The effects of an insoluble surfactant on the linear stability of a two-fluid core-annular flow in the thin annulus limit, for axisymmetric disturbances with wavelengths large compared to the annulus thickness, h0, are the focus of this investigation. A base shear flow affects the interfacial surfactant distribution, thereby inducing Marangoni forces that, along with capillary forces, affect the fluid-fluid interface stability. The resulting system's stability differs markedly from that of the same system with zero base flow. In the thin-annulus limit (the ratio ε of the undisturbed annulus thickness to core radius tends to zero), common in applications, a scaling and asymptotic analysis yields a coupled set of equations for the perturbed fluid-fluid interface shape and surfactant concentration. The linear dynamics of the annular film fully determine these equations, i.e. the core dynamics are slaved to the film dynamics. The theory provides a unified view of the mechanism of stability in three different regimes of capillary number Ca (defined as the product of the core viscosity, μ1, and the centreline velocity, W0 divided by the interface tension, σ0*, that corresponds to an undisturbed (signified by the subscript 0) uniform surfactant concentration, Γ0*. In the absence of a base flow or in the limit of small Ca (≪ε2, Marangoni forces deriving from non-uniformities in the interface concentration of insoluble surfactants oppose the net capillary forces. These latter forces normally stabilize the longitudinal curvature and destabilize the circumferential curvature of perturbations to the interface. In the limit of large Ca(≫ε2, Marangoni forces destabilize disturbances with wavelengths that are large compared to the annulus thickness. For moderately small Ca(∼ε2, increasing the Marangoni number Ma (defined as the product of (-∂σ*/ ∂Γ*)0 and Γ0*, divided by μ1W0 from zero increases the growth rates of all disturbances (with wavelengths ≫h0) and, consequently, reduces the marginal wavelength below that typical of the capillary instability. However a further increase in Ma eventually reverses these trends. A very large value for Ma stiffens the interface, which opposes any local variation of the tangential velocity along the interface, and this is true whether or not there is a base flow. In the limit of infinite Ma, the growth rate of the instability is 1/4 of that of the clean interface and the marginal wavenumber, non-dimensionalized by the undisturbed core circumference, returns to its clean interface (capillary) value of 1. All trends are explained physically.
All Science Journal Classification (ASJC) codes
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering