2.1.1 Mechanism of foam formation
Foam is a kind of gas-liquid dispersion medium, in which gas is the dispersed phase (discontinuous phase) and liquid is the dispersion medium (continuous phase). The foam is formed by the accumulation of bubbles on many sides. As shown in Fig. 1, the intersection of each bubble in the foam is defined as the Plateau boundary (Wei et al. 2020). Suppose the radius of the bubble is R, the pressure inside the bubble is pA, the pressure outside the bubble is pP, and the surface tension of the vacuolar membrane is σ. As the volume of bubbles increases dV, the surface area increases dA. Without considering other external forces, according to the first law of thermodynamics, there are (Jiang 1990):
$$\left( {{p_P} - {p_A}} \right){\text{d}}V+\sigma {\text{d}}A=0$$
1
If the shape of the foam is spherical, there are:
$$\frac{{{\text{d}}A}}{{{\text{~d}}V}}=\frac{{8\pi R{\text{d}}R}}{{4\pi {R^2}{\text{d}}R}}=\frac{2}{R}$$
2
According to formula (1) and formula (2), there are:

It is well known that bubbles are convex, and there will be
. It can be seen from formula (3) that the pressure at point P in the liquid film is less than point A, so the liquid film automatically flows from point A to point P, and the liquid film gradually becomes thinner, which is the process of foam drainage. When the liquid film becomes thinner to a certain extent, it will lead to the rupture of the film and the destruction of the foam. Pure liquid cannot form stable foam, and some surface-active agents (foam stabilizers) are often added to the foam formulation, which can reduce the surface tension σ of the liquid film to relieve the time of foam drainage and make the generated foam more durable.
2.1.2 Mathematical model of foam coalescence
Foams are different in size at the beginning. More importantly, at this time, the foam is in a state of high surface free energy and extremely unstable, and there will be coalescence between the foam and the foam. The process of coalescence of large foam into small foam will reduce the surface-free energy of foam, thus forming a more stable foam. For the convenience of research, two adjacent foams are taken as research objects, and a foam coalescence model is established, as shown in Fig. 2. And make the following assumptions about the model (Ren 2009, Ren 2013):
①The two foam contact surfaces are parallel flat plates, and gravity is not considered.
②The liquid forming the liquid film is axisymmetric when flowing;
③Ignore the other movements of the liquid forming the liquid film, only consider the radial movement, and the interface flows completely;
④The flow field distribution of the fluid in the membrane is flat;
⑤The fluid is a Newtonian fluid, which is incompressible and its viscosity value is certain.
⑥The liquid film thinning rate is independent of radius r, and the liquid film pressure change is independent of z.
Therefore, when the two foams coalesce, the differential equation when only radial momentum is considered is:
$$\rho \left( {\frac{{\partial {u_r}}}{{\partial t}}+{u_r}\frac{{\partial {u_r}}}{{\partial r}}+\frac{{{u_\theta }}}{r}\frac{{\partial {u_r}}}{{\partial r}} - \frac{{u_{\theta }^{2}}}{r}+{u_z}\frac{{\partial {u_r}}}{{\partial z}}} \right)= - \frac{{\partial p}}{{\partial r}} - \left( {\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{\tau _{rr}}} \right)+\frac{1}{r}\frac{{\partial {\tau _{r\theta }}}}{{\partial \theta }} - \frac{{{\tau _{\theta \theta }}}}{r}+\frac{{\partial {\tau _{rz}}}}{{\partial z}}} \right)$$
4
According to hypothesis ② and hypothesis ④, formula (4) can be simplified as follows:
$$\rho \left( {\frac{{\partial {u_r}}}{{\partial t}}+u\frac{{\partial {u_r}}}{{\partial r}}} \right)= - \frac{{\partial p}}{{\partial r}}+u\left[ {\frac{\partial }{{\partial r}}\left( {\frac{1}{r}\frac{\partial }{{\partial r}}r{u_r}} \right)+\frac{{\partial {\tau _{rz}}}}{{\partial z}}} \right]$$
5
For the flow field distribution of a flat plate, the influence of viscous force can be neglected, so the formula (5) can be further simplified as:
$$\rho \left( {\frac{{\partial {u_r}}}{{\partial t}}+u\frac{{\partial {u_r}}}{{\partial r}}} \right)= - \frac{{\partial p}}{{\partial r}}$$
6
The formula (6) is integrated along with the film thickness (z-direction), so there is:
$$\rho \left( {\frac{{\partial {u_r}}}{{\partial t}}+\frac{1}{2}\frac{{\overline {{\partial u_{r}^{2}}} }}{{\partial r}}} \right)= - \frac{{\partial p}}{{\partial r}}$$
7
Where,
$$\overline {{u_{r}^{2}}} =\frac{1}{h}\mathop \smallint \nolimits_{{ - h/2}}^{{h/2}} u_{r}^{2}{\text{~d}}z=u_{r}^{2}$$
8
Because the liquid film is a parallel plate, the continuity equation of the two foam coalescence processes is:
$$- \pi {r^2}\frac{{{\text{d}}h}}{{{\text{~d}}t}}=2\pi rh{u_r}$$
9
Substituting formula (9) into formula (6), there is:
$$\frac{\partial }{{\partial t}}\left( { - \frac{r}{{2h}}\frac{{{\text{d}}h}}{{{\text{d}}t}}} \right)+\left( { - \frac{r}{{2h}}\frac{{{\text{d}}h}}{{{\text{d}}t}}} \right)\frac{\partial }{{\partial r}}\left( { - \frac{r}{{2h}}\frac{{{\text{d}}h}}{{{\text{d}}t}}} \right)= - \frac{1}{\rho }\frac{{\partial p}}{{\partial r}}$$
10
Simplify formula (10) and integrate it along the radial direction, and there is:

Therefore, by combining formula (3), the kinetic equation of spherical foam coalescence can be obtained as follows:
$$\frac{{{{\text{d}}^2}h}}{{{\text{~d}}{t^2}}}=\frac{3}{{2h}}{\left( {\frac{{{\text{d}}h}}{{{\text{d}}t}}} \right)^2} - \frac{{4h}}{{\rho {R^2}}}\frac{{2\sigma }}{R}$$
12
As can be seen from formula (12), At the initial stage, the gas is dispersed in the liquid to form foams of different sizes. Since the newly formative foam is in a state of high surface free energy, the whole system is still in an unbalanced stage, at which time the foam will coalesce spontaneously. Big foams merge small foams, i.e., merging between unstable foams. In this way, the foam system reduces the surface free energy as much as possible to reach a relatively balanced state system and form a relatively stable foam.