*3.1. Settling of a Circular Particle in an Infinitely Long Tube*

The circular particle accelerates to settle down in the infinite tube due to gravity. As the speed increases, the resistance of the ball increases. When the three forces of resistance, gravity, and buoyancy are balanced, the particle falls at a uniform speed. Glowinski pointed out in the literature [18] that, for a circular particle settling in an infinitely long pipe, when its physical parameters are determined, the settling speed is also determined at a steady state.

When the liquid is at a low Reynolds number, the resistance Ff to circular particles moving in an infinitely long pipeline is directly proportional to the settlement speed [19] and can be expressed as:

$$F\_f = 4\pi K \eta v$$

where *v* is the falling velocity of circular particles, η is the dynamic viscosity of the fluid, *K* is the correction factor, and its value reflects the influence of the resistance of the pipe wall facing the particles. *K* is expressed as:

$$\mathcal{K} = \frac{1}{\ln \mathcal{W}\_{\text{i}} - 0.9157 + 1.7244 \left(\mathcal{W}\_{\text{i}}\right)^{-2} - 1.7302 \left(\mathcal{W}\_{\text{i}}\right)^{-4} + 2.4056 \left(\mathcal{W}\_{\text{i}}\right)^{-6} - 4.5913 \left(\mathcal{W}\_{\text{i}}\right)^{-8}} \tag{38}$$

where *Wi* = *W*/*D*, *W* is the width of a long square tube and *D* is the diameter of the circular particle. When particles are in a state of three forces equilibrium, the resistance can be obtained as:

$$F\_f = \frac{1}{4}\pi D^2 (\rho\_S - \rho\_L) \text{g} \tag{39}$$

ρ*<sup>S</sup>* and ρ*<sup>L</sup>* are solid density and liquid density, and g is gravity acceleration. According to Equations (37)–(39), the final falling speed of the circular particle can be calculated as:

$$v = \frac{D^2(\rho\_S - \rho\_f)\mathbf{g}}{16K\eta} \tag{40}$$

The size of the circular pipe in this paper is 4 cm × 8 cm. The side length of each grid is 0.01 cm, with a total of 400 <sup>×</sup> 800 grids. The particle diameter is taken as 0.48 cm, the fluid density is <sup>ρ</sup>*<sup>L</sup>* = 1.0 g/cm3 and the solid density is <sup>ρ</sup>*<sup>S</sup>* = 1.02 g/cm3, and the fluid viscosity is <sup>η</sup> = 0.33 g/(cm·s). The relaxation time τ is taken as 0.8, and the gravity acceleration is *g* = 980.0 cm/s2. At the initial moment, the circular particle are placed at points (2 cm, 6 cm), and they are at rest before the calculation starts.

The simulation results are shown in Figure 3. The particle dropped due to gravity, and two symmetrical vortices formed on both sides of the particle, which is consistent with what Do-Quang [5] described in his literature. According to the simulation value of particle settlement speed, it can be seen that the simulation results in this paper agree well with the theoretical analytical solution. Therefore, the method used in this paper can be used to calculate the moving boundary problem.

**Figure 3.** Diagram of circular particle settlement: (**a**) Flow field; (**b**) Settlement velocity.
