The equation
where the orthogonal Cartesian coordinates is called the Laplace equation. The expression on its left side is called the Laplacian of the function and, and the rule by which the expression is formed is called the Laplace operator. The Laplace operator is usually denoted by the symbol, as a result of which equation (1) can be written in the form
Inhomogeneous equation
where the given function is called the Poisson equation.
The form of differential expressions in the left parts of the Laplace and Poisson equations is the same in all orthogonal Cartesian coordinates. When passing to curvilinear coordinates, it changes and, for orthogonal curvilinear coordinates, it can be determined using the relations of § 7 of the previous chapter. In particular, using formulas (54), (48), and (49) of Chap. XVIII we find that in cylindrical coordinates
in spherical coordinates
Numerous problems of the theory of heat conduction, electrostatics, hydrodynamics, etc. lead to the Laplace and Poisson equations. Let us consider, for example, the formulation of some problems for the Laplace equation.
1. The problem of the stationary thermal state of a homogeneous body. Let's say we have some
a homogeneous isotropic body isolated from external space, the thermal state of which does not change with time. Let us denote by V the part of space occupied by it, by its surface, and by and the temperature at the point
Let us prove that at any interior point x of the body we have taken, the function satisfies the Laplace equation.
To this end, we select from the body a certain area bounded by an arbitrarily taken surface and consider the amount of heat that passes per unit time through the surface element. According to the Fourier principle, it is proportional to the area of the element and the normal derivative where denotes the direction of the outward normal to the surface. In other words, this amount of heat is equal to the product
The coefficient of proportionality is called the coefficient of internal thermal conductivity of the body.
Consider the movement of heat in a body. It is known from thermodynamics that heat flows from points of higher temperature to points of lower temperature. Therefore, with a negative derivative, the heat flow will occur from the inner part of the body, bounded by the surface, to the area external to this surface. If the indicated derivative is positive, then the heat propagation will represent the opposite picture.
This implies that the double integral
gives the algebraic sum of the amount of heat that has passed through the surface per unit time, with a negative sign attributed to the outflowing heat, and a positive sign to the incoming heat.
If we assume that both sources of heat and points of its absorption are absent inside the body, then integral (5) must equal zero. Indeed, if this were not the case, then heat would be accumulated or lost inside the body, and, consequently, the temperature of the body would change over time, which contradicts the assumption that the thermal state of the body is unchanged.
So, in this case, the following equality should take place:
Let us apply Green's formula (7) Ch. XVIII:
and put in it
Then, taking into account that the integral (5) is equal to zero, we find that
Hence, in view of the arbitrariness of the domain, it follows that
i.e., the function satisfies the Laplace equation.
Suppose now that we know the temperature distribution on the surface of the body and we want to determine the temperature of any point inside the body.
Obviously, we will solve this problem if we find a solution to the Laplace equation that would satisfy the boundary condition
where denotes the temperature at point x of the surface
2. The problem of the equilibrium of electrical masses on the surface of a conductor. Consider a stationary electrostatic field created in space by some system of electric charges. If the charges are located discretely at points, then the field potential at the point x
where is the distance from the charge to point x. If the charges are continuously distributed on some line or surface or in the volume Y, then the field potential is respectively expressed by one of the integrals:
where is the distance from the line element (surface, volume) to the field point with the potential u. In these formulas, the quantities denote the linear, surface or volume charge density:
where is the charge of the line element L (surface S, volume V). In the general case, the field potential is equal to the sum of the potentials created by each of these types of charge distribution separately.
Let us assume that the finite region V of space is occupied by a conducting medium - a conductor, i.e., a medium in which charges can move freely, and the rest of the space is a dielectric, i.e., a medium in which the movement of charges is impossible.
In a stationary state, the field potential at all points of the region V, including its boundary, is the same, since otherwise there would be a movement of electric charges, seeking to equalize the potential, and the field would change. From this it is directly obvious that in the region V the field potential u satisfies the Laplace equation:
Inside the conductor, charges of different signs must be mutually neutralized. In fact, the excess charges of any sign remaining inside the conductor under the action of repulsion between like charges would move until all of them were at the boundary of the conductor and were properly distributed on it. Consequently, if a stationary state is reached, then the excess charges are located at the conductor boundary in the form of an infinitely thin electrical layer.
