scholarly journals On contact-line dynamics with mass transfer

2015 ◽  
Vol 26 (5) ◽  
pp. 671-719 ◽  
Author(s):  
J. M. OLIVER ◽  
J. P. WHITELEY ◽  
M. A. SAXTON ◽  
D. VELLA ◽  
V. S. ZUBKOV ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Surface Tension ◽  
Asymptotic Analysis ◽  
Contact Line ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Leading Order ◽  
Vapour Transport ◽  
Gravity Surface ◽  
Matched Asymptotic Analysis

We investigate the effect of mass transfer on the evolution of a thin, two-dimensional, partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, other effects, such as gravity, surface tension gradients, vapour transport and heat transport, are neglected in favour of mathematical tractability. Our focus is on a matched-asymptotic analysis in the small-slip limit, which reveals that the leading-order outer formulation and contact-line law depend delicately on both the sign and the size of the mass transfer flux. This leads, in particular, to novel generalisations of Tanner's law. We analyse the resulting evolution of the drop on the timescale of mass transfer and validate the leading-order predictions by comparison with preliminary numerical simulations. Finally, we outline the generalisation of the leading-order formulations to prescribed non-uniform rates of mass transfer and to three dimensions.

On thin evaporating drops: When is the -law valid?

2016 ◽  
Vol 792 ◽  
pp. 134-167 ◽  
Author(s):  
M. A. Saxton ◽  
J. P. Whiteley ◽  
D. Vella ◽  
J. M. Oliver
Keyword(s):  
Mass Transfer ◽  
Asymptotic Analysis ◽  
Numerical Simulations ◽  
Viscous Dissipation ◽  
Square Root ◽  
Vapour Transport ◽  
Asymptotic Results ◽  
Rate Of Evaporation ◽  
And Diffusion ◽  
Matched Asymptotic Analysis

We study the evolution of a thin, axisymmetric, partially wetting drop as it evaporates. The effects of viscous dissipation, capillarity, slip and diffusion-dominated vapour transport are taken into account. A matched asymptotic analysis in the limit of small slip is used to derive a generalization of Tanner’s law that takes account of the effect of mass transfer. We find a criterion for when the contact-set radius close to extinction evolves as the square root of the time remaining until extinction – the famous $d^{2}$-law. However, for a sufficiently large rate of evaporation, our analysis predicts that a (slightly different) ‘$d^{13/7}$-law’ is more appropriate. Our asymptotic results are validated by comparison with numerical simulations.

Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions

2009 ◽  
Vol 465 (2109) ◽  
pp. 2725-2749 ◽  
Author(s):  
M. J. Ablowitz ◽  
T. S. Haut
Keyword(s):  
Surface Tension ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Asymptotic Series ◽  
High Order ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Small Surface ◽  
Petviashvili Equation ◽  
Dimensional Series

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

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.

Second-order Wagner theory for two-dimensional water-entry problems at small deadrise angles

2007 ◽  
Vol 572 ◽  
pp. 59-85 ◽  
Author(s):  
J. M. OLIVER
Keyword(s):  
Experimental Data ◽  
Asymptotic Analysis ◽  
Incompressible Liquid ◽  
Water Entry ◽  
Second Order ◽  
Two Dimensional ◽  
Predictive Capability ◽  
Wagner’S Theory ◽  
Wagner Theory ◽  
Matched Asymptotic Analysis

The theory of Wagner from 1932 for the normal symmetric impact of a two-dimensional body of small deadrise angle on a half-space of ideal and incompressible liquid is extended to derive the second-order corrections for the locations of the higher-pressure jet-root regions and for the upward force on the impactor using a systematic matched-asymptotic analysis. The second-order predictions for the upward force on an entering wedge and parabola are compared with numerical and experimental data, respectively, and it is concluded that a significant improvement in the predictive capability of Wagner's theory is afforded by proceeding to second order.

Nonlinear electroviscoelastic potential flow instability theory of two superposed streaming dielectric fluids

2014 ◽  
Vol 92 (10) ◽  
pp. 1249-1257 ◽  
Author(s):  
M.F. El-Sayed ◽  
N.T. Eldabe ◽  
M.H. Haroun ◽  
D.M. Mostafa
Keyword(s):  
Surface Tension ◽  
Electric Field ◽  
Potential Flow ◽  
Flow Instability ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Physical Parameters ◽  
Two Dimensional ◽  
Dielectric Fluids ◽  
Fluid Velocities

