Laplace equation solution

In the majority of the Laplace equation solutions, the electrical potential, subject to appropriate boundary conditions, is determined. These primary or secondary current-distribution problems may appear to be particularly relevant for electrodeposition, where useful deposit properties are obtained at small fractions of the limiting current. However, the fact that industry has paid considerable attention to fluid flow in reactor design suggests that flow effects can be important, even at a relatively small fraction of the limiting current density. ... 

Returning to equilibrium shapes, these have been determined both experimentally and by solution of the Young-Laplace equation for a variety of situations. Examples... 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

In the case of the free jet, the solution for the Aaberg exhaust system can be found by solving the Laplace equation by the method of separation of variables and assuming that there is no fluid flow through the surface of the workbench. At the edge of the jet, which is assumed to be at 0—0, the stream function is given by Eq. (10.113). This gives rise to... 

As the corrosion rate, inclusive of local-cell corrosion, of a metal is related to electrode potential, usually by means of the Tafel equation and, of course, Faraday s second law of electrolysis, a necessary precursor to corrosion rate calculation is the assessment of electrode potential distribution on each metal in a system. In the absence of significant concentration variations in the electrolyte, a condition certainly satisfied in most practical sea-water systems, the exact prediction of electrode potential distribution at a given time involves the solution of the Laplace equation for the electrostatic potential (P) in the electrolyte at the position given by the three spatial coordinates (x, y, z). 

The solution of the Laplace equation is not trivial even for relatively simple geometries and analytical solutions are usually not possible. Series solutions have been obtained for simple geometries assuming linear polarisation kinetics "" . More complex electrode kinetics and/or geometries have been dealt with by various numerical methods of solution such as finite differencefinite elementand boundary element. ... 

The numerical approaches to the solution of the Laplace equation usually demand access to minicomputers with fast processing capabilities. Numerical methods of this sort are essential when the electrolyte is unconfined, as for an off-shore rig or a submarine hull. However, where the electrolyte is confined, as within essentially cylindrical equipment such as pipework and heat-exchangers, or for restricted electrolyte depths, a simpler modelling procedure may be adopted in the case of electrolytes of good conductivity, such as sea-water . This simpler procedure enables computation to be carried out on small, desk-top microcomputers. 

G. Lippmann introduced the capillary electrometer to measure the surface tension of mercury (Fig. 4.10). A slightly conical capillary filled with mercury under pressure from a mercury column (or from a pressurized gas) is immersed in a vessel containing the test solution. The weight of the mercury column of height h is compensated by the surface tension according to the Laplace equation... 

Lagrange multipliers 255-256 Lagrange s moan-value theorem 30-32 Lagperre polynomials 140, 360 Lambert s law 11 Langevin function 61n Laplace transforms 279—286 convolution 283-284 delta function 285 derivative of a function 281-282 differential equation solutions 282-283... 

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... 

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,... 

Matrix D has I s along the diagonal, reflecting the use of a normalized discrete Laplace equation with a = —k /[2 h + 1 )], and B is a multiple of the identity matrix with j8 = —h /[2(h -I-1 )]. Matrix A displays a sparse block structure whose off-diagonal coefficients must be less than 1 to converge to a solution. 

Relaxation methods involve iteratively seeking a convergent solution to the Laplace equation. In the present case, for instance, if we rewrite the coefficient matrix A = I + E, where the latter matrix consists of elements that are all small compared to 1, the matrix Laplace equation takes the form = EU + b. One begins the calculation with values U = b [or, equivalently, U = 0] and iteratively computes successive values The calculation terminates when a specified limit of accuracy is achieved. One such measure involves calculating the proportional differences ... 

Consider a one dimensional oil droplet pressed against a solid surface and surrounded by an aqueous solution of surfactant (Fig. 8). The shape of the oil-water interface far away from the sohd surface is governed by the Laplace equation [18] and close to the sohd surface is augmented by an additional term relating to the film energy given by the structural disjoining pressure [12] ... 

The subject of kinetics is often subdivided into two parts a) transport, b) reaction. Placing transport in the first place is understandable in view of its simpler concepts. Matter is transported through space without a change in its chemical identity. The formal theory of transport is based on a simple mathematical concept and expressed in the linear flux equations. In its simplest version, a linear partial differential equation (Pick s second law) is obtained for the irreversible process, Under steady state conditions, it is identical to the Laplace equation in potential theory, which encompasses the idea of a field at a given location in space which acts upon matter only locally Le, by its immediate surroundings. This, however, does not mean that the mathematical solutions to the differential equations with any given boundary conditions are simple. On the contrary, analytical solutions are rather the, exception for real systems [J. Crank (1970)]. 

Solutions to the Laplace equation are called harmonic functions. Some harmonic functions are given below for particular boundary conditions. 

Using the solution of the Laplace equation for diffusion in cylindrical coordinates given by Eq. 5.10, fitting it to the boundary conditions given by Eq. 16.80, and employing Eq. 13.3 for the flux, the total diffusion current of atoms (per unit pipe length) passing radially from R,n to f out is... 

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 ... 

Recent theoretical studies indicate that thermal fluctuation of a liquid/ liquid interface plays important roles in chemical/physical properties of the surface [34-39], Thermal fluctuation of a liquid surface is characterized by the wavelength of a capillary wave (A). For a macroscopic flat liquid/liquid interface with the total length of the interface of /, capillary waves with various A < / are allowed, while in the case of a droplet, A should be smaller than 2nr (Figure 1) [40], Therefore, surface phenomena should depend on the droplet size. Besides, a pressure (AP) or chemical potential difference (An) between the droplet and surrounding solution phase increases with decreasing r as predicted by the Young-Laplace equation AP = 2y/r, where y is an interfacial tension [33], These discussions indicate clearly that characteristic behavior of chemical/physical processes in droplet/solution systems is elucidated only by direct measurements of individual droplets. 

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