Framework Of Lubrication Theory Research Articles

Dynamics of the interaction of a pair of thin evaporating droplets on compliant substrates

The dynamics of the interaction of a system of two thin volatile liquid droplets resting on a soft viscoelastic solid substrate are investigated theoretically. The developed model fully considers the effect of evaporative cooling and the generated Marangoni stresses due to the induced thermal gradients, while also accounting for the effect of the gas phase composition and the diffusion of vapour in the atmosphere of the droplets. Using the framework of lubrication theory, we derive evolution equations for both the droplet profile and the displacement of the elastic solid, which are solved in combination with Laplace's equation for the vapour concentration in the gas phase. A disjoining-pressure/precursor-film approach is used to describe contact-line motion. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis to investigate the physical properties and mechanisms that affect the dynamics of droplet interactions. The results show that the droplets interact through both the soft substrate and the gas phase. In the absence of thermocapillary phenomena, the combined effect of non-uniform evaporation due to the increased vapour concentration between the two droplets and elastocapillary phenomena determines whether the drop–drop interaction is attractive or repulsive. The Marangoni stresses suppress droplet attraction at the early stages of the drying process and lead to longer droplet lifetimes. For substrates with intermediate stiffness, the emergence of spontaneous symmetry breaking at late stages of evaporation is found. The rich dynamics of this complex system is explored by constructing a detailed map of the dynamic regimes.

Numerical simulation of hydraulic fracturing: A hybrid FEM‐based algorithm

AbstractIn this paper, a problem of numerical simulation of hydraulic fractures is considered. An efficient algorithm of solution is proposed for the plain strain model of hydraulic fracturing. The algorithm utilizes a FEM‐based subroutine to compute deformation of the fractured material. The flow of generalized Newtonian fluid in the fracture is modeled in the framework of lubrication theory. In this way, the architecture of the computational scheme is relatively simple and enables one to deal with advanced cases of the fractured material properties and configurations as well as various rheological models of fluid. In particular, the problems of poroelasticity, plasticity, and spatially varying properties of the fractured material can be analyzed. The accuracy and efficiency of the proposed algorithm are verified against analytical benchmark solutions. The algorithm capabilities are demonstrated using the example of the hydraulic fracture propagating in complex geological settings.

Wetting transition of the confined receding meniscus with tailing bead formation

Wetting transition of the confined receding meniscus is determined by two independent mechanisms, related to the apparent receding contact angle and the inclined contact lines, respectively. The coexistence of the two mechanisms impacts the geometry of the meniscus. An experiment is designed to observe the shape of a confined droplet when it is horizontally stretched by a plate. It is found that wetting transition is delayed with a tailing bead formed in the rear of the confined receding meniscus, and occurs at the critical point determined by the microscopic physical characteristics in the inner region of the contact line. Through experimental and numerical analysis, the confined droplet is found to be characterized similarly to the classical models of the liquid film on a plate vertically dragged out of a liquid reservoir and the droplet sliding on an inclined plane. The formation of the tailing bead is owing to the balance of viscosity and capillarity, and its shape can be described in the framework of lubrication theory. However, the tailing bead also shows some unique features. The formation of tailing bead marks that the dominant features of the meniscus convert from two-dimensional ones to three-dimensional ones; besides, it is irrelevant to gravity. The tailing bead formation and the selection of critical point reflect that the physical principles of the inner contact line region have a stronger influence on wetting transition than those of the outer contact line region. • The tailing bead of the confined receding meniscus is observed and investigated. • Wetting transition is delayed by the tailing bead and inclined contact lines. • Meniscus shapes are respectively modeled for the low and high capillary number. • The dominant meniscus characteristics turn from two-dimensional into three-dimensional. • Microscopic physical characteristics near the contact line strongly impact wetting transition.

