A recent paper by Needham, Riley & Smith (1988) considers the flow of a jet of inviscid, incompressible fluid emerging from a circular pipe into a weak crossflow. In this paper we extend their results. In particular we show how the Kutta condition at the pipe lip may be satisfied, and we present details of the flow field obtained from both the formal solution and a field calculation.

We present a Newtonian, one-dimensional, differential analysis for capillary breakup rheometry (CBR) to determine the surface tension to viscosity ratio $\unicode[STIX]{x1D6FC}$. Our local differential analysis does not require specific assumptions for the axial force to preclude its measurement. Our analysis indicates that measuring gradients in filament curvature is necessary to accurately determine $\unicode[STIX]{x1D6FC}$ when axial force is not measured. CBR experiments were performed on five silicone oils ($0.35~\text{Pa}~\text{s}<\unicode[STIX]{x1D707}<10~\text{ Pa}~\text{s}$), three sample volumes, and three strains to evaluate the operating range of the differential analysis and compare its performance to that of a standard integral method from literature. We investigate the role of filament asymmetry, caused mainly by gravity, on the performance of the differential method for the range of conditions studied. Experimental and analytical details for resolving gradients of curvature are also given.

Vortex-induced vibration of a circular cylinder that is free to move in the transverse ($Y$) and streamwise ($X$) directions is investigated at subcritical Reynolds numbers ($1500\lesssim Re\lesssim 9000$) via three-dimensional (3-D) numerical simulations. The mass ratio of the system for all the simulations is $m^{\ast }=10$. It is observed that while some of the characteristics associated with the $XY$-oscillation are similar to those of the $Y$-only oscillation (in line with the observations made by Jauvtis & Williamson (J. Fluid Mech., vol. 509, 2004, pp. 23–62)), notable differences exist between the two systems with respect to the transition between the branches of the cylinder response in the lock-in regime. The flow regime between the initial and lower branch is characterized by intermittent switching in the cylinder response, aerodynamic coefficients and modes of vortex shedding. Similar to the regime of laminar flow, the system exhibits a hysteretic response near the lower- and higher-$Re$ end of the lock-in regime. The frequency spectrum of time history of the cylinder response shows that the most dominant frequency in the streamwise oscillation on the initial branch is the same as that of the transverse oscillation.

Experiments on high-pressure vessel decompression have shown that vaporization occurs in ‘boiling shocks’ moving with a velocity of . To explain this phenomenon, a model accounting for bubble breakup was suggested (Ivashnyov, Ivashneva & Smirnov, J. Fluid. Mech., vol. 413, 2000, pp. 149–180). It was shown that the explosive boiling was caused by chain bubble fragmentation, which led to a sharp increase in the interface area and instantaneous transformation of the mixture into an equilibrium state. In the present study, this model is used to simulate nozzle flows with no change in the free parameters chosen earlier for modelling a tube decompression. It is shown that an advanced model ensures the best correspondence to experiments for flashing flows in comparison with an equilibrium model and with a model of boiling at a constant number of centres. It is also shown that the formation of a boiling shock in a critical nozzle flow leads to autovibrations.

Benilov, O'Brien & Sazonov (2003) and Benilov (2004) describe “a new type of instability” in a liquid film inside a rotating cylinder. Though their linear systems support only neutrally stable modes, they find explosive disturbances which become singular after a finite time. They suggest that this result casts doubt on the reliability of modal analysis for prediction of instability; and they claim that “Such cases have never been described in the literature, and they are probably extremely rare”. Here, other examples are given, some of which have been known (though not well-known) for many years. A common feature of all these singularities is a local phase synchronization of short-wave modes; but the configuration of Benilov et al. has the additional feature of eigenfunctions that exhibit very large changes in amplitude within the spatial domain. The relevance, or not, of such singularities to real physical systems is discussed.

