MATHEMATICAL METHODS IN THE APPLIED SCIENCES Math. Meth. Appl. Sci. 2004; 27:1197–1220 (DOI: 10.1002/mma.489) MOS subject classication: 35 Q 30; 35 R 55; 35 R 35; 76 M 55 On the evolution of thin viscous jets: lament formation Marco A. Fontelos ∗;† Escuela Superior de Ciencias Experimentales y Tecnologia; Universidad Rey Juan Carlos. C=Tulipan S=N; Mostoles 28933; Madrid; Spain Communicated by M. Renardy SUMMARY In this paper, we have studied the evolution of thin uid jets, paying special attention on the limit of very large viscosity. Local well-posedness of the one-dimensional system describing this evolution as well as the existence of break-up (at least as t →∞) under quite general conditions is proved. In addition, we have proved the well-known experimental fact that in the limits of very large viscosities the solutions develop very long and thin laments previous to break-up and a complete detailed description of their structure is given. Copyright ? 2004 John Wiley & Sons, Ltd. KEY WORDS: uid dynamics; singularities; viscous jets; degenerate parabolic systems 1. INTRODUCTION This paper includes results regarding the evolution and break-up of thin uid jets with a free boundary. The evolution of a uid jet and its fragmentation into drops has attracted the attention of scientists since the early 19th Century. The formation of drops is a commonly observed phenomenon and it is then natural to ask ourselves whether the equations of hydro- dynamics are able to predict it or should the equations be modied in order to accommodate the observations. In 1833, Savart [1] performed experiments in order to measure the size of the resulting jet fragmentation drops. In 1879, Rayleigh [2] presented the rst analytical study in relation to this problem. He showed that a stationary jet, which is a solution for both Euler and Navier–Stokes systems, is unstable and computed the dispersion relation for small perturbations. This dispersion relation is such that it attains a maximum at a wavelength coherent with the size of the drops measured by Savart. Rayleigh’s theory, nevertheless, fails to show that break-up follows from the equations. At the end of the last century, the problem was dealt with by using the theoretical, computational and experimental tools available at the time. A close experimental observation of the evolution, and break-up processes revealed their ∗ Correspondence to: Marco A. Fontelos, Escuela Superior de Ciencias Experimentales y Tecnologia, Universidad Rey Juan Carlos. C=Tulipan S=N, Mostoles 28933, Madrid, Spain. † E-mail: mafontel@escet.urjc.es Contract=grant sponsor: Spanish Ministry of Education Published online 7 April 2004 Copyright ? 2004 John Wiley & Sons, Ltd. Received 7 October 2002