Chemical and hydrodynamical analysis of stability of a spherical interface

Abstract

The hydrodynamical and chemical stability of deformation of the interface of a spherical drop suspended in an infinite amount of another immiscible liquid is investigated by the methods of linear, hydrodynamical stability theory. The two bulk fluids are homogeneous and continuous throughout. A general determinantal dispersion relation is evaluated between the complex frequency of the perturbation and the number characterising the surface harmonic normal mode of perturbation in the case of an arbitrary number of fluctuating and reacting species on the interface. The coupling between chemical reactions, surface diffusion, and hydrodynamics is effected by the interfacial through the equation of state of the interface and by convection motions on the interface. Surface shear and dilatational viscosity are taken into account assuming the surface fluid to be Newtonian. The stability of a stationary state of the interfacial chemical reaction with the bulk fluids in hydrodynamical rest with regard to small perturbations in the surface concentrations and in the velocity of the fluid is then studied. The fluxes from the bulk fluids to the interface remain constant, or otherwise they are perturbed proportional to the fluctuations in the surface concentrations. The case of one fluctuating species and small drop radii is treated in detail. The necessary condition for the system to be unstable is that the surface chemical reaction is unstable itself. In addition, the coefficient of autocatalysis of the surface reactions has to exceed a threshold value composed by the quenching effects of surface diffusion and of the bulk and surface viscosities. In cases with more than one fluctuating species there exist possibilities for the total system to be unstable even for stable surface reactions. The present theory is an extension of the theory of oscillations of a viscous drop due to capillary forces. It is thought to be an introduction to the study of “kicking drops” and motile events connected with the deformation of the biological cell membrane.