Derive Laplace Equation for Incompressible Irrotational Continuous Flow

Laplaces Equation

The Laplace equation is a mixed boundary problem which involves a boundary condition for the applied voltage on the electrode surface and a zero-flux condition in the direction normal to the electrode plane.

From: Computer Aided Chemical Engineering , 2019

Motivating Ideas and Governing Equations

Wilson C. Chin , in Quantitative Methods in Reservoir Engineering (Second Edition), 2017

Validity of Laplace's Equation

Since Laplace's equation, that is, Eq. (1.13) for the Darcy pressure and Eq. (1.30) for the inviscid aerodynamic potential, arise in both problems as a result of different physical limits, it is of interest to ask when the approximate models apply and why. This understanding is crucial to the translation process alluded to earlier, so that "fixes" used in aerodynamics, which may be inappropriate to Darcy flows, can be removed if and when they are present. It is especially important because the analogies presupposed by nonspecialists are sometimes not analogous at all.

Fig. 1.2 shows a typical streamline in the Darcy flow beneath a dam; the sketch is based on photographs of sand model experiments, with sheet pilings at "heel alone" and "heel plus toe" (e.g., see Muskat, 1937). The complete streamline pattern can be predicted quite well using the planar, liquid limit of Eq. (1.13), so that the solutions apply to all oncoming flow angles up to a sharp 180^° (refer to Muskat's work for detailed drawings). Thus, Eq. (1.13) appears to be generally valid for all low Reynolds number flows. Now, Fig. 1.3 shows a flat plate airfoil at a not-so-small "angle of attack" or flow inclination relative to the oncoming fluid. The creation of eddies at the trailing edge, which increase in size with increasing angle, is indicated.

Fig. 1.2. Darcy flow streamline beneath dam (un-separated).

Fig. 1.3. Inviscid flow streamlines past thin airfoils (separated).

The viscous effects in Fig. 1.3 require an analysis using the full unsteady Navier-Stokes equations; they cannot be modeled using Eq. (1.30). But Eq. (1.30) is meaningful for small flow inclinations, say, less than 10 degrees. When this is the case, it admits an infinity of solutions, each corresponding to a different position of the aft stagnation point C. This position is fixed and the solution rendered unique by forcing C to coincide with the trailing edge location D. This so-called Kutta-Joukowski theorem allows Eq. (1.30) to mimic solutions of the more rigorous Navier-Stokes model. That solutions to Laplace's equation are not unique may not be well known to petroleum engineers, who are accustomed to dealing with log r solutions. This nonuniqueness is related to the existence of θ solutions, usually reserved for advanced math courses. These elementary solutions, discussed briefly next and in Chapter 3, are important to modeling impermeable flow barriers like shale lenses.

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Rotations of Asymmetric Molecules and the Hydrogen Atom in Free and Confined Configurations

Ricardo Méndez-Fragoso , Eugenio Ley-Koo , in Advances in Quantum Chemistry, 2011

2.6 Common Generating Function for Three-Dimensional Hydrogen Atom Complete Wavefunctions

The Laplace equation and the Schrödinger equation for the free particle, the harmonic oscillator, and the hydrogen atom, in different dimensions, are superintegrable, which means that they admit separable and integrable solutions in different coordinate systems [14]. In general, the complete solutions of such equations in the respective coordinate systems can be obtained from common generating functions, as already illustrated in Sections 2.4 and 2.5 for the spherical and spheroconal solutions of the Laplace and Helmholtz equations. Our own works on Mathieu functions [35], spheroidal functions [36], D-dimensional harmonic oscillators [37], two-dimensional Hydrogen atom [38], and three-dimensional hydrogen atom [7] have identified or constructed such common generating functions including their expansions in the complete wavefunctions.

Here, we discuss the last one with focus on the rotations of the hydrogen atom. It is well known that the corresponding Schrödinger equation is separable in spherical, parabolic, and spheroidal prolate coordinates, and also in spheroconal coordinates [14]. In this case, the identification of the common generating function is based on the recognition that the Schrödinger equation separates in parabolic coordinates into equations with the same structure as the radial equations of two-dimensional harmonic oscillators in circular coordinates. Correspondingly, the common generating function for the three-dimensional hydrogen atom in such coordinates is the product of two known generating functions of two-dimensional harmonic oscillators and their expansions in the respective eigenfunctions in parabolic coordinates, with angular dependencies that guarantee the generation of the proper angular eigenfunctions e^imφ .

The transformations from parabolic coordinates into spherical, spheroconal, and prolate spheroidal coordinates allow the rewriting of the generating function in each one of them. Its respective expansions for each case are straightforward, but laborious. For the spherical and spheroidal cases, the common angular eigenfunctions e^imφ serve as guides to complete the job. In particular, in the spherical case, the complete angular dependence can be expressed as $P_{\ell }({\widehat{r}}_0·\widehat{r})$ through the addition theorem, reflecting the rotational invariance of the system. The latter leads to the extension for the spheroconal coordinates, just as in Section 2.4 and 2.5, with common radial functions. The interested reader may follow the details in Ref. [7].

