Modelling the foil rolling process

A review of the main rolling models is conducted to assess their suitability for modelling the foil rolling process. Two such models are Fleck and Johnson's Hertzian model and Fleck, Johnson, Mear and Zhang's Influence Function model. Both of these models are approximated through the use of perturbation methods. Decrease in the computation time resulted when compared with the numerical solution. The Hertzian model was approximated using the ratio of the yield stress of the strip to the plane-strain Young's Modulus of the rolls as the small perturbation parameter. The Influence Function model approximation takes advantage of the solution of the well-known Aerofoil Integral Equation to gain an insight into how the choice of interior boundary points affects the stability of numerical solution of the model's equations. These approximations require less computation than their full models and, in the case of the Hertzian approximation, only introduces a small error in the predictions of roll force roll torque. Hence the Hertzian approximate method is suitable for on-line control. The predictions from the Influence Function approximation underestimates the predictions from the numerical results. Better approximation of the pressure in the plastic reduction regions is the main source of this error.

Contracting bubbles in Hele-Shaw cells with a power-law fluid

The problem of bubble contraction in a Hele-Shaw cell is studied for the case in which the surrounding fluid is of power-law type. A small perturbation of the radially symmetric problem is first considered, focussing on the behaviour just before the bubble vanishes, it being found that for shear-thinning fluids the radially symmetric solution is stable, while for shear-thickening fluids the aspect ratio of the bubble boundary increases. The borderline (Newtonian) case considered previously is neutrally stable, the bubble boundary becoming elliptic in shape with the eccentricity of the ellipse depending on the initial data. Further light is shed on the bubble contraction problem by considering a long thin Hele-Shaw cell: for early times the leading-order behaviour is one-dimensional in this limit; however, as the bubble contracts its evolution is ultimately determined by the solution of a Wiener-Hopf problem, the transition between the long-thin limit and the extinction limit in which the bubble vanishes being described by what is in effect a similarity solution of the second kind. This same solution describes the generic (slit-like) extinction behaviour for shear-thickening fluids, the interface profiles that generalise the ellipses that characterise the Newtonian case being constructed by the Wiener-Hopf calculation.

The influence of Monte Carlo source parameters on detector design and dose perturbation in small field dosimetry

To obtain accurate Monte Carlo simulations of small radiation fields, it is important model the initial source parameters (electron energy and spot size) accurately. However recent studies have shown that small field dosimetry correction factors are insensitive to these parameters. The aim of this work is to extend this concept to test if these parameters affect dose perturbations in general, which is important for detector design and calculating perturbation correction factors. The EGSnrc C++ user code cavity was used for all simulations. Varying amounts of air between 0 and 2 mm were deliberately introduced upstream to a diode and the dose perturbation caused by the air was quantified. These simulations were then repeated using a range of initial electron energies (5.5 to 7.0 MeV) and electron spot sizes (0.7 to 2.2 FWHM). The resultant dose perturbations were large. For example 2 mm of air caused a dose reduction of up to 31% when simulated with a 6 mm field size. However these values did not vary by more than 2 % when simulated across the full range of source parameters tested. If a detector is modified by the introduction of air, one can be confident that the response of the detector will be the same across all similar linear accelerators and the Monte Carlo modelling of each machine is not required.