Lubricating motion of a sphere towards a thin porous slab with Saffman slip condition

Near-contact hydrodynamic interactions between a solid sphere and a plane porous slab are investigated in the framework of lubrication theory. The size of pores in the slab is small compared with the slab thickness so that the Darcy law holds there. The slab is thin: that is, its thickness is small compared with the sphere radius. The considered problem involves a sphere translating above the slab together with a permeation flow across the slab and a uniform pressure below. The pressure is continuous across both slab interfaces and the Saffman slip condition applies on its upper interface. An extended Reynolds-like equation is derived for the pressure in the gap between the sphere and the slab. This equation is solved numerically and the drag force on the sphere is calculated therefrom for a wide range of values of the slab interface slip length and of the permeability parameter $\unicode[STIX]{x1D6FD}=24k^{\ast }R/(e\unicode[STIX]{x1D6FF}^{2})$, where $k^{\ast }$ is the permeability, $e$ is the porous slab thickness, $R$ is the sphere radius and $\unicode[STIX]{x1D6FF}$ is the gap. Moreover, asymptotics expansions for the pressure and drag are derived for high and low $\unicode[STIX]{x1D6FD}$. These expansions, which agree with the numerics, are also handy formulae for practical use. All results match with those of other authors in particular cases. The settling trajectory of a sphere towards a porous slab in a fluid at rest is calculated from these results and, as expected, the time for reaching the slab decays for increasing slab permeability and upper interface slip length.

Dynamics and stability of a power-law film flowing down a slippery slope

A power-law fluid flowing down a slippery inclined plane under the action of gravity is deliberated in this research work. A Newtonian layer at a small strain rate is introduced to take care of the divergence of the viscosity at a zero strain rate. A low-dimensional two-equation model is formulated using a weighted-residual approach in terms of two coupled evolution equations for the film thickness h and a local velocity amplitude or the flow rate q within the framework of lubrication theory. Moreover, a long-wave instability is shown in detail. Linear stability analysis of the proposed two-equation model reveals good agreement with the spatial Orr-Sommerfeld analysis. The influence of a wall-slip on the primary instability has been found to be non-trivial. It has the stabilizing effect at larger values of the Reynolds number, whereas at the onset of the instability, the role is destabilizing which may be because of the increase in dynamic wave speed by the wall slip. Competing impressions of shear-thinning/shear-thickening and wall slip velocity on the primary instability are captured. The impact of slip velocity on the traveling-wave solutions is discussed using the bifurcation diagram. An increasing value of the slip shows a significant effect on the traveling wave and free surface amplitude. Slip velocity controls both the kinematic and dynamic waves of the system, and thus, it has the profound passive impact on the instability.

Dewetting of the residual layer of annealed nanoimprinted polystyrene films

The presence of a residual layer at the end of a classical NanoImprint Lithography (NIL) process is at the root of many issues when NIL is used to manufacture resist patterns used later as an etching mask. In this paper, we investigate how the spontaneous break-up of annealed polystyrene films, a phenomenon called dewetting, can produce residual-layer-free polymer patterns. Although the dewetting of flat polymer films has been characterized for decades, little interest has been focused on patterned films. We present exploratory experiments conducted on polystyrene 30kg/mol ultra-thin films. The samples are embossed using thermal NIL with a specific spatially modulated space/line patterned mold, and then annealed at 50°C above the glass transition temperature. The resulting patterns are then interpreted as a competition between surface tension and van der Waals forces. Finally, a quantitative analysis is made in the framework of lubrication theory, with the help of a finite volume simulation tool.

Dewetting films with inclined contact lines.

A partially wetting plate withdrawn from a liquid reservoir causes the deposition of a liquid film that is characterized by inclined contact lines. It has been experimentally indicated that the normal component of the contact-line velocity relative to the plate remains constant and is independent of the inclination angles, a fact that has never theoretically been justified. We demonstrate, in the framework of lubrication theory, that the speed-angle independence is only approximate and the normal velocity actually exhibits a weak decrease with the inclination angle of the contact line. This correlation is attributed to the variation of the effective separation of microscopic and macroscopic length scales. In addition, the inclination of the contact line results in a tangential flux of the liquid, which is confined in the vicinity of the contact line. Simple scaling relations are provided for both the normal velocity and the tangential flux.

