Equation of capillarity

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. 

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. 

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. 

A necessary condition for mechanical equilibrium of a fluid interface is the Laplace equation of capillarity" " ... 

Capillarity A general term referring either to the general subject of or to the various phenomena attributable to the forces of surface or interfacial tension. The Young—Laplace equation is sometimes referred to as the equation of capillarity. 

Young-Laplace Equation The fundamental relationship giving the pressure difference across a curved interface in terms of the surface or interfacial tension and the principal radii of curvature. In the special case of a spherical interface, the pressure difference is equal to twice the surface (or interfacial) tension divided by the radius of curvature. Also referred to as the equation of capillarity. 

In his classic investigation of capillarity, Laplace [76] explained the adhesion of liquids to solids in terms of central fields of force between the volume elements of a continuous medium. This approach was illuminating about the origin of surface tension and energy and their relation to the internal pressure, and it resulted in the fundamental differential equation of capillarity which has been the basis of all... 

In brief, via the CCD camera (1) with objective (2) and the frame grabber (3), an image of the shape of a drop (9) is transferred to a computer, where by using the ADSA software the coordinates of this drop are determined and compared to profiles calculated from the Gauss-Laplace equation of capillarity. The only free parameter in this equation, die interfaeial tension Y, is obtained at optimum fitting of the drop-shape coordinates. The dosing system (7) allows one to change the drop volume and hence the drop surface area. This possibility is used in dilational relaxation experiments as outlined in Sec. VI. 

For even better results, drop profile analysis can be applied instead of measuring the contact angle directly (axisymmetric drop shape analysis, ASDA Fig. 4.20). This technique extracts experimental drop profiles from video images while slowly increasing or decreasing the droplet volume [42, 43]. The best fit of experimental data with theoretical assumptions based on the Laplace equation of capillarity allows one to calculate the surface/inter-facial tension and subsequently the contact angle. Also droplet radius, droplet volume, and the contact area are computed. ADSA can therefore reveal... 

To describe wettability in a porous reservoir rock requires inclusion of both the fluid surface interaction and curvature of pore walls. Both are responsible for the capillary rise seen in porous media. The fundamental equation of capillarity is given by the equation of Young and Laplace [2]... 

Equation (2.34) is the well-known Young-Laplace equation of capillarity, which provides the condition for mechanical equilibrium of a curved interface. 

By using the Young-Laplace equation of capillarity, -(P" - P) = 2a/r, the expression for becomes... 

Effect of wettability. Our attention to the effect of curvature on the saturation pressure was directed to a bubble or, equivalently, when the liquid wets the solid surface. This is often true in hydrocarbon reservoirs in which gas is the nonwetting phase and oil is the wetting phase. However, in some other systems, gas may be the wetting phase and oil (or the liquid) may be the nonwetting phase. Even in rocks, liquid mercury is the nonwetting phase and air is the wetting phase. In such cases, the Young-Laplace equation of capillarity for a tube should be written as... 

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 ... 

All drop and bubble methods are based on the Laplace equation of capillarity. In order to study dynamic aspects of adsorption, the growing drop or bubble and the expanded drop methods are suitable (3). In Figure 12.13, the schematic of a static or growing drop instrument is shown. In applications of capillary pressure tensiometry, an equation which is equivalent... 

The Laplace equation of capillarity and the Young equation are two principal equations in contact angle measurements. In essence, the shape of a liquid drop is determined by a combination of surface tension and gravity effects. Surface forces tend to make drops spherical, whereas gravity tends to elongate a pendant drop or flatten a sessile drop. The shape of the drop is governed by the Laplace equation of capillarity, as follows ... 

Figure 14.5. Comparison between experimental points and a Laplacian curve. The curve u = u s) is a theoretical profile based on the Laplace equation of capillarity, while points Vi, i = 1,2,N, are points selected from the meridian of an experimental drop profile. The deviation of the ith point from the Laplacian curve, di, can be calculated. The purpose of the objective function of ADSA is to minimize the sum of the square of the minimum distance, di. Since the coordinate systems of the experimental profile and the predicted Laplacian curve do not necessarily coincide, their offset xq, zo) and rotation angle (a) must be considered. Both of the latter are optimization parameters. Since the program does not require the coordinates of the drop apex as input, the drop can be measured from any convenient reference frame, and all measured points on a drop profile are equally important (from ref. (36))...

The second equation in Eq. (105) represents a form of the Laplace equation of capillarity. For the so-called surface of tension, P = 0 by definition [143,208] then, 7 = 0 and Eq. (105) coincides with Eq. (95). On the other hand, if the Gibbs dividing surface is defined as the equimolecular dividing surface (the surface for which the adsorption of solvent is equal to zero see Refs. 8 and 207), then B is not zero and the generalized Laplace equation should be used. It is interesting to note that for a flat intermolecular dividing surface, B... 

Larkin BK (1967) Numerical solution of the equation of capillarity. J Colloid Interface 8d 23 305-312... 

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