Capillarity, equations

FIGURE 13.8 The geometry used to calculate the capillarity equation. 

This ratio, which is constant for all of the surfaces of a crystal at equilibrium, is termed the capillarity constant, analogous to that defined in the well-known Laplace capillarity equation. 

Equation (8) can also be derived differently, e.g., as presented in Ref. 19, where gravitational effect is directly included in terms of the Bond number (defined later on in this section). Equation (8) was first introduced in 1805 by Young and Laplace (hence Young Laplaee equation) and is eonsidered as the basic equation of capillarity. Equation (8) is equivalent to Eq. (6b) with the influence of air accounted for in Eq. (8). 

The phenomena of capillary penetration of liquids into pore spaces filled with a fluid are related to the rise of liquids in capillary tubes where the flow is driven by interfacial pressure differences, as described by the Laplace equation of capillarity (equation (7.1)). The magnitude of the pressure difference across each liquid-fluid interface (meniscus) depends on the local curvature which is determined by the local wetting properties and pore geometry. In a cylindrical capillary, where the capillary wall is completely wettable by the liquid, the liquid-vapour interface can be assumed to be a hemisphere and = R2 = r r << a, cf. equation (7.48) below), where r is the capillary radius. Then, equation (7.1) becomes particularly simple and reduces to the following ... 

Equation 11-3 is a special case of a more general relationship that is the basic equation of capillarity and was given in 1805 by Young [1] and by Laplace [2]. In general, it is necessary to invoke two radii of curvature to describe a curved surface these are equal for a sphere, but not necessarily otherwise. A small section of an arbitrarily curved surface is shown in Fig. II-3. The two radii of curvature, R and / 2[Pg.6]

Equation II-7 is the fundamental equation of capillarity and will recur many times in this chapter. 

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

We now describe the conditions that correspond to the interface surface. Eor stationary capillarity flow, these conditions can be expressed by the equations of continuity of mass, thermal fluxes on the interface surface and the equilibrium of all acting forces (Landau and Lifshitz 1959). Eor a capillary with evaporative meniscus the balance equations have the following form ... 

There is another important law that follows from the classical theory of capillarity. This law was formulated by J. Thomson [16], and was based on a Clausius-Clapeyron equation and Gibbs theory, formulating the dependence of the melting point of solids on their size. The first known analytical equation by Rie [17], and Batchelor and Foster [18] (cited according to Refs. [19,20]) is... 

Equation 9.6 and Equation 9.9 through Equation 9.12 are the basis of the classic theory of capillarity [9], The moderate surface curvature that was assumed for these equations follows the fundamental Gibbs Equation 9.1 and Equation 9. lb. However, there was a problem of application of the classic theory of capillarity to the region of high surface curvatures that corresponds to the nanoparticles (down to 2 nm). 

Thus, the set of stated arguments allows one application of the equations of classic theory of capillarity in the whole range of nanoparticle size without accounting the dependence of a on curvature. This makes it possible to use these equations for the description of basic typical mechanisms of catalysts texture genesis. 

For the purpose of illustration, in this paper we use a viscosity-capillarity model (Truskinovsky, 1982 Slemrod, 1983) as an artificial "micromodel",and investigate how the information about the behavior of solutions at the microscale can be used to narrow the nonuniqueness at the macroscale. The viscosity-capillarity model contains a parameter -Je with a scale of length, and the nonlinear wave equation is viewed as a limit of this "micromodel" obtained when this parameter tends to zero. As we show, the localized perturbations of the form x /-4I) can influence the choice of attractor for this type of perturbation, support (but not amplitude) vanishes as the small parameter goes to zero. Another manifestation of this effect is the essential dependence of the limiting solution on the... 

Schofield Phil. Mag. March, 1926) has recently verified this relation by direct experiment. In order to appreciate the significance of this result, it is necessary to consider in more detail the electrical potential difference V and the manner in which it arises. Instead of regarding the phenomenon from the point of view of the Gibbs equation, it has been, until recently, more usual to discuss the subject of electro-capillarity from the conceptions developed by Helmholtz and Lippmann. These views, together with the theory of electrolytic solution pressure advanced by Nemst, are not in reality incompatible with the principles of adsorption at interfaces as laid down by Gibbs. 

The left-hand side of the latter equation is related to the liquid inertia, whereas both terms in the right-hand side are related to capillarity (the driving force), and viscous resistance, respectively. Under steady conditions, capillarity is balanced by the viscous drag of the liquid, and the famous Lucas-Washbum s equation can be derived (De Geimes et al., 2002) ... 

Equation 20.66 is the general solution for the problem, evaluated at the interface of the perturbed sphere and written for a single harmonic of order n (i.e., the summation has been dropped to simplify comparison with the capillarity condition as given in Eq. 20.61). These two equations can be equated to evaluate the unknown constants B0 and Bn in Eq. 20.66. From terms not involving 8 Y ,... 

These equations for B0 and Bn are now substituted into Eq. 20.64, giving the general solution which matches the capillarity boundary condition on the interface of the perturbed sphere and which is valid everywhere in the matrix,... 

Equations (1) and (2) hold true when the curvature radius, Rcurv, of the body taken out of the liquid paste is much greater than the capillarity constant, i.e. when ... 

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

With the aid of the straight capillaric model and according to the Hagen-Poiseuille equation and Darcy s law, a is proportional to the viscosity of the fluid. Thus, Eq. (5.322) becomes... 

This equation was obtained by fitting a curve to the plotted values of Cd for a great many experiments and is purely empirical. Capillarity is accounted for by the second term, while velocity of approach (assumed to be uniform) is responsible for the last term. Rehbock s formula has been found to be accurate within 0.5% for values of Z of 0.33 to 3.3 ft and for values of H of 0.08 to 2.0 ft with the ratio H/Z not greater than 1.0. It is even valid for greater ratios than 1.0 if the bottom of the discharge channel is lower than that of the approach channel, so that backwater does not affect the head. 

The possibilities of the Poiseuille equation for explaining capillary phenomena in soils have had a certain amount of appeal to soil physicists. The theory of capillarity is so simple and so well established that if suitable soil parameters to express a bundle of capillaries could be found, the problem would be solved in the most direct fashion. Unfortunately, however, there has as yet been no completely satisfactory formula for the capillary bundle theory, although it has often given results of the proper magnitude in certain instances. We shall here develop the Poiseuille equation for special capillary conditions and point out the fallacies of this approach. 

Equation (3) is the fundamental equation of Capillarity. It permits the calculation of the forms of liquid surfaces when the weight of the liquid is not negligible, for the pressures can be expressed in terms of the height above a fixed point in the fluids, and their densities. 

This equation has been known for over a century it was given by Young2 (without proof ) and by Dupre 3 it can be deduced also from Laplace s theory of Capillarity, or indeed from any theory of the cohesive forces, since it can be obtained from consideration of energies only. Until recent years it has been little noticed, which is unfortunate, as the meaning of the contact angles is much clarified when the work of adhesion is introduced, and the surface tensions of the solid surfaces, which are not measurable, are eliminated. Most authors are now, however, expressing their results in terms of the work of adhesion or of closely related expressions. 

The exact calculation of the weight of liquid lifted, in terms of the surface tension and density, is difficult and requires usually special solutions of the fundamental equation of Capillarity, for figures which often are not figures of revolution. The pull may reach a maximum some distance before the object is completely detached and the measurement of this maximum is considered more satisfactory than that of the pull at the moment of detachment.7 In most cases, however, the pull is applied by means of a torsion balance, and the upward motion of the object cannot be checked after the maximum pull is past, so that the detachment takes place almost immediately the maximum pull is reached. 

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