Boundary conditions for a one-sided numerical model of evaporative instabilities in sessile drops of ethanol on heated substrates

Abstract

The work is focused on obtaining boundary conditions for a one-sided numerical model of thermoconvective instabilities in evaporating pinned sessile droplets of ethanol on heated substrates. In the one-sided model, appropriate boundary conditions for heat and mass transfer equations are required at the droplet surface. Such boundary conditions are obtained in the present work based on a derived semiempirical theoretical formula for the total droplet's evaporation rate, and on a two-parametric nonisothermal approximation of the local evaporation flux. The main purpose of these boundary conditions is to be applied in future three-dimensional (3D) one-sided numerical models in order to save a lot of computational time and resources by solving equations only in the droplet domain. Two parameters, needed for the nonisothermal approximation of the local evaporation flux, are obtained by fitting computational results of a 2D two-sided numerical model. Such model is validated here against parabolic flight experiments and the theoretical value of the total evaporation rate. This study combines theoretical, experimental, and computational approaches in convective evaporation of sessile droplets. The influence of the gravity level on evaporation rate and contributions of different mechanisms of vapor transport (diffusion, Stefan flow, natural convection) are shown. The qualitative difference (in terms of developing thermoconvective instabilities) between steady-state and unsteady numerical approaches is demonstrated.

Modeling Unsteady Bénard-Marangoni Instabilities in Drying Volatile Droplets on a Heated Substrate

“…Semenov et. al derive in [26] and employ in [25] analytical equations for the unsteady diffusionlimited evaporation from sessile droplets, which agree with the numerical results of [28]. However, these unsteady effects are small on time scales larger than the time required for the Brownian particle to pass the characteristic length R.…”

“…The authors have observed Marangoni instability in an ethanol droplet of capillary size on a heated and highly conductive substrate. They also performed numerical simulations by one-sided model described in [26], where they took into account, in particular, an unsteady diffusion-limited evaporation for non-isothermal pinned sessile droplet and the Stefan flow in the gas. After solving the heat transfer equation in the droplet bulk and carrying out the numerical simulation, the results obtained have agreed well with the experiment for the contact angle θ ≈ 29.3 • .…”

Joint effect of advection, diffusion, and capillary attraction on the spatial structure of particle depositions from evaporating droplets

“…In order to further refine the suggested model, one has to consider J, which depends on time t, as well. For example, dependencies J(r, t) are suggested in [63][64][65][66].…”

Droplet Evaporation on Hot Micro-Structured Superhydrophobic Surfaces: Analysis of Evaporation from Droplet Cap and Base Surfaces

“…( 11) with the three boundary conditions Eqs. ( 10) - (12) , we obtain the temperature distribution inside the water layer as:…”