The strategy of modeling and solving the problems described by Laplace’s equation with uncertainly defined boundary shape and boundary conditions

2022 ◽  
Vol 582 ◽  
pp. 439-461
Author(s):  
Eugeniusz Zieniuk ◽  
Marta Czupryna
Keyword(s):  
Boundary Conditions ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Boundary Shape

Moments of the Flow Variables, Part II : The Effective Conductivity

2003 ◽  
Author(s):  
Yoram Rubin
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Effective Conductivity ◽  
Effective Property ◽  
Flow Equation ◽  
Laplace’S Equation ◽  
Flow Conditions ◽  
Laplace's Equation ◽  
Head Gradient ◽  
The Difference

Many applications require primary information such as average fluxes as a prelude to more complex calculations. In water balance calculations one may be interested only in the average fluxes. For both cases the concept of effective conductivity is useful. The effective hydraulic conductivity is defined by where the angled brackets denote the expected value operator. The local flux fluctuation is defined by the difference qi(x) — (qi(x)). Its statistical properties as well as those of the velocity will be investigated in chapter 6. To qualify as an effective property in the strict physical sense, Kef must be a function of the aquifer’s material properties and not be influenced by flow conditions such as the head gradient and boundary conditions (Landauer, 1978). Our goal in this chapter is to explore the concept of the effective conductivity Kef and to relate it to the medium’s properties under as general conditions as possible. Additionally, we shall explore the conditions where this concept is irrelevant and applicable, the important issue being that Kef is defined in an ensemble sense, but for applications we need spatial averages. Several methods for deriving Kef will be described below. The general approach for defining Kef includes the following steps. First, H is defined as an SRF and is expressed with the aid of the flow equation in terms of the hydro-geological SRFs (conductivity, mostly) and the boundary conditions. The H SRF is then substituted in Darcy’s law and an expression in the form equivalent to (5.1) is sought. If and only if the coefficient in front of the mean head gradient is not a function of the flow conditions will it qualify as Kef. The derivation of the effective conductivity employs the flow equation. In steady-state incompressible flow, for example, Laplace’s equation is employed. Solutions derived under Laplace’s equation are applicable, under appropriate conditions, for other physical phenomena governed by the same mathematical model. For example, the electrical field in steady state is also described by Laplace’s equation.

An Electrooptic Analog for the Solution of the Laplace Equation

1967 ◽  
Vol 34 (2) ◽  
pp. 452-456
Author(s):  
R. O’Regan
Keyword(s):  
Boundary Conditions ◽  
Potential Gradient ◽  
Mathematical Physics ◽  
Experimental Methods ◽  
Organic Dye ◽  
Voltage Gradient ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Physical Problems ◽  
Flow Heat

Laplace’s equation occurs frequently in mathematical physics for problems relating to fluid flow, heat transfer, and so on. For some simple cases, the boundary-value problem can be solved; but more often, the differential equation proves intractable, and numerical analysis or experimental methods are used. The electrooptic analog is an experimental method based upon the fact that an organic dye solution becomes birefringent in an electric field. This effect enables one to determine voltage gradient throughout a two-dimensional field. The boundary conditions most readily applied are prescribed constant values of the electric potential φ on conducting segments of a boundary and ∂φ/∂n = 0 on insulated segments of a boundary. With the known conditions on the boundary and the potential gradient found from experiment, the problem is solved. This analog can be used for all physical problems for which the boundary conditions are applicable, and which satisfy Laplace’s equation.

scholarly journals The Block-Grid Method for Solving Laplace's Equation on Polygons with Nonanalytic Boundary Conditions

2010 ◽  
Vol 2010 (1) ◽  
pp. 468594 ◽  
Author(s):  
AA Dosiyev ◽  
S Cival Buranay ◽  
D Subasi
Keyword(s):  
Boundary Conditions ◽  
Grid Method ◽  
Laplace’S Equation ◽  
Laplace's Equation
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