A new and systematic approach is proposed to determine the migration of torque-free spherical bubbles immersed in a steady and non-uniform unbounded Stokes flow and subject to arbitrary capillary effects. The advocated procedure appeals to only a very few quantities on the surface of each bubble and is therefore suitable for a future numerical treatment of arbitrary clusters of bubbles. For a single bubble, the theory allows a straightforward analytical implementation and the predicted results agree well with Hetsroni & Haber (1970), Hetsroni et al. (1971) and Subramanian (1985). The thermocapillary motion of non-conducting spherical bubbles freely suspended in a quiescent liquid in the presence of an arbitrary ambient temperature $T_{\infty}$ is considered and it is shown that it is futile to determine the disturbed temperature field, whatever $T_{\infty},$ once bubbles are equivalent (i.e. experience the same velocity in a given uniform temperature gradient ${\bm \nabla}T_{\infty},$ as obtained by Acrivos et al. 1990 and Wang et al. 1994).

A group of three multiscale inhomogeneous grids have been tested to generate different types of turbulent shear flows with different mean shear rate and turbulence intensity profiles. Cross hot-wire measurements were taken in a wind tunnel with Reynolds number $Re_{D}$ of 6000–20 000, based on the width of the vertical bars of the grid and the incoming flow velocity. The effect of local drag coefficient $C_{D}$ on the mean velocity profile is discussed first, and then by modifying the vertical bars to obtain a uniform aspect ratio the mean velocity profile is shown to be predictable using the local blockage ratio profile. It is also shown that, at a streamwise location $x=x_{m}$, the turbulence intensity profile along the vertical direction $u^{\prime }(y)$ scales with the wake interaction length $x_{\ast ,n}^{peak}=0.21g_{n}^{2}/(\unicode[STIX]{x1D6FC}C_{D}w_{n})$ ($\unicode[STIX]{x1D6FC}$ is a constant characterizing the incoming flow condition, and $g_{n}$, $w_{n}$ are the gap and width of the vertical bars, respectively, at layer $n$) such that $(u^{\prime }/U_{n})^{2}\unicode[STIX]{x1D6FD}^{2}(C_{D}w_{n}/x_{\ast ,n}^{peak})^{-1}\sim (x_{m}/x_{\ast ,n}^{peak})^{b}$, where $\unicode[STIX]{x1D6FD}$ is a constant determined by the free-stream turbulence level, $U_{n}$ is the local mean velocity and $b$ is a dimensionless power law constant. A general framework of grid design method based on these scalings is proposed and discussed. From the evolution of the shear stress coefficient $\unicode[STIX]{x1D70C}(x)$, integral length scale $L(x)$ and the dissipation coefficient $C_{\unicode[STIX]{x1D716}}(x)$, a simple turbulent kinetic energy model is proposed that describes the evolution of our grid generated turbulence field using one centreline measurement and one vertical profile of $u^{\prime }(y)$ at the beginning of the evolution. The results calculated from our model agree well with our measurements in the streamwise extent up to $x/H\approx 2.5$, where $H$ is the height of the grid, suggesting that it might be possible to design some shear flows with desired mean velocity and turbulence intensity profiles by designing the geometry of a passive grid.

Stably stratified sheared flows are ubiquitous in geophysical flows from the ocean to the stars, and the route to turbulence in these flows remains an open question. The article by Lefauve et al. (J. Fluid Mech., vol. 848, 2018, pp. 508–544) is an invitation to this journey. With impressive experimental precision mastered by few teams in the world, the nature of the coherent structure that dominates the flow on the verge of turbulent breakdown is revealed and analysed through one- or two-dimensional modern stability analysis of an experimentally obtained base flow. The effect of confinement is surprisingly strong, advocating for leaving the textbook flows, inhomogeneous in only one direction, for the more complex shores of real flows, now accessible to analysis of multidimensional stability problems. The route explored by Lefauve et al. (2018) renews with the long tradition of the supercritical bifurcation scenario, it revisits the linear stability theory with possibility of resonances, critical layers and more to be imagined, since complex base flows are now available to explore both experimentally and analytically.

The fluid velocity far from a translating body in two-dimensional irrotational flow is generally dipolar. This is a classical result. Here we ask when the dipolar component vanishes. Lamb (1945, § 126) provides symmetry conditions on the virtual mass tensor for this to be the case. We show that these conditions are not necessary and obtain the sufficient and necessary condition in terms of the shape of the body using conformal maps. Some explicit examples are constructed based on this condition.