The equal standing of spherical harmonics and spheroconal harmonics is manifested in this case. The rotations of the hydrogen atom are familiarly described through eigenfunctions of ${\widehat{L}}^2$ and ${\widehat{L}}_z$ , but they also admit eigenfunctions of ${\widehat{L}}^2$ and Ĥ*. The same holds for any central potential quantum system. Some of the consequences are illustrated in the following section.

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Boundary Value Problems

George Lindfield , John Penny , in Numerical Methods (Fourth Edition), 2019

6.7 Elliptic Partial Differential Equations

The solution of a second-order elliptic partial differential equation is determined over a closed region, and the shape of the boundary and its condition at every point must be specified. Some important second-order elliptic partial differential equations, which arise naturally in the description of physical systems, are

(6.34) $\text{Laplace's equation:}{\mathrm{\nabla }}^{2}z=0$

(6.35) $\text{Poisson's equation:}{\mathrm{\nabla }}^{2}z=F(x,y)$

(6.36) $\text{Helmholtz's equation:}{\mathrm{\nabla }}^{2}z+G(x,y)z=F(x,y)$

where ${\mathrm{\nabla }}^{2}z={\partial }^{2}z/\partial x^2+{\partial }^{2}z/\partial y^2$ and $z(x,y)$ is an unknown function. Note that the Laplace and Poisson equations are special cases of Helmholtz's equation. In general, these equations must satisfy boundary conditions that are specified in terms of either the function value or the derivative of the function which is normal to the boundary. Furthermore, a problem can have mixed boundary conditions. If we compare (6.34), (6.35), and (6.36) to the standard second-order partial differential equation in two variables, that is

$A(x,y)\frac{{\partial }^{2}z}{\partial x^2}+B(x,y)\frac{{\partial }^{2}z}{\partial x\partial y}+C(x,y)\frac{{\partial }^{2}z}{\partial y^2}+f(x,y,z,\frac{\partial z}{\partial x},\frac{\partial z}{\partial y})=0$

we see that in each case $A=C=1$ and $B=0$ , so that $B^2-4AC<0$ , confirming that the equations are elliptic.

Laplace's equation is homogeneous, and if a problem has boundary conditions that are also homogeneous then the solution, $z=0$ , will be trivial. Similarly in (6.35), if $F(x,y)=0$ and the problem boundary conditions are homogeneous, then $z=0$ . However, in (6.36) we can scale $G(x,y)$ by a factor λ, so that (6.36) becomes

(6.37) ${\mathrm{\nabla }}^{2}z+\lambda G(x,y)z=0$

This is a characteristic or eigenvalue problem, and we can determine values of λ and corresponding non-trivial values of $z(x,y)$ .

The elliptic equations (6.34) through (6.37) can only be solved in a closed form for a limited number of situations. For most problems, it is necessary to use a numerical approximation. Finite difference methods are relatively simple to apply, particularly for rectangular regions. We will now use the finite difference approximation for ${\mathrm{\nabla }}^{2}z$ , given by (6.12) or (6.13), to the solve some elliptic partial differential equations over a rectangular domain.

Example 6.3

Laplace's equation. Determine the distribution of temperature in a rectangular plane section, subject to a temperature distribution around its edges as follows:

$x=0,T=100y;x=3,T=250y;y=0,T=0;\text{and}y=2,T=200+(100/3)x^2$

The section shape, the boundary temperature distribution and the two chosen nodes where we require the values of the temperature to be computed, are shown in Fig. 6.12.

Figure 6.12. Temperature distribution around a plane section. Nodes 1 and 2 are shown.

The temperature distribution is described by Laplace's equation. Solving this equation by the finite difference method, we apply (6.13) to nodes 1 and 2 of the mesh shown in Fig. 6.12. This gives

$\begin{array}{c}\hfill (233.33+T_2+0+100-4T_1)/h^2=0\\ \hfill (333.33+250+0+T_1-4T_2)/h^2=0\end{array}$

where $T_1$ and $T_2$ are the unknown temperatures at nodes 1 and 2, respectively, and $h=1$ . Rearranging these equations gives

$\left[\begin{array}{cc}\hfill -4& \hfill 1\\ \hfill 1& \hfill -4\end{array}\right]\left[\begin{array}{c}\hfill T_1\\ \hfill T_2\end{array}\right]=\left[\begin{array}{c}\hfill -333.33\\ \hfill -583.33\end{array}\right]$

Solving this equation, we have $T_1=127.78$ and $T_2=177.78$ .

