Dynamics of complex capillary flows: stability, rupture, and influence of surfactants

Mención Internacional en el título de doctorFluid-fluid interfaces are ubiquitous in nature and everyday life, where they are found across scales and material properties, as for instance in many engineering, biological, and physiological applications and processes. In particular, cylindrical interfaces and, in general, the spontaneous tendency of surface-tension-driven flows to break up into drops, have fascinated naturalists and scientists throughout history; a fascination that lasts to date due to its crucial relevance in many phenomena of fundamental and applied interest. This is the reason for the huge research effort devoted to understand the behavior and dynamics of these filaments, namely elongated vesicles and membranes which are frequent in biological environments, or liquid jets that are routinely used for additive manufacturing applications. In most of these scenarios, the interface is usually populated with surface-active molecules, macromolecules, proteins, or contaminated with particles, which eventually form a complex microstructure that endows the interface with a rheologically complex behavior. The interaction between this structure and the hydrodynamic forces is traduced macroscopically into nonlinear interfacial rheological properties and nontrivial constitutive equations relating the surface stress with the deformation of the surface. An interface that possesses these kinds of properties is usually referred to as a complex interface, and the particular field of study is typically denoted by interfacial rheology. Nonetheless, despite of this complexity, these material cylinders share the same intrinsic instability induced by the interfacial tension known as the Plateau-Rayleigh instability, where disturbances of sufficiently long wavelength trigger the instability by decreasing the surface energy at constant volume. The complex interactions between the bulk fluids and the surface layer complicate the theoretical modelling and the experimental protocols and measurements of the material properties associated with the interface. A vast number of issues regarding the behavior and dynamics of such complex fluid threads are yet not understood. In particular, this thesis aims to unravel fundamental aspects of the linear and nonlinear dynamics of liquid filaments whose interface is endowed with complex surface rheology, which can be elastic and/or viscous. We first deduce the components of the surface stress balance modified by interfacial elastic and viscous forces, which is necessary for the derivation of leading-order and second-order one-dimensional models. The performance of these approximations is then evaluated in the linear regime by comparing their associated growth rate of small perturbations with the one obtained from the complete conservation equations. To this end, we use Rayleigh’s temporal linear stability analysis to deduce the corresponding dispersion relation of a liquid filament with interfacial rheology. Additionally, by performing simulations of the full conservation equations, we then investigate the nonlinear dynamics of these complex filaments. In particular we study the effect of Marangoni and surface viscous stresses on the natural breakup and thinning of threads, and the subsequent formation of satellite droplets. Finally, we study the linear and nonlinear dynamics of a capillary jet injected in the direction of gravity and confined between the nozzle and a bath of the same fluid.This doctoral dissertation was supported by Ministerio de Educación, Cultura, y Deporte through the fellowship FPU16/02562, and its associated program Ayudas a la Movilidad 2017 and 2018 during my stays with the group of Prof. Benoit Scheid (TIPs) at the University of Brussels, and with the group of Prof. Howard A. Stone at Princeton University. This work also had financial support by Ministerio de Economía y Competitividad, Subdirección General de Gestión de Ayudas a la Investigación, under the projects DPI2014-59292-C3-1-P, DPI2014-59292-C3-3-P, DPI2015-71901-REDT, and by Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación through the project DPI2017-88201-C3-3-R, partly financed through FEDER European funds. Support from the Red Nacional para el Desarrollo de la Microfluídica, RED2018-102829-T, is also acknowledged.Programa de Doctorado en Mecánica de Fluidos por la Universidad Carlos III de Madrid, la Universidad de Jaén, la Universidad de Zaragoza, la Universidad Nacional de Educación a Distancia, la Universidad Politécnica de Madrid y la Universidad Rovira i VirgiliPresidente: Jens Eggers,.- Secretario: Nicolas Bremond.- Vocal: José M. Gordillo Arias de Saavedr

Universal Thinning of Liquid Filaments under Dominant Surface Forces

Theory and numerical simulations of the thinning of liquid threads at high superficial concentration of surfactants suggest the existence of an asymptotic regime where surface tension balances surface viscous stresses, leading to an exponential thinning with an $e$-fold time $F(\Theta)\,(3\mu_s + \kappa_s)/\sigma$, where $\mu_s$ and $\kappa_s$ are the surface shear and dilatational viscosity coefficients, $\sigma$ is the interfacial tension, $\Theta=\kappa_s/\mu_s$, and $F(\Theta)$ is a universal function. The potential use of this phenomenon to measure the surface viscosity coefficients is discussed.Comment: 6 pages, 3 figures. Published in Physical Review Letter

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

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

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

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

Fingering instability in spreading epithelial monolayers: roles of cell polarisation, substrate friction and contractile stresses

Collective cell migration plays a crucial role in many developmental processes that underlie morphogenesis, wound healing, or cancer progression. In such coordinated behaviours, cells are organised in coherent structures and actively migrate to serve different biological purposes. In some contexts, namely during epithelial wound healing, it is well known that a migrating free-edge monolayer develops finger-like instabilities, yet the onset is still under debate. Here, by means of theory and numerical simulations, we shed light on the main mechanisms driving the instability process, analysing the linear and nonlinear dynamics of a continuum compressible polar fluid. In particular, we assess the role of cell polarisation, substrate friction, and contractile stresses. Linear theory shows that it is crucial to analyse the perturbation transient dynamics, since we unravel a plethora of crossovers between different exponential growth rates during the linear regime. Numerical simulations suggest that cell-substrate friction could be the mechanism responsible for the formation of complex finger-like structures at the edge, since it triggers secondary fingering instabilities and tip-splitting phenomena. Finally, we obtain a critical contractile stress that depends on cell-substrate friction and the initial-to-nematic length ratio, characterising an active wetting-dewetting transition. In the dewetting scenario, the monolayer retracts and becomes stable without developing finger-like structures.Comment: 15 pages, 9 figures, 1 table, submitted to Soft Matte

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

A Digitally Programmable Analog Quadrature Sine Oscillator for on-chip Lock-In Measurement Systems

This paper presents a CMOS 1.8V-180nm analog quadrature sine oscillator. Thanks to a custom 12-bit bidirectional DAC-based architecture, the frequency can be digitally programmed over two decades with high accuracy, making it suitable as the actuation system in low-cost high-performance embedded lock-in measurement systems