Rigorous results in steady finger selection in viscous fingering

This thesis concerns existence of steadily translating finger solution in a Hele-Shaw cell for non-zero surface tension ($e2$). Though there are numerous numerical and formal asymptotic results for this problem, we know of no mathematically rigorous results to date. By showing equivalence of the problem to a certain nonlinear integro differential equation in a complex plane strip, we show that analytic symmetric finger solutions exist for sufficiently small surface tension for a discrete set of values of (λ − ½)/$e43$, where λ is the relative finger width.; The methodology consists of proving existence and uniqueness of analytic solutions for a weaker half strip problem for any λ. This is followed by Borel plane analysis near appropriate complex turning points that determine exponentially small terms in $e$, as $e$ → 0+. We prove that the symmetry condition is satisfied when (λ − ½)/$e43$ attains a discrete set of values. When the symmetry condition is satisfied, the weaker half strip problem is shown to be equivalent to the original problem.