Annular flows can keep unstable films from breakup: Nonlinear saturation of capillary instability

Frenkel, A.; Babchin, A.J.; Levich, B. G.; Shlang, T.; Sivashinsky, G. I.

doi:10.1016/0021-9797(87)90027-0

Cited by 90 publications

(86 citation statements)

References 17 publications

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“…The integral term represents the influence of viscosity stratification; when m = 1 that term disappears. Note that for m = 1 and β 0 = 0, the equation reduces to the Kuramoto-Sivashinsky equation (see Frenkel et al (1987), Papageorgiou et al (1990)). Finally for simplicity, the surfactant equation (3.16) is written as 37) where Γ = Γ 0 +Γ.…”

Section: Asymptotic Expansions and The Final Equationsmentioning

confidence: 99%

“…In general, consistent asymptotic theories for thin annuli can be developed when the interfacial amplitude is in the weakly nonlinear regime (an exception is Kerchman 1995 who allows disturbances scaled on the annulus but requires asymptotically small shear in the film). In the absence of viscosity differences Frenkel, Babchin, Levich, Shlang & Sivashinsky (1987) derive a Kuramoto-Sivashinsky equation for the spatio-temporal evolution and argue that capillary instability is saturated nonlinearly. Papageorgiou, Maldarelli & Rumschitzki (1990) have shown that when viscosity differences are present (as in lubricated pipelining for example), there is a coupling between the film and core dynamics with the solutions of the Stokes or linearised Navier-Stokes equations in the core required to close the problem.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear dynamics of core-annular film flows in the presence of surfactant

2009

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The nonlinear stability of two-phase core-annular flow in a cylindrical pipe is studied. A constant pressure gradient drives the flow of two immiscible liquids of different viscosities and equal densities, and surface tension acts at the interface separating the phases. Insoluble surfactants are included and we assess their effect on the flow stability and ensuing spatio-temporal dynamics. We achieve this by developing an asymptotic analysis in the limit of a thin annular layer -this is usually the relevant regime in applications -to derive a coupled system of nonlinear evolution equations that govern the dynamics of the interface and the local surfactant concentration on it. In the absence of surfactants the system reduces to the Kuramoto-Sivashinsky (KS) equation and its modifications due to viscosity stratification (present when the phases have unequal viscosities) are derived elsewhere. We report on extensive numerical experiments to evaluate the effect of surfactants on KS dynamics (including chaotic states, for example), both in the absence and presence of viscosity stratification. We find that chaos is suppressed in the absence of viscosity differences and that the new flow consists of successive windows (in parameter space) of steady-state travelling waves separated by time-periodic attractors. The intricate structure of the travelling pulses is also explained physically. When viscosity stratification is present we observe a transition from time-periodic dynamics, for instance, to steady-state travelling wave pulses of increasing amplitudes and speeds. Numerical evidence is presented that indicates that the transition occurs through a reverse Feigenbaum cascade in phase space.

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Section: Asymptotic Expansions and The Final Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of core-annular film flows in the presence of surfactant

2009

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“…saturation of the primary capillary instability at O( a) amplitudes under a balance between the effects of wave steepening due to gravity and the stabilizing effects of surface tension associated with axial interfacial curvature (8). For B < B * however, drainage due to gravity has a destabilizing effect on collars generated by capillary instability, since small collars can grow and possibly coalesce to form large ones (1,(5)(6)(7).…”

Section: Introductionmentioning

confidence: 98%

Draining Collars and Lenses in Liquid-Lined Vertical Tubes

Jensen

2000

Journal of Colloid and Interface Science

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“…Equivalently, as Plateau showed, only axisymmetric wavelengths larger than the circumference 2 E (see Figure 1) are unstable [57]. Frenkel, Babchin, Levich, Shlang and Sivashinsky showed that gravity-driven ow can keep the ÿlm from rupturing (in the form of droplets) as a result of the non-linear saturation of the instability, which is generated by the coupling between instability growth and the driving force (gravity or pressure as in the present problem) [58]. QuÃ erÃ e examined experimentally the conditions for instability of ÿlm ow down ÿbres (under gravity).…”

Section: Thin-ÿlm Equations and Boundary Conditionsmentioning

confidence: 98%

Influence of inertia, topography and gravity on transient axisymmetric thin‐film flow

Khayat

Kim

Delosquer

2004

Numerical Methods in Fluids

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SUMMARYThis study examines theoretically the development of early transients for axisymmetric ow of a thin ÿlm over a stationary cylindrical substrate of arbitrary shape. The uid is assumed to emerge from an annular tube as it is driven by a pressure gradient maintained inside the annulus, and/or by gravity in the axial direction. The interplay between inertia, annulus aspect ratio, substrate topography and gravity is particularly emphasized. Initial conditions are found to have a drastic e ect on the ensuing ow. The ow is governed by the thin-ÿlm equations of the 'boundary-layer' type, which are solved by expanding the ow ÿeld in terms of orthonormal modes in the radial direction. The formulation is validated upon comparison with the similarity solution of Watson (J. Fluid Mech 1964; 20:481) leading to an excellent agreement when only 2-3 modes are included. The wave and ow structure are examined for high and low inertia. It is found that low-inertia uids tend to accumulate near the annulus exit, exhibiting a standing wave that grows with time. This behaviour clearly illustrates the di culty faced with coating high-viscosity uids. The annulus aspect is found to be in uential only when inertia is signiÿcant; there is less ow resistance for a ÿlm over a cylinder of smaller diameter. For high inertia, the free surface evolves similarly to two-dimensional ow. The substrate topography is found to have a signiÿcant e ect on transient behaviour, but this e ect depends strongly on inertia. It is observed that the ow of a high-inertia uid over a step-down exhibits the formation of a secondary wave that moves upstream of the primary wave. Gravity is found to help the ÿlm (coating) ow by halting or prohibiting the wave growth. The initial ÿlm proÿle and velocity distribution dictate whether the uid will ow downstream or accumulate near the annulus exit.

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Annular flows can keep unstable films from breakup: Nonlinear saturation of capillary instability

Cited by 90 publications

References 17 publications

Nonlinear dynamics of core-annular film flows in the presence of surfactant

Nonlinear dynamics of core-annular film flows in the presence of surfactant

Draining Collars and Lenses in Liquid-Lined Vertical Tubes

Influence of inertia, topography and gravity on transient axisymmetric thin‐film flow

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