This paper addresses peristaltic flow induced in a non-axisymmetric annular tube by a periodic small-amplitude wave of arbitrary shape propagating axially along its inner surface, assumed to be a circular cylinder. The study is motivated by recent in vivo experimental observations pertaining to the flow of cerebrospinal fluid along the perivascular spaces of cerebral arteries. The analysis employs the lubrication approximation, describing low-Reynolds-number peristaltic flow in the long-wavelength approximation. Closed-form analytic expressions are derived for the average pumping rate in infinitely long tubes and also in tubes of finite length. Consideration is also given to the transverse motion arising in non-axisymmetric tubes. For small-amplitude waves, the solution is reduced to the integration of a parameter-free Stokes-flow problem, which is solved for relevant cross-sectional shapes, with closed-form analytical results derived for thin canals.

Motivated by observations of turbulence in the strongly stratified ocean thermocline, we use direct numerical simulations to investigate the interaction of a sinusoidal shear flow and a large-amplitude internal gravity wave. Despite strong nonlinearities in the flow and a lack of scale separation, we find that linear ray-tracing theory is qualitatively useful in describing the early development of the flow as the wave is refracted by the shear. Consistent with the linear theory, the energy of the wave accumulates in regions of negative mean shear where we observe evidence of convective and shear instabilities. Streamwise-aligned convective rolls emerge the fastest, but their contribution to irreversible mixing is dwarfed by shear-driven billow structures that develop later. Although the wave strongly distorts the buoyancy field on which these billows develop, the mixing efficiency of the subsequent turbulence is similar to that arising from Kelvin–Helmholtz instability in a stratified shear layer. We run simulations at Reynolds numbers Re of 5000 and 8000, and vary the initial amplitude of the internal gravity wave. For high values of initial wave amplitude, the results are qualitatively independent of $Re$. Smaller initial wave amplitudes delay the onset of the instabilities, and allow for significant laminar diffusion of the internal wave, leading to reduced turbulent activity. We discuss the complex interaction between the mean flow, internal gravity wave and turbulence, and its implications for internal wave-driven mixing in the ocean.

We present a detailed comparison of the rheological behaviour of sheared sediment beds in a pressure-driven, straight channel configuration based on data that were generated by means of fully coupled, grain-resolved direct numerical simulations and experimental measurements previously published by Aussillous et al. (J. Fluid Mech., vol. 736, 2013, pp. 594–615). The highly resolved simulation data allow us to compute the stress balance of the suspension in the streamwise and vertical directions and the stress exchange between the fluid and particle phases, which is information needed to infer the rheology, but has so far been unreachable in experiments. Applying this knowledge to the experimental and numerical data, we obtain the statistically stationary, depth-resolved profiles of the relevant rheological quantities. The scaling behaviour of rheological quantities such as the shear and normal viscosities and the effective friction coefficient are examined and compared to data coming from rheometry experiments and from widely used rheological correlations. We show that rheological properties that have previously been inferred for annular Couette-type shear flows with neutrally buoyant particles still hold for our set-up of sediment transport in a Poiseuille flow and in the dense regime we found good agreement with empirical relationships derived therefrom. Subdividing the total stress into parts from particle contact and hydrodynamics suggests a critical particle volume fraction of 0.3 to separate the dense from the dilute regime. In the dilute regime, i.e. the sediment transport layer, long-range hydrodynamic interactions are screened by the porous medium and the effective viscosity obeys the Einstein relation.

Microorganisms may exhibit rich swimming behaviours in anisotropic fluids, such as liquid crystals, which have direction-dependent physical and rheological properties. Here we construct a two-dimensional computation model to study the undulatory swimming mechanisms of microswimmers in a solution of rigid, rodlike liquid crystal polymers. We describe the fluid phase using Doi's $Q$-tensor model, and treat the swimmer as a finite-length flexible fibre with imposed propagating travelling waves on the body curvature. The fluid–structure interactions are resolved via an immersed boundary method. Compared with the swimming dynamics in Newtonian fluids, we observe non-Newtonian behaviours that feature both enhanced and retarded swimming motions in lyotropic liquid crystal polymers. We reveal the propulsion mechanism by analysing the near-body flow fields and polymeric force distributions, together with asymptotic analysis for an idealized model of Taylor's swimming sheet.