Viscoelastic effects and anomalous transient levelling exponents in thin films

We study theoretically the profile evolution of a thin viscoelastic film supported onto a no-slip flat substrate. Due to the nonconstant initial curvature at the free surface, there is a flow driven by Laplace pressure and mediated by viscoelasticity. In the framework of lubrication theory, we derive a thin-film equation that contains local viscoelastic stress through the Maxwell model. Then, considering a sufficiently regular small perturbation of the free surface, we linearise the equation and derive its general solution. We analyse and discuss in details the behaviour of this function. We then use it to study the viscoelastic evolution of a Gaussian initial perturbation through its transient levelling exponent. For initial widths of the profile that are smaller than a characteristic length scale involving both the film thickness and the elastocapillary length, this exponent is shown to reach anomalously high values at the elastic-to-viscous transition. This prediction should in particular be observed at sufficiently short times in experiments on thin polymer films.

Wavy regime of a power-law film flow

AbstractWe consider a power-law fluid flowing down an inclined plane under the action of gravity. The divergence of the viscosity at zero strain rate is taken care of by introducing a Newtonian plateau at small strain rate. Two-equation models are formulated within the framework of lubrication theory in terms of the exact mass balance and an averaged momentum equation, which form a set of evolution equations for the film thickness$h$, a local velocity amplitude or the flow rate$q$. The models account for the streamwise diffusion of momentum. Comparisons with Orr–Sommerfeld stability analysis and with direct numerical simulation (DNS) show convincing agreement in both linear and nonlinear regimes. The influence of shear-thinning or shear-thickening on the primary instability is shown to be non-trivial. A destabilization of the base flow close to threshold is promoted by the shear-thinning effect, whereas, further from threshold, it tends to stabilize the base flow when the viscous damping of short waves becomes dominant. A reverse situation is observed in the case of shear-thickening fluids. Shear-thinning accelerates solitary waves and promotes a subcritical onset of travelling waves at larger wavenumber than the linear cut-off wavenumber. A conditional stability of the base flow is thus observed. This phenomenon results from a reduction of the effective viscosity at the free surface. When compared with DNS, simulations of the temporal response of the film based on weighted residual models satisfactorily capture the conditional stability of the film.

Relaxation of a dewetting contact line. Part 1. A full-scale hydrodynamic calculation

The relaxation of a dewetting contact line is investigated theoretically in the so-called ‘Landau–Levich’ geometry in which a vertical solid plate is withdrawn from a bath of partially wetting liquid. The study is performed in the framework of lubrication theory, in which the hydrodynamics is resolved at all length scales (from molecular to macroscopic). We investigate the bifurcation diagram for unperturbed contact lines, which turns out to be more complex than expected from simplified ‘quasi-static’ theories based upon an apparent contact angle. Linear stability analysis reveals that below the critical capillary number of entrainment, Cac, the contact line is linearly stable at all wavenumbers. Away from the critical point, the dispersion relation has an asymptotic behaviour σ∝|q| and compares well to a quasi-static approach. Approaching Cac, however, a different mechanism takes over and the dispersion evolves from ∼|q| to the more common ∼q2. These findings imply that contact lines cannot be described using a universal relation between speed and apparent contact angle, but viscous effects have to be treated explicitly.

High-velocity penetration of a melting solid by a slender body

The high-velocity penetration of a melting solid by a thermally insulated slender body is considered. Under certain constraints on the dimensionless melting parameters the flow in the molten layer can be described within the framework of lubrication theory. The local angle of inclination of the body and the surfaces of the molten layer with respect to the velocity is assumed to be small and is taken into account in the linear approximation. The heat flow into the solid is found by simulating the body and the molten layer by means of a segment with distributed heat sources. Within the framework of this simple formulation a closed solution of the problem of the fusion zone around a moving slender body is constructed. The dependence of the shape of the molten layer and the structure of the temperature and longitudinal velocity fields in the layer on the shape of the body and the other governing parameters of the problem is investigated. The results obtained also give a solution of the problem of the melting of a solid rubbing at high velocity against a thermally insulated rough substrate, when the characteristic height of the roughness is of the order of the thickness of the layer and the characteristic length of the order of the contact length.