Temporal stability of free liquid threads with surface viscoelasticity

We analyse the effect of surface viscoelasticity on the temporal stability of a free cylindrical liquid jet coated with insoluble surfactant, extending the results of Timmermans & Lister (J. Fluid Mech., vol. 459, 2002, pp. 289-306). Our development requires, in particular, deriving the correct expressions for the normal and tangential stress boundary conditions at a general axisymmetric interface when surface viscosity is modelled with the Boussinesq-Scriven constitutive equation. These stress conditions are applied to obtain a new dispersion relation for the liquid thread, which is solved to describe its temporal stability as a function of four governing parameters, namely the capillary Reynolds number, the elasticity parameter, and the shear and dilatational Boussinesq numbers. It is shown that both surface viscosities have a stabilising influence for all values of the capillary Reynolds number and elasticity parameter, the effect being more pronounced at low capillary Reynolds numbers. The wavenumber of maximum amplification depends non-monotonically on the Boussinesq numbers, especially for very viscous threads at low values of the elasticity parameter. Finally, two different lubrication approximations of the equations of motion are derived. While the validity of the leading-order model is limited to small enough values of the elasticity parameter and of the Boussinesq numbers, a higher-order parabolic model is able to accurately capture the linearised behaviour for the whole range of values of the four control parameters.The authors thank the Spanish MINECO, Subdireccion General de Gestion de Ayudas a la Investigacion, for its support through projects DPI2014-59292-C3-1-P, DPI2015-71901-REDT and DPI2017-88201-C3-3-R. These research projects have been partly financed through European funds. A.M.-C. also acknowledges support from the Spanish MECD through the grant FPU16/02562

Non-linear dynamics and self-similarity in the rupture of ultra-thin viscoelastic liquid coatings

The influence of viscoelasticity on the dewetting of ultrathin polymer films is unraveled by means of theory and numerical simulations in the inertialess limit. Three viscoelastic models are employed to analyse the dynamics of the film, namely the Oldroyd-B, Giesekus, and FENE-P models. We revisit the linear stability analysis first derived by [Tomar et al., Eur. Phys. J. E., 2006, 20, 185–200] for a Jeffrey's film to conclude that all three models formally share the same dispersion relation. For times close to the rupture singularity, the self-similar regime recently discovered [Moreno-Boza et al., Phys. Rev. Fluids, 2020, 5, 014002], where the dimensionless minimum film thickness scales with the dimensionless time until rupture as hmin = 0.665τ1/3, is asymptotically established independently of the rheological model. The spatial structure of the flow is characterised by a Newtonian core and a thin viscoelastic boundary layer at the free surface, where polymeric stresses become singular as τ → 0. The Deborah number and the solvent-to-total viscosity ratio affect the rupture time but not the length scale of the resulting dewetting pattern and asymptotic flow structure close to rupture, which is thus shown to be universal. Our three-dimensional simulations lead us to conclude that bulk viscoelasticity alone does not explain the experimental observations of dewetting of polymeric films near the glass transition

The role of inertia in the rupture of ultrathin liquid films

Theory and numerical simulations of the Navier–Stokes equations are used to unravel the influence of inertia on the dewetting dynamics of an ultrathin film of Newtonian liquid deposited on a solid substrate. A classification of the self-similar film thinning regimes at finite Ohnesorge numbers is provided, unifying previous findings. We reveal that, for Ohnesorge numbers smaller than one, the structure of the rupture singularity close to the molecular scales is controlled by a balance between liquid inertia and van der Waals forces, leading to a self-similar asymptotic regime with hmin ∝ τ2/5 as τ → 0, where hmin is the minimum film thickness and τ is the time remaining before rupture. The flow exhibits a three-region structure comprising an irrotational core delimited by a pair of boundary layers at the wall and at the free surface. A potential-flow description of the irrotational core is provided, which is matched with the vortical layers, allowing us to present a complete parameter-free asymptotic description of inertia-dominated film rupture.This research was funded by the Spanish MINECO, Subdirección General de Gestión de Ayudas a la Investigación, through Project No. RED2018-102829-T and by the Spanish MCIU-Agencia Estatal de Investigación through Project No. DPI2017-88201-C3-3-R, partly financed through FEDER European funds. A.M.-C. also acknowledges support from the Spanish MECD through the Grant No. FPU16/02562.Publicad

The effect of wall slip on the dewetting of ultrathin films on solid substrates: linear instability and second-order lubrication theory

