Abstract: In this paper, the solidification process of a liquid drop attached to a vertical cold wall is investigated by direct numerical simulations. The numerical method used is a front-tracking method that represents the interface separating different phases by connected elements. The non-slip velocity boundary condition at the solid surface is satisfied by an interpolaton technique. Because of the vertical wall temperature lower than the fusion value of the drop liquid, a solid layer formed at the wall evolves to the top of the drop when the solidification proceeds. The numerical results show that because of gravity, the drop is deformed, resulting in an asymmetric drop with its tip shifting downwards after complete solidification. The solidification process is analyzed (through the drop shape, tip shift and solidification time) under the influence of some parameters such as the Bond number, the growth angle or the shape of the initial liquid drop.
Solid–liquid phase change of drops is of crucial importance because the process widely exists in nature and engineering phenomena. For instance, water drops freeze on leaves, airplane, or wind turbine blades. Metal drops solidify in deposition manufacturing or in atomization. Molten semiconductor drops crystallized during falling or on cold substrates have been used for solar cell applications. Accordingly, understanding the complex heat transfer and solidification phenomena is extremely important to advance the above-mentioned technologies.
Concerning liquid drops solidifying on a horizontal plate, many works have been conducted. Experimentally, one can find this problem investigation in Anderson et al. , in which the authors froze a water drop to demonstrate the accuracy of the proposed dynamic contact angle model at the tri-junction. An interesting feature observed from the experiment is the formation of an apex at the drop top after complete solidification. After then, a few experiments focusing on this singularity formed on frozen water drops have been done, e.g. [2–4]. Recently, this problem has been rapidly growing and getting much of attention [5, 6]. Focusing on not only water but also some other semiconductor materials (e.g. silicon, germanium and indium antimonide), Satunkin  reported the formation of conical solidified drops induced by volume expansion and growth angles at the tri-junction. Basing on theoretical analysis supported by the experiments, Satunkin found that the growth angle at the tri-junction is almost constant except near the end of the solidification process. Another work concerning the crystallization of a molten silicon drop can be found in Itoh et al. . Theoretical studies on these problems can be found in [9,10], in which the authors mostly paid attention to the solidified drop profile and the temporal evolution of the solidification front assumed to be flat. Concerning numerical simulations of drops solidifying on a plate, a few studies have been performed by a boundary integral method , a finite element method , an enthalpy-based method , a volume of fluid method , and a front-tracking method in our recent works [15–18]. However, in these works, the author considered only drops attached to a horizontal wall.
Concerning drops on a vertical wall, Podgorski et al.  report controlled experiments to show various shapes with remarkable temporal patterns of a water drop on a vertical plane. Smolka and SeGall  presented the formation of a fingering pattern, which tends to break up into drops, on the surface outside of a vertical cylinder. Li and Chen  experimentally formed frozen drops on a vertical wall and used the ultrasonic vibration to remove them from the wall. Numerically, Schwartz et al.  used a long-wave or lubrication approximation-based model to simulate three-dimensional unsteady motion of a drop on a vertical wall. Tilehboni et al.  used a lattice Boltzmann method to simulate a liquid drop moving on a vertical wall. However, the above-mentioned works have not considered solidification heat transfer.
Even though there have been many numerical studies on the phase change heat transfer of liquid drops, but mostly focused on the horizontal plate. The drop solidification on a vertical wall is rarely found. Accordingly, filling this gap is the main purpose of the present study because of its importance in academic and engineered applications [24–26]. We here use a two-dimensional front-tracking method, for tracking interfaces, combined with an interpolation technique, for dealing with the non-slip boundary on the solid surface, [15,27] to investigate the deformation with the solidification of a liquid drop attached a vertical cold wall. Various parameters are investigated to reveal their effects on the solidification process.
We have presented a fully resolved two-dimensional numerical investigation of a liquid drop solidifying on a vertical cold wall by the front-tracking method to track the temporal movement of the interfaces. The non-slip velocity boundary condition on the solid surface is satisfied by a linear interpolation technique. The gravitational force results in an asymmetric drop whose conical tip induced by volume expansion is shifted down after complete solidification. We varied the Bond number Bo (in the range of 0.1 – 3.16), the growth angle fgr (in the range of 0–200) and the contact angle f0 at the wall (in the range of 45–1350) to show their influences on the shape, height and tip shift of the solidified drop and the solidification time. The numerical results show that the solidified drop shape is strongly influenced by Bo, fgr and f0. The location of the solidified drop tip shifts more to the bottom as we increase any one of Bo and f0. For instance, the tip shift et varies with respect to Bo by et/d ≈ 0.1723Bo–0.0149. The height Hs of the solidified drop strongly increases with an increase in fgr (Hs/d ≈ 0.3556fgr + 0.8188 with fgr measured in radian) and f0 (Hs/d ≈ 0.3385f0 + 0.1855). Concerning time ts for completing the solidification process, the most affecting parameters are Bo (it increases with Bo by ts ≈ 0.0676Bo2–0.0445Bo+2.4156), fgr (it increases with fgr, in radian, by ts ≈ 1.0524fgr + 2.4428) and f0 (it increases with f0, in radian, by ts ≈ 0.4679 –1.3536 +2.6273f0–0.7035).
Even though our study is very detailed, many questions are still unresolved. For instance, the present calculations are merely two-dimensional, and indeed for a ridge rather than a drop, and thus three-dimensional simulations would circumvent the present limitation. The drop might slide on the wall before starting solidification and thus the simulation requires the dynamic contact angle and nucleation models. In some cases, the drop might pinch off before completing solidification, and thus further investigations need to be done to reveal the range in which the drop pinch-off happens.