Rigorous Results in Steady Finger Selection in Viscous Fingering
Author
Xie, Xuming
Year
2001
Advisor
Tanveer, Saleh
Abstract
This dissertation concerns existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface tension ($\epsilon^2 $).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 $(\lambda -\frac{1}{2})/\epsilon^{\frac{4}{3}}$, where $\lambda$ is the relative finger width. The methodology consists of proving existence and uniqueness of analytic solutions for a weaker half strip problem for any $\lambda$ where the symmetric condition is relaxed. This is followed by analysis near appropriate complex turning points that determine exponentially small terms in $\epsilon$,as $\epsilon \to 0^+$. We prove that for $\lambda-\frac{1}{2}$ small the symmetry condition is satisfied if only if $(\lambda -\frac{1}{2})/\epsilon^{\frac{4}{3}} $ 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 steady finger problem.
Xie, Xuming
Year
2001
Advisor
Tanveer, Saleh
Abstract
This dissertation concerns existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface tension ($\epsilon^2 $).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 $(\lambda -\frac{1}{2})/\epsilon^{\frac{4}{3}}$, where $\lambda$ is the relative finger width. The methodology consists of proving existence and uniqueness of analytic solutions for a weaker half strip problem for any $\lambda$ where the symmetric condition is relaxed. This is followed by analysis near appropriate complex turning points that determine exponentially small terms in $\epsilon$,as $\epsilon \to 0^+$. We prove that for $\lambda-\frac{1}{2}$ small the symmetry condition is satisfied if only if $(\lambda -\frac{1}{2})/\epsilon^{\frac{4}{3}} $ 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 steady finger problem.