The influence of wall slip on the instability of a non-wetting liquid film placed on a solid substrate is analyzed in the limit of negligible inertia. In particular, we focus on the stability properties of the film, comparing the performance of the three lubrication models available in the literature, namely, the weak, intermediate, and strong slip models, with the Stokes equations. Since none of the aforementioned leading-order lubrication models is shown to be able to predict the growth rate of perturbations for the whole range of slipping lengths, we develop a parabolic model able to accurately predict the linear dynamics of the film for arbitrary slip lengths.This research was funded by the Spanish MINECO, Subdirección General de Gestión de Ayudas a la Investigación, through Project No. RED2018-102829-T and by the Spanish MCIU-Agencia Estatal de Investigación through Project No. DPI2017-88201-C3-3-R, partly financed through FEDER European funds. A.M.-C. also acknowledges support from the Spanish MECD through Grant No. FPU16/02562.Publicad

Dripping dynamics and transitions at high Bond numbers

We report experiments on the dripping dynamics and jetting transitions that take place when a liquid is injected vertically downwards at a constant flow rate, for wide ranges of the liquid viscosity and injector radius. We explore values of the Bond number significantly larger than in previous works, revealing the existence of period-2 dripping regimes with satellite formation that do not exist at small Bond numbers. In addition, we quantify the influence of liquid viscosity on the hysteresis associated with the dripping-jetting transition, that had previously been studied only for the particular case of water.The authors thank the financial support of the Spanish MINECO through projects nos. DPI2014-59292-C03-01-P, DPI2014-59292-C03-03-P, DPI2015-71901-REDT, DPI2017-88201-C3-2-R and DPI2017-88201-C3-3-R. These research projects have been partly financed through European funds.Publicad

Natural break-up and satellite formation regimes of surfactant-laden liquid threads

We report a numerical analysis of the unforced break-up of free cylindrical threads of viscous Newtonian liquid whose interface is coated with insoluble surfactants, focusing on the formation of satellite droplets. The initial conditions are harmonic disturbances of the cylindrical shape with a small amplitude , and whose wavelength is the most unstable one deduced from linear stability theory. We demonstrate that, in the limit e → 0, the problem depends on two dimensionless parameters, namely the Laplace number, La = ρσ0R¯ /µ2, and the elasticity parameter, β = E/σ0, where ρ, µ and σ0 are the liquid density, viscosity and initial surface tension, respectively, E is the Gibbs elasticity and R¯ is the unperturbed thread radius. A parametric study is presented to quantify the influence of La and β on two key quantities: the satellite droplet volume and the mass of surfactant trapped at the satellite’s surface just prior to pinch-off, Vsat and Σsat, respectively. We identify a weak-elasticity regime, β . 0.05, in which the satellite volume and the associated mass of surfactant obey the scaling law Vsat = Σsat = 0.0042La1.64 for La . 2. For La & 10, Vsat and Σsat reach a plateau of about 3 % and 2.9 %, respectively, Vsat being in close agreement with previous experiments of low-viscosity threads with clean interfaces. For La < 7.5, we reveal the existence of a discontinuous transition in Vsat and Σsat at a critical elasticity, βc(La), with βc →0.98 for La . 0.2, such that Vsat and Σsat abruptly increase at β = βc for increasing β. The jumps experienced by both quantities reach a plateau when La . 0.2, while they decrease monotonically as La increases up to La = 7.5, where both become zero.A.M.-C. and A.S. thank the Spanish MINECO, Subdirección General de Gestión de Ayudas a la Investigación, for its support through project DPI2015-71901-REDT, and the Spanish MCIU-Agencia Estatal de Investigación through project DPI2017-88201-C3-3-R. These research projects have been partly financed through FEDER European funds. A.M.-C. also acknowledges support from the Spanish MECD through grant FPU16/02562, and its associated programme Ayudas a la Movilidad 2017 during his stay at TIPs–ULB, Brussels. J.R.-R. and B.S. thank the FRS-FNRS for financial support, in particular under the umbrella of the Wolflow project. The authors wish to express their deep gratitude to one anonymous reviewer for making insightful comments, which led to a significant improvement of the present work

Modeling of the bubbling process in a planar co-flow configuration

This work presents an analytical model developed to describe the bubbling regime resulting from the injection of an air sheet of thickness 2H(o) with a mean velocity u(a) between two water streams of thickness H-w - H-o, moving at a uniform velocity u(w). Based on previous experimental and numerical characterizations of this flow, the gas stream is modeled as a two-dimensional sheet divided into three different parts in the streamwise direction: a neck that moves downstream at the water velocity, a gas ligament attached to the injector upstream of the neck, and a forming bubble downstream of the neck, whose uniform dimensionless half-thicknesses are eta(n)(tau), eta(l)(tau), eta(b)(tau) respectively, and the corresponding pressures are given by Pi(n)(tau), Pi(l)(tau), and Pi(b)(tau) Pi(n)(tau). Lengths are made dimensionless with H-o, and pressures with rho(a)u(a)(2) where rho(a) is the air density.This work has been supported by the Spanish MINECO (Subdi-rección General de Gestión de Ayudas a la Investigación), Junta de Andalucía, and European Funds under projects numbers DPI2014-59292-C3-1-P and DPI2014-59292-C3-3-P, and P11-TEP7495. Financial support from the University of Jaén, project UJA2013/08/05, is also acknowledged

