Regions of high curvature of a material line as they evolve in a chaotic flow are considered. In such a region, the curvature as a function of arclength along the line is found to have a universal form with the peak curvature the only parameter involved. The alignment of the principal axes of the strain tensor with respect to the local tangent vector of the curve and the ratio of the two largest finite-time Lyapunov exponents play a key role. Numerical experiments with ABC flow demonstrate the result.

Nonlinear effects on the receptivity of cross-flow in the swept Hiemenz boundary layer are investigated. Numerical simulations are generated using a vorticity form of the Navier–Stokes equations. Steady perturbations are established using surface suction and blowing distributed along the spanwise direction as either a periodic strip or a band of small holes. The method of excitation, the size and the location of the prescribed forcing are shown to have a significant influence on the receptivity of the boundary layer. Blowing holes are found to excite perturbations with considerably larger magnitudes than those generated using a periodic suction and blowing strip. A semi-logarithmic relationship is derived that relates the initial amplitude of the linear-only disturbances with the location at which the absolute magnitude of the chordwise primary Fourier harmonic attains a stationary point or a size of approximately one-tenth of the free-stream spanwise velocity. Furthermore, the size of the physical chordwise velocity perturbation about this position can be estimated directly from the linear-only solutions. This would suggest that, for sufficiently small initial amplitudes, the onset of some nonlinear flow development properties can be predicted directly from a linear receptivity analysis.

We develop a bifurcation theory describing the conditions under which vortices are created or destroyed in a two-dimensional incompressible flow. We define vortices using the $Q$-criterion and analyse the vortex structure by considering the evolution of the zero contours of $Q$. The theory identifies topological changes of the vortex structure and classifies these as four possible types of bifurcations, two occurring away from boundaries, and two occurring near no-slip walls. Our theory provides a description of all possible codimension-one bifurcations where time is treated as the bifurcation parameter. To illustrate our results, we consider the early stages of boundary layer eruption at moderate Reynolds numbers in the range from $Re=750$ to $Re=2250$. By analysing numerical simulations of the phenomenon, we show how to describe the eruption process as sequences of the four possible bifurcations of codimension one. Our simulations show that there is a single codimension-two point within our parameter range. This codimension-two point arises at $Re=1817$ via the coalescence of two codimension-one bifurcations which are associated with the creation and subsequent destruction of one of the vortices that erupt from the boundary layer. We present a theoretical description of this process and explain how the occurrence of this phenomenon separates the parameter space into two regions with distinct evolution of the topology of the vortices.

Observations in experiments and simulations show that the kinematic behaviour of an elastic capsule, suspended and rotating in shear flow, depends upon the flow strength, the capsule membrane material properties and its at-rest shape. We develop a linear stability description of the periodically rotating base state of this coupled system, as represented by a boundary integral flow formulation with spherical harmonic basis functions describing the elastic capsule geometry. This yields Floquet multipliers that classify the stability of the capsule motion for elastic capillary numbers $Ca$ ranging from $Ca=0.01$ to 5. Viscous dissipation rapidly damps most perturbations. However, for all cases, a single component grows or decays slowly, depending upon $Ca$, over many periods of the rotation. The transitions in this stability behaviour correspond to the different classes of rotating motion observed in previous studies.

Equal coaxial symmetrically located helical vortices translate and rotate steadily while preserving their shape and relative position if they move in an unbounded inviscid incompressible fluid. In this paper, the linear and angular velocities of this set of vortices ($U$ and $\unicode[STIX]{x1D6FA}$ respectively) are computed as the sum of the mutually induced velocities found by Okulov (J. Fluid Mech., vol. 521, 2004, pp. 319–342) and the self-induced velocities found by Velasco Fuentes (J. Fluid Mech., vol. 836 2018). Numerical computations of the velocities using the Helmholtz integral and the Biot–Savart law, as well as numerical simulations of the flow evolution under the Euler equations, are used to verify that the theoretical results are accurate for $N=1,\ldots ,4$ vortices over a broad range of values of the pitch and radius of the vortices. An analysis of the flow topology in a reference system that translates with velocity $U$ and rotates with angular velocity $\unicode[STIX]{x1D6FA}$ serves to determine the capacity of the vortices to transport fluid.

