The fastest smooth Taylor bubble

The complicated nature of singularities associated with topological transit ion in the plane Taylor-bubble problem is briefly discussed in the context of estimating the speed of the fastest smooth Taylor-bubble in the absence of surface tension. Previous numerical studies were able to show the presen ce 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 bubbl es at the tip for values of F > 0.29 due to difficulties in obtaining numer ical solutions with well-rounded profiles at the apex. These difficulties r aise the question whether the bubbles rising at a speed F is an element of (0.29, 0.357) are smooth, pointed or spurious. This issue has led us to car efully scrutinize certain asymptotic behavior of the Fourier spectrums of t he 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 we ll-rounded profiles at the apex. The purpose of this paper is to describe o ur approach and its use in arriving at the above conclusion. (C) 2000 IMACS . Published by Elsevier Science B.V. All rights reserved.