We investigate the dynamics of cohesive particles in homogeneous isotropic turbulence, based on one-way coupled simulations that include Stokes drag, lubrication, cohesive and direct contact forces. We observe a transient flocculation phase, followed by a statistically steady equilibrium phase. We analyse the temporal evolution of floc size and shape due to aggregation, breakage and deformation. Larger turbulent shear and weaker cohesive forces yield smaller elongated flocs. Flocculation proceeds most rapidly when the fluid and particle time scales are balanced and a suitably defined Stokes number is $O(1)$. During the transient stage, cohesive forces of intermediate strength produce flocs of the largest size, as they are strong enough to cause aggregation, but not so strong as to pull the floc into a compact shape. Small Stokes numbers and weak turbulence delay the onset of the equilibrium stage. During equilibrium, stronger cohesive forces yield flocs of larger size. The equilibrium floc size distribution exhibits a preferred size that depends on the cohesive number. We observe that flocs are generally elongated by turbulent stresses before breakage. Flocs of size close to the Kolmogorov length scale preferentially align themselves with the intermediate strain direction and the vorticity vector. Flocs of smaller size tend to align themselves with the extensional strain direction. More generally, flocs are aligned with the strongest Lagrangian stretching direction. The Kolmogorov scale is seen to limit floc growth. We propose a new flocculation model with a variable fractal dimension that predicts the temporal evolution of the floc size and shape.

The theoretical existence of the granular monoclinal wave, based on the Saint-Venant equations for flowing granular matter, was reported recently by Razis et al. (J. Fluid Mech., vol. 843, 2018, pp. 810–846). The present paper focuses on the mathematical interpretation of its behaviour, treating the equation of motion that describes any granular waveform as a dynamical system, taking also into consideration the Froude number offset $\varGamma$ introduced by Forterre & Pouliquen (J. Fluid Mech., vol. 486, 2003, pp. 21–50). The critical value of the Froude number below which stable uniform flows are observed is determined directly from the stability analysis of the aforementioned dynamical system. It is shown that the granular monoclinal wave, represented as a heteroclinic orbit in phase space, can be categorized into two classes: (i) the mild class, for which the exact form of the waveform can be approximated by the non-viscous (first-order) adaptation of the granular Saint-Venant equations, and (ii) the steep class, for a description of which a second-order (viscous) term in the Saint-Venant equations is absolutely needed to capture the dynamics of the wave. The mathematical criterion that distinguishes the two classes is the changing sign of the trace of the Jacobian matrix evaluated at the fixed point corresponding to the waveform's lower plateau.

Viscous drops, subject to a linear flow in an immiscible viscous fluid, deform. When the resulting drop shape is simple the problem can be addressed by an asymptotic approach or by approximating the deformation using known simple shapes. When the resulting deformation is more complex, the problem is usually addressed numerically. In this paper, we address the problem of drops that are deforming in an axisymmetric compressional (bi-axial extensional) flow. Yielded shapes are flat drops, flat drops with dimples and toroidal drops. The latter two are highly unstable. We propose to approximate the solution of this problem, approximating the shapes by using generalized Cassini ovals, defined herein. The analysis reproduced the branches with shapes of stationary stable flat drops and stationary unstable toroidal drops, available from numerical calculation. Furthermore, it predicts the point of loss of stability of the flat drop to exhibit the transition branch that leads into the formation of the toroidal shapes, and shows that this branch shows stationary, yet unstable, flat drops with ever growing dimples up to collapse.

Oscillatory flows have the potential to overcome long-standing limitations encountered when using steady flows for inertial focusing due to low particle Reynolds numbers. The periodic displacement generated by oscillatory flow increases the total path length travelled by a suspended particle with no net displacement within a channel or the need for increased channel length. The effects of unsteady inertial forces on inertial focusing, however, have not been thoroughly examined. Here, we present a combined theoretical and experimental study on the effect of Womersley number on inertial focusing in planar pulsatile flows. The migration velocity for a small and weakly inertial particle was evaluated for different oscillation frequencies using the method of matched asymptotics. Using experiments in a custom-built microchannel, we show that oscillatory flows are remarkably efficient for inertial focusing, even at low particle Reynolds numbers. For appropriately selected oscillation amplitude and frequency, inertial focusing is achieved in only a fraction of the channel length (1 % to 10 %) compared to what would be required in a steady flow. We show that the Womersley number influences the equilibrium focusing position and the overall focusing efficiency. In fact, above a critical Womersley number, inertial focusing does not occur despite increasing particle Reynolds number. Lastly, the application of oscillatory inertial focusing for the direct measurement of particle migration velocity is demonstrated.

