The aim of the modelling is to determine the key parameters and fluid regimes underlying the nutrient mixing process, to ensure that required nutrient levels are maintained through- out the root zones, and to enable optimal scheduling of the nutrient and bubble flow.

Simple experiments were performed via the injection of dye into an operating Hydrosac⃝c that contained semi-mature plants. This enabled a basic understanding of the time and lengthscales of nutrient flow, and also the extent to which mixing occurs in different zones within the bag. Four different flow regimes are identified. At the scale of a single root, a Stokes-flow approximation may be used. At the scale of the individual plant, a so-called Brinkman flow regime may be employed which is describes a transition between slow porous- medium flow and fast channel flow. These equations may be homogenised into a 1D model that can be used to estimate the macro-scale flow of nutrients along the length of the bag.

A shear flow model is used to predict the extent to which this flow permeates into regions dominated by plant roots. This leads to the requirement to model the bubble-driven flow within a bag cross-section containing a plant. Simplified two-phase flow equations are de- rived and solved within the software COMSOL. The results suggest that the bubble flow is sufficient to drive recirculating flow, which is also found to be consistent with previous literature.

The overall conclusion is that both the periodic flow of nutrients and the aeration are re- quired in order to enable even nutrient spread in the Hydrosac⃝c . Wave effects can be ignored, as can the effect of stagnated nutrient diffusion. The longitudinal nutrient flow enables the whole sack to be reached on the time scale of several cycles of the main inlet flow, while the recirculation from the bubble flow enables enables nutrients to spread within the plant roots. Nevertheless, regions of stagnation can occur via this process near any sharp corners of the bag.

It is recommend that the various analyses are combined into a a reduced-order mathemat- ical model that can be used to optimise the dynamic operation of the Hydrosac⃝c , which can also be adaptable to other geometries and growing conditions.

There are three types of breakup: wavy sheet, perforation, and rim. In wavy sheet breakup, increasing viscosity or surface tension increases droplet size. To investigate further, we carry out direct numerical simulations of jet breakup, which show that suface tension has little effect, but increasing viscosity leads to fewer droplets. Decreasing the jet velocity also results in fewer droplets, with a wider size distribution.

Each type of breakup involves primary breakup into cylinders of fluid, then secondary breakup into droplets. We thus consider the breakup of a cylinder of fluid. Direct numerical simulations suggest that within the tested parameter range viscosity has little impact on droplet size, however it does influence the timescale on which the instability evolves considerably. Linear stability analysis suggests that increasing viscosity increases the wavelength of the most unstable mode, which we expect leads to larger droplets, and that it reduces the rate of breakup.

Perforations - holes in the sheet - also lead to breakup. We find how the length fraction of the sheet that is void changes with time.

After breakup, the droplets continue to evolve. We develop a model, based on a transport equation, for this process. A key parameter is the breakup rate constant - larger values lead to more breakup, fewer large droplets, and a narrower size distribution.

Together, these mathematical approaches improve our understanding of how droplets form, and can be used to guide experimental work.

The group recognised that capillary suction was the dominant process by which the contaminant spreads in the porous substrate. Therefore, in the first instance the absorption of the contaminant was modelled using Darcy’s law. At the next level of complication a diffuse interface model based on Richards’ equation was employed. The results of the two models were found to agree at early times, while at later times we found that the diffuse interface model predicted the more realistic scenario in which the contaminant has seeped deeper into the substrate even in the absence of further contaminant being supplied at the surface.

The decontamination process was modelled in two cases; first, where the product of the decontamination reaction was water soluble, and the second where the reaction product formed soluble in the contaminant phase and of similar density. These simple models helped explain some of the key physics involved in the process, and how the decontamination process might be optimised. We found that decontamination was most effective in the first of these two cases.

The group then sought to incorporate hydrodynamic effects into the reaction model. In the long wavelength limit, the governing equations reduced to a one-dimensional Stefan model similar to the one considered earlier. More detailed approximations and numerical simulations of this model were beyond the scope of this study group, but provide an entry point for future research in this area.

The team has developed three complementary models, each with different strengths and weaknesses so that, depending on the information desired, one model may be more useful than another. The three models are:

1. A continuum model giving a macroscopic description of the filter. The governing equations are derived from first-principle consider- ations of conservation of mass and momentum. Constitutive relations for this model are derived by considering the processes going on in the filter at a microscopic level.

2. A stochastic model based on a Markov Decision Process. Each droplet is modelled as a single entity that can merge or move stochastically. This leads to a Markov simulation of the filter and the computation of average quantities.

3. A Lattice-Boltzmann model. The droplets are modelled to interact with each other and with the filter, using a Boltzmann distribution for their speed. This simulates the hydrodynamic behaviour of the droplet inside the filter.

This report describes the results of simple modelling. The approach was to consider first how the rock cooled as a result of water moving through the network of cracks and then to consider second how this cooling might change the rock stresses and water viscosity, and hence the effective permeability of the cracks, thereby altering the flow pattern from inlet to outlet bore-hole.

modeled by a damped spring. One step further we also allow for pitch, a swinging motion around a horizontal axis perpendicular to the ship. It is recommended to investigate the way waves may directly drive this mode and to determine the amount of energy that flows along this path towards the roll mode. Since at sea waves are a superposition of waves with different wavelengths, we also pay attention to the properties of such a type of forcing containing stochastic elements. It is recommended that as a measure for the

occurrence of large deflections of the roll angle one should take the expected time for which a given large deflection may occur instead of the mean amplitude of the deflection.

reservoirs is studied. Some remarks regarding sensitivity with respect to the time horizon, terminal cost and forecast of inflow are made.

1) 1968-1971

2) 1974-1977

3) 1978-1980

4) 1980-1983

5) 1984-1986

6) 1987-1988

After the metallic liner has been wrapped, the composite vessel is subjected to a process termed Autofrettage. In that process the vessel is internally pressurized to the point that the ductile metal liner undergoes a small amount of plastic deformation (unlike elastic deformation which disappears upon removal of stress, plastic deformation remains after the stress has been removed). Upon de-pressurization of the vessel, the metallic liner remains under compression and the FRP under tension.

Acoustic emissions associated with fiber breakage are being developed currently as a non-destructive means of assessing the structural integrity of metal-lined continuous FRP over-wrapped vessels. Laboratory experiments have been carried out with flaws such as cracks and saw cuts of varying dimensions oriented in an axial-radial plane and located in the metallic liner, in the FRP or in both. Pressurization of the flawed vessel leads to fiber breakage, the extent of which is being examined with the intent that it will be a measure of structural integrity of the vessel. The results, however, suggest that the acoustic emissions attain a maximum for an intermediate flaw size. Low emissions are recorded when on the one hand the vessel has insignificant flaws, or on the other if the vessel has serious flaws. This non-monotonic variation of acoustic emission occurs whether the flaws are located in the metallic liner, in the FRP or in both.

The experiments suggest that the stress intensity at the discontinuity (crack tip) attains a maximum at an intermediate flaw size. A mathematical corroboration is desired.

Two problems on flows in low permeability reservoirs were posed. One of the problems is on radial axi-symmetric flows with a threshold pressure gradient and the other is on radial flows in a compressible medium. The main objective of the exercise is to obtain exact or approximate solutions. We summarize the discussion on one of the two problems, flows in a slightly compressible medium.