Interdisciplinary Applied Mathematics

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where h0 is the film thickness, and the capillary number Ca is defined here as Ca = rho/(2j). In practice, this instability is realizable, but because of surface defects or impurities the rivulets are not stationary, with a large variation in the value of Xc up to 30%; see (Kataoka and Troian, 1999), and references therein.

In order to drive the flow along specified pathways it is possible to pattern the surface with microstripes that have a different contact angle from that of the rest of the surface. This was attempted in (Kataoka and Troian, 1999), where alternating    stripes    of    bare    or    oxidized    SiO2    coated    with    a

bonded monolayer of octadecyltrichlorosilane (OTS) were patterned on a wafer using a spin-coating technique; see the sketch of Figure 8.6. Of fundamental interest here is to examine whether the wavelength imposed by fabrication dominates over the wavelength of the instability Ac. In (Kataoka and Troian, 1999), the thermocapillary spreading of a silicone oil was investigated on stripes varying from 100 p,m to 500 p,m. A typical result from the experiments is    shown    in    Figure    8.6    displaying    visualizations    of    the    sili

cone oil on alternating 200 p,m stripes subject to т = 0.8 dyn/cm2. The film thickness observed was less than 1 p,m. Unlike spreading over homogeneous substrates that requires a long incubation period, spreading over heterogeneous substrates is spontaneous, and the fluid is immediately channeled into the hydrophilic stripes. The average speed of tips and troughs for the flow visualization shown in Figure 8.6 was 2.60 p,m/s, and 1.73 p,m/s on the hydrophilic stripe with the troughs advancing more slowly on the OTS stripes. This is typically three orders of magnitude less than the induced speed in an electrocapillary; see the next section. It was found in (Kataoka and Troian, 1999),    that    on    the    patterned    surface    the    periodicity    of    the

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