The potential of this layer at a point is expressed by the integral:
where is the distance from the variable point of the conductor surface to the point x.
If the point x is outside the conductor, then the function y satisfies the Laplace equation. Indeed,
Therefore, the potential u defined by formula (12) also satisfies the Laplace equation. To prove this statement, it suffices to apply the rule of differentiation with respect to the parameter to the integral (12), which we have the right to do, since, according to
Assuming that the point x is outside the surface, therefore, the integrand in expression (12) does not go to infinity anywhere.
So, at each point x lying outside the conductor, the potential and also satisfies the Laplace equation.
Let us now turn to the elucidation of the circumstances that take place at infinitely distant points of space filled with a dielectric, and on the very surface of the conductor.
As we will find out below, the integral (12) vanishes at infinitely distant points (together with its first-order partial derivatives), and moreover, in such a way that the products
remain bounded when the distance from point x to the origin increases to infinity. As regards the circumstances that take place on the surface of the conductor, it will be proved that the potential and remains limited and continuous when the point x passes through the surface of the conductor. On the contrary, the normal derivatives of the potential undergo a finite discontinuity even under such a transition, and this discontinuity is characterized by the equality
where the limit values of the expression
when the point x approaches the point, respectively, along the internal and external normals to at the point
Let us use equation (13) to formulate the so-called electrostatic problem: to find the density of the electric layer continuously distributed on the surface of a given conductor, if the latter is in a state of electrical equilibrium.
Let us assume that such a state has come for the given conductor. Then, according to the above explanations, the potential inside the conductor will be a constant value, and, therefore, the equality will take place
From this equality and from formula (13) it follows that
i.e., the desired density of the layer will be found if we determine the potential of this layer at points lying outside the conductor.
This theory was not written in mathematical symbols and therefore could not show a quantitative relationship between the attraction of individual particles and the end result. Leslie's theory was later revised using Laplacian mathematical methods by James Ivory in an article on capillary action, under "Fluids, Elevation of", in the supplement to the 4th edition of the Encyclopaedia Britannica, published in 1819.
Theories of Jung and Laplace.
In 1804, Thomas Young substantiated the theory of capillary phenomena on the principle of surface tension. He also observed the constancy of the wetting angle of a solid surface (contact angle) with a liquid and found a quantitative relationship relating the contact angle to the surface tension coefficients of the corresponding interfacial boundaries. In equilibrium, the contact line should not move along the surface of a solid, which means that he said
where sSV, sSL, sLV are the coefficients of surface tension of interphase boundaries solid - gas (steam), solid - liquid, liquid - gas, respectively, q - contact angle. This ratio is now known as Young's formula. This work, however, did not have such an impact on the development of science in this direction, which was published a few months later by Laplace's article (Pierre Simon Laplace). This seems to be due to the fact that Jung avoided the use of mathematical notation, but tried to describe everything verbally, which makes his work seem confusing and unclear. Nevertheless, he is considered today one of the founders of the quantitative theory of capillarity.
The phenomena of cohesion and adhesion, the condensation of vapor into a liquid, the wetting of solids by liquids, and many other simple properties of matter - all indicated the presence of attractive forces many times stronger than gravity, but acting only at very small distances between molecules. As Laplace said, the only condition that follows from observable phenomena and is imposed on these forces is that they are "imperceptible at sensible distances."
The repulsive forces created more trouble. Their presence could not be denied - they must balance the forces of attraction and prevent the complete destruction of matter, but their nature was completely unclear. The question was complicated by the following two erroneous opinions. First, it was often believed that heat was the acting force of repulsion (as a rule, the opinion of supporters of the caloric theory), since (such was the argument) when a liquid first expands and then boils when heated, so that the molecules are separated over much greater distances than in a solid body. The second misconception arose from the notion, back to Newton, that the observed pressure of a gas is due to static repulsion between molecules, and not due to their collisions with the walls of the vessel, as Daniel Bernoulli argued in vain.