The impact of a drop train, a series of identical liquid drops separated by a constant distance, on a liquid pool initially generates a long slender cavity. However, the cavity soon collapses and turns into a shallow funnel. Here we theoretically model the dynamic profile of the elongated cavity and the steady shape of the funnel, which are then shown to agree well with experiment. When the liquid inertia plays a dominant role, the cavity assumes a parabolic profile that depends only on the drop diameter and the centre-to-centre spacing of adjacent drops. We consider the capillary forces as well as the drop impact force to obtain the shape of the funnel that persists once the elongated cavity collapses. Our study allows for predicting the interfacial morphology by the impact of a drop train and designing impact conditions useful for semiconductor cleaning processes.

We prove a new explicit inequality for the non-dimensional flow force constant, significantly improving the Benjamin and Lighthill conjecture about irrotational steady water waves. As a corollary, we prove a bound for the wave amplitude in terms of the Bernoulli constant. We show that the amplitude decays as $r^{-2}$ when $r \to +\infty$, where $r$ is the non-dimensional Bernoulli constant. We explain that the latter limit corresponds to deep water waves and the bound for the amplitude is sharp. In terms of physical parameters the result states that the amplitude $a$ of an arbitrary Stokes wave is bounded by $C m^2 g/Q^2$, where $m$ is the relative mass flux, $g$ is the gravitational constant, $Q$ is the total head and $C$ is an absolute constant given explicitly. In particular, this implies that $a < C c^2 g^{-1}$, where $c$ is the wave speed. The latter inequality is valid for all Stokes waves, irrespective of wavelength or amplitude, including extreme waves.

A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a ‘pure’ point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter $A$ in the hybrid solutions approaches the limits $A\to 0,\infty$. While $A\to 0$ reproduces the input equilibrium, $A\to \infty$ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by $A$ as a ‘Liouville link’. In some cases, the emergent point vortex equilibrium as $A\to \infty$ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a ‘Liouville chain’. We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717) and by Krishnamurthy et al. (J. Fluid Mech., vol. 874, 2019, R1) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (Proc. R. Soc. A, vol. 476, 2020, 20200310) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the two-dimensional Euler equation.

A remarkable property of geophysical fluids is that, even for nonlinear flows, a slow component can sometimes evolve independently of the fast-wave components. The dry Boussinesq equations, for instance, are known to exhibit this property for small Froude ($Fr$) and Rossby ($Ro$) numbers (i.e. strong stratification and rapid rotation). Here, we ask: Do the moist Boussinesq equations also exhibit this property, even if clouds are included as changes of water between different phases (vapour and liquid)? To investigate, the authors recently performed an asymptotic analysis and identified several ways in which phase changes could possibly induce coupling between the slow component and fast waves; however, these possibilities were not clearly settled from theoretical considerations alone. Here, to investigate further, a suite of numerical simulations is conducted, using a sequence of small values $Fr=Ro=1, 10^{-1}, 10^{-2}, 10^{-3}$. For $Fr=Ro=10^{-1}$, the influence of waves on the slow component is relatively small, but does not decrease proportional to $Fr$ and $Ro$, as $Fr$ and $Ro$ are decreased to $10^{-2}$ and $10^{-3}$. As an explanation and physical interpretation, it is shown that, while linear waves have a time average of zero, the piecewise-linear waves that arise due to phase changes actually have a non-zero time-averaged component.

