# Interdisciplinary Applied Mathematics

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r(t) oc (—V3t h

which shows a weak dependence on time. However, careful analysis shows that a thin precursor film of molecular dimensions is advancing at a rate proportional to л/t, i.e., it follows standard diffusion dynamics.

In nonsmooth surfaces the spreading of liquids follows different dynamics, since capillary wicking of small amounts of liquids into microgrooves occurs.

Capillary wicking is a well-known phenomenon that has been studied extensively, first in the pioneering work of (Washburn, 1921). Also, (Romero and Yost, 1996) performed a systematic analytical study of capillary flow into a V-shaped microgroove. A typical configuration is shown in Figure 8.5 with    the    flow out of    the    page; the groove    has height    h0,    the    height    of

the liquid is denoted by h(x,t), and the equilibrium contact angle is 90.

The pressure    drop    along    the    groove is    Ap    = p(x)    — p0    = yre(x),    where

k(x)    is    the    curvature and    p0    is    the    constant    pressure above    the    liquid.

This expression is valid if the capillary number Ca = Up/j ^ 1, which implies that surface tension forces dominate over viscous forces. For a long microgroove the curvature parallel to the flow direction is neglected, and thus

1    sin(a — 9) tan(a)

» * R(x)    h(x,t)

computed from the sketch of Figure 8.5. Following a quasi-one-dimensional flow analysis, Romero and Yost (1996) found that the flowrate is

dp

dx

Q

FIGURE 8.5. Sketch for liquid spreading in a V microgroove.

h4(x,t)

V    T{9, a)

P

which is an expression similar to that for Poiseuille flow. Also, Г(9, a) is a positive function that can be approximated numerically by

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