Against this background, it was natural that the first attempts to explain capillarity, or in general the cohesion of liquids, were based on the static aspects of matter. Mechanics was a well understood theoretical branch of science; thermodynamics and kinetic theory were still in the future. In mechanical considerations, the key was the assumption of large but short-range attractive forces. Liquids at rest (whether in a capillary tube or outside it) are obviously in equilibrium, and therefore these attractive forces must be balanced by repulsive forces. Since even less could be said about them than about the forces of attraction, they were often passed over in silence, and, in the words of Rayleigh, "the forces of attraction were left to perform the unthinkable trick of balancing themselves." Laplace was the first to satisfactorily solve this problem, believing that the repulsive forces (thermal, as he assumed) can be replaced by internal pressure, which acts everywhere in an incompressible fluid. (This assumption leads at times to ambiguity in 19th-century writings as to what is strictly meant by "pressure in a fluid.") Here is Laplace's calculation of internal pressure. (This conclusion is closer to the conclusions of Maxwell and Rayleigh. The conclusion is given by.)
It must balance the cohesive forces in the fluid, and Laplace identified this with the force per unit area that resists the separation of an infinite fluid body into two far-separable semi-infinite bodies bounded by flat surfaces. The derivation below is closer to those of Maxwell and Rayleigh than to Laplace's original form, but there is no significant difference in the argument.
Consider two semi-infinite liquid bodies with strictly flat surfaces, separated by an interlayer (of thickness l) of vapor with a negligible density (Fig. 1), and in each of them we single out the volume element. The first is located in the upper body at a height r above the flat surface of the lower body; its volume is dxdydz. The second is in the lower body and has a volume 
, where the origin of polar coordinates coincides with the position of the first elementary volume. Let f(s) be the force acting between two molecules separated by a distance s, and d be the radius of its action. Since this is always an attractive force, we have
If r is the density of the number of molecules in both bodies, then the vertical component of the interaction force of two volume elements is equal to
The above conclusion is based on the implicit assumption that the molecules are uniformly distributed with a density r, i.e. the liquid does not have a discernible structure on a scale of dimensions commensurate with the range of forces d. Without this assumption, it would be impossible to write expressions (2) and (3) in such a simple form, but it would be necessary to find out how the presence of a molecule in the first volume element affects the probability of the presence of a molecule in the second.
The surface tension force acts on the surface of the liquid in the capillary, which will be the resultant of the forces acting on the molecules of the surface layer adjacent to the vessel wall, for wetting liquids it will be directed outward (up), and for non-wetting liquids - inward (down). Under the action of these forces the liquid surface near the vessel wall takes a curvilinear (curved) shape, called the meniscus. The meniscus will be concave if the liquid wets the vessel wall (Fig. 8, a) and convex if it does not wet (Fig. 8, b).
Formula derivation (optional). By determining the surface tension coefficient, one can determine the pressure inside a spherical drop liquids or pressure inside gas bubble in liquid.
If R is the pressure inside a spherical liquid drop or inside a gas bubble, σ is the surface tension of the liquid, r is the radius of the ball, then to increase the radius r ball by Δ r (r1 =r + Δ r ) (Fig. 9 a) or increase its surface area S to Δ S it is necessary to expend work equal to the increment of surface energy: Δ W = Δ A = σ Δ S , where the area of the ball (recall from the school geometry course) is equal to S=4π r2 .
Then Δ A = σ Δ S = σ = σ ,
which means: Δ A = 4πσ [(r + Δ r) 2 - r 2 ] .
The square of the sum is known to be (a + b) 2 = a 2 + 2ab + b 2 , then:
Δ A = 4πσ [(r + Δ r) 2 - r 2 ] = 4πσ [(r 2 + 2r ּΔ r + (Δ r) 2) - r 2] = 4πσ ּ [r 2 + 2r ּΔ r +
(Δ r) 2 - r 2] = 4πσ [ 2r ּΔ r + (Δ r) 2 ] = 4πσ [ 2r ּΔ r + (Δ r) 2 ]
Insofar as (Δ r) 2 << 2r ּΔ r , term containing (Δ r) 2 can be neglected. Therefore, to change the work, we use: Δ A = σ ּ 8 π r ּΔ r .
