Young-Laplace relationship

We have obtained the Young Laplace relationship from normal-stress balance (2-135) by invoking the limiting case of no motion in the fluids. One might be tempted to suppose that the condition of no motion is independent of the interface, e.g., that it is determined by whether there is some source of fluid motion in the bulk-phase fluids away from the interface. However, this is not correct. In fact, if (V n) / constant on the interface (i.e., independent of position on S), the Young-Laplace equation cannot be satisfied, and there must be fluid motion so that the balance of normal stresses includes viscous contributions. Utilizing (2 137), we see that the condition (V n) = constant requires the sum of Rf1 and R2 1 to be constant on S. Examples of surfaces that satisfy this requirement are a sphere, where R = Ih = R (the radius) a circular cylinder, where R = R, R2 = 00 and a flat interface, where Ri = R2 = 00. 

Fundamentally, surface tension is of critical importance to both normal and pathophysiological pulmonary mechanics because it provides, at the microscale level, a cohesive property that functions to minimize the size of the air-liquid interface (Figure 16.2). As a result, surface tension serves to reduce the overall surface area of the liquid lining that coats the alveoli. To compensate, a static pressure difference across the air-liquid interface must be established to maintain the alveolar structure. This pressure difference follows the Young-Laplace relationship ... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

Finally, we note that the surface body force term constitutes the sum of the surface-excess body force and the bulk-phase body force vector densities. The surface-excess body force is the 2D analog of continuum body forces in 3D fluids (e.g., gravitational force, electromagnetic force, etc). This force is often neglected. The bulk-phase body force has no counterpart for 3D fluids, as it denotes the stresses applied intimately at the interface by the surrounding 3D bulk phases. The normal component of this force equals the pressure difference between the two bulk phases, a relationship often referred to as the Young-Laplace equation. 

Young—Laplace Equation The fundamental relationship giving the pres-... 

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. 

This relationship among the two bulk phase pressures, the mean curvature, and the interfacial tension is fimdamental to the study of fluid interfaces. It is often referred to as the Laplace or the Young-Laplace equation and is the basis for several methods of measuring interfacial tension. Note that with the interface cnrved in the manner shown in Figure 1.2, H is negative and Equation 1.22 implies that Pa > Pb- Thus the pressure inside a drop or bubble always exceeds the pressure outside. 

The Young-Laplace equation (equation 5) describes the force balance in terms of capillary pressure for two fluid phases in contact with each other and a surface. If one of the phases is present as a thin film, the equilibrium relationship that accounts for the thin film is the augmented Young-Laplace equation [6-S],... 

An example is provided in enhanced oil recovery. In an oil-bearing reservoir the relative oil and water saturations depend on the distribution of pore sizes in the rock, the pressure in a pore, the interfacial tension, and the contact angle according to the Young-Laplace and Young equations. The same relationships determine how water or other fluids can be injected to change pressure, interfacial tension. 

Unlike the phenomena and characterization techniques described in Sections 5.3.1 through 5.3.3, the water transport in the PTL lacks adequate theoretical foundations. The current literature questions the assumptions in the original Young-Laplace formulations, and it emphasizes the lack of reliable models. The clarification and consolidation of the relevant terminology, theory, and experimental techniques has been aided by the recent efforts, but finding predictive relationships to PTL performance, degradation, and durability remains a challenge. 

The pressure drop AP between the entrance (x = 0) and the advancing meniscus (x = L(t)) can be obtained from the Young s-Laplace relationship as... 

The pressure required to force water to enter the pore is called the breakthrough pressure. Bubble point as measured and reported in the literature is the air pressure needed to push out liquid imbibed in the pore of the membrane. The procedure for a bubble point test is described in ASTM Method F-316. The relationship between pore size and bubble point pressure is based on the application of the Young-Laplace equation. The smaller the... 

The variation of the surface area in Equation 2.5 can be calculated with the tools of differential geometry (see, e.g., references [32,33]) and a relationship is obtained which links the geometrical properties of the interface with the pressure field (the Young-Laplace equation) ... 

Because so many applications of surfactants involve surfaces and interfaces with high degrees of curvature, it is often important to understand the effect of curvature on interfacial properties. What is usually considered to be the most accurate procedure for the determination of the surface tension of liquids, the capillary rise method, depends on a knowledge of the relationship between surface curvature and the pressure drop across curved interfaces. Because of the existence of surface tension effects, there will develop a pressure differential across any curved surface, with the pressure greater on the concave side of the interface that is, the pressure inside a bubble will always be greater than that in the continuous phase. The Young-Laplace equation... 

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]

A very important thermodynamic relationship is that giving the effect of surface curvature on the molar free energy of a substance. This is perhaps best understood in terms of the pressure drop AP across an interface, as given by Young and Laplace in Eq. II-7. From thermodynamics, the effect of a change in mechanical pressure at constant temperature on the molar h ee energy of a substance is... 

It is now possible to insert this dynamic contact angle condition back into the spreading relationship in Eq. 15 to obtain an explicit power law for the spreading dynamics of a drop lying on a horizontal substrate [7]. If the drop can be assumed to be thin, then the axisymmetric Laplace-Young equation in the lubrication limit reads... 

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