Ship wakes are fascinating. They can be observed by the human eye and appear to have a V shape when the ship is advancing at constant speed along a straight trajectory. Under idealized conditions, Kelvin found that the angle between the two branches of the V is ${\sim }39^\circ $. However, in a number of cases, this angle appears to be smaller. This phenomenon has been studied by various authors, and several explanations have been suggested. The most elegant one, which is based on the amplitude of the ship waves rather than their phase, has recently been revisited by Darmon, Benzaquen & Raphaël (J. Fluid Mech., vol. 738, 2014, R3).

We consider an axisymmetric steady flow of a viscous incompressible conducting fluid. The flow is induced by the point source of momentum and point electrode discharging the electric current, both of which are located at the end of a thin semi-infinite insulated wire. We seek the solution in the conical self-similar class where the velocity and magnetic field decrease as the inverse distance from the origin. The solution is obtained for various parameters of the problem, namely the Reynolds number, dimensionless electric current and Batchelor number (magnetic Prandtl number). A reverse flow along the wire occurs, leading to the confinement of the current density in the direction of the jet. If the Batchelor number is zero, the solution obtains a singularity at finite values of the current leading to its breakdown; otherwise, the solution exists at all parameter values. We derive the boundary-layer equations near the wire for large current values and obtain the solution. The pitchfork bifurcation with non-zero poloidal magnetic field occurs and causes the rotation of the fluid, which eliminates the current confinement effect. We describe the conditions when the solution for the swirling jet exists. The connection of this problem to the ones considered previously is discussed.

It is usually believed that wall slip contributes small effects to macroscopic flow characteristics. Here we demonstrate that this is not the case for the thermocapillary migration of a long bubble in a slippery tube. We show that a fraction of the wall slip, with the slip length $\lambda $ much smaller than the tube radius $R$, can make the bubble migrate much faster than without wall slip. This speedup effect occurs in the strong-slip regime where the film thickness $b$ is smaller than $\lambda $ when the Marangoni number $S= \tau _{T} R/\sigma _{0}~ (\ll 1)$ is below the critical value $S^* \sim (\lambda /R)^{1/2}$, where $\tau _{T}$ is the driving thermal stress and $\sigma _{0}$ is the surface tension. The resulting bubble migration speed is found to be $U_{b} \sim (\sigma _{0}/\mu )S^{3}(\lambda /R)$, which can be more than a hundred times faster than the no-slip result $U_{b} \sim (\sigma _{0}/\mu )S^{5}$ (Wilson, J. Eng. Math., vol. 29, 1995, pp. 205–217; Mazouchi & Homsy, Phys. Fluids, vol. 12, 2000, pp. 542–549), with $\mu $ being the fluid viscosity. The change from the fifth power law to the cubic one also indicates a transition from the no-slip state to the strong-slip state, albeit the film thickness always scales as $b\sim RS^{2}$. The formal lubrication analysis and numerical results confirm the above findings. Our results in different slip regimes are shown to be equivalent to those for the Bretherton problem (Liao, Li & Wei, Phys. Rev. Lett., vol. 111, 2013, 136001). Extension to polygonal tubes and connection to experiments are also made. It is found that the slight discrepancy between experiment (Lajeunesse & Homsy, Phys. Fluids, vol. 15, 2003, pp. 308–314) and theory (Mazouchi & Homsy, Phys. Fluids, vol. 13, 2001, pp. 1594–1600) can be interpreted by including wall slip effects.

Based on the fast spectral approximation to the Boltzmann collision operator, we present an accurate and efficient deterministic numerical method for solving the Boltzmann equation. First, the linearized Boltzmann equation is solved for Poiseuille and thermal creep flows, where the influence of different molecular models on the mass and heat flow rates is assessed, and the Onsager–Casimir relation at the microscopic level for large Knudsen numbers is demonstrated. Recent experimental measurements of mass flow rates along a rectangular tube with large aspect ratio are compared with numerical results for the linearized Boltzmann equation. Then, a number of two-dimensional microflows in the transition and free-molecular flow regimes are simulated using the nonlinear Boltzmann equation. The influence of the molecular model is discussed, as well as the applicability of the linearized Boltzmann equation. For thermally driven flows in the free-molecular regime, it is found that the magnitudes of the flow velocity are inversely proportional to the Knudsen number. The streamline patterns of thermal creep flow inside a closed rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearized Poiseuille and thermal creep flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearized Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel.

