A New Wording of the Laplace Equation: Variational Numerical and Analytical Approach of the Liquid Capillary Rise Effect

Abstract

This chapter focuses on the computational aspects involved in the liquid capillary rise effect. It is well known that in the neighbourhood of continuous media interfaces the molecular interaction generates surface tension forces.

Appendix

Let us consider two parallel infinitely-long walls symmetrical with respect to the yoz plane and separated by the capillary distance 2s, Fig. 10.A.1 The assumptions made for the cylindrical tube hold here true, too. The meniscus would be obtained via Eqs. (10.8), (10.9) and (10.4), minimizing the functional (Staicovici 1998b):

$$ F\left( y \right) = \int\limits_{0}^{s} {f\left( {y,y^{\prime} } \right)dx} = \int\limits_{0}^{s} {\left( {l_{c}^{2} \sqrt {1 + y^{\prime 2} } + y^{2} } \right)} dx = \text{min}. $$

(10.A.1)

Fig. 10.A.1

Scheme of the capillary effect in an infinite parallel walls structure

The Euler equation for integrand $f\left( {y,y^{\prime} } \right)$ has the form (Silov 1989):

$$ f - y^{\prime} f_{{y^{\prime} }} = C = const. $$

(10.A.2)

and the extremals are found solving the first order nonlinear differential equation:

$$ \frac{{l_{c}^{2} }}{{\sqrt {1 + y^{\prime 2} } }} = C - y^{2} $$

(10.A.3)

provided that:

$$ y\left( 0 \right) = y_{0} ;\quad y^{\prime} \left( 0 \right) = 0;\quad y^{\prime} \left( s \right) = c {\tan} \theta $$

(10.A.4)

The solution of Eq. (10.A.3) is expressed by the elliptic integral (Landau 1971):

$$ x\left( y \right) = \int\limits_{{y_{0} }}^{y} {\frac{{\left( {A - \frac{{t^{2} }}{{l_{c}^{2} }}} \right)dt}}{{\sqrt {1 - \left( {A - \frac{{t^{2} }}{{l_{c}^{2} }}} \right)^{2} } }}} $$

(10.A.5)

where

$$ A = \frac{C}{{l_{c}^{2} }} = 1 + \frac{{y_{0}^{2} }}{{l_{c}^{2} }} $$

(10.A.6)

and $y \in \lfloor{y_{0} ,l_{c} \sqrt {A - \sin \theta } } \rfloor$.

The projection of force, Eq. (10.7), on the oy axis gives the integral-differential equation:

$$ \int\limits_{0}^{x} {y\left( t \right)dt} = \frac{1}{2}l_{c}^{2} \frac{{y^{\prime} }}{{\sqrt {1 + y^{\prime 2} } }} $$

(10.A.7)

Differentiating it totally with respect to x yields the second-order differential equation:

$$ \frac{y}{{l_{c}^{2} }} = \frac{{y^{\prime\prime} }}{{2\left( {1 + y^{\prime 2} } \right)^{3/2} }} $$

(10.A.8)

Equation (10.A.8) is exactly Eq. (10.14) for const. = 0. The right-hand member represents the mean curvature of the surface. Multiplying it by $y^{\prime}$ and integrating once, Eq. (10.A.3) is obtained, checking the equivalence of energy, force and pressure equations, in this case, also.

The application of the successive differentiation method, mentioned in Sect. 10.4., to Eq. (10.A.3) for Cauchy initial conditions (10.A.4), gives the following analytical expression of the liquid meniscus shape in case of the infinite parallel walls capillary:

$$ y\left( x \right) = y_{0} + 2\frac{{y_{0} }}{{l_{c}^{2} }}\frac{{x^{2} }}{2!} + 4\frac{{y_{0} }}{{l_{c}^{2} }}\left[ {\frac{1}{{l_{c}^{2} }} + 6\left( {\frac{{y_{0} }}{{l_{c}^{2} }}} \right)^{2} } \right]\frac{{x^{4} }}{4!} + 8\frac{{y_{0} }}{{l_{c}^{2} }}\left[ {\frac{1}{{l_{c}^{4} }} + \frac{66}{{l_{c}^{2} }}\left( {\frac{{y_{0} }}{{l_{c}^{2} }}} \right)^{2} + 180\left( {\frac{{y_{0} }}{{l_{c}^{2} }}} \right)^{4} } \right]\frac{{x^{6} }}{6!} + \cdots $$

(10.A.9)

where, 0 ≤ x ≤ s. Equation (10.A.9) must be supplemented with:

$$ y^{\prime} \left( s \right) = c {\tan} \theta $$

(10.A.10)

Identifying the true meniscus shape given by Eq. (10.A.9) with an assimilated circular cylindrical surface, described by Eq. (10.35), implies that the curvature radius would be $r = l_{c}^{2} /\left( {2y_{0} } \right)$ for the two first terms, but this holds not true for the higher-order terms. For this reason, the assimilation with a circular cylindrical meniscus is not recommended.

Finally, the functional (10.A.1) describes, also, the conditions of equilibrium of the liquid contained in an infinite reservoir and curved near a straight wall, Fig. 10.A.2. Here, Eq. (10.A.3) must be solved together with the new boundary conditions (the origin of the xoy coordinate system is moved at the wall):

$$ y^{\prime} \left( 0 \right) = - c{\tan} \theta ;\quad y\left( \infty \right) = y^{\prime} \left( \infty \right) = 0 $$

(10.A.11)

Fig. 10.A.2

Capillary effect near a straight wall in an infinite liquid reservoir

This time A from Eq. (10.A.6) equals unit, A = 1, and Eq. (10.A.5) has the primitive:

$$ x\left( y \right) = - \frac{{l_{c} }}{2\sqrt 2 }arg\,\text{tan}\sqrt {1 - \frac{1}{2}\left( {\frac{y}{{l_{c} }}} \right)^{2} } + \sqrt {2l_{c}^{2} - y^{2} } + x_{0} $$

(10.A.12)

where the integration constant x ₀ is found for x = 0 and $y = l_{c} \sqrt {1 - \sin \theta }$.