Faraday instability in a linear viscoelastic fluid

– The onset of surface waves in a vibrated layer of a viscoelastic fluid is investigated theoretically. For vibration frequencies close to the inverse relaxation time of the non-Newtonian fluid, surface waves respond harmonically instead of being subharmonic as is usually the case in Newtonian fluids. This prediction has been made on the basis of the Maxwell model for viscoelastic fluids, but the result is robust and has been confirmed by using the empiric viscosity spectrum for a mixture of 2% polyisobutylene in primol. The parameter range is determined, where harmonic surface waves are favored. Introduction. – Our understanding of pattern formation has experienced enormous progress during the recent decades [1]. Important insights were contributed by studies of Newtonian fluid systems, such as Rayleigh-Bénard convection [2], Taylor-Couette flow [3] and the Faraday instability [4-7]. Recently, pattern formation in viscoelastic fluids came into the focus of nonlinear science too. With the memory of viscoelastic fluids an additional time scale is introduced, which gives rise to a number of interesting phenomena. For example, the interplay between the viscoelastic relaxation time and the thermal diffusion time lead to an oscillatory onset of Rayleigh-Bénard convection [8], which exhibits also a multi-critical bifurcation behavior [9]. The competition between the elastic and inertial instability in Taylor-vortex flow leads to a number of novel phenomena [10], e.g. localized structures [11]. The well-founded theory of Newtonian fluids is one reason of the popularity of fluid systems in pattern formation. It allows ab initio calculations and hence quantitative comparisons between theoretical concepts in pattern formation and experimental measurements. For a number of examples, especially for Rayleigh-Bénard convection and Taylor-vortex flow, a very precise agreement has been achieved between both approaches. This accelerated the progress in pattern formation considerably. For viscoelastic fluids the theoretical basis is less advanced. A generally accepted description for the large scale motion of viscoelastic fluids —as the Navier-Stokes equations for Newtonian fluids— is not available yet. However, for small displacement gradients the “general () Permanent address: Theoretische Physik, Universität des Saarlandes D-66041 Saarbrücken, Germany.