When a small amount of marked solute is released into a stratified fluid, there is lateral dispersion of the marked solute and a larger lateral dispersion of any density anomaly. The method of moments is used to calculate the two dispersion coefficients. The excess dispersion for the density is shown to be proportional to the fractional density decrease from the bed to the free surface and to the cube of the water depth, and inversely proportional to the vertical mixing for lateral momentum. For weak turbulent mixing the stratification-induced lateral dispersion for the density anomaly can be several orders of magnitude greater than the lateral turbulent mixing for marked solute

This paper analyses a numerical model of a flow around an obstacle bounded in a channel with arbitrary walls. The model is based on the usual two-dimensional model of ideal incompressible weightless fluid. It is validated here by comparison with analytical findings and experimental data, which consisted of the velocity field determined by particle image velocimetry (PIV). Despite the simplicity of our wake model, the numerical data are usually in good agreement with the analytical and experimental data.

Steady incompressible flow down a slowly curving circular pipe is considered, analytically and numerically. Both real and complex solutions are investigated. Using high-order Hermite–Padé approximants, the Dean series solution is analytically continued outside its circle of convergence, where it predicts a complex solution branch for real positive Dean number, $K$. This is confirmed by numerical solution. It is shown that other previously unknown solution branches exist for all $K>0$, which are related to an unforced complex eigensolution. This non-uniqueness is believed to be generic to the Navier–Stokes equations in most geometries. By means of path continuation, numerical solutions are followed around the complex $K$-plane. The standard Dean two-vortex solution is shown to lie on the same hypersurface as the eigensolution and the four-vortex solutions found in the literature. Elliptic pipes are considered and shown to exhibit similar behaviour to the circular case. There is an imaginary singularity limiting convergence of the Dean series, an unforced solution at $K=0$ and non-uniqueness for $K>0$, culminating in a real bifurcation.

The linear stability of inviscid non-diffusive density-stratified shear flow in a rotating frame is considered. A temporally periodic base flow, characterized by vertical shear S, buoyancy frequency N and rotation frequency f, is perturbed by infinitesimal inertia–gravity waves. The temporal evolution and stability characteristics of the disturbances are analysed using Floquet theory and the growth rates of unstable solutions are computed numerically. The global structure of solutions is addressed in the dimensionless parameter space (N/f, S/f, φ) where φ is the wavenumber inclination angle from the horizontal for the wave-like perturbations. Both weakly stratified rapidly rotating flows (N<f) and strongly stratified slowly rotating flows (N>f) are examined. Distinct families of unstable modes are found, each of which can be associated with nearby stable solutions of periodicity T or 2T where T is the inertial frequency 2π/f. Rotation is found to be a destabilizing factor in the sense that stable non-rotating shear flows with N2/S2>1/4 can be unstable in a rotating frame. Morever, instabilities by parametric resonance are found associated with free oscillations at half and integer multiples of the inertial frequency.

This study is concerned with the stability of a flow of viscous conducting liquid driven by a pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Although the magnetic field has a strong stabilizing effect, this flow, similarly to its hydrodynamic counterpart – plane Poiseuille flow – is known to become turbulent significantly below the threshold predicted by linear stability theory. We investigate the effect of the magnetic field on two-dimensional nonlinear travelling-wave states which are found at substantially subcritical Reynolds numbers starting from $\mathit{Re}_{n}=2939$ without the magnetic field and from $\mathit{Re}_{n}\sim 6.50\times 10^{3}\mathit{Ha}$ in a sufficiently strong magnetic field defined by the Hartmann number $\mathit{Ha}$. Although the latter value is a factor of seven lower than the linear stability threshold $\mathit{Re}_{l}\sim 4.83\times 10^{4}\mathit{Ha}$, it is still more than an order of magnitude higher than the experimentally observed value for the onset of turbulence in magnetohydrodynamic (MHD) channel flow.

