Capillary instability of a liquid thread

We begin with capillary instability of a liquid thread. This is a problem that was discussed qualitatively already in Chap. 2. It is a problem with a physically clear mechanism for instability and thus provides a good framework for introducing the basic ideas of linear stability theory. This problem is one of several examples in which the viscosity of the fluid plays no role in determining stability, but only influences the rate of growth or decay of the infinitesimal disturbances that are analyzed in a linear theory. 

Problem 12-3. Capillary Instability for a Thread in a Second Immiscible Fluid. In this problem, we consider the effect on capillary instability if, instead of being surrounded by air, the thread of liquid is surrounded by a second viscous immiscible fluid that is assumed to be unbounded in the radial direction. Derive a condition from which you could in principle, calculate the growth-rate parameter for an axisymmetric disturbance, a = a(k, Re, 7.) where k is the axial wave number and 7. is the ratio of the external fluid viscosity to the viscosity of the liquid thread. This condition can be simplified if either Re I or the thread is inviscid (though viscous effects still remain in the outer fluid). Evaluate a for several k values in each of these two cases. What is the qualitative effect of the second viscous fluid For example, is the range of unstable k values changed Is the fastest-growing linear mode changed relative to the case of a thread in air ... 

Newhouse, L. A., C. Pozrikidls The capillary instability of annular layers and liquid threads, J. Fluid Mech. 242, 193-209 (1992). 

The disintegration of liquid sheets as studied by many authors [40-42] is generally a two-step process. First, liquid threads and/or ligaments disintegrate from the sheet, which secondly break into droplets due to capillary instabilities. This can be induced either through perforations or destabilising waves in the liquid sheet [40]. 

Considerations and measurements of interfacial tension between polymer melts dates to the 1960s and 1970s [54 to 57]. Several different methods have been used to measure interfacial tension. Extensive use has been made of 1) the shape of drops emerging from a capillary into a second phase (falling drop) [56 to 60] and 2) thread breakage, the breakup of stationary hlaments in a second liquid phase by a capillary instability [59 to 61 ]. The latter analysis is based on the work of Tomotika [62]. Other methods have been used. 

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