Interdisciplinary Applied Mathematics

Скачать в pdf «Interdisciplinary Applied Mathematics»


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

Скачать в pdf «Interdisciplinary Applied Mathematics»

Метки