In every turbulent flow with non-zero viscosity, heat is generated by viscous friction. This heat is then mixed by the velocity field. We consider how heat fluctuations generated in this way are injected and distributed over length scales in isotropic turbulence. A triadic closure is derived and numerically integrated. It is shown how the heat fluctuation spectrum depends on the Reynolds and Prandtl numbers.

A fully developed, turbulent Poiseuille flow with wall transpiration, i.e. uniform blowing and suction on the lower and upper walls correspondingly, is investigated by both direct numerical simulation (DNS) of the three-dimensional, incompressible Navier–Stokes equations and Lie symmetry analysis. The latter is used to find symmetry transformations and in turn to derive invariant solutions of the set of two- and multi-point correlation equations. We show that the transpiration velocity is a symmetry breaking which implies a logarithmic scaling law in the core of the channel. DNS validates this result of Lie symmetry analysis and hence aids establishing a new logarithmic law of deficit type. The region of validity of the new logarithmic law is very different from the usual near-wall log law and the slope constant in the core region differs from the von Kármán constant and is equal to 0.3. Further, extended forms of the linear viscous sublayer law and the near-wall log law are also derived, which, as a particular case, include these laws for the classical non-transpiring case. The viscous sublayer at the suction side has an asymptotic suction profile. The thickness of the sublayer increase at high Reynolds and transpiration numbers. For the near-wall log law we see an indication that it appears at the moderate transpiration rates ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}0.05<v_0/u_{\tau }<0.1$) and only at the blowing wall. Finally, from the DNS data we establish a relation between the friction velocity $u_{\tau }$ and the transpiration $v_0$ which turns out to be linear at moderate transpiration rates.

We consider the propagation of an air finger into a wide fluid-filled channel with a spatially varying depth profile. Our aim is to understand the origin of the multiple coexisting families of both steady and oscillatory propagating fingers previously observed in experiments in axially uniform channels each containing a centred step-like occlusion. We find that a depth-averaged model can reproduce all the finger propagation modes observed experimentally. In addition, the model reveals new modes for symmetric finger propagation. The inclusion of a spatially variable channel depth in the depth-averaged equations leads to: (i) a variable mobility coefficient within the fluid domain due to variations in viscous resistance of the channel; and (ii) a variable transverse curvature term in the dynamic boundary condition that modifies the pressure jump over the air–liquid interface. We use our model to examine the roles of these two distinct effects and find that both contribute to the steady bifurcation structure, while the transverse curvature term is responsible for the distinctive oscillatory propagation modes.

Direct numerical simulations are used to examine the pressure fluctuations generated by fully developed turbulence in a Mach 2.5 turbulent boundary layer, with an emphasis on the acoustic fluctuations radiated into the free stream. Single- and multi-point statistics of computed surface pressure fluctuations show good agreement with measurements and numerical simulations at similar flow conditions. Consistent with spark shadowgraphs obtained in free flight, the quasi-homogeneous acoustic near field in the free-stream region consists of randomly spaced wavepackets with a finite spatial coherence. The free-stream pressure fluctuations exhibit important differences from the surface pressure fluctuations in amplitude, frequency content and convection speeds. Such information can be applied towards improved modelling of boundary layer receptivity in conventional supersonic facilities and, hence, enable a better utilization of transition data acquired in such wind tunnels. The predicted acoustic characteristics are compared with the limited available measurements. Finally, the numerical database is used to understand the acoustic source mechanisms, with the finding that the supersonically convecting eddies that can directly radiate to the free stream are confined to the buffer zone within the boundary layer.

In the literature, it has been conceptualized that a group of dense particles released instantaneously into homogeneous stagnant water would form a circulating vortex cloud and descend through the water column as a thermal. However, Wen & Nacamuli (Hydrodynamics: Theory and Applications, 1996, pp. 1275–1280) observed the formation of particle clumps characterized by a narrow, fast-moving core shedding particles into the wake. They found clump formation to be possible even for particles in the non-cohesive range as long as the source Rayleigh number was large ($\mathit{Ra} > {10^3}$) or, equivalently, the source cloud number was small ($\mathit{Nc} < 3.2 \times 10^{-2}$). This physical phenomenon has not been investigated further since the experiments of Wen and Nacamuli. In the present study, the relationship between Nc and the formation process is examined more systematically. The theoretical support for cloud number dependence is explored by considering flows passing a porous sphere. Here $\mathit{Nc}$ values ranging from $2.9 \times 10^{-3}$ to $5.9 \times 10^{-2}$ are tested experimentally using particles with different initial masses and grain sizes, from non-cohesive to marginally cohesive. The formation processes are categorized into cloud formation, a transitional regime and clump formation, and their distinct features are presented through qualitative description of the flow patterns and quantitative assessment of the gross characteristics.

