Three-dimensional Laplace equation

This is Laplace s equation, which describes potential flow. It is widely used in heat flow and electrostatic field problems an enormous number of solutions to Laplace s equation are known for various geometries. These can be used to predict the two-dimensional flow in oil fields, underground water flow, etc. The same method can be used in three dimensions, but solutions are more difficult. The solutions to the two-dimensional Laplace equation for common problems in petroleum reservoir engineering are summarized by Muskat [3]. The analogous solutions for groundwater flow are shown in the numerous texts on hydrology, e.g., Todd [4]. See Chap. 10 for more on potential flow. 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

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. 

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... 

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. 

What is of further interest here, as a model of the hydrogen atom and its angular momentum, is the vibration of a three-dimensional fluid sphere in a central field. As in 2D the wave equation separates into radial and angular parts, the latter of which determines the angular momentum and is identical with the angular part of Laplace s equation. 

The summation is over all the spanwise modes. One can use the above ansatz in three-dimensional Navier-Stokes equation and linearize the resultant equations after making a parallel flow approximation to get the following Orr-Sommerfeld equation for the Fourier- Laplace transform f of v ... 

For calculating one-dimensional flow behavior in a packed bed, the Ergun equation is generally sufficient. For single-phase, constant-density flow in two or three dimensions, Darcy s law can be substituted into the continuity equation (V v = 0) and simplified to obtain the Laplace equation ... 

To gain some feeling for the idea of a potential flow, we will show what kind of flows are described by various choices of (f>. We restrict our attention to two-dimensional flows, because they are mathematically much easier than three-dimensional flows. In general, will be < > = (x, y), but not every such function satisfies Laplace s equation, so not every such function represents a potential flow. You may verify that = and = sin jc... 

The important idea of an irrotational flow is that at any point in the fluid the angular velocity about any axis is zero. This is shown for zero angular velocity about any axis perpendicular to the xy plane in Figs. 10.10 and 10.11. And it can be shown that for any three-dimensional flow which obeys Laplace s equation the angular velocity is zero about any axis. 

As noted in the chapter on Volume Conductor Theory, most bioelectric field problems can be formulated in terms of either the Poisson or the Laplace equation for electrical conduction. Since Laplace s equation is the homogeneous counterpart of the Poisson equation, we will develop the treatment for a general three-dimensional Poisson problem and discuss simplifications and special cases when necessary. 

For a three-dimensional system (x, y, z), the potential variation in the eleetrolyte volume ean eorrespondingly be expressed by the Laplace equation... 

Special linear systems arise from the Poisson equation, d uldx + d uldy = f x, y) on a rectangle, 0 Laplace equation of Section II.A is a special case where fix, y) = 0.] If finite differences with N points per variable replace the partial derivatives, the resulting linear system has equations. Such systems can be solved in 0(N log N) flops with small overhead by special methods using fast Fourier transform (FFT) versus an order of AC flops, which would be required by Gaussian elimination for that special system. Storage space also decreases from 2N to units. Similar saving of time and space from O(N ) flops, 2N space units to 0(N log N) flops and space units is due to the application of FFT to the solution of Poisson equations on a three-dimensional box. 

Considering capillary dynamics, the pressure drop term is often described by the Laplace equation, AP = 2yH, where y represents liquid surface tension and H represents the mean curvature of the liquid-gas interface associated with aU curves, C, passing through the surface. Furthermore, the character of a sufficiently smooth surface is through the invariant from differential geometry, the principal curvature of each curve, kj. The radii of curvature are the inverse of each principal curvature, A , = 1/r,. Considering the maximum and minimum radii of curvature at a point on a three-dimensional surface, the mean curvature can be calculated explicitly (see Appendix for more thorough derivation of the mean curvature parameter) ... 

Curvature is usually assumed to follow geometries that can be fit to a circle (or sphere in the three-dimensional case) in the Laplace equation this is not typically the case but usually results in a decent approximation. Further, for straight paths, r2 can be considered as infinite, and approximations can be made for rj [3] ... 

Now we return to the example of Fig. 4a and analyze it in more detail. We take, as shown in Fig. 7, a problem with 256 grid points arranged as a 16 x 16 mesh. We decompose it onto 16 processors. We replace the wave equation of a seismic simulation by the similar iterative solution to Laplace s equation where potential (/, j) is to be determined at grid points (/, j). Iteration is not the best way to tackle this particular problem. However, more sophisticated iterative techniques are probably the best approach to large three-dimensional finite-difference or finite-element calculations. Thus, although the simple example in Fig. 7 is not real or interesting itself, it does illustrate important issues. The computational solution consists of a simple algorithm... 

Specifically these are the associated Legendre polynomials of the first kind and are usually written as fimctions of cosO rather than 6. They are named after Adrien-Marie Legendre (1752-1833), who discovered them as a general family of solutions to differential equations in spherical coordinates while he was working on a mathematical description of the motions of stars. His colleague, Simon-Pierre Laplace (1749-1827), then drew on the Legendre polynomials to formulate the three-dimensional spherical harmonics. 

Obviously, the above transformed governing equations for a linear viscoelastic material (Eqs. 9.33- 9.36) are of the same form as the governing equations for a linear elastic material (Eqs. 9.25 - 9.28) except they are in the transform domain. This observation leads to the correspondence principle for three dimensional stress analysis For a given a viscoelastic boundary value problem, replace all time dependent variables in all the governing equations by their Laplace transform and replace all material properties by s times their Laplace transform (recall, e.g., G (s) = sG(s)),... 

So far we have treated two-dimensional planar flows. However, many three-dimensional problems are also amenable to analytical solution. To proceed, we introduce the notion of the point spherical source. Actually, the concept is best taught through global mass conservation considerations. We consider two-dimensional flows first. First, the radial Darcy velocity is proportional to dp/dr. This, times the area 2cr in planar problems, must be constant hence, in such flows, dp/dr goes like 1/r, which on integration leads to the expected logarithmic pressure. In three dimensions, dp/dr x 4 7t r must remain constant thus, dp/dr goes like l/i, so that p(r) varies like 1/r. This describes the point spherical source. We could also have started more formally with the spherically symmetric form of Laplace s equation,... 

Illustration 2.9 Laplace s Equation, Steady-State Diffusion in Three-Dimensional Space Emissions from Embedded Sources... 

The primary information obtained from Laplace s equation concerns the three-dimensional concentration distributions, which are of no direct practical use. It is common practice to convert these results into an equivalent rate equation based on a linear driving force. For conduction, it takes the form... 

Equation (2) is recognized as Laplace s equation in four dimensions and in alternative form is known as the wave equation in three-dimensional space. 

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