Laplace equation, applied

The Laplace equation applied specifically to spherical surfaces can be derived in a variety of ways. Example 6.1 considers an alternative derivation that points out the thermodynamic character of the result quite clearly. 

With this idea in mind, the horizontal surface in Figure 6.3b can be taken as a reference level at which Ap = 0. Just under the meniscus in the capillary the pressure is less than it would be on the other side of the surface owing to the curvature of the surface. The fact that the pressure is less in the liquid in the capillary just under the curved surface than it is at the reference plane causes the liquid to rise in the capillary until the liquid column generates a compensating hydrostatic pressure. The capillary possesses an axis of symmetry therefore at the bottom of the meniscus the radius of curvature is the same in the two perpendicular planes that include the axis. If we identify this radius of curvature by b, then the Laplace equation applied to the meniscus is Ap = 2y/b. Equating this to the hydrostatic pressure gives... 

In Chap. 2 steady-state heat transfer was calculated in systems in which the temperature gradient and area could be expressed in terms of one space coordinate. We now wish to analyze the more general case of two-dimensional heat flow. For steady state, the Laplace equation applies. 

Steady-state electric conduction in a homogeneous material of constant resistivity is analogous to steady-state heat conduction in a body of similar geometric shape. For two-dimensional electric conduction the Laplace equation applies ... 

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]. 

The Laplace equation represents the basic law in the theory of capillary action. The generalized expression of the Laplace equation applied to non-spherical surfaces can be written as... 

Equation (51) is the Laplace equation applied to the reactant concentration. It is identical in form to the primary distribution and the characteristics of the primary distribution discussed above apply here as well. The approximation stated by (51) pertains particularly to small features such as plating within narrow trenches or vias. Often on these scales, convective transport is not effective, and the ionic transport progresses mainly through diffusion. Furthermore, the ohmic drop on these scales is negligible, rendering (51) the relevant controlling equation. 

The trace vanishes because only p- and /-electrons contribute to the EFG, which have zero probability of presence at r = 0 (i.e. Laplace s equation applies as opposed to Poisson s equation, because the nucleus is external to the EFG-generating part of the electronic charge distribution). As the EFG tensor is symmetric, it can be diagonalized by rotation to a principal axes system (PAS) for which the off-diagonal elements vanish, = 0. By convention, the principal axes are chosen such that... 

The Laplace equation also applies to the distribution of electrical potential and current flow in an electrically conducting medium as well as the temperature distribution and heat flow in a thermally conducting medium. For example, if => E, V => i, and fi/K => re, where re is the electrical resistivity (re = RA/Ax), Eq. (13-22) becomes Ohm s law ... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

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. 

Consider Eq. (15.1) with no sources (Laplace) applied to a square plate with U defined everywhere on the boundary. If the problem specification is symmetric under interchange of the x and y directions, the Laplace equation may then be separable with solutions of the general form U(x, y) = X(x)L(y) namely,... 

The pressure applied produces work on the system, and the creation of the bubble leads to the creation of a surface area increase in the fluid. The Laplace equation relates the pressure difference across any curved fluid surface to the curvature, 1/radius and its surface tension y. In those cases where nonspherical curvatures are present, the more universal equation is obtained ... 

Apply the Laplace equation to determine the pressure across a curved surface. 

The Laplace equation in this form is general and applies equally well to geometrical bodies whose radii of curvature are constant over the entire surface to more intricate shapes for which the Rs, are a function of surface position. In the instance of constant radii of curvature across the surface, Eq. (2.68) reduces for several common cases. For spherical surfaces, R = R2 = R, where R is the radius of the sphere, and Eq. (2.68) becomes ... 

As noted above, it is possible that a different pair of radii of curvature applies at different locations on a surface. In this case the Laplace equation shows that Ap also varies with location. This is the reason for the variation of the pressure with z in the meniscus shown in the capillary in Figure 6.3b. As is often true of pressures, it is convenient to define pressure variations relative to some reference plane. 

In Section 6.4 we discussed the pressure difference that exists across a curved surface. The Laplace equation, in the form provided by Equation (35), gives a general description of this pressure difference Ap. Our objective is to apply this relationship to the meniscus formed by a liquid surface at a flat solid wall. The first thing to notice about this is that one of the R s in Equation (35) becomes infinite since the support is planar hence this R l term disappears from the Laplace equation. 