If we require a more accurate solution of Laplace's equation, then we must use more nodes and the computation burden increases rapidly. The following Matlab function ellipgen uses the finite difference approximation (6.12) to solve the general elliptic partial differential equations (6.34) through (6.37) for a rectangular domain only. The function is also limited to problems in which the boundary value is specified by values of the function $z(x,y)$ , not its derivative. If the user calls the function with 10 arguments, the function solves (6.34) through (6.36); see Examples 6.4 and 6.5. Calling it with the first six arguments causes it to solve (6.37); see Example 6.6.

function [a,om] = ellipgen(nx,hx,ny,hy,G,F,bx0,bxn,by0,byn)

% Function either solves:

% nabla^2(z)+G(x,y)*z = F(x,y) over a rectangular region.

% Function call: [a,om]=ellipgen(nx,hx,ny,hy,G,F,bx0,bxn,by0,byn)

% hx, hy are panel sizes in x and y directions,

% nx, ny are number of panels in x and y directions.

% F and G are (nx+1,ny+1) arrays representing F(x,y), G(x,y).

% bx0 and bxn are row vectors of boundary conditions at x0 and xn.

% each beginning at y0. Each is (ny+1) elements.

% by0 and byn are row vectors of boundary conditions at y0 and yn.

% each beginning at x0. Each is (nx+1) elements.

% a is an (nx+1,ny+1) array of sol'ns, inc the boundary values.

% om has no interpretation in this case.

% or the function solves

% (nabla^2)z+lambda*G(x,y)*z = 0 over a rectangular region.

% Function call: [a,om]=ellipgen(nx,hx,ny,hy,G,F)

% hx, hy are panel sizes in x and y directions,

% nx, ny are number of panels in x and y directions.

% G are (ny+1,nx+1) arrays representing G(x,y).

% In this case F is a scalar and specifies the

% eigenvector to be returned in array a.

% Array a is an (ny+1,nx+1) array giving an eigenvector,

% including the boundary values.

% The vector om lists all the eigenvalues lambda.

nmax = (nx-1)*(ny-1); r = hy/hx;

a = zeros(ny+1,nx+1); p = zeros(ny+1,nx+1);

if nargin==6

ncase = 0; mode = F;

end

if nargin==10

test = 0;

if F==zeros(nx+1,ny+1), test = 1; end

if bx0==zeros(1,ny+1), test = test+1; end

if bxn==zeros(1,ny+1), test = test+1; end

if by0==zeros(1,nx+1), test = test+1; end

if byn==zeros(1,nx+1), test = test+1; end

if test==5

disp('WARNING - problem has trivial solution, z = 0.')

disp('To obtain eigensolution use 6 parameters only.')

return

end

bx0 = bx0(1,ny+1:-1:1); bxn = bxn(1,ny+1:-1:1);

a(1,:) = byn; a(ny+1,:) = by0;

a(:,1) = bx0'; a(:,nx+1) = bxn'; ncase = 1;

end

for i = 2:ny

for j = 2:nx

nn = (i-2)*(nx-1)+(j-1);

q(nn,1) = i; q(nn,2) = j; p(i,j) = nn;

end

end

C = zeros(nmax,nmax); e = zeros(nmax,1); om = zeros(nmax,1);

if ncase==1, g = zeros(nmax,1); end

for i = 2:ny

for j = 2:nx

nn = p(i,j); C(nn,nn) = -(2+2*r^2); e(nn) = hy^2*G(j,i);

if ncase==1, g(nn) = g(nn)+hy^2*F(j,i); end

if p(i+1,j)~=0

np = p(i+1,j); C(nn,np) = 1;

else

if ncase==1, g(nn) = g(nn)-by0(j); end

end

if p(i-1,j)~=0

np = p(i-1,j); C(nn,np) = 1;

else

if ncase==1, g(nn) = g(nn)-byn(j); end

end

if p(i,j+1)~=0

np = p(i,j+1); C(nn,np) = r^2;

else

if ncase==1, g(nn) = g(nn)-r^2*bxn(i); end

end

if p(i,j-1)~=0

np = p(i,j-1); C(nn,np) = r^2;

else

if ncase==1, g(nn) = g(nn)-r^2*bx0(i); end

end

end

end

if ncase==1

C = C+diag(e); z = C\g;

for nn = 1:nmax

i = q(nn,1); j = q(nn,2); a(i,j) = z(nn);

end

else

[u,lam] = eig(C,-diag(e));

[om,k] = sort(diag(lam)); u = u(:,k);

for nn = 1:nmax

i = q(nn,1); j = q(nn,2);

a(i,j) = u(nn,mode);

end

end

We now give examples of the application of the ellipgen function.

