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.

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.

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.

Interaction of Levitating Microdroplets with Moist Air Flow in the Contact Line Region

2017 ◽  
Vol 21 (2) ◽  
pp. 60-69 ◽  
Author(s):  
Oleg A. Kabov ◽  
Dmitry V. Zaitsev ◽  
Dmitry P. Kirichenko ◽  
Vladimir S. Ajaev
Keyword(s):  
Contact Line ◽  
Air Flow ◽  
Moist Air ◽  
Line Region

Droplet spreading and absorption on rough, permeable substrates

2015 ◽  
Vol 784 ◽  
pp. 465-486 ◽  
Author(s):  
Leonardo Espín ◽  
Satish Kumar
Keyword(s):  
Surface Roughness ◽  
Contact Line ◽  
Nonlinear Evolution ◽  
Precursor Film ◽  
Radial Coordinate ◽  
Droplet Spreading ◽  
Liquid Spreading ◽  
Line Region ◽  
Influence Of Substrate ◽  
Contact Line Pinning

Wetting of permeable substrates by liquids is an important phenomenon in many natural and industrial processes. Substrate heterogeneities may significantly alter liquid spreading and interface shapes, which in turn may alter liquid imbibition. A new lubrication-theory-based model for droplet spreading on permeable substrates that incorporates surface roughness is developed in this work. The substrate is assumed to be saturated with liquid, and the contact-line region is described by including a precursor film and disjoining pressure. A novel boundary condition for liquid imbibition is applied that eliminates the need for a droplet-thickness-dependent substrate permeability that has been employed in previous models. A nonlinear evolution equation describing droplet height as a function of time and the radial coordinate is derived and then numerically solved to characterize the influence of substrate permeability and roughness on axisymmetric droplet spreading. Because it incorporates surface roughness, the new model is able to describe the contact-line pinning that has been observed in experiments but not captured by previous models.

An obstacle problem for elliptic membrane shells

2018 ◽  
Vol 24 (5) ◽  
pp. 1503-1529 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare ◽  
Paolo Piersanti
Keyword(s):  
Vector Field ◽  
Asymptotic Analysis ◽  
Obstacle Problem ◽  
Displacement Vector ◽  
Middle Surface ◽  
Two Dimensional ◽  
Displacement Vector Field ◽  
Membrane Shells ◽  
Signorini Condition ◽  
Confinement Condition

Our objective is to identify two-dimensional equations that model an obstacle problem for a linearly elastic elliptic membrane shell subjected to a confinement condition expressing that all the points of the admissible deformed configurations remain in a given half-space. To this end, we embed the shell into a family of linearly elastic elliptic membrane shells, all sharing the same middle surface [Formula: see text], where [Formula: see text] is a domain in [Formula: see text] and [Formula: see text] is a smooth enough immersion, all subjected to this confinement condition, and whose thickness [Formula: see text] is considered as a “small” parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as [Formula: see text] approaches zero, the corresponding “limit” two-dimensional variational problem. This problem takes the form of a set of variational inequalities posed over a convex subset of the space [Formula: see text]. The confinement condition considered here considerably departs from the Signorini condition usually considered in the existing literature, where only the “lower face” of the shell is required to remain above the “horizontal” plane. Such a confinement condition renders the asymptotic analysis substantially more difficult, however, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field.

scholarly journals divand-1.0: <i>n</i>-dimensional variational data analysis for ocean observations

2014 ◽  
Vol 7 (1) ◽  
pp. 225-241 ◽  
Author(s):  
A. Barth ◽  
J.-M. Beckers ◽  
C. Troupin ◽  
A. Alvera-Azcárate ◽  
L. Vandenbulcke
Keyword(s):  
Cost Function ◽  
Dimensional Analysis ◽  
Computational Cost ◽  
Posteriori Error ◽  
Cholesky Factorization ◽  
Ocean Current ◽  
Posteriori Error Estimate ◽  
Two Dimensional ◽  
A Posteriori ◽  
A Posteriori Error

Abstract. A tool for multidimensional variational analysis (divand) is presented. It allows the interpolation and analysis of observations on curvilinear orthogonal grids in an arbitrary high dimensional space by minimizing a cost function. This cost function penalizes the deviation from the observations, the deviation from a first guess and abruptly varying fields based on a given correlation length (potentially varying in space and time). Additional constraints can be added to this cost function such as an advection constraint which forces the analysed field to align with the ocean current. The method decouples naturally disconnected areas based on topography and topology. This is useful in oceanography where disconnected water masses often have different physical properties. Individual elements of the a priori and a posteriori error covariance matrix can also be computed, in particular expected error variances of the analysis. A multidimensional approach (as opposed to stacking two-dimensional analysis) has the benefit of providing a smooth analysis in all dimensions, although the computational cost is increased. Primal (problem solved in the grid space) and dual formulations (problem solved in the observational space) are implemented using either direct solvers (based on Cholesky factorization) or iterative solvers (conjugate gradient method). In most applications the primal formulation with the direct solver is the fastest, especially if an a posteriori error estimate is needed. However, for correlated observation errors the dual formulation with an iterative solver is more efficient. The method is tested by using pseudo-observations from a global model. The distribution of the observations is based on the position of the Argo floats. The benefit of the three-dimensional analysis (longitude, latitude and time) compared to two-dimensional analysis (longitude and latitude) and the role of the advection constraint are highlighted. The tool divand is free software, and is distributed under the terms of the General Public Licence (GPL) (http://modb.oce.ulg.ac.be/mediawiki/index.php/divand).

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