Forces Laplace

An important structural element in a foam is the Plateau border, i.e., the channel having three cylindrical surfaces that is formed between any three adjacent bubbles. Similar channels are formed where two bubbles meet the wall of the vessel containing the foam. Figure 11.3a shows a cross section, and it follows that the Laplace pressure inside the Plateau border is smaller than that in the adjacent films. This means that liquid is sucked from the films to the Plateau borders, whence it can flow away (drain), because the Plateau borders form a connected network. The curvature in the Plateau border is determined by a balance of forces, Laplace pressure versus... 

Despite this interesting experimental work, Lavoisier s resulting explanations of the nature of heat and light seem forced. Laplace favored a mechanical explanation of heat as the motion of particles of matter (as it is currently understood), but Lavoisier described heat as a substance. This material he called caloric, the matter of fire, and described it as weightless (or too difficult to weigh), which made it reminiscent of phlogiston. But unlike the phlogistonists he could... 

Capillary Forces Laplace Equation (Liquid Curvature AND Pressure) (Mechanical Definition)... 

Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. 

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

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. 

Surfactants aid dewatering of filter cakes after the cakes have formed and have very Httle observed effect on the rate of cake formation. Equations describing the effect of a surfactant show that dewatering is enhanced by lowering the capillary pressure of water in the cake rather than by a kinetic effect. The amount of residual water in a filter cake is related to the capillary forces hoi ding the Hquids in the cake. Laplace s equation relates the capillary pressure (P ) to surface tension (cj), contact angle of air and Hquid on the soHd (9) which is a measure of wettabiHty, and capillary radius (r ), or a similar measure appHcable to filter cakes. 

The emulsification process in principle consists of the break-up of large droplets into smaller ones due to shear forces (10). The simplest form of shear is experienced in lamellar flow, and the droplet break-up may be visualized according to Figure 4. The phenomenon is governed by two forces, ie, the Laplace pressure, which preserves the droplet, and the stress from the velocity gradient, which causes the deformation. The ratio between the two is called the Weber number. We, where Tj is the viscosity of the continuous phase, G the velocity gradient, r the droplet radius, and y the interfacial tension. 

These do not contain the variable t (time) exphcitly accordingly, their solutions represent equihbrium configurations. Laplace s equation corresponds to a natural equilibrium, while Poisson s equation corresponds to an equilibrium under the influence of an external force of density proportional to g(x, y). 

The calculations show that the liquid pressure monotonically decreases along the heating region. Within the evaporation region a noticeable difference between the vapor and liquid pressures takes place. The latter is connected with the effect of the Laplace force due to the curvature of the interface surface. In the superheated region the vapor pressure decreases downstream. 

When the sphere and plane are separated by a small distance D, as shown in Figure 4, then the force due to the Laplace pressure in the liquid bridge may be calculated by considering how the total surface free energy of the system changes with separation [1] ... 

Capillary phenomena are due to the curvature of liquid surfaces. To maintain a curved surface, a force is needed. In Eq. (8), this force is related to the second term in the integral. The so-called Laplace pressure due to the force to maintain the curved surface can be expressed as, Pl = 2-yIr, where r is the curvature radius. The combination of the capillary effects and disjoining pressure can make a liquid film climb a wall. 

In a small-diameter capillary tube, wetting forces produce a distortion of the free liquid surface, which takes a curvature. Across a curved liquid surface, a difference of pressure exists and the variation of pressure across the surface is the Laplace or capillary pressure, given by ... 

Bubble Point Large areas of microfiltration membrane can be tested and verified by a bubble test. Pores of the membrane are filled with liquid, then a gas is forced against the face of the membrane. The Young-Laplace equation, AF = (4y cos Q)/d, relates the pressure required to force a bubble through a pore to its radius, and the interfacial surface tension between the penetrating gas and the liquid in the membrane pore, y is the surface tension (N/m), d is the pore diameter (m), and P is transmembrane pressure (Pa). 0 is the liquid-solid contact angle. For a fluid wetting the membrane perfectly, cos 0 = 1. 

Liquid between the surface of two solid bodies gives rise to boundary forces. A pressure difference arises and is known as the capillary pressure (Pc). This can be calculated from Laplace s equation. 

Using deterministic kinetics, one can force-fit the time evolution of one species—for example, eh but then those of other yields (e.g., OH) will be inconsistent. Stochastic kinetics can predict the evolutions of radicals correctly and relate these to scavenging yields via Laplace transforms. 

Concentration Profiles In the general case but with a linear isotherm, the concentration profile can be found by numerical inversion of the Laplace-domain solution of Haynes and Sarma [see Lenhoff, J. Chromatogr., 384, 285 (1987)] or by direct numerical solution of the conservation and rate equations. For the special case of no axial dispersion, an explicit time-domain solution is also available in the cyclic steady state for repeated injections of arbitrary duration tE followed by an elution period tE with cycle time tc = tE + tE [Carta, Chem. Eng. Sci, 43, 2877 (1988)]. For the linear driving force mechanism, the solution is... 

Laplace s equation, V2V = 0, in any number of dimensions, describes a system of balanced forces in a potential field. The equation is satisfied by a variety of functions, such as... 

The relations between the intrinsic pressure and other physical constants developed in the foregoing paragraphs have been found from theoretical considerations based on Laplace s theory, that is, on the assumption of cohesive forces acting over very small distances. They are of interest to us inasmuch as there is a necessary connection between intrinsic pressure and surface tension. While no numerical expression has so far been found for this, it is obvious that high intrinsic pressures must be accompanied by high surface tensions, since the surface tension is a manifestation of the same cohesive force as causes intrinsic pressure. (See, however, equation 3, p. 27, for an empirical relation between the two.)... 

Let us start with the action of Young-Laplace law (Equation 9.6), which determines the equilibrium configuration of the fluids (liquid and liquid-like phases) and the driving force of mass transfer that cause the spontaneous formation of equilibrium configurations. 

Laplace could not express any opinion on the absolute values of the attraction assumed by him. At present, our knowledge of interatomic, interionic, and analogous forces is much greater than 170 years ago, and attempts to calculate surface energies and surface tensions are possible. 

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