scholarly journals Asymptotic analysis of the evaporation dynamics of partially wetting droplets

2017 ◽  
Vol 824 ◽  
pp. 574-623 ◽  
Author(s):  
Nikos Savva ◽  
Alexey Rednikov ◽  
Pierre Colinet
Keyword(s):  
Asymptotic Analysis ◽  
Evolution Equations ◽  
Slip Length ◽  
Vapour Phase ◽  
Droplet Radius ◽  
Type Model ◽  
Line Region ◽  
Volatile Liquid ◽  
Kinetic Resistance ◽  
Evaporation Dynamics

We consider the dynamics of an axisymmetric, partially wetting droplet of a one-component volatile liquid. The droplet is supported on a smooth superheated substrate and evaporates into a pure vapour atmosphere. In this process, we take the liquid properties to be constant and assume that the vapour phase has poor thermal conductivity and small dynamic viscosity so that we may decouple its dynamics from the dynamics of the liquid phase. This leads to a so-called ‘one-sided’ lubrication-type model for the evolution of the droplet thickness, which accounts for the effects of evaporation, capillarity, gravity, slip and kinetic resistance to evaporation. By asymptotically matching the flow near the contact line region and the bulk of the droplet in the limit of a small slip length and commensurably small evaporation and kinetic resistance effects, we obtain coupled evolution equations for the droplet radius and volume. The predictions of our asymptotic analysis, which also include an estimate of the evaporation time, are found to be in excellent agreement with numerical simulations of the governing lubrication model for a broad range of parameter regimes.

scholarly journals Validation and modification of asymptotic analysis of slow and rapid droplet spreading by numerical simulation

2013 ◽  
Vol 715 ◽  
pp. 283-313 ◽  
Author(s):  
Yi Sui ◽  
Peter D. M. Spelt
Keyword(s):  
Numerical Simulation ◽  
Asymptotic Analysis ◽  
Contact Line ◽  
Slip Length ◽  
Interface Shape ◽  
Droplet Spreading ◽  
Moving Contact Line ◽  
Moving Contact ◽  
Line Region ◽  
Modified Theory

AbstractUsing a slip-length-based level-set approach with adaptive mesh refinement, we have simulated axisymmetric droplet spreading for a dimensionless slip length down to $O(1{0}^{\ensuremath{-} 4} )$. The main purpose is to validate, and where necessary improve, the asymptotic analysis of Cox (J. Fluid Mech., vol. 357, 1998, pp. 249–278) for rapid droplet spreading/dewetting, in terms of the detailed interface shape in various regions close to the moving contact line and the relation between the apparent angle and the capillary number based on the instantaneous contact-line speed, $\mathit{Ca}$. Before presenting results for inertial spreading, simulation results are compared in detail with the theory of Hocking & Rivers (J. Fluid Mech., vol. 121, 1982, pp. 425–442) for slow spreading, showing that these agree very well (and in detail) for such small slip-length values, although limitations in the theoretically predicted interface shape are identified; a simple extension of the theory to viscous exterior fluids is also proposed and shown to yield similar excellent agreement. For rapid droplet spreading, it is found that, in principle, the theory of Cox can predict accurately the interface shapes in the intermediate viscous sublayer, although the inviscid sublayer can only be well presented when capillary-type waves are outside the contact-line region. However, $O(1)$ parameters taken to be unity by Cox must be specified and terms be corrected to ${\mathit{Ca}}^{+ 1} $ in order to achieve good agreement between the theory and the simulation, both of which are undertaken here. We also find that the apparent angle from numerical simulation, obtained by extrapolating the interface shape from the macro region to the contact line, agrees reasonably well with the modified theory of Cox. A simplified version of the inertial theory is proposed in the limit of negligible viscosity of the external fluid. Building on these results, weinvestigate the flow structure near the contact line, the shear stress and pressure along the wall, and the use of the analysis for droplet impact and rapid dewetting. Finally, we compare the modified theory of Cox with a recent experiment for rapid droplet spreading, the results of which suggest a spreading-velocity-dependent dynamic contact angle in the experiments. The paper is closed with a discussion of the outlook regarding the potential of using the present results in large-scale simulations wherein the contact-line region is not resolved down to the slip length, especially for inertial spreading.

Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: the small-solid-fraction singularity

2017 ◽  
Vol 820 ◽  
pp. 580-603 ◽  
Author(s):  
Ory Schnitzer
Keyword(s):  
Shear Flow ◽  
Asymptotic Analysis ◽  
Solid Fraction ◽  
Asymptotic Expansions ◽  
Slip Length ◽  
Longitudinal Shear ◽  
Unidirectional Flow ◽  
Effective Slip Length ◽  
Effective Slip ◽  
Scaling Arguments

We study the effective slip length for unidirectional flow over a superhydrophobic mattress of bubbles in the small-solid-fraction limit $\unicode[STIX]{x1D716}\ll 1$. Using scaling arguments and utilising an ideal-flow analogy we elucidate the singularity of the slip length as $\unicode[STIX]{x1D716}\rightarrow 0$: relative to the periodicity it scales as $\log (1/\unicode[STIX]{x1D716})$ for protrusion angles $0\leqslant \unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}/2$ and as $\unicode[STIX]{x1D716}^{-1/2}$ for $0<\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D6FC}=O(\unicode[STIX]{x1D716}^{1/2})$. We continue with a detailed asymptotic analysis using the method of matched asymptotic expansions, where ‘inner’ solutions valid close to the solid segments are matched with ‘outer’ solutions valid on the scale of the periodicity, where the bubbles protruding from the solid grooves appear to touch. The analysis yields asymptotic expansions for the effective slip length in each of the protrusion-angle regimes. These expansions overlap for intermediate protrusion angles, which allows us to form a uniformly valid approximation for arbitrary protrusion angles $0\leqslant \unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x03C0}/2$. We thereby explicitly describe the transition with increasing protrusion angle from a logarithmic to an algebraic small-solid-fraction slip-length singularity.

Energy dissipation and the contact-line region of a spreading bridge

2012 ◽  
Vol 703 ◽  
pp. 111-141 ◽  
Author(s):  
H. B. van Lengerich ◽  
P. H. Steen
Keyword(s):  
Energy Dissipation ◽  
Flow Field ◽  
Contact Line ◽  
Sliding Friction ◽  
Slip Length ◽  
Cox Model ◽  
Hydrophobic Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Line Region

AbstractA drop on a circular support spontaneously spreads upon contact with a substrate. The motion is driven by a loss of surface energy. The loss of recoverable energy can be expressed alternatively as work done at the liquid–gas interface or dissipation through viscosity and sliding friction. In this paper we require consistency with the energy lost by dissipation in order to infer details of the contact-line region through simulations. Simulations with the boundary integral method are used to compute the flow field of a corresponding experiment where polydimethylsiloxane spreads on a relatively hydrophobic surface. The flow field is used to calculate the energy dissipation, from which slip lengths for local slip and Navier slip boundary conditions are found. Velocities, shear rates and pressures along the interface as well as interface shapes in the microscopic region of the contact line are also reported. Angles, slip length and viscous bending length scale allow a test of the Voinov–Hocking–Cox model without free parameters.

An Accurate and Simple Approach to Allow for the Transient Temperature Gradient in a Vaporizing Droplet

2012 ◽  
Author(s):  
Alexander Snegirev ◽  
Victor Talalov
Keyword(s):  
Temperature Gradient ◽  
Life Time ◽  
Droplet Evaporation ◽  
Transient Temperature ◽  
Internal Temperature ◽  
Ambient Temperatures ◽  
Wide Range ◽  
Order Polynomial ◽  
Volatile Liquid ◽  
Evaporation Dynamics

The purpose of this work is to analyze the importance of considering internal temperature gradient in modeling droplet evaporation, and to demonstrate performance of simplified methods in which the temperature gradient is approximately taken into account. Based on three characteristic time scales, two dimensionless criteria have been identified which determine magnitude of the internal temperature gradient and its effect on the evaporation dynamics. Numerical values of these criteria in a wide range of ambient temperatures show that the effect of the internal temperature gradient is more pronounced in more volatile liquid at higher ambient temperatures. Although droplet life time predictions are not sensitive to the internal temperature gradient, its effect might be strong at the initial stages of droplet evaporation, and this substantiates the need in robust and computationally inexpensive methods to take it into account. Two simple and yet accurate approaches (quasi-steady higher order polynomial approximation and the integral balance method) have been favourably tested and recommended for use in CFD spray modeling.