On the other hand, the expended work of the gas at a constant temperature is equal to: Δ A
= R
Δ V
, where the change in the volume of the ball as a function differential is equal to 
.
Then Δ A = R Δ V = R ּ 4 π r2 ּΔ r . Equating both expressions, we get:
Δ A
= σ
ּ 8
π r
ּΔ r
= R
ּ 4
π r2
ּΔ r
.
As a result, we get: σ ּ 2 = R ּ r , which can be converted as follows: .
This formula is called Laplace's formula for additional pressure under a curved liquid surface.
Laplace formula it reads as follows: the additional pressure under the curved surface of the liquid due to the action of surface tension forces is directly proportional to the coefficient of surface tension σ , inversely proportional to the radius r drops of a liquid or a gas bubble in a liquid and is directed towards the concavity (towards the center of curvature).
Note that since the pressure is inversely proportional to the radius of a liquid drop or gas bubble in a liquid , the greater the pressure, the smaller the radius of the spherical drop.
The Laplace formula is also valid for capillary phenomena.
Under the action of surface tension forces, the surface layer of the liquid is curved, forming a meniscus, and exerts an additional pressure Δ in relation to the external one. R . In the capillary, the external pressure is atmospheric pressure (hydrostatic pressure of the atmospheric column above us), due to gravity and equal to 760 mm Hg on the sea surface. or 1.0135 10 5 Pa.
The resulting force of surface tension of a curved surface is directed towards the concavity (toward the center of curvature). In the case of a spherical surface, the radius of curvature of which is r , additional pressure according to the Laplace formula: .
With good wetting, a concave meniscus is formed. The forces of the additional Laplace pressure are directed outward from the liquid, i.e. up.
Additional Laplace pressure acts against atmospheric pressure, reducing it, causing the liquid to rise in the capillary.
The liquid will rise in the capillary until the additional pressure Δ p (Laplace pressure), due to surface tension forces and directed upwards (toward the center of the meniscus circle), will not be balanced by hydrostatic (weight) pressure p hydrost = ρ gh acting downward (Δ p= p hydrost ).
But the meniscus radius is equal to the capillary radius ( R = r) only with complete wetting, when Θ=0 0 . In all other cases, it is not easy to find the meniscus radius experimentally, so we express r across R is the capillary radius. From fig. 9b shows that .
Therefore, taking into account Laplace's law, we obtain the equality: 
, whence the height of liquid rise in the capillary 
(*), i.e. depends on the properties of the liquid and the material of the capillary, as well as on its radius.
In case of poor wetting (non-wetting) cosΘ< 0 and formula (*) will show the height of the lowering of the liquid in a capillary.
The same formula makes it possible to determine the surface tension of a liquid by the height of the rise of the liquid in the capillary and the value of the contact angle between the meniscus of the liquid and the walls of the vessel ( capillary method ):
.
In the case of complete wetting (angle Θ = 0 °, which means cosΘ = 1 ) and complete nonwetting (angle Θ = 180° , which means cos Θ = -1 ) the formula is much simpler.
There are other methods for determining the surface tension coefficient σ : a) droplet separation method, b) ring and frame separation methods, c) air bubble separation method (Rebinder). They will be discussed below.
THERMOPHYSICS OF HIGH TEMPERATURES, 2010, volume 48, no. 2, p. 193-197
THERMOPHYSICAL PROPERTIES OF THE SUBSTANCE
UDC 532.6:004.932
IMPROVED SIDDLE DROP METHOD FOR DETERMINING THE SURFACE TENSION OF LIQUIDS
L. B. Director, V. M. Zaichenko, and I. L. Maikov
Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow Received May 25, 2009
An improved technique for processing images of the meridional section of a liquid drop obtained by implementing the sessile drop method for determining the surface tension of a liquid has been developed. The technique provides scanning of a digital image of a drop, numerical solution of the Young-Laplace equation, as well as calculation of surface tension, contact angle, and drop volume.
INTRODUCTION
The sessile (hanging) or stationary drop method is considered the most reliable static method for studying the surface tension of metal melts, salt, polymer and other liquids.
