# Interdisciplinary Applied Mathematics

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Г(9, a) « Г(а, a)

 hc(9, a) 3 A(6, a) ho _ cot(a)

where hc is the height of the fluid at the middle of the groove given by

hc(0, a)

ho

1 + cot a

cos(a — 9) — 1 sin(a — 9)

The constant Г(а, a) was approximated analytically in (Romero and Yost, 1996), for the case (9 < a) as follows:

Г(а, a)

1    cot3a + 3.4 cot4a + cot5a

6 1 + 3.4 cot a + 4 cot2a + 3.4 cot3a + cot4a

Finally, the function A(9, a) is related to the cross-sectional area A(x, t) of the liquid inside the groove, i.e.,

A(x,t) = h2(x,t)A(9(x,t),a),

where

A(9, a)

sin2 (a — 9) tan a — (a — 9)+ sin(a — 9) cos(a — 9) tan2a sin2 (a — 9)

The following equation for the height of the liquid in the groove is derived from the quasi one-dimensional continuity and momentum equations:

dh2{x,t)

dt

D d

ho dx

h2(x,t)

dh(x,t) dx

(8.17)

where the diffusion coefficient is D = jh0//лк(90, a) with

k(9o, a)

Г(90, a) sin(a — 90) tan a A(90, a)

This is a nonlinear diffusion equation for h2(x,t). The diffusion coefficient is positive if a > 90, which is equivalent to having a concave free surface; for a < 90 no capillary wicking takes place.

Equation (8.17) was solved in (Romero and Yost, 1996), using similarity variables for various conditions. For the simple case h(0, t) = h0 we have

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