Two-dimensional Laplace equation

The most common technique for the derivation of fundamental solutions is to use integral transforms, such as, Fourier, Laplace or Hankel transforms [29, 39]. For simple operators, such as the Laplacian, direct integration and the use of the properties of the Dirac delta are typically used to construct the fundamental solution. For the case of a two-dimensional Laplace equation we can use a two-dimensional Fourier transform, F, to get the fundamental solution as follows,... 

Consider the two-dimensional Laplace equation for temperature in the domain presented in... 

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... 

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. 

For example, substitution of f(x,y) = X(x).Y(y) in the two-dimensional Laplace equation and solution of the two resulting one-dimensional equations gives a complete set. The trial function has the form... 

This is the two-dimensional Laplace equation for steady state flow in an isotropic porous medium. The partial differential equation governing the two-dimensional unsteady flow of water in an anisotropic aquifer can be written as ... 

The field between the planes with the given potentials will be electrostatic and plane-parallel and satisfies the two-dimensional Laplace s equation... 

In this paper, an inverse problem for galvanic corrosion in two-dimensional Laplace s equation was studied. The considered problem deals with experimental measurements on electric potential, where due to lack of data, numerical integration is impossible. The problem is reduced to the determination of unknown complex coefficients of approximating functions, which are related to the known potential and unknown current density. By employing continuity of those functions along subdomain interfaces and using condition equations for known data leads to over-determined system of linear algebraic equations which are subjected to experimental errors. Reconstruction of current density is unique. The reconstruction contains one free additive parameter which does not affect current density. The method is useful in situations where limited data on electric potential are provided. 

Therefore, both the real and imaginary parts of an analytic function are solution of the two-dimensional Laplace s equation, known as harmonic functions. This can be verified for m(x, y) = x2 — y2 and u(x, y) = 2xy, given in tbe above example of the analytic function w(z) = z. ... 

The idea of the method of finite differences of solving boundary conditions for a two-dimensional Laplace or Poisson equation is as follows ... 

FMM was initially introduced by Rokhlin [2] as a fast solution method for inte al equations for two dimensional Laplace s equation and then developed by Greengard [3]. FMM has been plied to problems in various fields such as BEM and Molecular Dynamics. 

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, /. 

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.. 

Green s identities for a 2D Laplace s equation (heat conduction) Here, we will demonstrate how to develop Green s identities for a two-dimensional heat conduction problem, which for a material with constant properties is described by the Laplace equation for the temperature, i.e.,... 

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 ... 

We now intend to examine the modifying effect of the elliptic hole on the distribution of stress in the plate. A straightforward but tedious solution of the Laplace equation for the displacement vector field leads (See section 1.2.2(a), for the equivalent solution of the scalar potential problem in a two-dimensional conductor with an elliptic dielectric hole inside), to the largest concentration of stress at the point C, where... 

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 ... 

Airy Stress Eunction and the Biharmonic Equation The biharmonic equation in many instances has an analogous role in continuum mechanics to that of Laplace s equation in electrostatics. In the context of two-dimensional continuum mechanics, the biharmonic equation arises after introduction of a scalar potential known as the Airy stress function f such that... 

The direct integration of the Laplace equation in the case of a one-dimensional Cartesian problem renders a linear function of distance between the two electrodes, when the current density is the same for the two electrodes. 

The parabolic barrier plays a special role in rate theory. The GLE (with space-independent friction) may be solved analytically using Laplace transforms. The two-dimensional Fok-ker-Planck equation derived from the Langevin equation may be solved analytically, as was done by Kramers in his famous paper of 1940. In this section we present some of the analytic results for the parabolic barrier dynamics. These results are important from both a conceptual and a practical point of view. Later we shall see how one returns to the parabolic barrier case as a source of comprehension, approximation, etc. 

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... 

Potential Flow around a Gas Bubble Via the Scalar Velocity Potential, An incompressible fluid with constant approach velocity (i.e., S Vapproach) flows upward past a stationary nondeformable gas bubble of radius R. This two-dimensional flow is axisymmetric about the scalar velocity potential spherical coordinates because this coordinate system provides the best match with the macroscopic boundary at r = / . The appropriate partial differential equation for is... 

This motion of the bubble induces axisymmetric two-dimensional flow in the liquid phase. In the potential flow regime, one calculates the scalar velocity potential tb(r, 0) via Laplace s equation. The general solution in spherical coordinates is... 

Potential Flow Transverse to a Long Cylinder Via the Scalar Velocity Potential. The same methodology from earlier sections is employed here when a long cylindrical object of radius R is placed within the flow field of an incompressible ideal fluid. The presence of the cylinder induces Vr and vg within its vicinity, but there is no axis of symmetry. The scalar velocity potential for this two-dimensional planar flow problem in cylindrical coordinates must satisfy Laplace s equation in the following form ... 

Hence, cos6> is a good function that satisfies the angular part of Laplace s equation for (r, 0) via separation of variables for axisynunetric flow in spherical coordinates, and two-dimensional planar flow transverse to a long cylinder. At any position with the incompressible liquid, one postulates that... 

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