An oblate spheroid, the respective hemispheroids of which are kept at different uniform temperatures, placed in a rarefied gas at rest is considered. The explicit formula for the force acting on the spheroid (radiometric force) is obtained for small Knudsen numbers. This is a model of a vane of the Crookes radiometer. The analysis is performed for a general axisymmetric distribution of the surface temperature of the spheroid, allowing abrupt changes. Although the generalized slip flow theory, established by Sone (Rarefied Gas Dynamics, vol. 1, 1969, pp. 243–253), is available for general rarefied gas flows at small Knudsen numbers, it cannot be applied to the present problem because of the abrupt temperature changes. However, if it is combined with the symmetry relations for the linearized Boltzmann equation developed recently by Takata (J. Stat. Phys., vol. 136, 2009, pp. 751–784), one can bypass the difficulty. To be more specific, the force acting on the spheroid in the present problem can be generated from the solution of the adjoint problem to which the generalized slip flow theory can be applied, i.e. the problem in which the same spheroid with a uniform surface temperature is placed in a uniform flow of a rarefied gas. The analysis of the present paper follows this strategy.

When a salt-stratified body of fluid is heated from the side a series of almost horizontal convective layers can develop with well-mixed interiors. These layers can propagate into the interior of the stratified fluid. This behaviour is also observed with intrusions at fronts between stratified bodies of fluid where their composition varies. We look at a simplified model of intrusion growth where the mechanism behind the creation of their well-mixed interiors is neglected, and look at how a stack of such intrusions will propagate away from the wall or front. We find that there is a transition from a regime where the propagation is essentially inviscid and the intrusion length is proportional to time, to one where viscosity is important and the propagation rate slows down, with the length being proportional to the square-root of time.

The classical Stokes’ problem describing the fluid motion due to a steadily moving infinite wall is revisited in the context of dense granular flows of mono-dispersed beads using the recently proposed $\unicode[STIX]{x1D707}(I)$-rheology. In Newtonian fluids, molecular diffusion brings about a self-similar velocity profile and the boundary layer in which the fluid motion takes place increases indefinitely with time $t$ as $\sqrt{\unicode[STIX]{x1D708}t}$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. For a dense granular viscoplastic liquid, it is shown that the local shear stress, when properly rescaled, exhibits self-similar behaviour at short time scales and it then rapidly evolves towards a steady-state solution. The resulting shear layer increases in thickness as $\sqrt{\unicode[STIX]{x1D708}_{g}t}$ analogous to a Newtonian fluid where $\unicode[STIX]{x1D708}_{g}$ is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular medium, such as grain diameter $d$, density $\unicode[STIX]{x1D70C}$ and friction coefficients, but also on the applied pressure $p_{w}$ at the moving wall and the solid fraction $\unicode[STIX]{x1D719}$ (constant). In addition, the $\unicode[STIX]{x1D707}(I)$-rheology indicates that this growth continues until reaching the steady-state boundary layer thickness $\unicode[STIX]{x1D6FF}_{s}=\unicode[STIX]{x1D6FD}_{w}(p_{w}/\unicode[STIX]{x1D719}\unicode[STIX]{x1D70C}g)$, independent of the grain size, at approximately a finite time proportional to $\unicode[STIX]{x1D6FD}_{w}^{2}(p_{w}/\unicode[STIX]{x1D70C}gd)^{3/2}\sqrt{d/g}$, where $g$ is the acceleration due to gravity and $\unicode[STIX]{x1D6FD}_{w}=(\unicode[STIX]{x1D70F}_{w}-\unicode[STIX]{x1D70F}_{s})/\unicode[STIX]{x1D70F}_{s}$ is the relative surplus of the steady-state wall shear stress $\unicode[STIX]{x1D70F}_{w}$ over the critical wall shear stress $\unicode[STIX]{x1D70F}_{s}$ (yield stress) that is needed to bring the granular medium into motion. For the case of Stokes’ first problem when the wall shear stress $\unicode[STIX]{x1D70F}_{w}$ is imposed externally, the $\unicode[STIX]{x1D707}(I)$-rheology suggests that the wall velocity simply grows as $\sqrt{t}$ before saturating to a constant value whereby the internal resistance of the granular medium balances out the applied stresses. In contrast, for the case with an externally imposed wall speed $u_{w}$, the dense granular medium near the wall initially maintains a shear stress very close to $\unicode[STIX]{x1D70F}_{d}$ which is the maximum internal resistance via grain–grain contact friction within the context of the $\unicode[STIX]{x1D707}(I)$-rheology. Then the wall shear stress $\unicode[STIX]{x1D70F}_{w}$ decreases as $1/\sqrt{t}$ until ultimately saturating to a constant value so that it gives precisely the same steady-state solution as for the imposed shear-stress case. Thereby, the steady-state wall velocity, wall shear stress and the applied wall pressure are related as $u_{w}\sim (g\unicode[STIX]{x1D6FF}_{s}^{2}/\unicode[STIX]{x1D708}_{g})f(\unicode[STIX]{x1D6FD}_{w})$ where $f(\unicode[STIX]{x1D6FD}_{w})$ is either $O(1)$ if $\unicode[STIX]{x1D70F}_{w}\sim \unicode[STIX]{x1D70F}_{s}$ or logarithmically large as $\unicode[STIX]{x1D70F}_{w}$ approaches $\unicode[STIX]{x1D70F}_{d}$.

