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

Abstract

Using 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(10 −4 ). The main purpose is to validate -and where necessary improve -the asymptotic analysis of Cox (1998) 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, Ca. Before presenting results for inertial spreading, simulation results are compared in detail with the theory of Hocking & Rivers (1982) for slow spreading, showing these to 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 (1998) 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 in Cox (1998) must be specified and terms be corrected to 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 (1998). A simplified version of the inertial theory is proposed in the limit of negligible viscosity of the external fluid. Building on these results, we investigate 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 (1998) 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.

Dynamic wetting failure in surfactant solutions

“…The parameter λ cox = l mic /l M (assumed small) is the ratio between an arbitrary macroscopic length scale l M (10 µm-1 mm) and a microscopic length scale l mic (∼1 nm) (Hocking 1976;Blake 2006;Sui, Ding & Spelt 2014). Although λ cox has a similar meaning to the dimensionless slip length λ in the Navier-slip boundary condition (2.10), the values for these two parameters may not be the same in general (Kistler 1993;Sui & Spelt 2013). In addition, θ ap need not equal θ M .…”

Inertial coalescence of droplets on a partially wetting substrate

“…10 We present here results obtained with two independent methods that allow us to assess the sensitivity of the simulation results to the contact-line model. One of these approaches is based on a diffuse-interface (DI) method, 11,12 with newly incorporated adaptive mesh refinement function; 13,14 in the following, we shall indicate by "n levels of refinement" that the finest grid spacing is 1/2 n−1 times the coarsest grid spacing. The DI method naturally regulates the stress singularity at the contact line by considering the fluid interface as a diffused layer.…”

Non-isothermal droplet spreading/dewetting and its reversal

“…For DNS beyond the lubrication limit, the finite-volume-based level-set method of Sui & Spelt (2013a) has been extended. The continuous surface tension formulation of the momentum equations is, as under isothermal conditions (e.g.…”

“…In order to simulate flows at more or less realistic values of a dimensionless slip length,λ/R = O(10 −4 ), we have incorporated an adaptive mesh refinement tool. Details and extensive validation test results for isothermal conditions can be found in Sui & Spelt (2013a). For non-isothermal conditions we have validated the code by studying thermocapillary migration of a deformable droplet in an infinite quiescent flow with a constant temperature gradient.…”

“…For this purpose we extend our computational method (Sui & Spelt 2013a) to non-isothermal spreading/dewetting problems. The main objective there is to establish the role of hot and cold surfaces in promoting or retarding spreading for large contact angles.…”