# 13.8.2: Solid With Lighter Density $$\rho_S< \rho$$ and With Gravity
$\dfrac{\pi\,D^3\,g(\rho_S - \rho_L)}{6} = \dfrac{{C_D}_\infty\,\pi \, D^2 \rho_L \left( U_S-U_L\right)^2 }{8} \label{phase:eq:ligthSolid} \tag{54}$
From equation 13.54, it can observed that increase of the liquid velocity will increase the solid particle velocity at the same amount. Thus, for large velocity of the fluid it can be observed that $$U_L/U_S \rightarrow 1$$. However, for a small fluid velocity the velocity ratio is very large, $$U_L/U_S \rightarrow 0$$. The affective body force "seems'' by the particles can be in some cases larger than the gravity. The flow regimes will be similar but the transition will be in different points. The solid–liquid horizontal flow has some similarity to horizontal gas–liquid flow. Initially the solid particles will be carried by the liquid to the top. When the liquid velocity increase and became turbulent, some of the particles enter into the liquid core. Further increase of the liquid velocity appear as somewhat similar to slug flow. However, this author have not seen any evidence that show the annular flow does not appear in solid–liquid flow.