Thin-film flows of granular suspensions on a solid surface

TL;DR

This paper addresses the problem of thin-film free-surface flows of granular suspensions on rigid substrates, where gravity, capillarity, and particle confinement interact across multiple scales.It surveys canonical simple-fluid theories for drop spreading and dip-coating, and then connects these frameworks to suspension rheology under confinement, highlighting where effective viscosity descriptions succeed or fail.Key findings show that classical continuum models can be extended with an effective viscosity in many regimes, but strong confinement reveals discrete, microstructural effects such as particle layering and reduced η_eff relative to bulk, affecting both spreading and film entrainment.The review synthesizes extensive experimental data, including Cox–Voinov–type dynamics for the contact line and LLD/WT-type laws for film thickness, and discusses how particle size, volume fraction, and wettability modulate these behaviors, with implications for coating and printing technologies.

Abstract

This review article examines the complex dynamics of thin-film flows of granular suspensions spreading over rigid solid substrates with free air interfaces. Such systems feature an involved coupling of the free-surface dynamics with the flow and microstructure of the suspension. In particular, we develop two canonical thin-film situations: drop spreading and dip-coating. In drop spreading, confinement of the particulate phase near the advancing contact line alters both the spreading rate and the interface shape. In dip-coating, understanding the entrainment of fluid and particles becomes challenging as the film thickness approaches the particle size.

Figures (13)

• Figure 1: Drop spreading on a solid surface.
• Figure 2: Drop spreading: length scales and regions.
• Figure 3: Validation of the Cox-Voinov law adapted from Refs. pelosse2023probingpelosse2023ecoulements: $\theta_\text{app}^3/9$ as a function of the capillary number Ca, during the spreading of 300-µL drops made of Newtonian fluids: glycerol ($\eta =$ 1.2Pas, $\gamma=$ 63mNm), poly(ethylene glycol)-ran-poly(propylene glycol) monobutyl ether melt (PEG-PPG, $\eta =$ 2.3Pas, $\gamma=$ 35mNm), V1000 silicone oil ($\eta =$ 1.0Pas, $\gamma=$ 21mNm). The fitted value of the factor $\log(x/\lambda)$ is provided in the legend assuming $\theta_\mathrm{m}=0$.
• Figure 4: (a) Parameters returned by the fit of experimental results, $R(t) = A(t-t_0)^n$ for drops of viscous fluids. Main graph: factor $A$, inset: exponent $n$. Silicone oil: experimental radius growths extracted from the publications gathered in Table \ref{['tab:spreadingt-tanner-exp']} . (b) Factor $k_\text{c/g}$ of the spreading law as a function of the drop Bond number, $\text{Bo}=(3V_0/4\pi)^{2/3}/\ell_c^2$, computed according to the spreading regime, i.e., $k_\text{c}=A\left[\eta/(\gamma V_0^3)\right]^{1/10}$ for $\text{Bo}<1$ and $k_\text{g}=A\left[\eta/(\rho g V_0^3)\right]^{1/8}$ for $\text{Bo}>1$. Capillary regime: $k_\text{c}= 0.90\pm = 0.03$, gravity regime: $k_\text{g}= 0.62\pm0.02$.
• Figure 5: Sketch of the dip-coating experiment.