We investigate the mechanisms that affect the formation of columnar vortices for spin-up in a cylinder where the temperatures at the horizontal walls are prescribed. Numerical results from the three-dimensional Navier–Stokes equations show that a short-lived instability, suppressed by the combined effect of rotation and stratification, generates small temperature variations in the azimuthal direction. Temperature-gradient anomalies produce vorticity, and these vortices stir the fluid at the interface of the central vortex core thus reinforcing the temperature gradients. For sufficiently strong temperature gradients, the central vortex core breaks up into several columnar vortices. It is found, in particular, that small aspect ratios (height over radius of the cylindrical fluid layer) tend to inhibit the instability, while larger ones, , have the opposite effect. The main source of instability is the baroclinic vorticity production and not the presence of a solid sidewall since, counter-intuitively, the flow is more unstable for a free-slip boundary than for a no-slip one. Finally the effect of the temperature boundary conditions (isothermal versus adiabatic) on the horizontal boundaries has been investigated. The adiabatic boundaries help to preserve for longer times the sloping density interfaces that feed, with their potential energy, the baroclinic vorticity production; this results in more unstable flows.

Maintaining the stratification of the ocean requires deep mixing. Part of the energy that provides this mixing is transferred over topography from the surface tides into the internal tides. Numerous theoretical, numerical and experimental studies have aimed at quantifying the energy transfer of this conversion mechanism. Maas (J. Fluid Mech., this issue, vol. 684, 2011, pp. 5–24) constructs model topographies with localized response and no energy transfer into the propagating internal tide, a surprising result that raises questions about models of tidal conversion.

Stochastic roughness is a widespread feature of natural surfaces and is an inherent byproduct of most fabrication techniques. In view of the rapid development of microfluidics, the important question is how this inevitable problem affects the low-Reynolds-number flows that are common for micro-devices. Moreover, one could potentially turn the flaw into a virtue and control the flow properties by means of specially ‘tuned’ random roughness. In this paper we investigate theoretically the statistics of fluctuations in fluid velocity produced by the waviness irregularities at the surface of a no-slip wall. Particular emphasis is laid on the issue of the universality of our findings.

In this paper the qualitative nonlinear influence of advection and continuity on the resonance characteristics of co-oscillating coastal basins is investigated. For this purpose a weakly nonlinear analysis was carried out on the shallow-water equations describing a coastal basin resonating with exterior water-level oscillations. It extends previous work on an almost-enclosed basin with one single (Helmholtz) mode to arbitrarily shaped ‘shallow’ basins with an infinite number of modes. In line with that work, it is necessary to assume friction to be sufficiently weak such that it is in balance with the nonlinear effects, instead of occurring in the linearized equations. The main result of this paper is the system of Landau equations describing the slow evolution of the amplitudes of the oscillatory eigenmodes of the basin, disregarding a zero-frequency eigenmode, should it exist. The dynamics of the zero-frequency mode neglecting oscillatory eigenmodes have been discussed by others, but a consistently balanced model for small amplitudes incorporating both the zero-frequency and oscillatory modes is not yet available. The behaviour of this system, describing the dynamics of the oscillatory eigenmodes only, is analysed. On the longer time scale, it gives rise to a ‘bent resonance curve’, multiple equilibria (several tidal regimes under the same tidal forcing), sudden regime changes and even chaotic dynamics (when these regime changes occur in an irregular way).

Fully developed flows are often used to describe fluid motion in complex geometrical systems, including the human macrocirculation. In fact they may frequently be quite inappropriate even for geometrically simple pipes, owing to the unfeasibly large viscous entry lengths required. Inviscid adjustment to changes in geometry, however, occurs on the lengthscale of the pipe diameter. Inviscid idealizations are therefore more likely to apply in relatively short arterial sections. We aim to quantify the distances involved by calculating the rates of spatial decay for a general disturbance superimposed on an idealized base flow. Both irrotational and rotational base flows are examined, although in the latter case there can exist non-decaying inertial waves, so that an arbitrary inflow need not attain an inviscid state independent of the downstream coordinate. In the rotational case, we therefore restrict attention to those flows which settle down to perturbations of such a state, whereas the potential flows can be regarded as developing from an arbitrary input.

