Laplace s equation

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. 

Due to the conservation law, the diffiision field 5 j/ relaxes in a time much shorter than tlie time taken by significant interface motion. If the domain size is R(x), the difhision field relaxes over a time scale R Flowever a typical interface velocity is shown below to be R. Thus in time Tq, interfaces move a distanc of about one, much smaller compared to R. This implies that the difhision field 6vj is essentially always in equilibrium with tlie interfaces and, thus, obeys Laplace s equation... 

Equation (A3.3.73) is referred to as the Gibbs-Thomson boundary condition, equation (A3.3.74) detemiines p on the interfaces in temis of the curvature, and between the interfaces p satisfies Laplace s equation, equation (A3.3.71). Now, since ] = -Vp, an mterface moves due to the imbalance between the current flowing into and out of it. The interface velocity is therefore given by... 

The solution of Laplace s equation, (A3.3.71), with these boundary conditions is, for [Pg.749]

Lapachic acid Lapacol [84-79-7 Laparoscope Lapelloy Lapis Lazuli Lapis lazuli [1302-85-8] Laplace pressure Laplace s equation... 

Equation 9 is Laplace s equation which also occurs in several other fields of mathematical physics. Where the flow problem is two-dimensional, the velocities ate also detivable from a stream function, /. 

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. 

Although the assumption of electroneutrahty is appropriate for many systems, electroneutrahty does not imply that Laplace s equation holds for the... 

When concentration gradients in the solution can be ignored, equations 25 through 29 show that the electric potential is governed by Laplace s equation... 

Laplace s equation is appHcable to many electrochemical systems, and solutions are widely available (8). The current distribution is obtained from Ohm s law... 

The distribution of current (local rate of reaction) on an electrode surface is important in many appHcations. When surface overpotentials can also be neglected, the resulting current distribution is called primary. Primary current distributions depend on geometry only and are often highly nonuniform. If electrode kinetics is also considered, Laplace s equation stiU appHes but is subject to different boundary conditions. The resulting current distribution is called a secondary current distribution. Here, for linear kinetics the current distribution is characterized by the Wagner number, Wa, a dimensionless ratio of kinetic to ohmic resistance. 

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). 

Harmonic Functions Both the real and the imaginary )arts of any analytic function/= u + iij satisfy Laplaces equation d /dx + d /dy = 0. A function which possesses continuous second partial derivatives and satisfies Laplace s equation is called a harmonic function. 

For solutions of uniform composition, Eq. (22-21) reduces to Laplace s equation for the potential ... 

For many particles, the diffuse-charge layer can be characterized adequately by the value of the zeta potential. For a spherical particle of radius / o which is large compared with the thickness of the diffuse-charge layer, an electric field uniform at a distance from the particle will produce a tangential electric field which varies with position on the particle. Laplace s equation [Eq. (22-22)] governs the distribution... 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

Substituting the conditions imposed by continuity gives Laplace s equation ... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Outside the jet and away from the boundaries of the workbench the flow will behave as if it is inviscid and hence potential flow is appropriate. Further, in the central region of the workbench we expect the airflow to be approximately two-dimensional, which has been confirmed by the above experimental investigations. In practice it is expected that the worker will be releasing contaminant in this region and hence the assumption of two-dimensional flow" appears to be sound. Under these assumptions the nondimensional stream function F satisfies Laplace s equation, i.e.. 

Laplace s equation, 146 Least action, principle of, 69, 304 Line of heterogeneous states, 172 Liquefaction of gases, 167, 173 of mixtures, 428... 

Newman, J. Determination of Current Distributions Governed by Laplace s Equation 23... 

Determination of Current Distributions Governed by Laplace s Equation Direct Methanol Fuel Cells From a Twentieth Century Electrochemist s Dream to a Twenty-First Century Emerging Technology West, A. C. Newman, J. Lamy, C. Ueger, J.-M. Srinivasan, S. 23... 

At the same time, in the vicinity of points where masses are absent, ( = 0), in place of Equation (1.41) we have Laplace s equation ... 

Both Poisson s and Laplace s equations describe the behavior of the potential at regular points where the first derivatives of the field exist. To characterize the behavior of the potential at the boundary of media with different densities, let us make use of Equation (1.39) according to which a component of the field along some direction / is equal to the derivative of the potential in this direction ... 

Now we return to Poisson s and Laplace s equations, which describe the behavior of the potential inside and outside masses, respectively. Earlier we have already derived an expression for the potential ... 

Therefore, if the function U satisfies the Laplace s equation, then it possesses a remarkable interesting feature, namely, its average value calculated around some point p is exactly equal to the value of the function at this point. A certain class of functions has this feature only, and such functions are called harmonic. Correspondingly, we conclude that the potential of the attraction field is a harmonic function outside the masses. In accordance with Laplace s equation the sum of the second derivatives along coordinate lines, v, y, and z, equals zero, provided that U(p) is a harmonic function. At the same time we know that in the one-dimensional case there is a class of functions for which the second derivative is equal to zero, that is. 

If U and U2 are solutions of Poisson s equation and obey the boundary condition (1.94), their difference = U2—U1 satisfies Laplace s equation and the condition on the surface S ... 

Indeed, it satisfies Laplace s equation everywhere except at the point p, since it describes up to a constant the potential of a point mass located at the point p. Also, it has a singularity at this point and provides a zero value of the surface integral over the hemisphere when its radius r tends to infinity. Correspondingly, we can write... 

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