The statistical properties of prograde spanwise vortex cores and internal shear layers (ISLs) are evaluated for a series of high-Reynolds-number turbulent boundary layers. The considered flows span a wide range of both Reynolds number and surface roughness. In each case, the largest spanwise vortex cores in the outer layer of the boundary layer have size comparable to the Taylor microscale $\lambda _T$, and the azimuthal velocity of these large vortex cores is governed by the friction velocity ${u_\tau }$. The same scaling parameters describe the average thickness and velocity difference across the ISLs. The results demonstrate the importance of the local large-eddy turnover time in determining the strain rate confining the size of the vortex cores and shear layers. The relevance of the turnover time, and more generally the Taylor microscale, can be explained by a stretching mechanism involving the mutual interaction of coherent velocity structures such as uniform momentum zones with the evolving shear layers separating the structures.

We present an asymptotic theory for analytical characterization of the high-Reynolds-number incompressible flow of a Newtonian fluid past a shear-free circular cylinder. The viscosity-induced modifications to this flow are localized and except in the neighbourhood of the rear stagnation point, behave like a linear perturbation of the inviscid flow. Our theory gives a highly accurate description of these modifications by including the contribution from the most significant viscous term in a correctional perturbation expansion about an inviscid base state. We derive the boundary layer equation for the flow and deduce a similarity transformation that leads to a set of infinite, shear-free-condition-incompatible, self-similar solutions. By suitably combining members from this set, we construct an all-boundary-condition-compatible solution to the boundary layer equation. We derive the governing equation for vorticity transport through the narrow wake region and determine its closed-form solution. The near and far-field forms of our wake solution are desirably consistent with the boundary layer solution and the well-known, self-similar planar wake solution, respectively. We analyse the flow in the rear stagnation region by formulating an elliptic partial integro-differential equation for the distortion streamfunction that specifically accounts for the fully nonlinear and inviscid dynamics of the viscous correctional terms. The drag force and its atypical logarithmic dependence on Reynolds number, deduced from our matched asymptotic analysis, are in remarkable agreement with the high-resolution simulation results. The logarithmic dependence gives rise to a critical Reynolds number below which the viscous correction term, counterintuitively, reduces the net dissipation in the flow field.

Cube arrays are one of the most extensively studied types of surface roughness, and there has been much research on cubical roughness with low-to-moderate surface coverage densities. In order to help populate the literature of flow over cube arrays with high surface coverage densities, we conduct direct numerical simulations (DNSs) of flow over aligned cube arrays with coverage densities $\lambda =0.25$ (for validation and comparison purposes), $0.5$, $0.6$, $0.7$, $0.8$ and $0.9$. The roughness are in the d-type roughness regime. Essential flow quantities, including the mean velocity profiles, Reynolds stresses, dispersive stresses and roughness properties, are reported. Special attention is given to secondary turbulent motions in the roughness sublayer. The spanwise-alternating pattern of the thin slots between two neighbouring cubes gives rise to spanwise-alternating regions of low- and high-momentum pathways above the cube crests. We show that the strength and spanwise location of these low- and high-momentum pathways depend on the surface coverage density, and that the high-momentum pathways are not necessarily located directly above the roughness elements. In order to determine the physical processes responsible for the generation and the destruction of these secondary turbulent motions, we analyse the dispersive kinetic energy (DKE) budget. The data shows that the secondary motions get their energy from the DKE-specific production term and the wake production term, and lose energy to the DKE-specific dissipation term.

The problem of defining oceanic mesoscale eddies remains generally unresolved because there is no unique local spatio-temporal filter that can be used for extracting the eddies, and it is unclear what part of the eddy field cannot be actually resolved and needs to be parameterized in a coarse-grid model. We propose using of the coarse-grid model itself for reconstructing dynamically unresolved eddies, which are actually field errors on the top of the dynamically resolved, large-scale reference flow. The novelties and strengths of the approach are that (i) no spatio-temporal filtering is ever needed, (ii) field errors are dynamically translated into the error-correcting forcing and (iii) the latter exactly augments the coarse-grid model solution towards the reference flow. After implementation of the proposed approach, we study statistical properties of the field errors, show their robustness and reveal their significant differences from the locally filtered eddies. We argue that dynamical effects of unresolved eddies can be ultimately parameterized by emulating field errors and closing them on the dynamically resolved flow. So far, our results are limited to the quasigeostrophic approximation, but this serves as a proof of concept and starting point for the follow-up extension into the primitive equations, which are used routinely in the comprehensive oceanic general circulation models.

