Table 1.

Summary of force analyses.

Category	Expression	Cause	Annotations	Reference
lift force
Inertial lift force	$F_L=f_L(R_e,x/h)\cdot \rho U^2{a}^4/H^2$	inertial focusing effect	fL: a nondimensional lift coefficient; h: height of the rectangular cross-section; ρ: density of fluid; U: maximum velocity of fluid; a: diameter of particle; H: hydraulic diameter;	[41]
Magnus lift force	$F_{LR}=\frac{1}{8}\pi a^3\rho U\times \mathsf{\Omega }$	when the rotational angular momentum Ω of the rotating object is not coincident with its velocity vector U, there is a lateral force	a: diameter of particle; ρ: density of fluid; U: relative velocity of the particle and the fluid; Ω: rotational angular momentum of particle	[53]
Saffman lift force	$F_S=KV{r}^2\sqrt{\gamma /\upsilon }$	A velocity gradient in the fluid will be generated under the effect of channel wall which further leads to the spinning of the particle under the effect of shear force	K: numerical constant, usually regarded as 81.2; V: relative velocity between the fluid and particles; γ: velocity gradient; ν: viscosity;	[55]
drag force
Dean drag force	$F_{Stokes}=3\pi \mu a_P{U}_D$	secondary flow	μ: fluid viscosity; ap: diameter of the particle; UD: Dean flow velocity	[51]
Viscous drag force	$F_d=S\cdot f_d=\pi a^2{f}_d/4$	shear effect on the contact surface between fluid and particles	S: cross-section of particles; a: diameter of particle; fd: viscous drag coefficient	[49]