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The breakup of fluid threads and films by capillary instability is ubiquitous in engineering, science, and nature. For example, an understanding of this phenomenon is essential to the technologies of ink-jet printing [1], the production of stable thin coatings (e.g., polymer films, coating optical fibers, and wires) [2], the morphology and stability of extruded polymer blends [3, 4], the formation of polymer fibers from polymer blends [5], the transportation of oil in pipelines[6–9], and tertiary oil recovery [10(a)]. In many of these technological applications, thread breakup occurs under tubular confinement conditions. Examples include the phase separation of blends and other fluid mixtures in porous media or in the presence of large quantities of filter particles where a near tubular geometry exists locally, oil recovery from porous rocks [10(a)] and synthetic tubular networks encountered in multicomponent fluid processing, ranging in scale from plant pipelines to microfluidic devices 9, 10(b)]. Moreover, the consequences of this type of instability are apparent in a multitude of natural phenomena such as the stability of liquid jets (e.g., kitchen faucets of garden hoses) [1], the beading of liquids on natural (e.g., spider webs) and synthetic fibers [11] and perhaps even in fundamental biological processes such as morphogenesis 12(a)]. Tubular confinement of multiphase fluids is known have important consequences for respiration and pulmonary disease [12(b)] and for transport processes occurring in host of biological structures in animals ranging in scale from the arteries, veins, and capillaries of the circulatory system to microtubules and other structures within animal cells. Hierarchically organized vascular structures with tubular structures containing multicomponent fluids are also characteristic structural features of plants, influencing the fluid distribution within these structures [12(a)] and transport processes vital for life.
The study of capillary breakup has a long history. Savart [13] gave the first scientific report of the breakup of liquid threads in 1833, followed by Magnus [14] in 1855. Plateau [15] and his assistants performed experiments on the breakup fluid threads, and Plateau provided the first theoretical explanation of the occurrence of this instability when thread length is greater than its circumference. For threads greater length, boundary undulations reduce the surface area for a fixed volume of fluid. Rayleigh [16] formulated the first theory of the dynamics of thread breakup in the absence viscosity effects in either the thread or the surrounding fluid medium. He estimated that the wavelength of the undulatory instability along the thread should be comparable to the stability length (now known as the Rayleigh-Plateau stability length) estimated using thermodynamic reasoning by Plateau. Later, Weber generalized the theory to describe combined effects of the viscosity of the fluid thread, density and surface tension [17] and Tomotika [18] included the effect of the matrix viscosity while excluding density effects. Lee and Flumerfelt [19] have given a unified treatment fluid thread breakup that includes viscosity mismatch, density and inertial effects. The simpler Tomitika theory is suitable for understanding the breakup of viscous fluid threads having nearly the same density as the surrounding fluid. This situation often applies well to polymer mixtures, but the non-Newtonian rheology of high molecular mass polymers is often a complication in interpreting measurements on this technologically important class of fluids [3].
In the present paper, we are concerned about how the presence of boundaries
influences the capillary breakup of viscous fluid threads and we focus
particularly on the nature of thread breakup within a coaxial tube in the
absence of imposed flow. Figure 1 illustrates a thread subjected to tube
confinement where Fig. 1(a) shows the initial stage in which
the thread has a cylindrical form, while Fig. 1(b) shows incipient
thread breakup by the Rayleigh-Plateau instability
[15,16]. Schematic images of the late-stage morphologies
observed after thread breakup, i.e., "plugs," plugs with "collars"
and "capsules" are indicated in Fig. 1(c). The collars on
the plugs are transient features and these structures relax into
plugs at long times. Capsules are sometimes observed to become
unstable to plug formation at long times and the plugs
tend to coalesce slowly at still longer times so that the latestage
evolution of the ruptured thread is characterized by a
succession of long-lived transient states. Geometrical parameters
that are important in specifying the simulation conditions
are also indicated in Fig. 1.
FIG. 1. Schematic illustration of thread breakup in a confining tube. (a) Initial configuration of a fluid thread confined to a tube. (b) Thread undergoing capillary undulations. The image corresponds to the simulation described below. (c) Schematic images of post-rupture structures-"plug." plug with "collar," and "capsule."
Since fluid threads embedded in films (e.g., spinodal decomposition in films) and between parallel plates [20, 21] are common in processing applications, we briefly compare our tube confinement simulations to those for a threads confined between two parallel plates. We also consider the breakup of arrays of threads subjected simultaneously to parallel plate confinement since multiple thread geometry illustrates some essential aspects of the influence of flexible boundaries on thread breakup.
In the absence of dispersion interaction effects and other non-hydrodynamic effects relevant to capillaries having a sub-micron scale [22], geometrical confinement of an infinite Newtonian thread surrounded by another liquid in a coaxial pipe does not provide thermodynamic stability against capillary breakup [9, 23]. The rate of breakup and fluid morphology is certainly influenced by confinement, however [24, 25]. Lubrication theory in a linearized approximation predicts that the rate of thread breakup relative to the bulk vanishes as thread radius approaches the tube radius [23, 25] and this effect has been qualitatively indicated in numerical boundary integral calculations [24]. These former simulations also indicate that the geometrical form of the thread breakup morphology changes under high confinement [24]. Notably, this type of "sharp interface" model has difficulty following the singular thread rupture and coalescence processes so that the simulations have primarily focused on phenomena occurring before thread breakup. This method also does not apply to fluids for which the interfacial width is comparable to the scale of confinement since this type of model assumes a vanishing interfacial width. (The interfacial width describing the composition profile or "interface" of two liquids at equilibrium can be appreciable in polymer and other complex fluids and even for "simple" fluids near their critical point for phase separation.) We can readily treat the long time evolution of thread breakup using the lattice Boltzmann (LB) method and this allows us to obtain a more complete picture of geometric and kinetic aspects of thread breakup in confined geometries, although this method also encounters difficulties when the scale of confinement is comparable to the interfacial width (see the discussion below). The LB model is advantageous because it allows for the incorporation of polymer-fluid thermodynamic interactions that allow for surface segregation of the fluid components, a physical affect apparently difficult to incorporate in finite element and boundary element simulations of thread confinement since these models implicitly assume that the fluids are perfectly immiscible.
The present work focuses on the geometrical character of the thread breakup instability due to finite size effects and the influence of the fluid-surface interaction on kinetic aspects of this process using the LB model of multiphase fluid dynamics. LB simulations allow for both the treatment of a diffuse interfacial width between the liquid phases and the thermodynamic interaction between the fluid and confining wall. These interactions are commonly important in polymeric blends and other fluid mixtures (e.g., surfactant solutions) that are only weakly immiscible and the boundary interactions lead to a compositional segregation of the energetically preferred coexisting phase to the walls, thereby considerably affecting the rate of thread breakup. This effect is found in our simulations and, moreover, recent measurements have demonstrated the crucial role of fluid "wetting" properties on the stability and form of two-phase flows of immiscible fluids in microchannels [10(b)].
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