Universal Free-Fall Law for Liquid Jets under Fully Developed Injection Conditions

We show that verticals lenderjets of liquid injected in air with a fully developed outlet velocity profile have a universal shape in the common case in which the viscous force is much smaller than the gravitational force. The theory of ideal flows with vorticity provides an analytical solution that, under negligible surface tension forces, predicts RjðZÞ¼½ð1þZ=4Þ1=2−ðZ=4Þ1=21=2, where Rj is the jet radius scaled with the injector radius and Z is the vertical distance scaled with the gravitational length, lg¼u2 o=2g, where uo is the mean velocity at the injector outlet and g is the gravitational acceleration. In contrast with Mariotte’s law, Rj¼ð1þZÞ−1=4, previously reported experiments employing long injectors collapse almost perfectly on to the new solution.The authors thank the Spanish MCIU-Agencia Estatal de Investigación through Project No. PID2020–115655GB-C22, partially financed through FEDER European funds

The necking time of gas bubbles in liquids of arbitrary viscosity

We report an experimental and theoretical study of the collapse time of a gas bubble injected into an otherwise stagnant liquid under quasi-static conditions and for a wide range of liquid viscosities.This work has been supported by the Spanish MINECO (Subdirección General de Gestión de Ayudas a la Investigación), Junta de Andalucía, and European Funds under Project Nos. DPI2014-59292-C3-1-P, DPI2014-59292-C3-3-P, and P11-TEP7495. Financial support from the University of Jaén, Project No. UJA2013/08/05, is also acknowledged

Start-up flow in shallow deformable microchannels

Microfluidic systems are usually fabricated with soft materials that deform due to the fluid stresses. Recent experimental and theoretical studies on the steady flow in shallow deformable microchannels have shown that the flow rate is a nonlinear function of the pressure drop due to the deformation of the upper soft wall. Here, we extend the steady theory of Christov et al. (J. Fluid Mech., vol. 841, 2018, pp. 267–286) by considering the start-up flow from rest, both in pressure-controlled and in flow-rate-controlled configurations. The characteristic scales and relevant parameters governing the transient flow are first identified, followed by the development of an unsteady lubrication theory assuming that the inertia of the fluid is negligible, and that the upper wall can be modelled as an elastic plate under pure bending satisfying the Kirchhoff–Love equation. The model is governed by two non-geometrical dimensionless numbers: a compliance parameter β, which compares the characteristic displacement of the upper wall with the undeformed channel height, and a parameter γ that compares the inertia of the solid with its flexural rigidity. In the limit of negligible solid inertia, γ → 0, a quasi-steady model is developed, whereby the fluid pressure satisfies a nonlinear diffusion equation, with β as the only parameter, which admits a self-similar solution under pressure-controlled conditions. This simplified lubrication description is validated with coupled three-dimensional numerical simulations of the Navier equations for the elastic solid and the Navier–Stokes equations for the fluid. The agreement is very good when the hypotheses behind the model are satisfied. Unexpectedly, we find fair agreement even in cases where the solid and liquid inertia cannot be neglected.The authors are grateful to J. Rivero-Rodríguez and B. Scheid for key numerical advice, to I. C. Christov for pointing out a mistake in figure 2 of an earlier version of the manuscript, and to R. Zaera for helpful discussions. A.M.-C. and A.S. thank the Spanish MINECO, Subdirección General de Gestión de Ayudas a la Investigación, for its support through projects DPI2014-59292-C3-1-P and DPI2015-71901-REDT, and the Spanish MCIU-Agencia Estatal de Investigación through project DPI2017-88201- C3-3-R. These research projects have been partly financed through FEDER European funds. A.M.-C. also acknowledges support from the Spanish MECD through the grant FPU16/02562 and to its associated programme Ayudas a la Movilidad 2018 during his stay at the Complex Fluids Group in Princeton. H.A.S. thanks the NSF for support via CMMI-1661672 and through Princeton University’s Material Research Science and Engineering Center DMR-1420541.Publicad