UC San Diego

Applied mathematics techniques are used in this investigation to gain insight into three different physical processes of current interest in combustion and fluid dynamics. The first problem addresses the propagation of spiral edge flames found in von Karman swirling flows induced in rotating porous-disk burners. In this configuration, a porous disk is spun at a constant angular velocity in an otherwise quiescent oxidizing atmosphere. Gaseous methane is injected through the disk pores and burns in a flat diffusion flame adjacent to the disk. Among other flame patterns experimentally found, a stable, rotating spiral flame is observed for sufficiently large rotation velocities and small fuel flow rates as a result of partial extinction of the underlying diffusion flame. The tip of the spiral can undergo a steady rotation for sufficiently large rotational velocities or small fuel flow rates, whereas a meandering tip in an epicycloidal trajectory is observed for smaller rotational velocities and larger fuel flow rates. A formulation of this problem is presented in the equidiffusional and thermodiffusive limits within the framework of one-step chemistry with large activation energies. Conditions for extinction of the underlying uniform diffusion flame are obtained by using activation energy asymptotics. Edge-flame propagation regimes are obtained by scaling analyses of the conservation equations and exemplified by numerical simulations of nearly straight two-dimensional edge flames near a cold porous wall in a von Karman boundary layer, for which lateral heat losses to the disk induce extinction of the trailing diffusion flame but are relatively unimportant in the front region, consistent with the existence of the cooling tail found in the experiments. The propagation dynamics of a steadily rotating spiral edge is studied in the large-core limit, for which the characteristic Markstein length is much smaller than the distance from the center at which the spiral tip is anchored. An asymptotic description of the edge tangential structure is obtained, spiral edge shapes are calculated, and an expression is found that relates the spiral rotational velocity with the rest of the parameters. A quasistatic stability analysis of the edge shows that the edge curvature at extinction in the tip region is responsible for the stable tip anchoring at the core radius. Finally, experimental results are analyzed, and theoretical predictions are tested. The second problem analyzes, in the limit of small Reynolds and ionic Peclet numbers and small clearances, the canonical problem of the forces exerted on a small solid spherical particle undergoing slow translation and rotation in an incompressible fluid moving parallel to an elastic substrate, subject to electric double-layer and van der Waals intermolecular forces, as a representative example of particle gliding and of the idealized swimming dynamics of more complex bodies near soft and sticky surfaces in a physiological solvent. The competition of the hydrodynamic, intermolecular and surface-deformation effects, induces a lift force, and drag-force and drift-force perturbations, which do not scale linearly with the velocities, and produces a non-additivity of the intermolecular effects by reducing the intensity of the repulsive forces and by increasing the intensity of the attractive forces. Reversible and irreversible elastohydrodynamic adhesion regimes are found, and elastohydrodynamic corrections are derived for the critical coagulation concentration of electrolyte predicted by the the Derjaguin-Landau Verwey- Overbeek (DLVO) standard theory of colloid stabilization. The third problem addresses the dynamics of pollen shedding from wind-pollinated plants, and establishes a fluid-dynamical framework for future refinements. A simple scaling analysis, supported by experimental measurements on typical wind-pollinated plant species, is used to estimate the suitability of previous resolutions of this process based on wind-gust aerodynamic models of fungal- spore liberation. According to this scaling analysis, unsteady boundary-layer forces produced by wind gusts are found to be mostly ineffective since the Stokes-Reynolds number is a small parameter for typical anemophilous species and wind streams. A hypothetical model of a stochastic aeroelastic mechanism, initiated by the atmospheric turbulence typical of the micrometeorological conditions in the vicinity of the plant, is proposed to contribute to wind pollination