# A water droplet-cleaning of a dusty hydrophobic surface: influence of dust layer thickness on droplet dynamics Effect of dust layer thickness on the droplet behavior is examined and droplet the fluid spreading rate on dust surface is determined. Wicking conditions for dust layer is assessed and the dust cleaning process by rolling droplet is evaluated.

### Hydrophobic surface and dust particles

$$frac{mu varepsilon }{K}hfrac{dh}{dt}+rho gh=frac{{gamma }_{L}}{{L}_{eff}}$$

(1)

The effective capillary length is evaluated using the SEM micrograph (Fig. 5); however, the capillary length varies along the different locations of the dust column because of the variation of the dust size, dust shape, and the dust orientation in the dust column. Hence the approximate effective capillary length is considered as same as the column length (0.012 m). The solution of Eq. (1) yields37:

$$h=frac{C left[Wleft(-expleft[frac{-({B}^{2}t+AC}{AC}right]right)+1right]}{B}$$

(2)

The Lambert (W) function has the series expansion, i.e.:

$$Wleft(xright)= sum_{n-1}^{infty }frac{{(-1)}^{n-1} {n}^{n-2}}{(n-1)!} {x}^{n}$$

(3)

The Lambert function takes the form: (Wleft(xright)=x-{x}^{2}+ {frac{3}{2}x}^{3}-{frac{8}{3}x}^{4}+{frac{125}{24}x}^{5}-{frac{54}{5}x}^{6}+{frac{16807}{720}x}^{7}+dots ). The coefficients in Eq. (2) are: (A= frac{mu varepsilon }{k}), (B=rho g), and (C=frac{{gamma }_{L}}{{L}_{eff}}) (Eq. 1). Incorporating the fluid viscosity, porosity, permeability, surface tension of fluid, and the effective capillary length, Eq. (2) can be solved and height (h) variation with time can be obtained. Figure 6 shows the wetting height (h) with time obtained from the experiment. The predicted wetting height is also included in Fig. 6. The wetting front of the liquid reaches 2.54 mm height in 0.04 s. Hence, the time for the droplet liquid infusing into the dust layer of 150 µm thickness on the hydrophobic surface becomes about 1 ms, which is less than the transition time of wetted length of the rolling droplet on the dusty surface (0.02 s). Consequently, the droplet fluid fully infuses and wets the dust layer on the hydrophobic surface during its rotational transition.