[1] Estimates of volcanic source mass flux, currently deduced from observations of plume height, are crucial for ash dispersion models for aviation and population hazard. This study addresses the role of the atmospheric wind in determining the height at which volcanic plumes spread in the atmosphere and the relationship between source mass flux and plume height in a wind field. We present a predictive model of volcanic plumes that describes the bending over of the plume trajectory in a crosswind and show that model predictions are in accord with a dataset of historic eruptions if the profile of atmospheric wind shear is described. The wind restricts the rise height of volcanic plumes such that obtaining equivalent rise heights for a plume in a windy environment would require an order of magnitude increase in the source mass flux over a plume in a quiescent environment. Our model calculations are used to calibrate a semi-empirical relationship between the plume height and the source mass flux that explicitly includes the atmospheric wind speed. We demonstrate that the model can account for the variations in plume height observed during the first explosive phase of the 2010 Eyjafjallajökull eruption using independently measured wind speeds and show that changes in the observed plume height are better explained by changing meteorology than abrupt changes in the source mass flux. This study shows that unless the wind is properly accounted for, estimates of the source mass flux during an explosive eruption are likely to be very significant underpredictions of the volcanic source conditions.

The effects of resistive forces on unsteady shallow flows over rigid horizontal boundaries are investigated theoretically. The dynamics of this type of motion are driven by the streamwise gradient of the hydrostatic pressure, which balances the inertia of the fluid and the basal resistance. Drag forces are often negligible provided the fluid is sufficiently deep. However, close to the front of some flows where the depth of the moving layer becomes small, it is possible for drag to substantially influence the motion. Here we consider three aspects of unsteady shallow flows. First we consider a regime in which the drag, inertia and buoyancy (pressure gradient) are formally of the same magnitude throughout the entire current and we construct a new class of similarity solutions for the motion. This reveals the range of solution types possible, which includes those with continuous profiles, those with discontinuous profiles and weak shocks and those which are continuous but have critical points of transition at which the gradients may be discontinuous. Next we analyse one-dimensional dam-break flow and calculate how drag slows the motion. There is always a region close to the front in which drag forces are not negligible. We employ matched asymptotic expansions to combine the flow at the front with the flow in the bulk of the domain and derive theoretical predictions that are compared to laboratory measurements of dam-break flows. Finally we investigate a modified form of dam-break flow in which the vertical profile of the horizontal velocity field is no longer assumed to be uniform. It is found that in the absence of drag it is no longer possible to find a kinematically consistent front of the fluid motion. However the inclusion of drag forces within the region close to the front resolves this difficulty. We calculate velocity and depth profiles within the drag-affected region, and obtain the leading-order expression for the rate at which the fluid propagates when the magnitude of the drag force is modelled using Chæ#169;zy, Newtonian and power-law fluid closures; this compares well with experimental data and provide new insights into dam-break flows

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Particle-size segregation within granular materials is of great technological significance yet it is still very poorly understood. There are several causes for segregation, but this paper focuses on kinetic sieving which is the dominant mechanism in dense gravity-driven shallow free-surface flows, or, granular avalanches. The segregation model is derived from a three-phase mixture theory composed of large particles, small particles and a passive interstitial fluid. Steady-state solutions are constructed for a normally graded inflow in a steady uniform flow field. This problem is of fundamental interest, because it shows how an unstably stratified layer readjusts into a stable configuration. Expansion fans and concentration shocks are generated and sufficiently far downstream inversely-graded segregated layers form, with the larger particles overlying the finer ones. The distance for complete segregation to occur is shown to increase with rising fluid density and tends to infinity as its density approaches that of the grains. If the particles are buoyant then the initial configuration is stable. An exact time-dependent two-dimensional solution is constructed for plug-flow, which exploits the uncoupling of material columns of grains in the absence of shear. This yields insight into the nature of more complex numerical solutions for strong shear, which are computed with a high resolution, shock-capturing numerical scheme.

The inertial migration of a small rigid spherical particle, suspended in a fluid flowing between two plane boundaries, is investigated theoretically to find the effect on the lateral motion. The channel Reynolds number is of order unity and thus both boundary-induced and Oseen-like inertial migration effects are important. The particle Reynolds number is small but non-zero, and singular perturbation techniques are used to calculate the component of the migration velocity which is directed perpendicular to the boundaries of the channel. The particle is non-neutrally buoyant and thus its buoyancy-induced motion may be either parallel or perpendicular to the channel boundaries, depending on the channel alignment. When the buoyancy results in motion perpendicular to the channel boundaries, the inertial migration is a first-order correction to the magnitude of this lateral motion, which significantly increases near to the boundaries. When the buoyancy produces motion parallel with the channel boundaries, the inertial migration gives the zeroth-order lateral motion either towards or away from the boundaries. It is found that those particles which have a velocity exceeding the undisturbed shear flow will migrate towards the boundaries, whereas those with velocities less than the undisturbed flow migrate towards the channel centreline. This calculation is of practical importance for various chemical engineering devices in which particles must be filtered or separated. It is useful to calculate the forces on a particle moving near to a boundary, through a shear flow. This study may also explain certain migration effects of bubbles and crystals suspended in molten rock flow flowing through volcanic conduits.