Surface- and interface oscillations of a rotating viscoelastic liquid column of immiscible liquids

Abstract

The natural damped coupled frequencies of a solidly rotating visco-elastic infinite liquid column with no axial dependency (∂/∂z=0, two-dimensional problem) have been determined. The frequency equation is numerically evaluated for a single visco-elastic liquid column, where the influence of the tension parameterTa/ϱv2, the relaxation parameter τ*/a2 and the rotational Reynoldsnumber\(\tilde Re = \Omega _0 a^2 /v(\tilde Re \equiv \sqrt {\tilde T} a,\tilde Ta\)=Taylornumber) has been determined. It was found that the liquid column becomes unstable for a rotational speed\(\Omega _0^2 \geqq \frac{T}{{\rho a^3 }}(m^2 - 1)\), which is much earlier than in the case of frictionless liquid, where\(\Omega _0^2 \geqq \frac{T}{{\rho a^3 }}m(m + 1)\). In addition the stability boundary does neither depend on the magnitude of the viscosity nor the Maxwell relaxation time τ* of the liquid. The complex frequencies are presented for the modem=2, where the cross-section of the liquid column assumes during its oscillation an elliptic shape.