Self-similar solutions for viscous capillary pinch-off

Abstract

The axisymmetric capillary pinch-off of a viscous fluid thread of viscosity $\lambda\mu$ and surface tension $\gamma$ immersed in a surrounding fluid of viscosity $\mu$ is studied. Similarity variables are introduced (with lengthscales decreasing like $\tau$, the time to pinch-off, in a rapidly translating frame) and the self-similar shape is determined directly by a combination of modified Newton iteration and a standard boundary-integral method. A large range of viscosity ratios is studied ($0.002\,{\leq}\,\lambda\,{\leq}\, 500$) and asymmetric profiles are observed for all $\lambda$, with conical shapes far from the pinching point, in agreement with previous time-dependent studies. The stability of the steady solutions is investigated and oscillatory instability is found for $\lambda \ge 32$. For $\lambda\,{\ll}\, 1$ an asymptotic scaling of $\lambda^{1/2}$ is suggested for the slopes of the far-field conical shapes. These compare well with the quantitative predictions of a one-dimensional theory based on Taylor's (1964) analysis of a slender bubble.

Dynamics of a viscous thread surrounded by another viscous fluid in a cylindrical tube under the action of a radial electric field: breakup and touchdown singularities

“…The dominant balance is between the capillary pressure and the viscous stresses due to the presence of the external fluid. Such findings have also been established in the studies of Cohen et al (1999) and Cohen & Nagel (2001); a more complete study of the self-similar equations and their solutions covering a much larger range of viscosity ratios, can be found in Sierou & Lister (2003) where a stability analysis of the constructed solutions is also carried out indicating oscillations when the viscosity of the outer fluid becomes sufficiently small (approximately when the inner fluid is 32 times more viscous than the outer one). It is important to note that the presence of an outer viscous fluid alters possible self-similar structures near pinching in a fundamental manner since the dominant axial and radial scales are now of the same order as opposed to the case of a passive outer fluid where long wave dynamics are in control (see Eggers (1993), Eggers & Dupont (1994), Papageorgiou (1995)).…”

“…Recent and ongoing analytical studies concentrate on describing the pinching process asymptotically by utilizing the separation of radial and axial scales and mapping the dynamics to a class of self-similar solutions which are universal when inertia is present; notable studies include the work of Eggers (1993), Eggers & Dupont (1994), Papageorgiou (1995), Brenner et al (1996) for jets surrounded by a passive medium; Craster et al (2002), Craster et al (2003), Craster et al (2005), for surfactant-covered or compound jets; Conroy et al (2010), for core-annular arrangements in the presence of electrokinetic effects. Significant computational work has also been carried out with the aim of simulating the phenomena and evaluating the asymptotic theories (the latter are considerably less demanding numerically) -see Newhouse & Pozrikidis (1992), Pozrikidis (1999), Lister & Stone (1998), Sierou & Lister (2003), who simulate Stokes flows using boundary integral methods, and Ambravaneswaran et al (2002), Chen et al (2002), Notz et al (2001), Notz & Basaran (2004), Collins et al (2007), Hameed et al (2008) who compute the flow at arbitrary Reynolds number and in some instances include the effects of surfactants and electric fields -the extensions and novel aspects of the present work are outlined later.…”

Breakup of Air Bubbles in Water: Memory and Breakdown of Cylindrical Symmetry

“…This therefore sets bubble detachment apart from all other breakup situations studied so far where one or both fluids are viscous, or where two inviscid fluids differ little in density, as well as from the inverse case of water in air. In those cases, the breakup is driven by surface tension and α h ≥ 2/3 [3,4,5,6,7,8,9,10,11,12,20]. Here we observe the consequences of a different driving mechanism on the breakup dynamics.…”

“…Each symmetry in nature implies an underlying conservation law, so that the symmetries of the singularity associated with pinch-off naturally have important consequences for its dynamics. It was previously believed [1,2,3,4,5,6,7,8,9,10,11,12] that the pinching neck of any drop or bubble would become cylindrically (i.e. azimuthally) symmetric in the course of pinch-off.…”

Pore-scale mass and reactant transport in multiphase porous media flows

“…We focus on the dynamics of capillary channels as they control the heat transfer beyond the melting front associated with heat diffusion and host the majority of the mass transport of non-wetting fluid through the porous matrix. Although elongated channels of fluid embedded in an unbounded and immiscible viscous or inviscid ambient fluid have been shown to be unstable to capillary forces (see for example Newhouse & Pozrikidis 1992;Eggers 1993;Papageorgiou 1995;Chen & Steen 1997;Day, Hinch & Lister 1998;Zhang & Lister 1999;Sierou & Lister 2003;Quan & Hua 2008), the effect of confining solid boundaries and steady flow through the channel have been suggested to slow down the breakup of the channel into bubbles or slugs (Tomotika 1935;Hagedorn et al 2004). In a dynamical setting such as when capillary channels invade a porous medium, however, these structures are known to be able to remain stable over the duration of laboratory or numerical experiments as long as the injection rate of the invading fluid remains constant.…”