Various extensions of our model are discussed, including materials with finite dilatational surface viscosity, linear and nonlinear (visco)elasticity, and large-amplitude deformations. Moreover, we show that the mechanism of instability is distinct from that of the famous Saffman-Taylor fingering instability. A shape instability arises above a critical surface capillary number (Ca) that relates surface-viscous stresses to line tension. Surface dilatational viscosity is shown to have a significant damping effect on the otherwise undamped inviscid oscillations. While the line tension at the domain border tends to maintain circular symmetry, surface-viscous stresses generated by surface compression tend to destabilize the perimeter. We then examine the linear stability of the domain shape to small, nonaxisymmetric perturbations when the surface is steadily compressed (i.e., the pressure difference across it is increased). Consequently, the curvature of the domain appears discontinuous from the rest of the surface under rapid oscillation. At small enough Pe, the surface deforms quasistatically, whereas at high Pe the curvature varies sharply within a thin boundary layer adjacent to the domain border. We find that the curvature relaxes diffusively, and thus define a Peclet number (Pe) relating the rate of diffusion to the oscillation frequency. The bulk viscosity,, is commonly defined as the coeffi- cient of Vu in Eq. We then analyze the frequency-dependent dynamics of the surface stress and curvature within the domain when the pressure difference across the surface is sinusoidally oscillated. The derived general rheological equation for insoluble surfactants, gave as limiting cases Voight and Maxwell type equations for fast and slow reactions, respectively. The greater the resistance to flow, the higher the viscosity, so for example, syrup has a higher viscosity than water. It can also be thought of as a measure of a fluid’s thickness or its resistance to objects passing through it.
We model the surface stress within the domain using the classical Boussinesq-Scriven constitutive equation, simplified for a near-spherical cap undergoing a small-amplitude curvature deformation. Viscosity is a measure of a fluid’s resistance to flow. Our aim is to understand how changes in surface curvature generate two-dimensional Stokes flows inside the domain, thereby resisting curvature deformation and distorting the domain shape. In this paper, we investigate a simple, dynamical model of a two-phase surface fluid on a curved interface: a condensed, surface-viscous domain embedded within a surface-inviscid, spherical interface of time-varying radius of curvature.
To date, most models of rheologically complex interfaces have focused on homogeneous systems of planar or fixed curvature. The mechanics of curved, heterogeneous, surfactant-laden surfaces and interfaces are important to a variety of engineering and biological applications.
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