Example 6.4

Use function ellipgen to solve Laplace's equation over a rectangular region subject to the boundary conditions shown in Fig. 6.12. The script e4s606.m calls the function to solve this problem using a $12\times 12$ mesh. The example is the same as Example 6.3, but a finer mesh is used in the solution.

% e4s606.m

Lx = 3; Ly = 2;

nx = 12; ny = 12; hx = Lx/nx; hy = Ly/ny;

by0 = 0*[0:hx:Lx];

byn = 200+(100/3)*[0:hx:Lx].^2;

bx0 = 100*[0:hy:Ly];

bxn = 250*[0:hy:Ly];

F = zeros(nx+1,ny+1); G = F;

a = ellipgen(nx,hx,ny,hy,G,F,bx0,bxn,by0,byn);

aa = flipud(a);   colormap(gray)

surfl(aa)

xlabel('x direction')

ylabel('y direction')

zlabel('Temperature')

axis([0 12 0 12 0 500])

The output from this script is the surface plot shown in Fig. 6.13. The actual temperatures can be obtained from aa.

Figure 6.13. Finite difference estimate for the temperature distribution for the problem defined in Fig. 6.12.

Example 6.5

Poisson's equation. Determine the deflection of a uniform square membrane, held at its edges and subject to a distributed load which can be approximated to a unit load at each node. This problem is described by Poisson's equation, (6.35), where $F(x,y)$ specifies the load on the membrane. We use the following script to determine the deflection of this membrane using the Matlab function ellipgen.

% e4s607.m

Lx = 1; Ly = 1;

nx = 18; ny = 18; hx = Lx/nx; hy = Ly/ny;

by0 = zeros(1,nx+1); byn = zeros(1,nx+1);

bx0 = zeros(1,ny+1); bxn = zeros(1,ny+1);

F = -ones(nx+1,ny+1); G = zeros(nx+1,ny+1);

a = ellipgen(nx,hx,ny,hy,G,F,bx0,bxn,by0,byn);

surfl(a)

axis([1 nx+1 1 ny+1 0 0.1])

xlabel('x-node nos.'), ylabel('y-node nos.')

zlabel('Displacement')

max_disp = max(max(a))

Running this script gives the output shown in Fig. 6.14 together with

Figure 6.14. Deflection of a square membrane subject to a distributed load.

max_disp =

0.0735

This compares with the exact value of 0.0737.

Example 6.6

Characteristic value problem. Determine the natural frequencies and mode shapes of a freely vibrating square membrane held at its edges. This problem is described by the eigenvalue problem (6.37). The natural frequencies are related to the eigenvalues, and the mode shapes are the eigenvectors. The following Matlab script, e4s608, determines the eigenvalues and vectors. It calls the function ellipgen and outputs a list of eigenvalues and provides Fig. 6.15, showing the second mode shape of the membrane.

Figure 6.15. Finite difference approximation of the second mode of vibration of a uniform rectangular membrane.

% e4s608.m

Lx = 1; Ly = 1.5;

nx = 20; ny = 30; hx = Lx/nx; hy = Ly/ny;

G = ones(nx+1,ny+1); mode = 2;

[a,om] = ellipgen(nx,hx,ny,hy,G,mode);

eigenvalues = om(1:5), surf(a)

view(140,30)

axis([1 nx+1 1 ny+1 -1.2 1.2])

xlabel('x - node nos.'), ylabel('y - node nos.')

zlabel('Relative displacement')

Running script e4s608.m gives

eigenvalues =

14.2318

27.3312

43.5373

49.0041

56.6367

These eigenvalues compare with the exact values given in Table 6.1.

Table 6.1. Finite difference (FD) approximations compared with exact eigenvalues for a uniform rectangular membrane

FD approximation	Exact	Error (%)
14.2318	14.2561	0.17
27.3312	27.4156	0.31
43.5373	43.8649	0.75
49.0041	49.3480	0.70
56.6367	57.0244	0.70

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Field Problems: A Multidimensional Finite Element Method

O.C. Zienkiewicz , ... J.Z. Zhu , in The Finite Element Method: its Basis and Fundamentals (Seventh Edition), 2013

5.3.6 Irrotational and free surface flows

The basic Laplace equation which governs the flow of viscous fluid in seepage problems is also applicable in the problem of irrotational fluid flow outside the boundary layer created by viscous effects. The seepage example given above is adequate to illustrate the general applicability in this context. Further examples for this class of problems are cited by Martin [26] and others [25, 27–32].

If no viscous effects exist, then it can be shown that for a fluid starting at rest the motion must be irrotational, i.e.,

(5.38) ${\omega }_z\equiv \frac{\partial u}{\partial y}-\frac{\partial v}{\partial x}=0$

where $u$ and $v$ are appropriate velocity components.

