We present the results of an experimental study on the solidification of aqueous solutions of potassium nitrate and sodium nitrate cooled from below. Upon cooling, two distinct mushy layers form, primary and cotectic, separated by an approximately planar horizontal interface. A density reversal between the two mushes causes the residual liquid in the upper, primary mush to be more buoyant than the melt overlying it, while the cotectic mush is compositionally stable. The unstable concentration gradient between the melt and primary mush causes convection that keeps the melt well-mixed and reduces the concentration gradient to zero after a finite time. At this point, the cotectic mush overtakes the primary mush and a transition from a convective regime to a diffusive regime occurs. Our measurements show that this transition is rapid and alters the growth rate of the single (cotectic) mush layer that remains. Concentration measurements taken from within the melt during convection and from within the mush during the diffusive regime show good agreement with the concentration evolution predicted by use of the equilibrium ternary phase diagram. We describe a global conservation model for solidification of a ternary alloy in this regime. Predictions from our model forced with empirical data for the heat and solute fluxes are in good agreement with the measured data for the interface positions of the two mushy layers. We also discuss how solid fractions vary with different melt concentrations in a non-convecting alloy and examine the influence of vertical solute transport in the convecting case. The identification of a density reversal in the solidification of a ternary alloy begins to address the complexities in solidification processes of multi-component alloys.

In this study we report discovery of a new type of water bell. This is formed by impinging a vertical liquid jet on to the underside of a large horizontal flat plate. After impact, the liquid spreads radially along the plate before falling at an abrupt unspecified radius. This falling liquid may then coalesce to form a curtain which encloses a volume of air. When the flow rate of the impinging jet is altered from the value at initial formation, a pronounced hysteretic effect in the water bell shape can be observed. We present detailed observations of these new phenomena, including the size and nature of the flow underneath the plate and the shape of the liquid curtain. These observations are interpreted theoretically in a companion paper (Part 2, Button et al. vol. 649, 2010, pp. 45–68).

In this paper we report results from some investigations on the use of ultrasonics in controlling travelling bubble cavitation. Control of this type of cavitation, generated using a venturi device, has been achieved by manipulation of potential nuclei using a piezoelectric device, termed the Ultrasonic Nuclei Manipulator (UNM). The performance of the UNM, activated in continuous-wave (CW) and pulsed modes, has been studied over a range of dissolved gas concentration ($C$). The performance under CW-excitation is found to depend sensitively on $C$, with lack of control in near-saturated water samples. Failure to control cavitation at $C \approx 1$ under CW-excitation is suggested to be a result of bubble growth by rectified diffusion under these conditions. The pulsed mode of excitation of the UNM, in such cases, seems to be a very promising alternative. Further improvement is observed by using two piezoelectric crystals, one driven in the CW-mode and the second in pulsed mode, as the UNM. Through carefully designed experimentation, this has been traced to the movement of nuclei under the influence of Bjerknes forces. Besides reduction of noise, other measures of control have been identified and investigated. For example, it has been found that the maximum velocity achievable at the venturi throat can be increased from about 15 m s$^{-1}$ to about 22 m s$^{-1}$ with nuclei manipulation using ultrasonics.

In this paper we solve two initial value problems for two-dimensional internal gravity waves. The waves are contained in a uniformly stratified, square-shaped domain whose sidewalls are tilted with respect to the direction of gravity. We consider several disturbances of the initial stream function field and solve both for its free evolution and for its evolution under parametric excitation. We do this by developing a structure-preserving numerical method for internal gravity waves in a two-dimensional stratified fluid domain. We recall the linearized, inviscid Euler–Boussinesq model, identify its Hamiltonian structure, and derive a staggered finite difference scheme that preserves this structure. For the discretized model, the initial condition can be projected onto normal modes whose dynamics is described by independent harmonic oscillators. This fact is used to explain the persistence of various classes of wave attractors in a freely evolving (i.e. unforced) flow. Under parametric forcing, the discrete dynamics can likewise be decoupled into Mathieu equations. The most unstable resonant modes dominate the solution, forming wave attractors.

