Linear instability of compound liquid threads in the presence of surfactant

This paper investigates the linear instability of compound liquid threads in the presence of surfactant. The limitation of the one-dimensional approximation in previous work [Craster, Matar, and Papageorgiou, Phys. Fluids 15, 3409 (2003)10.1063/1.1611879] is removed; hence the radial dependence of the axial velocity can be taken into account. Therefore both the stretching and the squeezing modes can be investigated. The disturbance growth rate is reduced with an increase of the dimensionless surface-tension gradient (whether in the stretching or squeezing mode). For the parameter range investigated, it is found that the squeezing mode is much more sensitive to the Marangoni effect than the stretching mode. The disturbance axial velocity and disturbance surfactant concentration for a typical case is investigated. It is found that the disturbance axial velocity is close to uniform in the stretching mode when the dimensionless surface-tension gradient and the wave number are small. In contrast, for wave numbers close to cutoff, or a large dimensionless surface-tension gradient, or in the squeezing mode, the disturbance axial velocity is not uniform. Analytical relations between growth rate and wave number valid in the long-wave limit are derived. In the stretching mode, the flow moves from an extension-dominated regime to a shear-dominated regime when β1+Rσβ2 increases through 1+Rσ, where β1 and β2 are the dimensionless surface-tension gradient of the inner and outer interface, respectively, R is the radius ratio, and σ is the surface tension ratio. In the squeezing mode, whatever the values of β1 and β2, the flow is always in the shear-dominated regime. The expressions of the leading-order axial perturbation velocity in the long-wave limit are derived and they explain the applicability of one-dimensional models. It is found that the leading-order axial velocity in the extension-dominated regime is always uniform and one-dimensional models work well in this regime. For the shear-dominated regime, the leading-order axial velocity can be either nonuniform or close to uniform, depending on the ratio between the dimensionless surfactant diffusivity d1 and the Laplace number La: when d1â‰"La the velocity profile is close to uniform and one-dimensional models work well; otherwise the velocity profile is nonuniform and one-dimensional models fail.