation tends to suppress the formation of relative periodic states. To examine this, a linear stability analysis of steady travelling waves is performed, indicating that the rotation has a stabilizing effect on the steady travelling waves. This result is adverse to the destabilizing effect of rotation on the linear stability of initially uniform films. A bifurcation analysis shows that the relative periodic state is born from the instability of steady travelling wave, which represents the coalescence and breakup process between a large droplet and a serial of much smaller droplets.

In the phase-field description of moving contact line problems, the two-phase system can be described by free energies, and the constitutive relations can be derived based on the assumption of energy dissipation. In this work we propose a novel boundary condition for contact angle hysteresis by exploring wall energy relaxation, which allows the system to be in non-equilibrium at the contact line. Our method captures pinning, advancing and receding automatically without the explicit knowledge of contact line velocity and contact angle. The microscopic dynamic contact angle is computed as part of the solution instead of being imposed. Furthermore, the formulation satisfies a dissipative energy law, where the dissipation terms all have their physical origin. Based on the energy law, we develop an implicit finite element method that is second order in time. The numerical scheme is proven to be unconditionally energy stable for matched density and zero contact angle hysteresis, and is numerically verified to be energy dissipative for a broader range of parameters. We benchmark our method by computing pinned drops and moving interfaces in the plane Poiseuille flow. When the contact line moves, its dynamics agrees with the Cox theory. In the test case of oscillating drops, the contact line transitions smoothly between pinning, advancing and receding. Our method can be directly applied to three-dimensional problems as demonstrated by the test case of sliding drops on an inclined wall.

The impact of a collapsing gas bubble above rigid, notched walls is considered. Such surface crevices and imperfections often function as bubble nucleation sites, and thus have a direct relation to cavitation-induced erosion and damage structures. A generic configuration is investigated numerically using a second-order accurate compressible multi-component flow solver in a two-dimensional axisymmetric coordinate system. Results show that the crevice geometry has a significant effect on the collapse dynamics, jet formation, subsequent wave dynamics and interactions. The wall-pressure distribution associated with erosion potential is a direct consequence of development and intensity of these flow phenomena.