Interdisciplinary Applied Mathematics

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x


h(x, t) = h0T(n) and n


An approximate solution was obtained for this case in the form


®(v) =    Vo)2 — j^^iv ~ Vof 4— ,


where no is the location where the solution goes to zero and stays zero. It was found    by    (Romero and    Yost,    1996)    that    for    small    positive    values    of


(a — 90), we have no ~ 2.272, while for large values of (a — 90) the value of


По can be increased up to 40%. The main result of the analysis of Romero and Yost for V-grooves as well as grooves of more complex cross-sectional areas is that diffusion dynamics dominate, and thus the leading edge (i.e., spreading front) of the fluid propagates as


xo(t) = no(So, a)VDt.


This expression shows that spreading increases proportionally with the depth of the groove h0, since D ж h0. Also, the self-similar solution implies that the free surface of the liquid spreading remains self-similar downstream.


Similar work was presented in (Darhuber et al., 2001), where the dynamics of capillary spreading along hydrophilic microstripes were studied numerically and experimentally. The surface was smooth but was processed chemically in order to create narrow hydrophilic stripes on a hydrophobic background. Following an analysis similar to the work of (Romero and Yost, 1996), it was found that self-similar solutions also exist for the microstripes, and these can be obtained from the equation


± (фЗ—+^—=0 dn V dn)    2 dn


where the similarity variables are defined in the same way as before. The approximate self-similar solution in this case is

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