A global analysis of steady states of low Mach number inviscid premixed reacting swirling flows in a straight circular finite-length open pipe is developed. We focus on modelling the basic interaction between the swirl and heat release of the reaction. For analytic simplicity, a one-step first-order Arrhenious reaction kinetics is considered in the limit of high activation energy and infinite Peclet number. Assuming a complete reaction with chemical equilibrium upstream and downstream of the reaction zone, a nonlinear partial differential equation is derived for the solution of the flow stream function downstream of the reaction zone in terms of the specific total enthalpy, specific entropy and circulation functions prescribed at the inlet. Several types of solutions of the nonlinear ordinary differential equation for the columnar flow case describe the outlet states of the flow in a long pipe. These solutions are used to form the bifurcation diagram of steady reacting flows with swirl as the inlet swirl level is increased at a fixed heat release from the reaction. The approach is applied to two profiles of inlet flows, the solid-body rotation and the Lamb–Oseen vortex, both with constant profiles of the axial velocity, temperature and mixture reactant mass fraction. The computed results provide theoretical predictions of the critical inlet swirl levels for the appearance of vortex breakdown states and for the size of the breakdown zone as a function of the inlet flow swirl level, Mach number and heat release of the reaction. For the inlet solid-body rotation, flow is decelerated to breakdown as the inlet swirl is increased above the critical swirl level, and there is a delay in the appearance of breakdown with the increase of the heat release of the reaction. For the inlet Lamb–Oseen vortex at low values of heat release, the critical swirl for breakdown is decreased with the increase of heat release while, at high values of heat release, the appearance of breakdown is delayed to higher incoming flow swirl levels with the increase of heat release. The analysis sheds light on the global dynamics of low Mach number reacting flows with swirl and vortex breakdown and on the interaction between vortex breakdown and heat release that affects the shape of the reaction zone in the domain.

This technique solves the two-dimensional Poisson equations in geometries involving cylindrical objects. The method uses three fundamental solutions, corresponding to a line force, a line couple and a pressure gradient, on each cylinder. Superposition of the fundamental solutions due to all the cylinders involved, while approximately satisfying the no-slip condition on each cylinder, yields a mobility matrix relating the various forces and motions of all the cylinders. Any specific problem can be solved by prescribing the motions of the cylinders and solving the matrix. For problems involving few cylinders or with a sufficient degree of symmetry this can be done analytically.

Once constructed, the general method is applied analytically to a series of specific problems. The permeability of an eccentric annulus is derived. The result is numerically indistinguishable from the exact solution to the problem, but unlike the exact solution the present one is obtained in closed form. The drag on two parallel rods moving past one another is also derived and compared to the exact solution. In this case the result is accurate for rod separations down to about 0.2 times the rod diameter. Finally the drag on a rod moving in a triangular array of identical rods is derived. Here it is shown that due to screening it is sufficient to include the six nearest neighbours, regardless of the rod separation. Although the present examples are all worked out analytically, the matrix can also be solved numerically, in which case any two-dimensional arrangement of cylindrical objects can be studied.