We consider two problems in low Reynolds-number, interfacial fluid mechanics: the rupture of thin liquid films on chemically patterned solid substrates, and the engulfment of foreign particles by a solidification front progressing through a binary alloy. First we investigate the stability and rupture of thin liquid films on patterned sub- strates. The behavior of thin films is important to many industrial processes such as optical coatings and semiconductor fabrication. It is well known that thin liquid films on uniform substrates can rupture and dewet, due to a long-wave spinodal instability caused by attractive intermolecular (van der Waals) forces. We show that striped patterning on a length scale comparable to that of the spinodal instability leads to a resonance effect and an imperfect bifurcation of equilibrium film shapes. Weakly nonlinear analysis yields predictions for film shapes, stability, growth rates, and rupture times, which are confirmed by numerical solution of the thin-film equation. Film behavior is qualitatively different in the resonant patterning regime, but with sufficiently large domains, rupture nonetheless occurs on a spinodal length scale regardless of patterning. Instabilities transverse to the patterning are examined and shown to behave similarly as disturbances to films on uniform substrates. Finally, we explain some previously reported effects in terms of the imperfect bifurcation. Next we examine the interaction of a foreign particle with a solidification front in binary alloys. Depending on the material properties and front velocity, the particle may be pushed ahead of the front, or it may be engulfed and incorporated into the solid phase. The outcome of this interaction plays a crucial role in issues such as the strength of composites and the survival or death of cryogenically preserved cells. We apply numerical boundary integral and continuation methods to obtain the dependence on system parameters of the critical velocity for particle capture. We reconcile two different theoretical critical velocity scalings, and show that many typical systems may obey yet another scaling. We also show that the presence of solute decreases particle velocities by an order of magnitude below those for the single-component system, due to constitutional undercooling.

Last modified

• 09/07/2018

Creator

• Justin Kao

DOI

• https://doi.org/10.21985/N2KM81

Subject

• Applied Mathematics

Keyword

• Mathematics

Date created

• 2008-05-08

Resource type

• Dissertation

Rights statement

• In Copyright