We report two-dimensional simulations of drop dynamics on a substrate subject to a wetting gradient and an external pressure gradient along the substrate. A phase-field formulation is used to represent the drop interface, and the moving contact line is modelled by Cahn–Hilliard diffusion. The Navier–Stokes–Cahn–Hilliard equations are solved by finite elements on an adaptively refined unstructured grid. For a single drop and a pair of drops, we consider three scenarios of drop motion driven by the wetting gradient only, by the external flow only, and by a combination of the two. Both the capillary force and the hydrodynamic drag depend strongly on the shape of the drop. Since the drop adapts its shape to the local wetting angles and to the external flow on a finite time scale, hysteresis is a prominent feature of the drop dynamics under opposing forces. For each wetting gradient, there is a narrow range of the magnitude of the external flow within which a single drop can achieve a stationary state. The equilibrium drop shape and position depend on its initial shape and the history of forcing. For a pair of drops, the wetting gradient or external flow alone tends to produce catch-up and coalescence. The flow-driven coalescence arises from a viscous shielding effect that relies on the asymmetric shape of the trailing drop once it is deformed by flow. This mechanism operates at zero Reynolds number, but is much enhanced by inertia. With the two forces opposing each other, the external flow favours separation while the wetting gradient favours coalescence. The outcome depends on their competition.

The opening of wind-driven coastal polynyas has often been investigated using idealised flux models. Polynya flux models postulate that the boundary separating the region of thin ice adjacent to the coast within the polynya from the thicker ice piling up downstream is a mathematical shock. To conserve mass, any divergence of the ice flux across the shock translates into a change in the shock’s position or, in other words, a change in the width of the thin-ice region of the polynya. Polynya flux models are physically incomplete in that, while they conserve ice mass, they do not conserve linear momentum. In this paper, we investigate the improvements that can be achieved in the simulation of polynyas by imposing conservation of momentum as well as mass. We start by adopting a mathematically solid formulation of the ice mass and momentum balances throughout the polynya region, from the coast to the pack ice. Hydrostatic and plastic versions of the ice internal forces are used in the model. Two different approaches are then explored. We first postulate the existence of a shock at the seaward edge of the thin-ice region of the polynya and derive jump conditions for the conservation of ice mass and momentum at the shock which are consistent with the continuous model physics. Polynyas simulated by this mass- and momentum-conserving shock model always reach a steady state if the polynya forcing is uniform in space and constant in time. This is also true for all polynya flux models presented previously in the literature, but the location of the steady-state polynya edge and the time required to reach it can greatly differ between shock formulations and more simplistic flux ones. We next relax the assumption that a shock exists and let the boundary between thin ice and piling up ice emerge naturally as part of the full solution of the continuous model equations. Polynyas simulated in this way are very different from those simulated by either shock or flux models. Most notably, we find that steady-state polynya solutions are not always attainable in the continuous model. We determine under which conditions this is so and explain how such unsteady solutions come about. We also show that, in those cases when a steady-state solution exists in the continuous model, the steady-state polynya width is considerably larger than in the shock model, and the time required to attain it is accordingly longer. The occurrence of such significant differences between the polynya solutions calculated with flux and shock models, on the one hand, and with more sophisticated continuous formulations, on the other hand, suggests that the former are, at best, incomplete, and should be used with caution.

Slender-body theory is used to study axisymmetric swimmers propelled by motions of their surfaces. To leading order, the locomotion speed is given by an integral involving the fluid velocity at the surface of the slender body. Locomotion speeds are calculated for fixed-shape swimmers with prescribed fluid surface velocities and for impermeable swimmers driven by propagating surface waves. Next, the internal mechanics is considered, modelling the swimmer as a viscous fluid bounded by an elastic shell. Prescribed forces are exerted on the shell to drive both the internal and external fluid flow and the surface waves. The internal fluid mechanics is determined using lubrication theory. Locomotion speeds are calculated for transverse and longitudinal waves of surface deformation, and the efficiency of the motions is determined. Transverse surface waves are both weaker and less efficient at driving locomotion than longitudinal waves. The results indicate how estimates of swimming speed based on nearly spherical swimmers with low-amplitude surface waves can be adapted for slender swimmers with nonlinear surface deformations.

