Linear stability analysis surface viscosity

Sternling and Scriven (1959). They applied linear stability analysis to systems where a solute is transferred through a non-deformable interface between two semi-infinite liquid layers. The instabihty can develop in systems far from partition equilibrium and its appearance depends mainly on the solvent properties, the surface activity of the solute and on the formation of critical solute concentration gradients in the normal to the interface direction (Kovalchuk et al. 2012). Unstable transfer is expected when the solute diffuses out of the phase in which its diffusivity is lower and kinematic viscosity is higher. 

The steady-state solution for fiber spinning (Newtonian and isothermal case) was presented in Section 9.1.1, and it consists of Eqs. 9.26 and 9.28. Linearized (small disturbances) stability analysis involves (Fisher and Denn, 1976) the study of finite amplitude disturbances, and we do not present it. Rather, we present the results of such an analysis. The value of Dr = 20.21 is considered to be the critical draw ratio beyond which the flow becomes unstable. Figure 9.13 (Donnelly and Weinberger, 1975) shows experimental data that confirm the theory. More specifically, silicone oil (of viscosity equal to 100 Pa-s), which seems to be Newtonian, was extruded and the ratio of maximum to minimum filament diameters was plotted against the draw ratio. An instability appears at a draw ratio of about 17, or about 22 if we take into consideration about 14% die swell. The value of the critical draw ratio of 22 compares well with the theoretical value of 20.21. Pearson and Shah (1974) extended the analysis to a power-law fluid and included surface tension, gravitational. 

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