We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh–Bénard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy–Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection in both the porous and the unconfined fluid regions, while the porous flow still remains strongly confined and governed by Darcy's law. We demonstrate that the heat flux and average thermal structure of the system can be predicted using previous results of convection in individual fluid or porous layers. We revisit a controversy about the role of subcritical ‘penetrative convection’ in the porous medium, and confirm that such induced flow does not contribute to the heat flux through the system. Lastly, we briefly study the temporal coupling between the two layers and find that the turbulent fluid convection above acts as a low-pass filter on the longer time-scale variability of convection in the porous layer.

The present study deals with the effect of micromagnetorotation (MMR) on a micropolar Poiseuille flow in the presence of a uniform magnetic field. Micromagnetorotation is associated with the impact of magnetization on magnetohydrodynamic (MHD) micropolar flows. Previously, magnetization was assumed to be parallel to the applied magnetic field and thus, its influence on the flow was ignored. This assumption is incorrect in the case of micropolar fluids, because their anisotropy affects magnetization. Here, the velocity and microrotation fields, as well as the skin friction coefficient are examined analytically by using a new MHD micropolar fluid theory that includes a constitutive equation for magnetization. Results reveale that MMR has a strong braking effect both on velocity and microrotation. Flow deceleration is found to be up to 16 %, while an increase in the skin friction coefficient is also observed. Moreover, the stability of the MHD micropolar flow is studied by introducing a modified version of the Orr–Sommerfeld equation, which incorporates MMR. The eigenvalue problem is solved with the use of the open-source Chebfun library. It is found that the MMR has a strong stabilizing effect on the MHD micropolar flow. Thus, the MMR is proved to be a mechanism similar to the Lorentz force, which dissipates additional magnetic energy to the flow via microrotation. In summary, the important effect of MMR, neglected by researchers so far, should be considered for industrial and bioengineering applications that involve micropolar fluids and magnetic fields.

The linear stability of a vertical interface separating two miscible fluid columns of different densities and viscosities under the influence of gravity is investigated. This flow possesses a time-dependent reference state (each column accelerates at different rates owing to their different densities) and the interface thickness grows as the square root of time (by diffusion). Numerical integration of the linear initial-value problem is carried out and discussed in detail as a function of vertical and spanwise wavenumbers and the flow parameters. Adjoint-based optimization is performed in order to determine initial conditions that lead to maximum growth of disturbances in finite time. Results indicate that the rate of growth of the perturbation energy at small wavenumbers (less affected by viscosity initially) is dominated by two-dimensional modes (no spanwise variation). Substantial transient growth is observed at higher wave modes initially, followed by asymptotic decay of the perturbations at large time. Sensitivity of perturbation growth with respect to initial time, density and viscosity ratios is investigated. This work is complementary to previous inviscid analysis of this configuration, which showed that the interface was unconditionally unstable at all wave modes, even in the presence of surface tension, and that instability grew as the exponential of time squared.

Oceanic mesoscale currents (‘eddies’) can have significant effects on the distributions of passive tracers. The associated inhomogeneous and anisotropic eddy fluxes are traditionally parametrised using a transport tensor ($K$-tensor), which contains both diffusive and advective components. In this study, we analyse the eddy transport tensor in a quasigeostrophic double-gyre flow. First, the flow and passive tracer fields are decomposed into large- and small-scale (eddy) components by spatial filtering, and the resulting eddy forcing includes an eddy tracer flux representing advection by eddies and non-advective terms. Second, we use the flux-gradient relation between the eddy fluxes and the large-scale tracer gradient to estimate the associated $K$-tensors in their entire structural, spatial and temporal complexity, without making any additional assumptions or simplifications. The divergent components of the eddy tracer fluxes are extracted via the Helmholtz decomposition, which yields a divergent tensor. The remaining rotational flux does not affect the tracer evolution, but dominates the total tracer flux, affecting both its magnitude and spatial structure. However, in terms of estimating the eddy forcing, the transport tensor prevails over its divergent counterpart because of the significant numerical errors induced by the Helmholtz decomposition. Our analyses demonstrate that, in general, the $K$-tensor for the eddy forcing is not unique, that is, it is tracer-dependent. Our study raises serious questions on how to interpret and use various estimates of $K$-tensors obtained from either observations or eddy-resolving model solutions.