The Fastest Smooth Taylor Bubble

The complicated nature of singularities associated with topological transition in the plane Taylorbubble problem is briefly discussed in the context of estimating the speed of the fastest smooth Taylorbubble in the absence of surface tension. Previous numerical studies were able to show the presence of a stagnation point at the tip of the bubbles for dimensionless speed F < 0.357 but were incomplete in characterizing the topology of these bubbles at the tip for values of F > 0.29 due to difficulties in obtaining numerical solutions with well-rounded profiles at the apex. These difficulties raise the question whether the bubbles rising at a speed F ∈ (0.29, 0.357) are smooth, pointed or spurious. This issue has led us to carefully scrutinize certain asymptotic behavior of the Fourier spectrums of the numerical solutions for a wide range of values of F and to extend these results in an appropriate limiting sense. Our findings indicate that these plane bubbles with F < 0.35784 (accurate up to four decimal places) have well-rounded profiles at the apex. The purpose of this paper is to describe our approach and its use in arriving at the above conclusion.