Applying the Young-Laplace equation

When applying the equation of Young and Laplace to simple geometries it is usually obvious at which side the pressure is higher. For example, both inside a bubble and inside a drop, the pressure is higher than outside (Fig. 2.7). In other cases this is not so obvious because the curvature can have an opposite sign. One example is a drop hanging between the planar ends of two cylinders (Fig. 2.7). Then the two principal curvatures, defined by 

For a bubble in a liquid environment the two principal curvatures are negative C = C2 = —1/R. The pressure difference is negative and the pressure inside the liquid is lower than inside the bubble. 

For a drop hanging between the ends of two cylinders (Fig. 2.7B) in a gaseous environment, one curvature is conveniently chosen to be C = l/R. The other curvature is negative, C2 = —l/R2. The pressure difference depends on the specific values of R and R2. 

The shape of a liquid surface is determined by the Young-Laplace equation. In large structures we have to consider also the hydrostatic pressure. Then the equation of Young and Laplace becomes 

g is the acceleration of free fall and h is the height coordinate. 

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Mercury porosimetry is the most suitable method for the characterization of the pore size distribution of porous materials in the macropore range that can as well be applied in the mesopore range [147-155], To obtain the theoretical foundation of mercury porosimetry, Washburn [147] applied the Young-Laplace equation... 

The treatment of capillary phenomena usually requires the mathematical analysis of curved fluid-fluid interfaces. As a prelimineuy to this chapter and to chapter 5, we shall now repeat and extend parts of sec. 1.2.23a, where this matter was introduced. The description of curvature is a prerequisite for applying the Young-Laplace equation [1.1.2]. 

Porosimeter An instrument for the determination of pore size distribution by measuring the pressure needed to force liquid into a porous medium and applying the Young-Laplace equation. If the surface tension and contact angle appropriate to the injected liquid are known, pore dimensions can be calculated. A common liquid for this purpose is mercury hence, the term mercury porosimetry. 

Fig. 8.24 Interfacial tension calculated by the real-time FPGA approach and a software reference implementation applying the Young-Laplace equation and a fourth order Runge-Kutta solver for 800 captured frames [68]...

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. 

This is the Young-Laplace equation applied to a spherical surface. A more general form of this equation is used when the curvature of the interface is not spherical [Gl]. 

Although the Young-Laplace equation can be applied in a rigorous sense only to systems in which there is no fluid motion, the concept that surface tension leads to a pressure jump across an interface with nonzero curvature can be used to qualitatively anticipate the nature of many capillary flows. This qualitative use of the Young-Laplace equation is similar in spirit to the use of the hydrostatic pressure distributions to anticipate the nature of gravity-driven flows, such as the gravity front, discussed earlier in this chapter. 

This is the derivation of the Young-Laplace equation for a spherical interface from surface thermodynamics. Now, if we consider the general case, where /q A jc2 and d( /q - K-f) A 0, for any curved figure, Equation (314) applies for this condition and it has been proved that... 

Wenzel s relation has been confirmed in terms of the first two laws of thermodynamics. Huh and Mason, in 1976, used a perturbation method for solving the Young-Laplace equation while applying Wenzel s equation to the surface texture. Their results can be reduced to Wenzel s equation for random roughness of small amplitude. They assume that hysteresis was caused by nonisotropic equilibrium positions of the three-phase contact line, and its movement was predicted to occur in jumps. On the other hand, in 1966, Timmons and Zisman attributed hysteresis to microporosity of solids, because they found that hysteresis was dependent on the size of the liquid molecules or associated cluster of molecules (like water behaves as an associated cluster of six molecules). 

Tortuosity In porosimetry evaluations, experimental data tend to be interpreted in terms of a model in which the porous medium is taken to comprise a bundle of cylindrical pores having radius r. If the Young-Laplace equation is then applied to the data, an effective value of r can be calculated, even though this model ignores the real distribution of irregular channels. The calculated r value is sometimes considered to represent the radius of an equivalent cylinder or, alternatively, is termed the tortuosity. 