We propose a procedure – partly analytical and partly numerical – to find the frequency and the damping rate of the small-amplitude oscillations of a massless elastic capsule immersed in a two-dimensional viscous incompressible fluid. The unsteady Stokes equations for the stream function are decomposed into normal modes for the angular and temporal variables, leading to a fourth-order linear ordinary differential equation in the radial variable. The forcing terms are dictated by the properties of the membrane and result in jump conditions at the interface between the internal and external media. The equation can be solved numerically, and excellent agreement is found with a fully computational approach that we have developed in parallel. Comparisons are also shown with results available in the scientific literature for drops, and a model based on the concept of entrained fluid is presented, which allows for a good representation of the present results and a consistent interpretation of the underlying physics.

A previous study by Forbes (ANZIAM J., vol. 53, 2011, pp. 87–121) has argued that, when a light fluid is injected from a point source into a heavier ambient fluid, the interface between them is most unstable to perturbations at the lowest spherical mode. This means that, regardless of initial conditions, the outflow from a point source eventually becomes a one-sided jet. However, two-sided (bi-polar) outflows are nevertheless often observed in astrophysics, in apparent contradiction to this prediction. While there are many possible explanations for this fact, the present paper considers the effect of a straining flow in the ambient fluid. In addition, solid-body rotation in the inner fluid is also accounted for, in a Boussinesq viscous model. It is shown analytically that there are circumstances under which straining flow alone is sufficient to convert the one-sided jet into a genuine bi-polar outflow, in linearized theory. This is confirmed in a numerical solution of a viscous model of the flow, based on a spectral solution technique that accounts for nonlinear effects. Rotation can also generate flows that are two-sided, and this is likewise revealed through an asymptotic analysis and numerical solutions of the nonlinear equations.

Flow driven through a planar channel having a finite-length membrane inserted in one wall can be unstable to self-excited oscillations. In a recent study (Xu, Billingham & Jensen J. Fluid Mech., vol. 723, 2013, pp. 706–733), we identified a mechanism of instability arising when the inlet flux and outlet pressure are held constant, and the rigid segment of the channel downstream of the membrane is sufficiently short to have negligible influence on the resulting oscillations. Here we identify an independent mechanism of instability that is intrinsically coupled to flow in the downstream rigid segment, which becomes prominent when the downstream segment is much longer than the membrane. Using a spatially one-dimensional model of the system, we perform a three-parameter unfolding of a degenerate bifurcation point having four zero eigenvalues. Our analysis reveals how instability is promoted by a 1:1 resonant interaction between two modes, with the resulting oscillations described by a fourth-order amplitude equation. This predicts the existence of saturated sawtooth oscillations, which we reproduce in full Navier–Stokes simulations of the same system.

We consider the dynamics of actively entraining turbulent density currents on a conical sloping surface in a rotating fluid. A theoretical plume model is developed to describe both axisymmetric flow and single-stream currents of finite angular extent. An analytical solution is derived for flow dominated by the initial buoyancy flux and with a constant entrainment ratio, which serves as an attractor for solutions with alternative initial conditions where the initial fluxes of mass and momentum are non-negligible. The solutions indicate that the downslope propagation of the current halts at a critical level where there is purely azimuthal flow, and the boundary layer approximation breaks down. Observations from a set of laboratory experiments are consistent with the dynamics predicted by the model, with the flow approaching a critical level. Interpretation in terms of the theory yields an entrainment coefficient $E\propto 1/\Omega $ where the rotation rate is $\Omega $. We also derive a corresponding theory for density currents from a line source of buoyancy on a planar slope. Our theoretical models provide a framework for designing and interpreting laboratory studies of turbulent entrainment in rotating dense flows on slopes and understanding their implications in geophysical flows.

We propose a simple, robust and efficient sloshing model that accounts for breaking phenomena evolving in rectangular tanks and in shallow-water conditions. The model has been obtained by applying Fourier analysis to Boussinesq-type equations and using an approximate analytic solution for the vorticity generated by wave breaking. The toe of the breaker and the intensity of the vorticity injected at the free surface are computed on the basis of literature results for coastal-type breakers. A first experimental campaign has been used to calibrate the turbulent viscosity of the sloshing model, while a second campaign has been run as final validation. The overall good agreement between the numerical outputs and the experimental data confirms the reliability and robustness of the proposed model.

