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The main purpose of this work is to prove short-time structural stability of compressible vortex sheets with surface tension. The main result can be summarised as follows. Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We assume the fluids are modelled by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density in each fluid such that the sound speed is positive. Then, for a short time, which may depend on the initial configuration, there exists a unique solution of the equations with the same structure, that is, two fluids with density bounded below flowing smoothly past each other, where the surface tension force across the common interface balances the pressure jump. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al in the setting of a compressible liquid-vacuum interface. Although already considered by Shkoller et al, we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.

Experimental studies have been conducted on the topic of the interaction of compressiblevortex rings on stationary surfaces. Throughout the campaign experimentswere carried out at pressure ratios of ! 4, 8, and 12. In the classical set up of airas both the driver and driven gas, these corresponded to theoretical incident Machnumbers Ms of 1.34, 1.54, and 1.61.Experiments were conducted on vortex rings impinging on a stationary surfacelocated at three (increasing) distances (1.66, 3.33, and 5.00 inner diameters) fromthe shock tube exit and on a stationary surface at a set distance but at three anglesinclinations (75, 60, and 45deg at 3.33 inner diameters). Results of the impingementof a vortex ring on a stationary solid surface perpendicular to the flow showed asymmetrical impingement process. A boundary layer is generated over the surfacewith an associated increase in pressure. An increase in velocity due to the radialexpansion causes the pressure over the surface to decrease. This expansion leads tothe development of azimuthal wave instabilities along the core. Pressure was seen toincrease with an increase in incident Mach number value. The variation in distanceresulted in an increase in pressure with an increase in distance. This counter-intuitiveresult can be explained by the higher translational velocity at impingement, alongwith the absence of the initial radial expansion of the counter-rotating vortex rings. The variation in surface angle inclination introduced several degrees of asymmetry. One core of the vortex ring impinges first on the surface due to its closerproximity to it, while the other core is still free to propagate. This process generatesan asymmetric boundary layer over the surface, and a higher rate of stretching ofthe lower core, resulting in its dissipation. At higher incident Mach numbers, theembedded rearward facing shock is reflected and propagates perpendicularly to thesurface. At the inclination angles of 60 and 45deg, the counter-rotating vortex ringsare fully deflected upwards and orbit around the main vortex. This phenomenonresult in a significant difference in pressure distribution between the upper and lowersections of the surface.

In this thesis we collect the study of two problems in the Calculus of Variations and Partial Differential Equations. Our first group of results concern the analysis of minimizers in a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques and an analysis of fine properties of minimizers within the class of sets of finite perimeter, we establish existence, convexity and symmetry of minimizers. In the case of smooth surface tensions, we obtain uniqueness of minimizers via an ODE characterization. In the second group of results discussed in this thesis, which is joint work with A. Vasseur, we treat a problem in compressible fluid dynamics, establishing a uniform bound from below on the temperature for a variant of the compressible Navier-Stokes-Fourier system under suitable hypotheses. This system of equations forms a mathematical model of the motion of a compressible fluid subject to heat conduction. Building upon the work of (Mellet, Vasseur 2009), we identify a class of weak solutions satisfying a localized form of the entropy inequality (adapted to measure the set where the temperature becomes small) and use a form of the De Giorgi argument for L[superscript infinity] bounds of solutions to elliptic equations with bounded measurable coefficients.

The role of compressibility on the convective velocity of large-scale structures in axisymmetric jets is studied using a home-built pulse burst laser system and newly developed high-repetition rate experimental diagnostics. A pulse burst laser system was designed and constructed with the ability to produce a burst of short duration (10 nsec), high energy (order of 10 -100 mJ/pulse) pulses over a ~150 microsecond period with inter-pulse timing as low as 1 microsecond (1 MHz). The application of the pulse burst laser for flow measurements was investigated through the development of MHz rate flow visualization and MHz rate planar Doppler velocimetry (PDV). MHz rate PDV is a spectroscopic technique that produces 28 time-correlated realizations of the velocity over a plane with a maximum repetition rate of up to 1 MHz and accuracies on the order of 5%. Space-time correlations were used to track structures within the flow field and determine their convective velocity. Data produced using flow visualization images agrees with previous research and indicates a strong departure of the convective velocity from theory. Data produced using velocity data, however, shows starkly different trends and does not produce the same measurements of convective velocity. This difference in measurement is attributed to a misinterpretation of the use of space-time correlation for tracking structures. The presence of a distinct boundary between the mixing layer and the jet core as well as the mixing layer and ambient air in the flow visualization data and some of the velocity data leads to a bias in the measurement. The space-time correlation is found to preferentially follow these boundaries, thus leading to faster and/or slower measurements of convective velocity. For the Mach 2.0 jet, velocity data was obtained with seed particles marking the jet core and the mixing layer, but not the ambient air. This lack of velocity measurements on the low-speed side of the jet’s mixing layer biased the space-time correlation results. An algorithm was developed to extrapolate the velocity into unseeded regions of the flow. Space-time correlation results based on the extrapolated data indicate a convective velocity close to the theoretical value.