This implies the existence of a velocity potential, giving

(5.39a) $u=-\frac{\partial \varphi }{\partial x}\text{,}v=-\frac{\partial \varphi }{\partial y}$

or

(5.39b) $\mathbf{u}=-\mathbf{\nabla }\varphi$

If, further, the flow is incompressible, the continuity equation [which is similar to Eq. (2.80)] has to be satisfied, i.e.,

(5.40) ${\mathbf{\nabla }}^{\text{T}}\mathbf{u}=0$

and therefore

(5.41) ${\mathbf{\nabla }}^{\text{T}}(\mathbf{\nabla }\varphi )={\nabla }^2\varphi =0$

Alternatively, for two-dimensional flow a stream function may be introduced defining the velocities as

(5.42) $u=-\frac{\partial \psi }{\partial y}\text{,}v=\frac{\partial \psi }{\partial x}$

and this identically satisfies the continuity equation. The irrotational condition must now ensure that

(5.43) ${\mathbf{\nabla }}^{\text{T}}(\mathbf{\nabla }\psi )={\nabla }^2\psi =0$

and thus problems of ideal fluid flow can be posed in either form. As the standard formulation is again applicable, there is little more that needs to be added, and for examples the reader can consult the literature cited. We also discuss this problem in more detail in Ref. [33].

The similarity with problems of seepage flow, which has already been discussed, is obvious [34, 35].

A particular class of fluid flow deserves mention. This is the case when a free surface limits the extent of the flow and this surface is not known a priori.

The class of problem is typified by two examples—that of a freely overflowing jet (Fig. 5.16a) and that of flow through an earth dam (Fig. 5.16b). In both, the free surface represents a streamline and in both the position of the free surface is unknown a priori but has to be determined so that an additional condition on this surface is satisfied. For instance, in the second problem, if formulated in terms of the potential for the hydraulic head $H$ , Eq. (5.36) governs the problem.

Figure 5.16. Typical free surface problems with a streamline also satisfying an additional condition of pressure $=$ 0: (a) jet overflow and (b) seepage through an earth dam.

The free surface, being a streamline, imposes the condition that

(5.44) $\frac{\partial H}{\partial n}=0$

be satisfied there. In addition, however, the pressure must be zero on the surface as this is exposed to atmosphere. As

(5.45) $H=\frac{p}{\gamma }+y$

where $\gamma$ is the fluid specific weight, $p$ is the fluid pressure, and $y$ is the elevation above some (horizontal) datum, we must have on the surface

(5.46) $H=y$

The solution may be approached iteratively. Starting with a prescribed free surface streamline the standard problem is solved. A check is carried out to see if Eq. (5.46) is satisfied and, if not, an adjustment of the surface is carried out to make the new $y$ equal to the $H$ just found. A few iterations of this kind show that convergence is reasonably rapid. Taylor and Brown [36] show such a process. Alternative methods including special variational principles for dealing with this problem have been devised over the years and interested readers can consult Refs. [37–45].

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Fluid Mechanics of Viscoelasticity

In Rheology Series, 1997

Problem 51.A

Show that the Laplace equation:

(51.A1) $\Delta u=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$

(51.A1)is elliptic in the square domain (0 ≤ x, y ≤ 1). Given the boundary conditions

$\begin{array}{ccc}\text{u}(x,0)=\sin \pi \text{x,}& \text{u(x,1)=sin}\pi \text{xexp(-}\pi \text{),}& \text{u(0,y)=0=u(1,y),}\end{array}$

find the solution by a separation of variables.

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Linear Systems of Equations (Computer Science)

Victor Pan , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.A Numerical Solution of the Laplace Equation

We consider the Laplace equation ∂² u/∂ x ²  +   ∂² u/∂ y ²  =   0 on the square region 0   ≤ x, y  ≤   1, provided that the function u(x, y) is given on the boundary of that region; u(x, y) models the temperature distribution through a square plate with fixed temperature on its sides. To compute u(x, y) numerically, we superimpose a mesh of horizontal and vertical lines over the region as shown by

so that u ₁ denotes the point ( $\frac{1}{3},\frac{2}{3}$ ); u ₂ denotes the point ( $\frac{2}{3},\frac{2}{3}$ ); u ₇ denotes ( $\frac{2}{3}$ , 1), and so on; u ₅, u ₆, …, u ₁₆ are given; and u ₁, u ₂, u ₃, u ₄ are unknowns. Then we replace the derivatives by divided differences,

