FIGURE1.3: Sketch of the different forms of rivulets that emerge at the advancing front of a (a) fully wetting and (b) partially wetting fluid Silvi and Dussan (1985).
The wavelength of the rivulet pattern that forms at the advancing front of a spread- ing film can be altered by regular patterned trench stripes with equal spacing (Kondic and Diez, 2002); if the spacing is less than the natural wavelength of the instability then rivulets are forced to grow in close proximity to one another enforcing merg- ing. If the spacing is greater than the natural wavelength then the rivulets form in regular spacing down the path of least resistance. For large spacing several rivulets form in each channel. Kondic and Diez (2004) further noted how small trench topographies could introduce a large enough disturbance to induce rivulet forma- tion and that the spacing directly influenced the wavelength of the instability that emerged. Similar observations were made when considering chemical heterogene- ity; experiments in Kondic and Diez (2004) (Figures 16,17 and 18 in their publica- tion) were performed using PDMS (polydimenthylsiloxane) to coat a glass surface; stripes of an oil-based paint were deposited on the substrate at controlled intervals. PDMS has a much higher contact angle with the paint (low wetting) so as the ad- vancing front approaches the stripes the fluid travels down the non-painted areas, thus the spacing between the two directly impacts the wavelength of the rivulets. The field of wetting and spreading, including hysteresis in droplet spreading, is a vastly researched area with still many opposing theories requiring validation. While neither the macro- or micro-scale science behind spreading is discussed in detail
here, the reader is directed to a recent thorough review by Bonn et al. (2009) and the references therein contained.
FIGURE 1.4: Top: Rivulet flow over randomly distributed heterogeneous patches from Zhao and Marshall (2006). The contact angle associated at each co-ordinate (x, y) using a random correlation function controlled by correlation length, l; (a) l = 6, (b) l = 13 and (c) l = 24. Three time snapshots are shown - (i) t = 12, (ii) t = 24 and (iii) t = 33. Flow is from left to right and contours shown at h = 0.2, 0.6, 1 and 1.4, grey shading is used when h > 1.4. Bottom: The network of patches created by the random function assigning the associated contact angle to the substrate; white areas are fully wetting, grey areas indicate where the contact angle is greater than 15o and in black shaded areas the contact angle is
less than 9o.
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Eley (1998); Schwartz (1998) employing a disjoining pressure model (Derjaguin et al., 1987) to imitate the effect of hysteresis and assigning regular patchwork areas of high contact angle. They found that although the spreading rates/time scale of the simulations were out by a large factor, the numerical solutions ob- tained matched well with experiments for a droplet spreading over a low wetting Teflon cross and splitting into four micro-droplets. Further work including droplet spreading onto mound topographies with low wettability/high contact angle was carried out in Gaskell, Jimack, Sellier and Thompson (2004) employing a multi- grid method with error-controlled variable time-stepping to solve the lubrication equations. They found that the wettability of the liquid on the surface impacted the spreading dynamics and shape of the drop; for example, a drop on a highly wet- ting surface that spread towards a low wetting square mound would recede from the low wetting area and spread on the highly wettable surface creating a bow shaped interface at the corner of the low wetting region. In the reverse case, a low wet- ting substrate containing a highly wetting mound, the droplet would preferentially spread and climb onto the mound.
With respect to rivulets, surface patterning with chemical heterogeneities has gar- nered much less attention than other aspects of thin film flow. Silvi and Dussan (1985) observed that when silicone oil spread on a perspex surface the rivulets would take a much different form to those seen when glycerin, which has a much larger contact angle on perspex than silicone oil, was used to coat the substrate; see Figure 1.3 for a sketch of the saw-tooth pattern of a fully wetting fluid and long, thin fingers of a partially wetting fluid that were observed. The elongation of the rivulets was also observed to be much faster when glycerin was used, with the bulk staying almost stationary compared to the rivulets. Similarly, Jerrett and Bruyn (1992) con- sidered three different liquids in their experiments; glycerin and two different types of mineral oil. The mineral oils had a static contact angle of 14o on the plexiglass substrate utilised, whereas glycerin had much higher contact angle of 60o. Their
ferences in pattern and shape between the higher and lower wetting liquids. They also correlated the wavelength and found two different expressions for mineral oil and glycerin; λ = 14.1Lcsin α0.12 (glycerin)
19.2Lcsin α0.21 (mineral oil),
where α is the inclination angle and the capillary length Lc = H0/(3Ca)1/3, where
H0is the asymptotic film thickness and Ca the capillary number. Clearly, the wave-
length is smaller when the fluid is less wetting (glycerin) than another one (mineral oil); however, the expressions do not include the contact angle (which indicates wettability).
Numerical investigations have mostly ignored the effect of wetting properties on the wavelength in the context of rivulet formation. Of those that attempt to assess their impact, Eres et al. (2000) found, for single rivulets, that an increase in contact angle made a rivulet longer and thinner. Marshall and Wang (2005) and Zhao and Marshall (2006) uncovered the subtle influence that both periodically and randomly distributed heterogeneous regions can have over the rivulet instability - see Figure 1.4 for an example of flow over randomly distributed patches of varying wettability - finding that when the regions were small (a small correlation length l, see Figure 1.4) the wavelength that emerged was similar to that calculated from linear stability theory. However, when the regions were large the wavelength varied significantly around the predicted value. Kondic and Diez (2004) showed experimentally that rivulets of PDMS form within the spacings between regularly arranged low wet- ting patches (created with an oil-based paint), as discussed previously. A similar observation to this was noted for a climbing film, a phenomena seen when a tem- perature gradient is applied to a vertically aligned substrate in such a manner that the Marangoni forces, induced by the surface tension gradient associated with the temperature profile, are large enough to overcome gravity (Cazabat et al., 1990; Kataoka and Troian, 1997, 1999). An example of this is explored in Kataoka and Troian (1999) using chemically striped silicone wafers. A flow of PDMS, driven
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by Marangoni forces, develops rivulets at the advancing front climbing the sub- strate. The rivulet pattern was seen to develop a wavelength that correlated with the spacing of the chemical heterogeneity, much like in the gravity-driven case.