Now is evaluated at the pressure exerted by the fluid at the initial flat interface, which is also the pressure in the sohd under these conditions if the Young-Laplace equation is assumed to apply. But onee the interface is deformed, as in Figure 6.19, the pressure in the solid near the interfaee becomes a funetion of position. Indeed, if pressure in the fluid phase is presumed to be uniform, we can use the Young-Laplace equation and the well-known dependence of fugadty upon pressure to write... 

Relation 9.77 is usually called the Washburn equation [55,237], One should consider it as a special case of the fundamental Young-Laplace equation [3,9-11], Washburn was the first to propose the use of mercury for measurements of porosity. Now, it is a common method [3,8,53-55] of psd measurements for a range of sizes from several hundreds of microns to 3 to 6 nm. The lower limit is determined by the maximum pressure, which is applied in a mercury porosimeter the limiting size of rWl = 3 nm is achieved under PHg = 4000 bar. The measurements are carried out after vacuum treatment of a sample and filling the gaps between pieces of solid with mercury. Further, the hydraulic system of a device performs the gradual increase of PHg, and the appropriate intmsion of mercury in pores of the decreasing size occurs. 

The equilibrium bubble size can be determined based on applying the pressure balance at the bubble interface. Gibbs-Duhem equation and Young-Laplace equation are written for the bubble [1],... 

Young-Laplace equation (pressure Applied in the derivation of the Kelvin... 

When measuring membrane pore size using the fluid displacement method, the membrane pores are first filled by a fluid, which can be either air or a wetting liquid. The fluid inside the pores is then displaced by another immiscible fluid under applied pressure. The principle of the fluid displacement method is based on Young-Laplace equation [171] ... 

Because ceramic powders usually have macropores, mercury porosimetry is more suitable than gas adsorption. The principle of the technique is the phenomenon of capillary rise, as shown schematically in Fig. 4.4 [19]. When a liquid wets the walls of a narrow capillary, with contact angle, 9 < 90°, it will climb up the walls of the capillary. If the liquid does not wet the walls of a capillary, with contact angle, 9 > 90°, it will be depressed. When a nonwetting liquid is used, it is necessary to force the liquid to flow up the capillary to the level of the reservoir by applying a pressure. For a capillary with principal radii of curvature rj and r2 in two orthogonal directions, the pressure can be obtained by using the Young and Laplace equation ... 

These conditions are satisfied by the soap film contained by the tetrahedral framework. The conditions given by Plateau apply to surfaces bounded by any frame. These surfaces do not have to be planar and the lines of soap film need not be straight. It is only recently that Frederick J. Almgren Jr. and Jean E. Taylor S have shown that these conditions follow from the mathematical analysis of minimum surfaces and surfaces containing bubbles of air or gas at different pressures, both of which can be described by the Laplace-Young differential equation. 

A soap film has two parallel surfaces. Thus applying the Laplace-Young equation to each surface the excess pressure, pf, across the film is... 

THE SCIENCE OF SOAP FILMS AND SOAP BUBBLES Applying the Laplace-Young equation at P,... 

In the Steiner problem, Chapter 3, and the minimum surface area problems, discussed earlier in this chapter, the Laplace-Young equation had a simple form. This resulted from the zero excess pressure across any point on the surface of the soap film. In the case of a bubble, or clusters of bubbles, the excess pressure across any surface is not in general zero. However the Laplace-Young equation can be applied under these more general conditions. Plateau s rules concerning the angles at which surfaces and lines of soap films intersect apply also to the surfaces and lines of soap film produced by clusters of bubbles. 

A uniformly stretched rubber membrane is similar in many respects to a soap film, or the interface between two fluids. It has a uniform tension and the thickness of the membrane is small compared with the dimensions of the surface area. The analysis of section 5.2, which derives the Laplace-Young equation for a fluid interface or soap film, applies equally to a uniformly stretched membrane with a transverse pressure load that is perpendicular to the surface. So the Laplace-Young equation for the membrane is... 

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