Electrohydrodynamic instability of a confined viscoelastic liquid film

Abstract

We study the surface instability of a confined viscoelastic liquid film under the influence of an applied electric field using the Maxwell and Jeffreys models for the liquid. It was shown recently for a Maxwell fluid in the absence of inertia that the growth rate of the electrohydrodynamic instability diverges above a critical value of Deborah number [L. Wu, S.Y. Chou, Electrohydrodynamic instability of a thin film of viscoelastic polymer underneath a lithographically manufactured mask, J. Non-Newtonian Fluid Mech. 125 (2005) 91] and the problem of pattern length selection becomes ill-defined. We show here that inclusion of fluid inertia removes the singularity and leads to finite but large growth rates for all values of Deborah number. The dominant wavelength of instability is thus identified. Our results show that the limit of small inertia is not the same as the limit of zero inertia for the correct description of the dynamics and wavelength of instability in a polymer melt. In the absence of inertia, we show that the presence of a very small amount of solvent viscosity (in the Jeffreys model) also removes the non-physical singularity in the growth rate for arbitrary Deborah numbers. Our linear stability analysis offers a plausible explanation for the highly regular length scales of the electric field induced patterns obtained in experiments for polymer melts. Further, the dominant length scale of the instability is found to be independent of bulk rheological properties such as the relaxation time and solvent viscosity.