Arch Appl Mech (2007) 77: 613–626 DOI 10.1007/s00419-007-0118-0 ORIGINAL M. F. El-Sayed · M. I. Syam Electrohydrodynamic instability of a dielectric compressible liquid sheet streaming into an ambient stationary compressible gas Received: 3 October 2006 / Accepted: 18 January 2007 / Published online: 22 February 2007 © Springer-Verlag 2007 Abstract The effect of compressibility of fluids on the linear electrohydrodynamic instability of a dielectric liquid sheet issued from a nozzle into an ambient dielectric stationary gas in the presence of a horizontal electric field is investigated. It is found that increasing the Mach number from subsonic to transonic causes the maximum growth rate and the dominant wavenumber of the disturbances to increase, and the increase is higher in the presence of the electric field. Liquid compressibility has been found to have a minimal effect on instability. At constant wavenumber and electric field values, the growth rate of disturbances increases as the gas Mach number tends to 1, and then begins to decrease with further increase in the gas Mach number. At small values of wavenumber, antisymmetrical disturbances grow faster than symmetrical ones, while the growth rate of both types of disturbances approach each other at large wavenumbers, which increases by increasing the electric field values. At small Weber numbers, antisymmetrical disturbances exhibit a higher maximum growth rate and a lower dominant wavenumber than symmetrical disturbances. However, the maximum growth rate and dominant wavenumber of the two types of disturbances are almost identical when both Weber number and electric field values become large. An increase in the gas to liquid density ratio enhances the instability, and this effect is enhanced for high electric field values. Surface tension and electric fields always oppose and increase the development of instability, respectively; and they have opposite effects for long wavelengths and high Weber numbers. Keywords Hydrodynamic stability · Electrohydrodynamics · Compressibility · Liquid sheets · Gas-liquid interface · Dielectric fluids · Muller and secant methods 1 Introduction The study of capillary instabilities and their causes goes back more than one century [1]. While the desta- bilization of planar liquid surfaces on a macroscopic scale requires an externally applied potential (such as electric or magnetic fields [2,3]), thin films are known to become spontaneously unstable. Recently, this topic was rediscovered, leading to a flurry of publications on the dewetting of thin polymer films, liquid metals, etc. This renewed interest was not only caused by technological importance of film stability (e.g., for coating, lubrication layers, etc.), but also because liquid dewetting leads to a rich pattern formation process [4]. On the other hand, the process of atomization is one in which a liquid sheet or jet is disintegrated into a multiplicity of small drops. Combustion of liquid fuels in diesel engines, gas turbines, and rocket engines is dependent on M. F. El-Sayed (B ) Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis (Roxy), Cairo, Egypt E-mail: mfahmye@yahoo.com M. I. Syam Department of Mathematical Sciences, Faculty of Science, UAE University, P. O. Box 17551 Al Ain, United Arab Emirates E-mail: m.syam@uaeu.ac.ae