We consider thin liquid films on the inside (rimming flows) or outside (coating flows) of a cylinder with horizontal axis, rotating about this axis. If the liquid’s net volume is small, such films are known to evolve towards a steady state with a smooth surface, whereas, for larger amounts, the flow develops a ‘shock’ similar to a tidal bore. In this work, smooth films are shown to be unstable. Since the strongest instability occurs at wavelengths comparable to the film’s thickness, our analysis is based on the full Navier–Stokes equations, not on the lubrication approximation (which has been traditionally used in this problem). It is also shown that, for cylinders of sufficiently small radii, the instability can be suppressed by surface tension.

The two-dimensional steady potential free surface flow due to a pressure distribution moving at a constant velocity at the surface of a fluid of infinite depth is considered. The effects of gravity and surface tension are included in the dynamic boundary condition. The fully nonlinear problem is solved numerically by a boundary integral equation method and the results are compared with those of the linear theory of Rayleigh (1883). It is found that for some values of the capillary number, the nonlinear solutions do not approach the linear solution of Rayleigh as the magnitude of the pressure distribution approaches zero. Appropriate linear and nonlinear solutions are constructed.

In order to gain insight into the hydraulics of rotating-channel flow, a set of initial-value problems analogous to Long's towing experiments is considered. Specifically, we calculate the adjustment caused by the introduction of a stationary obstacle into a steady, single-layer flow in a rotating channel of infinite length. Using the semigeostrophic approximation and the assumption of uniform potential vorticity, we predict the critical obstacle height above which upstream influence occurs. This height is a function of the initial Froude number, the ratio of the channel width to an appropriately defined Rossby radius of deformation, and a third parameter governing how the initial volume flux in sidewall boundary layers is partitioned. (In all cases, the latter is held to a fixed value specifying zero flow in the right-hand (facing downstream) boundary layer.) The temporal development of the flow according to the full, two-dimensional shallow water equations is calculated numerically, revealing numerous interesting features such as upstream-propagating shocks and separated rarefying intrusions, downstream hydraulic jumps in both depth and stream width, flow separation, and two types of recirculations. The semigeostrophic prediction of the critical obstacle height proves accurate for relatively narrow channels and moderately accurate for wide channels. Significantly, we find that contact with the left-hand wall (facing downstream) is crucial to most of the interesting and important features. For example, no instances are found of hydraulic control of flow that is separated from the left-hand wall at the sill, despite the fact that such states have been predicted by previous semigeostrophic theories. The calculations result in a series of regime diagrams that should be very helpful for investigators who wish to gain insight into rotating, hydraulically driven flow.

We consider flow in a finite-length channel, one wall of which contains a membrane under longitudinal tension. The upstream flux and downstream pressure are prescribed and an external linear pressure distribution is applied to the membrane such that the system admits uniform Poiseuille flow as a steady solution. The system is described using a one-dimensional model that accounts for viscous and inertial effects. A linear stability analysis reveals that the uniform state is unstable to static (or divergent) and oscillatory instabilities. Asymptotic analysis in the neighbourhood of a Takens–Bogdanov bifurcation point shows how, when the downstream rigid section of the channel is not substantially longer than the membrane, an oscillatory mode arises through an interaction between two static eigenmodes. Perturbations to the uniform state exhibit the dynamics of a weakly dissipative Hamiltonian system for which low-frequency self-excited oscillations are forced by the divergent instability of two nearby steady solutions, before ultimately growing to large amplitudes. Simulations show that the subsequent dynamics can involve slamming motion in which the membrane briefly comes into near-contact with the opposite rigid wall over short length scales.