$\begin{array}{l}{\partial }^2\text{u}/\partial {\text{x}}^2\text{by}\text{u}\left(\text{x}-\text{h},\text{y}\right)-2\text{u}\left(\text{x},\text{y}\right)+\text{u}\left(\text{x}+\text{h},\text{y}\right)\hfill \\ {\partial }^2\text{u}/\partial {\text{y}}^2\text{by}\text{u}\left(\text{x},\text{y}-\text{h}\right)-2\text{u}\left(\text{x},\text{y}\right)+\text{u}\left(\text{x},\text{y}+\text{h}\right)\hfill \end{array}$

where h  = $\frac{1}{3}$ . This turns the Laplace equation into a linear system that can be equivalently derived if we just assume that the temperature at an internal point of the grid equals the average of the temperatures at the four neighboring points of the grid;

$\begin{array}{l}-4{\text{u}}_1+{\text{u}}_2+{\text{u}}_3=-{\text{u}}_6-{\text{u}}_{16}\hfill \\ {\text{u}}_1-4{\text{u}}_2+{\text{u}}_4=-{\text{u}}_7-{\text{u}}_9\hfill \\ {\text{u}}_1-4{\text{u}}_3+{\text{u}}_4=-{\text{u}}_{13}-{\text{u}}_{15}\hfill \\ {\text{u}}_2+{\text{u}}_3-4{\text{u}}_4=-{\text{u}}_{10}-{\text{u}}_{12}\hfill \end{array}$

The coefficient matrix A of the system is the block tridiagonal of Eq. (3). With smaller spacing we may obtain a finer grid and compute the temperatures at more points on the plate. Then the size of the linear system will increase, say to N ² equations in N ² unknowns for larger N; but its N ²  × N ² coefficient matrix will still be block tridiagonal of the following special form (where blank spaces mean zero entries),

$\begin{array}{l}\left[\begin{array}{lllll}{\text{B}}_{\text{N}}\hfill & {\text{I}}_{\text{N}}\hfill & \hfill & \hfill & \hfill \\ {\text{I}}_{\text{N}}\hfill & {\text{B}}_{\text{N}}\hfill & {\text{I}}_{\text{N}}\hfill & \hfill & \hfill \\ \hfill & {\text{I}}_{\text{N}}\hfill & {\text{B}}_{\text{N}}\hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \ddots \hfill & \ddots \hfill & {\text{I}}_{\text{N}}\hfill \\ \hfill & \hfill & \hfill & {\text{I}}_{\text{N}}\hfill & {\text{B}}_{\text{N}}\hfill \end{array}\right]\hfill \\ {\text{B}}_{\text{N}}=\left[\begin{array}{llll}-4\hfill & 1\hfill & \hfill & \hfill \\ 1\hfill & -4\hfill & \ddots \hfill & \hfill \\ \hfill & \ddots \hfill & \ddots \hfill & 1\hfill \\ \hfill & \hfill & 1\hfill & -4\hfill \end{array}\right]\hfill \end{array}$

Here, B _N is an N  × N tridiagonal matrix, and I _N denotes the N  × N identity matrix (see Section II.D). This example demonstrates how the finite difference method reduces the solution of partial differential equations to the solution of linear systems [Eq. (1)] by replacing derivatives by divided differences. The matrices of the resulting linear systems are sparse and well structured.

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Example Problems Involving Two-Dimensional Fluid Flow in Soils

Victor N. Kaliakin , in Soil Mechanics, 2017

7.4 Solution of the Governing Equation

The exact solution of Laplace's equation is typically difficult to realize. The equation must thus be solved approximately. Approximations are commonly obtained using either computer solutions such as the finite difference or finite element method or using hand-drawn flow nets. The topic of computer solutions is discussed elsewhere. ³ Instead, the focus herein is on approximate solution obtained using flow nets.

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Mixed finite element–boundary element model for linear water–structure interactions

Jing Tang Xing , in Fluid-Solid Interaction Dynamics, 2019

8.1.1 Fundamental solution of Laplace equation

The solution of the Laplace equation in Eq. (6.48), when defined in an unbounded 3D region, for a point source of unit strength, is called the fundamental solution, or free-space Green's function, of the problem. For a homogeneous and isotropic water domain, the 3D fundamental solution must have spherical symmetry. Mathematically, it can be defined as a scalar function G that is at least twice differentiable with respect to the coordinates of any field point $x_i$ and that satisfies Laplace's equation at all points except at $x_i^{*}$ , the point of application of the source, that is,

(8.1) $\begin{array}{l}{\nabla }^{2}G=G_{,jj}\left(x_i^{*}\right)=\mathrm{\Delta }\left(x_i-x_i^{*}\right)=\{\begin{array}{c}\mathrm{0}{x}_i\ne x_i^{*},\\ \infty x_i=x_i^{*},\end{array}\hfill \\ G_{,j}{\nu }_j=0,x_i\to \infty .\hfill \end{array}$

The solution of Eq. (8.1) can be found by writing it in spherical coordinates and taking into account its central symmetrical character, which is

(8.2) ${\nabla }^{2}G=\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dG}{dr}\right)=0,$

where r is the distance between the source and field points. This is an ordinary differential equation whose solution is

(8.3) $G=c_0+\frac{c_1}{r},$

where $c_0$ is an arbitrary additive constant that corresponds to the reference level adopted for measuring velocity potential and that has no influence on the calculation of fluid velocities. The constant $c_1$ can be determined from mass conservation by considering a spherical control surface of radius $\epsilon$ centered at the source point $x_i^{*}$ , as shown in Fig. 8.1. For the convenience of derivation, it is assumed that a sink source is located at point $x_i^{*}$ .

