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

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.

Figure 1

Experimental setup based on previous work [19].

Figure 2

Visible and real average contact radii. The dashed line shows a circle with the same area as the one of droplet's base.

Figure 3

Liquid-gas interface.

Figure 4

Normalized evaporation rate $\beta$ from computer simulations, theory [Eqs. (11) and (12)] and experiment in microgravity (the one with $T_s-T_{\infty }=15.1$ K); $0g$ and $1g$ stand for zero and terrestrial gravity, respectively.

Figure 5

Normalized evaporation rate $\beta$ for different substrate temperatures.

Figure 6

Fitting local vapor flux $j$ at the liquid-gas interface from our 2D computer simulations (case of $T_s-T_{\infty }=15.1$ K) with approximation $j_{\mathrm{approx}}$ given by Eq. (20). Same parameters $B=0.45K^{-1}$ and $\mathrm{\Delta }h=0.019$ mm are used for three presented time moments.

Figure 7

Computer simulations $(T_s-T_{\infty }=15.1$ K, $t=14$ s): convective pattern in the gas phase and corresponding contours of vapor density at zero (a) and terrestrial (b) gravity.

Figure 8

Unsteady computer simulation: development of thermoconvective instabilities within the drop at $T_s-T_{\infty }=15.1$ K. Color represents temperature field, white streamlines and black arrows represent velocity in the drop (multimedia view).

Figure 9

Steady-state computer simulation: instantaneous distribution of fluxes in the course of droplet evaporation into open atmosphere. Color represents temperature field. Black, white, and blue arrows represent velocity of liquid and interface and evaporation flux, respectively.