The perturbation of a turbulent flow by an organized wavelike disturbance is examined using a dynamical, rather than phenomenological, approach. On the basis of the assumption that an infinitesimal perturbation results in a linear change in the statistics of the turbulence, and that the turbulence is either weak or that the turbulent perturbations are quasi-Gaussian, a method for predicting the perturbation turbulent Reynolds stresses is developed. The novel aspect of the analysis is that all averaging is delayed until the dynamical equations have been solved rather than attempting to find, apriori, equations for averaged quantities. When applied to long-wave perturbations the analysis indicates that the perturbation shear stress is of primary dynamical importance, and that this stress is determined by the principal component of mean shear through a relation which depends on the spectrum of the turbulent velocity component parallel to the gradient of the undisturbed mean velocity (the component perpendicular to the wall in a turbulent boundary layer). Theoretical arguments and observations are used to estimate the form of this spectrum in a constant stress shear layer. This results in a prediction of the constitutive law relating turbulent stress and the mean flow. The law is visco-elastic in nature, and is in agreement with the known constitutive relation for stress perturbations to a constant stress boundary layer; it resembles the eddy viscosity relation used successfully by others in describing perturbations in turbulent flows. The details of the constitutive law depend on how well the turbulence obeys Taylor's hypothesis that phase velocity equals mean flow velocity, and some insight into this question is given.

The three-soliton solution of the two-dimensional Korteweg-de Vries equation is analysed to show that the structure of the interaction can be represented in terms of the motion of two-soliton resonant interactions (resonant triads) as described by Miles (1977). The schematic development of the interaction with time is obtained and shown to approximate closely to computer calculations of the analytic solution. Similar results follow for interactions of more solitons and other equations.

An account is given of photographic and pressure observations made on the oblique shock waves occurring in the wake of self-sustaining detonation waves in hydrogen-oxygen mixtures initially at atmospheric pressure. Four explosion tubes were employed, of which three are of circular cross-section with internal diameters of 10, 5 and 1·6 cm and the fourth is a square-section tube of side 1·5 in.

On the assumption that the oblique shocks are sufficiently weak to be regarded as Mach waves, the flow Mach number relative to the detonation front is determined; these are found to be substantially higher than the values predicted by deal one-dimensional theory. The measured flow Mach numbers in the rarefaction are then used to calculate the pressure distribution in this region on the basis of the supersonic nozzle model due to Fay (1959, 1962). The predictions of this model are found to disagree with with the observed static pressure profiles. Moreover, the pressure following the initial peak persists at a higher value than the theoretical for distances of the order of 5–10 cm behind the front. This phenomenon implies that the wall boundary-layer pressure remains higher than the C-J value and it is suggested that the pressure difference across the boundary layer can account for the formation of the oblique waves.

The supersonic features of the flow can be accounted for by the turbulent-structure hypothesis of White (1961). Some validation of this hypothesis is provided here by the observation of the absence of the oblique shocks in overdriven detonation waves caused by the diminished effects of turbulence. This observation is consistent with the view that the oblique shocks are generated by the pressure difference across the boundary layer near the front as this difference would also be diminished in an over-driven wave.

The nonlinear character of the primary bifurcation is investigated for the flow around a flexibly mounted circular cylinder. We have considered the cases in which the cylinder can oscillate in the transverse direction only and in both transverse and in-line directions. Low and high values of mass ratio ($m^{\ast }=5$ and 50) were studied, and reduced velocity ($V_{r}$) values are chosen inside ($V_{r}=9$) and outside ($V_{r}=5$ and $V_{r}=13$) the lock-in range for low Reynolds numbers. For each combination of $m^{\ast }$ and $V_{r}$, a global linear stability analysis was applied to find the critical Reynolds number $Re_{c}$ of the fluid–structure system. For $V_{r}$ in the lock-in range, the values of $Re_{c}$ were noticeably less than the critical Reynolds number of the flow around a fixed circular cylinder ($Re_{c_{0}}\cong 47$). On the other hand, for $V_{r}$ outside the lock-in range, the values of $Re_{c}$ were close to $Re_{c_{0}}$. Next, nonlinear analyses were performed in the vicinity of $Re_{c}$ for each case. Subcritical character (with hysteresis) was observed for $V_{r}$ in the lock-in range, while for $V_{r}$ outside the lock-in region the bifurcations were found to be supercritical (without hysteresis). This shows that when the coupling between the structure and flow is strong, due to the proximity of the natural frequencies of the isolated systems, it significantly changes both the linear and nonlinear responses observed.