Singularity formation in three-dimensional motion of a vortex sheet

1995 ◽  
Vol 300 ◽  
pp. 339-366 ◽  
Author(s):  
Takashi Ishihara ◽  
Yukio Kaneda
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Integration ◽  
Finite Time ◽  
Fourier Coefficients ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Three Dimensions ◽  
Leading Order ◽  
Singularity Formation

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

scholarly journals DECOUPLED AND UNIDIRECTIONAL ASYMPTOTIC MODELS FOR THE PROPAGATION OF INTERNAL WAVES

2013 ◽  
Vol 24 (01) ◽  
pp. 1-65 ◽  
Author(s):  
VINCENT DUCHÊNE
Keyword(s):  
Initial Data ◽  
Internal Waves ◽  
Evolution Equations ◽  
Approximate Solutions ◽  
Type Model ◽  
Fluid System ◽  
Asymptotic Models ◽  
Counterpropagating Waves ◽  
Scalar Equations ◽  
Two Fluid

We study the relevance of various scalar equations, such as inviscid Burgers', Korteweg–de Vries (KdV), extended KdV, and higher order equations, as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified in two ways. The first method consists in approximating the flow by two uncoupled, counterpropagating waves, each one satisfying such an equation. One also recovers these equations when focusing on a given direction of propagation, and seeking unidirectional approximate solutions. This second justification is more restrictive as for the admissible initial data, but yields greater accuracy. Additionally, we present several new coupled asymptotic models: a Green–Naghdi type model, its simplified version in the so-called Camassa–Holm regime, and a weakly decoupled model. All of the models are rigorously justified in the sense of consistency.

scholarly journals General dynamical sharp-interface conditions for phase transformations in viscous heat-conducting fluids

2007 ◽  
Vol 581 ◽  
pp. 323-370 ◽  
Author(s):  
DANIEL M. ANDERSON ◽  
PAOLO CERMELLI ◽  
ELIOT FRIED ◽  
MORTON E. GURTIN ◽  
GEOFFREY B. MCFADDEN
Keyword(s):  
Phase Transformations ◽  
Evolution Equations ◽  
Vapour Phase ◽  
Sharp Interface ◽  
Thermal Dissipation ◽  
Configurational Forces ◽  
Heat Conducting ◽  
Thermodynamic Structure ◽  
Viscous Heat ◽  
Heat Conducting Fluids

We develop a complete set of equations governing the evolution of a sharp interface separating two fluid phases undergoing transformation. In addition to the conventional balances for mass, linear momentum and energy these equations include also a counterpart of the Gibbs–Thomson equation familiar from theories for crystal growth. This additional equation arises from a consideration of configurational forces within a thermodynamical framework. Although the notion of configurational forces is well-developed and understood for the description of materials, such as crystalline solids, that possess natural reference configurations, little has been done regarding their role in materials, such as viscous fluids, that do not possess preferred reference states. We therefore provide a comprehensive discussion of configurational forces, the balance of configurational momentum, and configurational thermodynamics that does not require a choice of reference configuration. The general evolution equations arising from our theory account for the thermodynamic structure of the bulk phases and the interface and for viscous and thermal dissipation in the bulk phases and for viscous dissipation on the interface. Because of the complexity of these equations, we provide a reduced system of equations based on simplified constitutive assumptions and approximations common in the literature on phase transformations. Using these reduced equations, we apply the theory to the radially symmetric problem for the condensation of a liquid drop into the vapour phase.

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