Rheol Acta 35:567-583 (1996) © Steinkopff Verlag 1996 Anuj Chauhan Charles Maldarelli David S. Rumschitzki Demetrios T. Papageorgiou Temporal and spatial instability of an inviscid compound jet Received: 25 July 1996 Accepted: 30 August 1996 Dedicated to the memory of Professor Tasos C. Papanastasiou A. Chauhan • Ch. Maldarelli D.S. Rumschitzki The Levich Institute and Department of Chemical Engineering City College of CUNY New York, New York 10031, USA Prof. Dr. D.T. Papageorgiou (EN) Department of Mathematics Center for Applied Mathematics and Statistics New Jersey Institute of Technology University Heights Newark, New Jersey 07102, USA Abstract This paper examines the linear hydrodynamic stability of an inviscid compound jet. We perform the temporal and the spatial analyses in a unified framework in terms of transforms. The two analyses agree in the limit of large jet velocity. The dispersion equation is explicit in the growth rate, af- fording an analytical solution. In the temporal analysis, there are two growing modes, stretching and squeezing. Thin film asymptotic ex- pressions provide insight into the instability mechanism. The spatial analysis shows that the compound jet is absolutely unstable for small jet velocities and admits a convec- tively growing instability for larger velocities. We study the effect of the system parameters on the tem- poral growth rate and that of the jet velocity on the spatial growth rate. Predictions of both the tem- poral and the spatial theories com- pare well with experiment. Key words Compound jet- capillary instability - temporal instability - spatial instability - absolute instability Introduction The breakup of a liquid jet due to a capillary-driven hydrodynamic instability is widely studied due to many potential applications in fuel injection, ink jet printing (Hertz and Hermanrud, 1983; Sweet, 1964), polymer processing, fiber spinning, particle sorting (Herzberg et al., 1976) and many others. The instability in the liquid jet may initiate from an initial disturbance, e.g., at the interface. This "temporal" instability (e ikz+st, k = k r , s = Sr+iSi) can grow in time (Sr> 0) but not in space. The instability can also initiate from an excitation in time, localized in space. This is the spatial instability (e ikz+st, k = kr + iki, s = isi). These spatial disturbances can grow (ki< 0) and convect with the jet so that, at a fixed spatial position, the disturbances may finally decay (Bogy, 1979). It is also possible that the disturbances that originate either from the initial disturbance or from a localized spatial excitation in time, grow in time and, in the latter case, in space as well so even at a fixed point in space the disturbances continue to grow. This is the absolute in- stability. Jets are usually produced by forcing fluids through a nozzle. If the instability is convective, the jet breaks downstream from the orifice. If the jet is absolute- ly unstable it breaks as soon it comes out of the nozzle, which could cause spraying. Rayleigh (1879) originally used normal mode analysis to solve the temporal problem for an inviscid jet. He con- cluded that the jet is unstable for axisymmetric waves longer than the undisturbed jet circumference. Keller et al. (1973) modified Rayleigh's analysis to solve for the spatial growth, i.e., the spatial (convective) instability. Keller also showed that for large jet velocities, one of the spatial modes approaches the temporal solution. Both Rayleigh (1879) and Keller et al. (1973) assumed the jet to be doubly infinite. In experiments the jets are produced at a nozzle tip; so, they are semi-infinite. Leib and Gold-