Atherosclerotic lesions in mammalian blood vessels show a definite spatial pattern, and it has been proposed that lesions occur preferentially in regions with a low wall shear stress. Near the entrance to an intercostal artery, lesions on the wall of the aorta occur, initially at least, downstream of the entrance. We model the flow in this region by a linear shear flow along a flat wall (the wall of the main tube/aorta) past an infinitely deep circular hole perpendicular to the wall (the side-branch/intercostal artery), with fluid being sucked into the hole. By assuming Stokes flow, the three-dimensional model problem is reduced to two independent problems on a two-dimensional domain. By addition, the solutions for any particular side-branch flow rate. We find that the wall shear stress in the main tube is elevated upstream of the side-branch entrance, and downstream as well is the side-branch flow rate is small. However, if the flow rate in the side branch is large enough, there will be regions of both elevated and reduced wall shear stress in the main tube downstream of the side-branch entrance, including a stagnation point. The wall shear is lower downsteam than upstream unless there is no net flow into the side branch.

The solutions given apply to the case with flow out of as well as into the hole. Also, the asymptotic structure for the flow in the hole when there is no net flow into the hole, and the analysis of three-dimensional flow near a sharp corner, are given.

In this paper, linear stability analysis on a Newtonian fluid film flowing under the effect of gravity over an inclined porous medium saturated with the same fluid in isothermal condition is carried out. The focus is placed on the effect of the anisotropic and inhomogeneous variations in the permeability of the porous medium on the shear mode and surface mode instabilities. The fluid–porous system is modelled by a coupled two-dimensional Navier–Stokes/Darcy problem. The perturbation equations are solved numerically using the Chebyshev collocation method. Detailed stability characteristics as a function of the depth ratio (the ratio of the depth of the fluid layer to that of the porous layer), the anisotropic parameter (the ratio of the permeability in the direction of the basic flow to that in the direction transverse to the basic flow) and the inhomogeneity functions are presented.

As the vortical disturbances of a shrouded jet pass the sharp edge of the shroud exit some of the energy is scattered into acoustic waves. Scattering into upstream-propagating acoustic modes is a potential mechanism for closing the resonance loop in the ‘howling’ resonances that have been observed in various shrouded jet configurations over the years. A model is developed for this interaction at the shroud exit. The jet is represented as a uniform flow separated by a cylindrical vortex sheet from a concentric co-flow within the cylindrical shroud. A second vortex sheet separates the co-flow from an ambient flow outside the shroud, downstream of its exit. The Wiener–Hopf technique is used to compute reflectivities at the shroud exit. For some conditions it appears that the reflection of finite-wavelength hydrodynamic vorticity modes on the vortex sheet defining the jet could be sufficient to reinforce the shroud acoustic modes to facilitate resonance. The analysis also gives the reflectivities for the shroud acoustic modes, which would also be important in establishing resonance conditions. Interestingly, it is also predicted that the shroud exit can be ‘transparent’ for ranges of Mach numbers, with no reflection into any upstream-propagating acoustic mode. This is phenomenologically consistent with observations that indicate a peculiar sensitivity of resonances of this kind to, say, jet Mach number.

Theoretical calculations show that the horizontal displacement of particles in the surface of steep solitary waves exceeds that predicted by the Korteweg–de Vries equation by as much as 100%.

Experimental evidence is given in support of the higher values.