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Register a new userParticle size selection in capillary instability of locally heated co-axial fiber
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Harnessing fluidic instabilities to produce structures with robust and regular properties has recently emerged as a new fabrication paradigm. This is exemplified in the work of Gumennik et al. [Nat. Comm. 4:2216, DOI: 10.1038/ncomms3216, (2013)], in which the authors fabricate silicon spheres by feeding a silicon-in-silica co-axial fiber into a flame. Following the localized melting of the silicon, a capillary instability of the silicon-silica interface induces the formation of uniform silicon spheres. Here, we try to unravel the physical mechanisms at play in selecting the size of these particles, which was notably observed by Gumennik et al. to vary monotonically with the speed at which the fiber is fed into the flame. Using a simplified model derived from standard long-wavelength approximations, we show that linear stability analysis strikingly fails at predicting the selected particle size. Nonetheless, nonlinear simulations of the simplified model do recover the particle size observed in experiments, without any adjustable parameters. This shows that the formation of the silicon spheres in this system is an intrinsically nonlinear process that has little in common with the loss of stability of the underlying base flow solution.
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We investigate the effective rheology of two-phase flow in a bundle of parallel capillary tubes carrying two immiscible fluids under an external pressure drop. The diameter of each tube varies along its length and the corresponding capillary threshold pressures are considered to be distributed randomly according to a uniform probability distribution. We demonstrate through analytical calculations that a transition from a linear Darcy regime to a non-linear behavior occurs while decreasing the pressure drop $Delta P$, where the total flow rate $langle Q rangle$ varies with $Delta P$ with an exponent $2$. This exponent for the non-linear regime changes when a lower cut-off $P_m$ is introduced in the threshold distribution. We demonstrate analytically that, in the limit where $Delta P$ approaches $P_m$, the flow rate scales as $langle Q rangle sim (|Delta P|-P_m)^{3/2}$. We have also provided some numerical results in support to our analytical findings.
A thin liquid film falling on a uniformly heated horizontal plate spreads into fingering ripples that can display a complex dynamics ranging from continuous waves, nonlinear spatially localized periodic wave patterns (i.e. rivulet structures) to modulated nonlinear wavetrain structures. Some of these structures have been observed experimentally, however conditions under which they form are still not well understood. In this work we examine profiles of nonlinear wave patterns formed by a thin liquid film falling on a uniformly heated horizonal plate. In this purpose, the Benney model is considered assuming a uniform temperature distribution along the film propagation on the horizontal surface. It is shown that for strong surface tension but relatively small Biot number, spatially localized periodic-wave structures can be analytically obtained by solving the governing equation under appropriate conditions. In the regime of weak nonlinearity, a multiple-scale expansion combined with the reductive perturbation method leads to a complex Ginzburg-Landau equation, the solutions of which are modulated periodic pulse trains which amplitude, width and period are expressed in terms of characteristic parameters of the model.
In this Letter we regard nonlinear gravity-capillary waves with parameter of nonlinearity being $varepsilon sim 0.1 div 0.25$. For this nonlinearity time scale separation does not occur and kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in emph{Kartashova, emph{EPL} textbf{97} (2012), 30004.} We compute for the first time an analytical expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function depending on the ratio of surface tension to the gravity acceleration. It is shown that its two limits - pure capillary and pure gravity waves on a fluid surface - coincide with the previously obtained results. We also discuss relations of the model of D-cascade with a few known models used in the theory of nonlinear waves such as Zakharovs equation, resonance of the modes with nonlinear Stokes corrected frequencies and Benjamin-Feir index. These connections are crucial in the understanding and forecasting specifics of the energy transport in a variety of multi-component wave dynamics, from oceanography to optics, from plasma physics to acoustics.
In this work we consider a new class of oscillatory instabilities that pertain to thermocapillary destabilization of a liquid film heated by a solid substrate. We assume the substrate thickness and substrate-film thermal conductivity ratio are large so that the effect of substrate thermal diffusion is retained at leading order in the long-wave approximation. As a result, system dynamics are described by a nonlinear partial differential equation for the film thickness that is nonlocally coupled to the full substrate heat equation. Perturbing about a steady quiescent state, we find that its stability is described by a non-self adjoint eigenvalue problem. We show that, under appropriate model parameters, the linearized eigenvalue problem admits complex eigenvalues that physically correspond to oscillatory (in time) instabilities of the thin film height. As the principal results of our work, we provide a complete picture of the susceptibility to oscillatory instabilities for different model parameters. Using this description, we conclude that oscillatory instabilities are more relevant experimentally for films heated by insulating substrates. Furthermore, we show that oscillatory instability where the fastest-growing (most unstable) wavenumber is complex, arises only for systems with sufficiently large substrate thicknesses.
A capillary jet falling under the effect of gravity continuously stretches while thinning downstream. We report here the effect of external periodic forcing on such a spatially varying jet in the jetting regime. Surprisingly, the optimal forcing frequency producing the most unstable jet is found to be highly dependent on the forcing amplitude. Taking benefit of the one-dimensional Eggers & Dupont (J. Fluid Mech., vol. 262, 1994, 205-221) equations, we investigate the case through nonlinear simulations and linear stability analysis. In the local framework the WKBJ formalism, established for weakly non-parallel flows, fails to capture the nonlinear simulation results quantitatively. However in the global framework, the resolvent analysis supplemented by a simple approximation of the required response norm inducing breakup, is shown to correctly predict the optimal forcing frequency at a given forcing amplitude and the resulting jet breakup length. The results of the resolvent analysis are found to be in good agreement with those of the nonlinear simulations.
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