Analysis and numerical computations of moving boundaries in fluid dynamics and materials science

0104350

Siegel

This proposal is concerned with the analysis and numerical computation of

moving boundaries in problems from fluid dynamics and materials

science. The proposed problems are motivated by different applications

but are remarkably similar from a mathematical point of view. One class

of problems concerns the deformation and breakup via tip streaming of

drops and bubbles in extensional flows. Tip streaming refers to the

phenomenon where a bubble develops a deformed shape featuring

cusplike ends, from which fine filaments or small bubbles are emitted into

the exterior fluid. Experiments suggest that interfacial tension gradients

such as those produced by surfactants are important for the onset of this

instability. The PI shall continue his development of an analytical theory

for the deformation and evolution of slender bubbles with surfactant,

focusing on the effects of soluble surfactant and its influence in tip

streaming. Mathematical difficulties in the analysis are associated with a

jump in boundary conditions at 'stagnant caps' of surfactant on the free

surface. Techniques employed to address these difficulties include

singular perturbation theory, theory of singular integral equations and

Riemann-Hilbert problems, and complex variable theory in moving

boundary problems. The theory will be complemented by accurate

numerical computations. The PI also proposes to investigate tip streaming

and air entrainment in simple models of fiber coating, as well as cusp

formation and a possible analogue of tip streaming which may occur

during the diffusion controlled evolution of voids in a stressed solid. The

mathematical statements of these free boundary problems are similar, and

the PI expects that there will be a certain synergism between them.

Moving boundary problems, such as the evolution of waves on water, the

propagation of flame fronts, or the growth of crystals, continue to

challenge applied scientists and engineers. The PI proposes to study a

class of fundamental moving boundary problems that are important in

technological applications. A common feature among the proposed

problems is that the moving interface develops very small-scale features,

such as cusps or very fine filaments, which can then greatly influence the

properties of the material or fluid. One application of the proposed work is

in the coating of materials (such as optical fibers) by pulling at high speed

through a liquid bath. Air filaments produced during the coating process

can snap off, leaving voids or other blemishes that adulterate the coating.

A second application is in the failure of materials. Small pores present in

materials evolve (via atomic diffusion) when the material is stressed. Cusp

development in the pores can initiate cracks or dislocations (misalignment

of atoms); this has been implicated as a prominent cause of failure in

microelectronic circuits. Other applications include emulsion formation

and mixing in multi-component fluid systems.