Instead of the actual network of irregular channels, the interpretation of this experiment is based on a model that imagines the plug to consist of a bundle of cylindrical pores of radius Rc. The model is represented by Figure 6.16c. The intrusion of the liquid into the cylindrical pores in response to the applied pressure follows the same mathematical description as the rise of a liquid in a capillary. In view of the approximate nature of the model, it is adequate to use the Laplace equation in the form given by Equation (3) to describe this situation ... 

The quasi-steady-state theory has been applied particularly where a condensed phase exists whose volume changes slowly with time. This is true, for example, in the sublimation of ice or the condensation of water vapor from air on liquid droplets (M3, M4). In the condensation of water vapor onto a spherical drop of radius R(t), the concentration of water vapor in the surrounding atmosphere may be approximated by the well-known spherically symmetric solution of the Laplace equation ... 

Similarly, with the 3-D liquid water distribution in the GDL micro structure available from the two-phase LB model and representatively shown in Fig. 15, the DNS model25,27 can be applied to solve the Laplace equation for oxygen transport given below.68... 

Mathematical modeling and computer simulation have been applied for various flow studies in rectangular microchannels (see Table 3.1). An equation to describe the flow in a rectangular channel has been given [124]. Simulation of fluid flow can be conducted by solving the coupled Poisson and Navier-Stokes equation for fluid velocity [532]. However, this complicated computation has been simplified by solving the Laplace equation for the electric field because it is proportional to fluid velocity [321]. 

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... 

Note that for 9 > 90°, ze is positive i.e., it corresponds to a depression for 9 < 90°, ze is negative and corresponds to a capillary rise. Equation (1.55) can also be derived by a mechanical approach, considering the hydrostatic pressure APh = pgz and the capillary pressure APC. Applying the Laplace equation (1.20) to the capillary configuration with R] = R2 = -r/cos0 (see Figure 1.37), APcis ... 

As shown previously, the product yLVcos can be measured from experimental values of the force f (equation (3.15)). To determine 0a and 0r, one must at first calculate yLV- To this end, the Laplace equation is introduced, which applies at each point Q of the liquid meniscus surface with a vertical coordinate z relative to the flat horizontal surface of the bulk liquid (Figure 3.19.b) ... 

The difficulty in the use of these equations in practice is the experimental one of determining. for fine powders. Bartell and Walton [96] developed a method in which a pressure was applied to prevent liquid from penetrating a plug of powder. The required pressure is given by the Laplace equation ... 

Let the lengths of the semi-major and -minor axes of the ellipse be 21 and 26, while the linear size of the conductor is L (>> / or 6). A potential difference EqL is applied across the conductor (say, in the p-direction). For obtaining the voltage distribution within the conductor, one has to solve the two-dimensional Laplace equation in the xy plane... 

A variable transformation for treating coupled diffusion and migration of multiple ionic species in stagnant solution was recently described by Baker et al. [53]. The problem is redefined in terms of a pseudo-potential which obeys the Laplace equation, enabling solution by a number of available methods. Although it has not yet been applied to electrochemical microfabrication problems, this transformation could be useful to treat cases that do not involve an excess of supporting electrolyte. 

Oil Entrapment Mechanisms. Enhanced oil recovery processes depend in large part on the elimination or reduction of capillary forces. Capillary forces are the strongest that occur under typical reservoir conditions, and are most responsible for oil entrapment. Viscous forces, which act to displace oil, are composed of the applied pressure gradient, gravity, density differences between phases, and viscosity ratio. In a permeable medium, capillary forces result when the pores constrain the oil-water interface to a high degree of curvature. From the Laplace equation, the capillary pressure in a capillary tube can be derived ... 

In order to imposed the effective boundary conditions on a eigenwave in CNT [1], the equation for scalar electromagnetic potential (the Laplace equation) is applied,... 

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]. 

The Washburn equation applies to ideal cylinders (constancy of radius a and contact angle a along its length) cmd is derived on the basis of the Laplace pressure Ap = (2y cosa)/a as the driving force for a PoiseuUle-type flow rate, dV/dt = dlna h / dt, where h is the penetration depth of the intruding liquid and rj the liquid viscosity. For obtuse angles dh / dt < 0. So,... 

These concepts have been routinely employed to determine psds of genuine porous media [88]. A difficulty arises when they are applied to PEMs, since these membranes do not possess an intrinsic porosity. Instead, pores in them are created by the water of hydration, whereas in the dry state the pore network collapses. Gas permeability of PEM is very small. Thus, only with a certain degree of tolerance can one speak about three-phase capillary equilibria, implied in the Laplace equation. It is rather a semiempirical phenomenology, that allows one to relate the liquid pressure (the driving force of the hydraulic permeation)... 

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