A series of analytical solutions are presented for viscous compacting flow past a rigid impermeable sphere. The sphere is surrounded by a two-phase medium consisting of a viscously deformable solid matrix skeleton through which a low-viscosity liquid melt can percolate. The flow of the two-phase medium is described by McKenzie’s compaction equations, which combine Darcy flow of the liquid melt with Stokes flow of the solid matrix. The analytical solutions are found using an extension of the Papkovich–Neuber technique for Stokes flow. Solutions are presented for the three components of linear flow past a sphere: translation, rotation and straining flow. Faxén laws for the force, torque and stresslet on a rigid sphere in an arbitrary compacting flow are derived. The analytical solutions provide instantaneous solutions to the compaction equations in a uniform medium, but can also be used to numerically calculate an approximate evolution of the porosity over time whilst the porosity variations remain small. These solutions will be useful for interpreting the results of deformation experiments on partially molten rocks.

We present experimental measurements of a wall-bounded gravity current, motivated by characterizing natural gravity currents such as oceanic overflows. We use particle image velocimetry and planar laser-induced fluorescence to simultaneously measure the velocity and density fields as they evolve downstream of the initial injection from a turbulent channel flow onto a plane inclined at $10^\circ $ with respect to horizontal. The turbulence level of the input flow is controlled by injecting velocity fluctuations upstream of the output nozzle. The initial Reynolds number based on the Taylor microscale of the flow, $R_{\lambda }$, is varied between 40 and 120, and the effects of the initial turbulence level are assessed. The bulk Richardson number $\mathit{Ri}$ for the flow is ${\sim }$0.3 whereas the gradient Richardson number $\mathit{Ri}_g$ varies between 0.04 and 0.25, indicating that shear dominates the stabilizing effect of stratification. Kelvin–Helmholtz instability results in vigorous vertical transport of mass and momentum. We present baseline characterization of standard turbulence quantities and calculate, in several different ways, the fluid entrainment coefficient $E$, a quantity of considerable interest in mixing parameterization for ocean circulation models. We also determine the properties of mixing as represented by the flux Richardson number $\mathit{Ri}_f$ as a function of $\mathit{Ri}_g$ and diapycnal mixing parameter $K_{\rho }$ versus the buoyancy Reynolds number $\mathit{Re}_b$. We find reasonable agreement with results from natural flows.

The changes in a turbulent channel flow subjected to sinusoidal oscillations of wall flush-mounted rigid discs are studied by means of direct numerical simulations (DNS). The Reynolds number is ${Re}_{\tau }=180$ , based on the friction velocity of the stationary-wall case and the half-channel height. The primary effect of the wall forcing is the sustained reduction of wall-shear stress, which reaches a maximum of 20 %. A parametric study on the disc diameter, maximum tip velocity, and oscillation period is presented, with the aim of identifying the optimal parameters which guarantee maximum drag reduction and maximum net energy saving, the latter computed by taking into account the power spent to actuate the discs. This may be positive and reaches 6 %. The Rosenblat viscous pump flow, namely the laminar flow induced by sinusoidal in-plane oscillations of an infinite disc beneath a quiescent fluid, is used to predict accurately the power spent for disc motion in the fully developed turbulent channel flow case and to estimate localized and transient regions over the disc surface subjected to the turbulent regenerative braking effect, for which the wall turbulence exerts work on the discs. The Fukagata–Iwamoto–Kasagi identity is employed effectively to show that the wall-friction reduction is due to two distinguished effects. One effect is linked to the direct shearing action of the near-wall oscillating-disc boundary layer on the wall turbulence, which causes the attenuation of the turbulent Reynolds stresses. The other effect is due to the additional disc-flow Reynolds stresses produced by the streamwise-elongated structures which form between discs and modulate slowly in time. The contribution to drag reduction due to turbulent Reynolds stress attenuation depends on the penetration thickness of the disc-flow boundary layer, while the contribution due to the elongated structures scales linearly with a simple function of the maximum tip velocity and oscillation period for the largest disc diameter tested, a result suggested by the Rosenblat flow solution. A brief discussion on the future applicability of the oscillating-disc technique is also presented.