Figure 8.1. Fundamental solution of Laplace equation.

From the principle of mass conservation, the total flux across the spherical surface must be equal to the mass flux into the unit sink source at point $x_i^{*}$ , thus

(8.4) ${\iint }_{{\Gamma }_{\epsilon }}{G}_{,j}{\nu }_{j}d\Gamma =1.$

Substituting Eq. (8.3) into Eq. (8.4) and noting ${()}_{,j}{\nu }_j=-{()}_{,r}$ on the spherical control surface gives the following equation:

(8.5) $c_1\frac{1}{{\epsilon }^2}{\iint }_{{\Gamma }_{\epsilon }}d\Gamma =1.$

Since the area of the spherical surface is $4\pi {\epsilon }^2$ , we obtain the constant $c_1=1/4\pi$ . Assuming $c_0=0$ , the expression for the fundamental solution of the Laplace equation in a 3D water domain of infinite extent is expressible as

(8.6) $\begin{array}{lll}G=\frac{1}{4\pi r},\hfill & r=\sqrt{\left(x_i-x_i^{*}\right)\left(x_i-x_i^{*}\right)},\hfill & i=1,2,3.\hfill \end{array}$

Similarly, for a 2D problem, the fundamental solution of the Laplace equation is obtained as

(8.7) $\begin{array}{lll}G=\frac{1}{2\pi }\ln r,\hfill & r=\sqrt{\left(x_i-x_i^{*}\right)\left(x_i-x_i^{*}\right)},\hfill & i=1,2.\hfill \end{array}$

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Kinematics, Composition, and Thermodynamics

Nikolaos D. Katopodes , in Free-Surface Flow, 2019

2.4 The Laplacian

The divergence of the gradient of a scalar field f produces a differential operator known as the Laplacian. Thus

(2.44) $\mathbf{\nabla }\cdot \mathbf{\nabla }f={\mathbf{\nabla }}^{2}f$

is a second-order operator that consists of the sum of the second spatial derivatives of f, i.e.

(2.45) ${\mathbf{\nabla }}^{2}f=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$

When the Laplacian of a function vanishes, the function is called harmonic, and the corresponding equation

(2.46) ${\mathbf{\nabla }}^{2}f=0$

is known as the Laplace equation, named after the French mathematician Pierre Simon Marquis de Laplace (1749–1827), who first applied it to the study of celestial mechanics.

If a function satisfies the Laplace equation, all of its spatial derivatives are also harmonic, thus a harmonic function is by definition continuous. The Laplace equation plays a major role in physics and engineering, as it represents the flux density of the gradient of the corresponding scalar field. An example of importance in environmental applications is the process of diffusion. If the mass of a solute per unit volume is not distributed uniformly, the gradient of the corresponding field generates a vector field due to the flux of solute mass. If this mass is to be conserved, the divergence of the flux must vanish leading to a form of Laplace's equation known as the diffusion equation, to which Chapter 3 is devoted.

Example 2.4.1

Find the Laplacian of the pressure field in a shallow lake that is described by the expression $p=x^{2}z$ .

The gradient of the pressure field is given by

$\mathbf{\nabla }p=\frac{\partial p}{\partial x}\mathbf{i}+\frac{\partial p}{\partial y}\mathbf{j}+\frac{\partial p}{\partial z}\mathbf{k}=2xz\mathbf{i}+x^2\mathbf{k}$

Next, the divergence of the gradient field is found as follows

$\mathbf{\nabla }\cdot \left(\mathbf{\nabla }p\right)={\mathbf{\nabla }}^{2}p=2z$

2.4.1 Curvilinear Coordinates

The Laplacian operator can be expressed in curvilinear coordinates as well although some care needs to be exercised in taking the divergence of the gradient, i.e. Eq. (2.44). As mentioned earlier, the base vectors ${\mathbf{e}}_r$ , ${\mathbf{e}}_{\theta }$ , etc., are not constant, thus when expanding the scalar product $\mathbf{\nabla }\cdot \mathbf{\nabla }$ , the derivative operators in the first factor must be applied to the unit vectors in the second factor before the dot product is taken. This approach is complicated, but can be avoided if the formulas for gradient and divergence in curvilinear coordinates are used directly.

