Interdisciplinary Applied Mathematics

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Ф(П) ~ (1 — n/no)1/3

where no ~ 0.87. Clearly, the spreading front advances as x(t) oc y/Dt, where D = (64yh3)/(35p,w2) with w the width of the microstripe. The average streamwise velocity, however, is proportional to

yw4 1

U oc—.

^ x

Therefore, the spreading speed is proportional to w4 and decays downstream; experimental results with microstripes of widths varying from 200 to 800 p,m verified the theoretical results; see (Darhuber et al., 2001), for more details.

8.5 Thermocapillary Pumping

We now discuss capilary spreading in the presence of thermal gradients. The development of thermocapillary stress at the air-liquid interface of a thin film gives rise to fluid motion. Since d^/dT is constant and negative

for most liquids, by applying a constant thermal gradient a constant shear stress is produced given by

dj    dj dT

dx    dT dx

along the    x-direction    from    cold to    hot    regions.    Such    a spreading    of    the

fluid on a homogeneous substrate is subject to fingering-type instabilities similar to the phenomena observed in flows driven by gravity or centrifugal forces. In particular, the spreading front develops a capillary ridge, which becomes unstable in the presence of infinitesimal disturbances of a certain wavelength. The parallel small rivulets that form after the instability occurs have a characteristic wavelength

27^n A 1/3 h0 3т )    * (ЗСа)1/3

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