Figure 1 shows the configuration of an axisymmetric molten drop solidifying on a cold plate with the presence of three interfaces meeting at a triple point. At this point, the growth angle

φ_{gr} is specified as

where

φ_{s} and

φ_{l} are the solid and liquid angles with

s and

l denoting solid and liquid, respectively [

19,

24,

25]. Initially, the drop is assumed to be a section of a sphere, which is specified by a contact angle

φ_{0} and the wetting radius

R_{w} at the plate. The plate is kept at constant temperature

T_{c} below the fusion value of the drop liquid

T_{m}. As a consequence, a thin solid layer forms on the plate at the start. As the solidification proceeds, the solidifying interface moves upwards with a normal velocity

V_{n} given as

where

ρ and

L_{h} is the density and latent heat, respectively.

${\dot{q}}_{f}$, the heat flux at the solidification interface, is given by

where

k is the thermal conductivity. It, as an interfacial heat source, is also introduced to the energy equation that is solved in the entire domain

where

δ(

**x** −

**x**_{f}) is the Dirac’s Delta function with

f denoting interface.

C_{p} is the heat capacity. Another interfacial source known as the interfacial tension force acting on the liquid–gas interface (the last term in the following equation), is accounted for in the momentum equation

here,

**u** = (

u,

v) is the velocity vector,

p is the pressure,

**g** is the acceleration due to gravity. The superscript

T denotes the transpose.

σ is the interfacial tension that is linearly varied with the temperature [

19], i.e.,

$\sigma ={\sigma}_{0}-{\beta}_{\sigma}(T-{T}_{m})$ (

σ_{0} and

β_{δ} are the surface tension coefficient at a reference temperature and the Marangoni tension coefficient).

κ is twice the mean curvature, and

**n**_{f} is the unit normal vector to the interface.

**f** is the forcing term used to impose the no-slip condition on the solid–fluid interface [

24,

25,

26]. The problem is closed by the following continuity equation with proper boundary conditions shown in

Figure 1,

We choose the effective radius of the drop

R =

${\left[3{V}_{0}/\left(4\pi \right)\right]}^{1/3}$ as a length scale and

${\tau}_{c}={\rho}_{l}{C}_{pl}^{}{R}_{}^{2}/{k}_{l}$ as a time scale (

V_{0} is the volume of the initial liquid drop). The velocity scale is

U_{c} =

R/

τ_{c}. The problem is governed by the Prandtl number

Pr, Stefan number

St, Bond number

Bo, Weber number

We, Marangoni number

Ma, dimensionless initial temperature of the liquid

θ_{0}, density ratios

ρ_{sl} and

ρ_{gl}, viscosity ratio

μ_{gl}, thermal conductivity ratios

k_{sl} and

k_{gl}, heat capacity ratios

C_{psl} and

C_{pgl}The dimensionless time and temperature are

τ =

t/

τ_{c} and

$\theta =\left(T-{T}_{c}\right)/\left({T}_{m}-{T}_{c}\right)$, respectively. The domain size is chosen as

W ×

H = 3

R × 3

R with a grid resolution of 482 × 482. In this paper, we are interested in the effects of the contact angle

φ_{0} for two solid-to-liquid density ratios and two growth angles, and thus other parameters are kept constant, i.e.,

St = 0.1,

Pr = 0.01,

Bo = 0.1,

Ma = 10,

We = 0.1,

ρ_{gl} =

μ_{gl} = 0.05,

k_{sl} =

C_{psl} =

C_{pgl} = 1.0,

k_{gl} = 0.005, and

θ_{0} = 1. As demonstrated in our previous works [

18,

19],

Ma in order of 10 has a minor effect on the solidification process, and thus the Marangoni effect with

θ_{0} = 1 can be negligible. These parameters correspond to a liquid drop of metals or semiconductor materials, such as silicon or germanium (i.e.,

Pr ≅ 0.01), with

R of a few millimeters [

10,

11].

Method validations have been carefully carried out in our previous works [

18,

19,

27]. Such some validations are shown in

Figure 2.

Figure 2a compares the predicted profiles with the experimental ones of a water drop reported by Anderson et al. [

5]. Details of this comparison can be found in Vu et al. [

19].

Figure 2b shows the results of the silicon drop crystallization, reproduced by the method, in comparison with the solidified drop reported in [

10] (for more details of this comparison, see our recent work [

27]). It is observed that the numerical results are in good agreement with the experimental data, indicating that the method can accurately predict the drop shape after complete solidification.