The nonlinear electrohydrodynamic Kelvin–Helmholtz instability of two superposed viscoelastic Walters B′ dielectric fluids in the presence of a tangential electric field is investigated in three dimensions using the potential flow analysis. The method of multiple scales is used to obtain a dispersion relation for the linear problem, and a nonlinear Ginzburg–Landau equation with complex coefficients for the nonlinear problem. The linear and nonlinear stability conditions are obtained and discussed both analytically and numerically. In the linear stability analysis, we found that the fluid velocities and kinematic viscosities have destabilizing effects, and the electric field, kinematic viscoelasticities, and surface tension have stabilizing effects; and that the system in the three-dimensional disturbances is more stable than in the corresponding case of two-dimensional disturbances. While in the nonlinear analysis, for both two- and three-dimensional disturbances, we found that the fluid velocities, surface tension, and kinematic viscosities have destabilizing effects, and the electric field, kinematic viscoelasticities have stabilizing effects, and that the system in the three-dimensional disturbances is more unstable than its behavior in the two-dimensional disturbances for most physical parameters except the kinematic viscosities.

Two-dimensional phoretic swimmers: the singular weak-advection limits

2017 ◽  
Vol 816 ◽  
Author(s):  
Ehud Yariv
Keyword(s):  
Three Dimensions ◽  
Order Kinetic ◽  
Two Dimensional ◽  
Leading Order ◽  
Logarithmic Divergence ◽  
First Order ◽  
Order Kinetic Model ◽  
Solute Absorption ◽  
And Diffusion ◽  
Asymptotic Matching

Because of the associated far-field logarithmic divergence, the transport problem governing two-dimensional phoretic self-propulsion lacks a steady solution when the Péclet number $\mathit{Pe}$ vanishes. This indeterminacy, which has no counterpart in three dimensions, is remedied by introducing a non-zero value of $\mathit{Pe}$, however small. We consider that problem employing a first-order kinetic model of solute absorption, where the ratio of the characteristic magnitudes of reaction and diffusion is quantified by the Damköhler number $\mathit{Da}$. As $\mathit{Pe}\rightarrow 0$ the dominance of diffusion breaks down at distances that scale inversely with $\mathit{Pe}$; at these distances, the leading-order transport represents a two-dimensional point source in a uniform stream. Asymptotic matching between the latter region and the diffusion-dominated near-particle region provides the leading-order particle velocity as an implicit function of $\log \mathit{Pe}$. Another scenario involving weak advection takes place under strong reactions, where $\mathit{Pe}$ and $\mathit{Da}$ are large and comparable. In that limit, the breakdown of diffusive dominance occurs at distances that scale as $\mathit{Da}^{2}/\mathit{Pe}$.

A formula for ‘wave damping’ in the drift of a floating body

1994 ◽  
Vol 275 ◽  
pp. 147-155 ◽  
Author(s):  
J. A. P. Aranha
Keyword(s):  
Potential Flow ◽  
Simple Formula ◽  
Wave Amplitude ◽  
Three Dimensions ◽  
Floating Body ◽  
Two Dimensional ◽  
Leading Order ◽  
Dimensional Problem ◽  
Wave Damping ◽  
Wave Celerity

A simple formula for ‘wave damping’ is derived, exact within the context of the proposed theory, namely: potential flow correct to second order in the wave amplitude and to leading order in U/c, where U is the drift velocity and c the wave celerity. The analysis is restricted to a two-dimensional problem although the extension to three dimensions seems possible.

Ring formation on an inclined surface

2015 ◽  
Vol 775 ◽  
Author(s):  
Xiyu Du ◽  
R. D. Deegan
Keyword(s):  
Asymptotic Analysis ◽  
Contact Line ◽  
Narrow Band ◽  
Two Dimensional ◽  
A Posteriori ◽  
Drop Volume ◽  
Line Region ◽  
Initial Drop ◽  
Inclined Surface ◽  
Lower Contact

A drop dried on a solid surface will typically leave a narrow band of solute deposited along the contact line. Here we examine variations of this deposit due to the inclination of the substrate using numerical simulations of a two-dimensional drop, equivalent to a strip-like drop. An asymptotic analysis of the contact line region predicts that the upslope deposit will grow faster at early times, but the growth of this deposit ends sooner because the upper contact line depins first. From our simulations we find that the deposit can be larger at either the upper or lower contact line depending on the initial drop volume and substrate inclination. For larger drops and steeper inclinations, the early lead in deposited mass at the upper contact line is wiped out by the earlier depinning of the upper contact line and subsequent continued growth at the lower contact line. Conversely, for smaller drops and shallower inclinations, the early lead of the upper contact line is insurmountable despite its earlier termination in growth. Our results show that it is difficult to reconstruct a posteriori the inclination of the substrate based solely on the shape of the deposit.

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