We focus on the last surviving mode of decay in simple uniform pipe geometries, in particular a straight pipe, part of a torus, and a helical pipe. In this way we are able to assess the effects of curvature and torsion on the inviscid entry lengths.

Principally, it is shown that the rate of decay is fastest in a straight pipe and slowest in a toroidal pipe, with that in a helical pipe somewhere in between. Core vorticity tends to reduce the decay rate. If an idealized flow occurs in a geometrically simple arterial portion, our results determine its domain of validity.

We analyse numerically a pinch-type instability in a semi-infinite planar layer of inviscid conducting liquid bounded by solid walls and carrying a uniform electric current. Our model is as simple as possible but still captures the salient features of the instability which otherwise may be obscured by the technical details of more comprehensive numerical models and laboratory experiments. Firstly, we show the instability in liquid metals, which are relatively poor conductors, differs significantly from the astrophysically relevant Tayler instability. In liquid metals, the instability develops on the magnetic response time scale, which depends on the conductivity and is much longer than the Alfvén time scale, on which the Tayler instability develops in well conducting fluids. Secondly, we show that this instability is an edge effect caused by the curvature of the magnetic field, and its growth rate is determined by the linear current density and independent of the system size. Our results suggest that this instability may affect future liquid-metal batteries when their size reaches a few metres.

We present the solution of the idealized steady-state gravity current of height $h$ and density $\unicode[STIX]{x1D70C}_{1}$ that propagates into an ambient motionless fluid of height $H$ and density $\unicode[STIX]{x1D70C}_{2}$ in a channel of general non-rectangular cross-section, with an upper surface open to the atmosphere, at high Reynolds number. The current propagates with speed $U$ and causes a depth decrease $\unicode[STIX]{x1D712}$ of the top surface. This is a significant extension of Benjamin’s (J. Fluid Mech., vol. 31, 1968, pp. 209–248) seminal solution for the gravity current in a rectangular (or laterally unbounded) channel with a fixed top ($\unicode[STIX]{x1D712}=0$). The determination of $\unicode[STIX]{x1D712}$ is a part of the problem. Supposing that the direction of propagation is $x$ and gravity acceleration $g$ acts in the $-z$ direction, the sidewalls are specified by $y=-f_{I}(z)$ and $y=f_{II}(z),~z\in [0.H]$, and the width is $f(z)=f_{I}(z)+\,f_{II}(z)$. The dimensionless parameters of the problem are $a=h/H\in (0,1)$ and $r=\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1}\in (0,1)$. We show that a control-volume analysis of the type used by Benjamin produces a system of algebraic equations for $\tilde{\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D712}/H$ and $Fr=U/(g^{\prime }h)^{1/2}$ as functions of $a$ and $r$, where $g^{\prime }=(r^{-1}-1)g$ is the reduced gravity. The geometry enters the equation via the width function $f(z)$. We present solutions for typical $f(z)$: rectangle, semi-circle, $\vee$ triangle and trapezoid $\text{}\underline{/~\backslash }$ . The results are physically acceptable and insightful. The non-negative dissipation condition defines the domain of validity $a\leqslant a_{max}(r)$ (also depending on $f(z)$); the equality sign corresponds to energy-conserving cases. The critical speed limitation (with respect to the characteristics) is also considered briefly and suggests a slightly smaller $a\leqslant a_{crit}(r)$. The open-top results in the Boussinesq limit $r\rightarrow 1$ coincide with the fixed-top solution. Upon the reduction of $r$, for a fixed thickness $a$, the value of $Fr$ decreases and $\unicode[STIX]{x1D712}$ increases, until the point of energy-conserving (non-dissipative) flow; for smaller $r$, a negative non-physical dissipation appears. The trends are more pronounced for a converging cross-section geometry (like $\text{}\underline{/~\backslash }$ ) than for the opposite shape (like $\vee$ triangle). The previously investigated Benjamin-type steady-state $Fr$ and dissipation results are particular cases of the new formulation: $f(z)=$ const. reproduces the two-dimensional results, and $\unicode[STIX]{x1D712}=0$ recovers the fixed-top solution.