Recalling the expressions of Eq. (2.38) and Eq. (2.39), for example, we obtain

(2.47) $\begin{array}{cc}\hfill {\mathbf{\nabla }}^{2}f& =\mathbf{\nabla }\cdot (\frac{\partial f}{\partial r}{\mathbf{e}}_r+\frac{1}{r}\frac{\partial f}{\partial \theta }{\mathbf{e}}_{\theta }+\frac{\partial f}{\partial z}{\mathbf{e}}_z)\hfill \\ \hfill & =\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial f}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}f}{\partial {\theta }^2}+\frac{{\partial }^{2}f}{\partial z^2}=0\hfill \end{array}$

Similarly, using Eq. (2.41) we obtain the Laplace equation for a scalar ϕ in spherical coordinates, as follows

(2.48) ${\mathbf{\nabla }}^{2}f=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2{\sin }^2\varphi }\frac{{\partial }^{2}f}{\partial {\theta }^2}+\frac{1}{r^2{\sin }^2\varphi }\frac{\partial }{\partial \varphi }(\sin \varphi \frac{\partial f}{\partial \varphi })=0$

Notice that if there exists spherical symmetry in the scalar field, so that f is a function of r only, Laplace's equation reduces to an ordinary differential equation, as follows

(2.49) ${\mathbf{\nabla }}^{2}f=\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0$

2.4.2 Vector Laplacian

The Laplacian operator can also be applied to a vector field. The result is another vector field that satisfies the following expression

(2.50) $\begin{array}{cc}\hfill {\mathbf{\nabla }}^2\mathbf{F}& =\mathbf{\nabla }(\mathbf{\nabla }\cdot \mathbf{F})-\mathbf{\nabla }\times (\mathbf{\nabla }\times \mathbf{F})\hfill \\ \hfill & ={\mathbf{\nabla }}^2{F}_x\mathbf{i}+{\mathbf{\nabla }}^2{F}_y\mathbf{j}+{\mathbf{\nabla }}^2{F}_z\mathbf{k}\hfill \end{array}$

Thus, the vector Laplacian is a vector whose components are given by the scalar Laplacian of the original vector components.

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Cell, Stack and System Modelling

Mohammad A. Khaleel , J. Robert Selman , in High Temperature and Solid Oxide Fuel Cells, 2003

11.8.4 Monte Carlo or Stochastic Electrode Structure Model

A third distinct type of electrode model developed in response to the need for modelling the composite structure of SOFC electrodes more accurately is the Monte Carlo, or stochastic structure, model. This model is based on a random number-generated 2- or 3-D structure of electrode particles, electrolyte particles, and holes (for gas pores). It has been shown to represent the composite conductivity quite well and may be able to model polarisation behaviour adequately [56–58]. This is of interest because microstructure, and in particular hard-to-control variations in local microstructure, may have an important effect on overall polarisation, perhaps more so than the intrinsic kinetic characteristics measured at an 'ideal' interface.

The Monte Carlo-type electrode model is also called the particle connectivity model because its physics is straightforwardly based on Kirchhoff's law for an electrical network, with particle resistance and interconnection resistances defined by a set of rules to mimic the current flow and electrochemical current generation within the microstructure. The electrochemical process is considered to take place with a constant resistance in agreement with intuitive notions about the mechanism. Variants of this concept attach correlated values to the resistances in the network to model polarisation more closely according to a percolation concept of active sites and passive connections [59]. Other specialised types of electrode models are mentioned briefly below.

11.8.4.1 Electrode or Cell Models Applied to Ohmic Resistance-Dominated Cells

These models start from solving Laplace's equation (Eq. (36)) with appropriate boundary conditions, sometimes including polarisation. The most important application is the correct design of test cells with reference electrodes because small deviations in reference electrode placement may cause appreciable deviations in polarisation readings [60–63].

11.8.4.2 Diagnostic Modelling of Electrodes to Elucidate Reaction Mechanisms

Because the electrode kinetics of both anode and cathode and their dependence on microstructure are so important for performance, much attention has been given to elucidating reaction mechanisms based on independent electrochemical measurements (usually with respect to a reference electrode). AC impedance measurements are particularly favoured. The interpretation of these measurements requires specialised models that reflect in part the hypothesised kinetics and in part the electrode structure. It seems certain that eventually the results will be integrated with both macro- and molecular modelling [64–69].

11.8.4.3 Models of Mixed Ionic and Electronic Conducting (MIEC) Electrodes

These specialised electrode models usually consider the MIEC electrode in combination with the electrolyte and focus on correlating performance with the semiconductor characteristics of the electrode (and sometimes electrolyte) [70–72]. Recent modelling of oxygen reduction and oxygen permeation at perovskite electrodes includes both MIEC effects and classical diffusion-type analysis [73–75].

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