Static methods are based on the solution of the Young-Laplace differential equation. Approximate solutions to this equation have been obtained by many authors, and the most common way to determine the surface tension coefficient is based on the use of Bashforth and Adams tables. The existing empirical dependencies are essentially an approximation of these tables. The disadvantages of such methods are low accuracy, as well as limitations associated with the size of the drop. The geometric parameters of the drop are determined by measuring its photographic image using a measuring microscope. The measurement process is quite laborious, and its results contain an error associated with the individual characteristics of the observer.
The aim of this work is to create a high-speed software package that allows one to process a digital image of a drop and carry out an optimization procedure for determining the surface tension coefficient of a liquid using both the sessile method and the method of drop detachment (hanging drop). The technique is based on the ideology of numerical integration of the Young-Laplace equation, presented in the work.
METHOD OF PROCESSING THE DROP IMAGE
The initial information is a graphic file in a standard dot form
mate BitMaP (BMP), which contains an image of the meridional section of the drop. The image has a black and white palette with grayscale from white to black (in hexadecimal representation from 000000 to FFFFFF) in RGB color (Fig. 1).
Determining the exact border of an image is a separate task. There are quite complex algorithms based on the level set function method and requiring the solution of partial differential equations of hyperbolic type. In this paper, to simplify numerical calculations, we use a simple algorithm described below and evaluate its accuracy.
At the first stage of processing, the gray image is converted into black and white monochrome as follows. The average color value is selected from the color palette (in hexadecimal representation, this corresponds to color 888888). The further processing is
Rice. 1. Image of a drop on a substrate (BMP format).
scanning the image for each pixel. All pixels with a color value less than the boundary value change their value to white, and more than the boundary value changes to black, as a result of which the boundary of white and black colors and, accordingly, the coordinates of the image contour points are determined (Fig. 2).
The choice of the boundary color when converting an image from gray to monochrome introduces a certain error into the result, which is illustrated by the curve of the dependence of the relative volume of the standard (calibrated steel ball) on the choice of the boundary color (Fig. 3).
When choosing the fifth part of the full palette (the colors of the palette from 666666 to LLLLA in hexadecimal representation correspond to colors from 1 to 4 in Fig. 3), the relative error in determining the volume is 0.2%. Palette color 888888 (the middle of the full palette) corresponds to a value of 3 on the x-axis and a relative volume of 1.
Relative volume 1.0010
color separation border
Rice. 3. Dependence of the relative volume of the standard on the choice of the boundary color.
NUMERICAL PROCEDURE FOR PROCESSING THE IMAGE OF A DROP
The shape of the drop lying on the substrate (Fig. 4) satisfies the Young-Laplace equation
(l + Y "2)3/2 Y (1 + Y
capillary constant; st - co -
surface tension coefficient; H is the height of the drop; [x, y(x)] - coordinates of the boundary of the meridional section of the drop (see Fig. 4); R0 is the radius of curvature at the top of the drop; Ap is the difference between the densities of the liquid and the surrounding gas.
For the numerical solution of equation (1), we will carry out its parametrization x = x(1),
Here I is the length of the arc of the curve from the top of the drop to the point with coordinates x(1), y(1). Then the Young-Laplace equation in parametric form will be written in the form
v a y Ro n - x + x + _2_
A y Roy with initial conditions x(0) = H, y(0) = 0, x(0) = 0, y(0) = -1.
Rice. 4. Meridional cross section of a sessile drop.
improved sessile drop method
The system of two second-order differential equations (2) can be represented as a system of four first-order equations
u = -v + ä + 2
" H - x , ü , 2 v = ü |-2--1---1--
with initial conditions x(0) = H, y(0) = 0,
and (0)=0, v(0)=-1.
To integrate the system of ordinary differential equations (3), a numerical method for solving stiff differential equations was used - a linear multi-step method with automatic step selection, implemented in the DIFSUB algorithm.
When processing the data obtained in the sessile drop method (drop separation), the inverse problem of determining the capillary constant a2, the drop height H, and its curvature radius R is solved using the dependence of the radius of the circle of the horizontal section of the drop on the distance of this section from the substrate.
Consider a functional representing the sum of squared deviations of the experimental points from the calculated curve
L \u003d K - X;) 2 + (Ye1 - Y,) 2),
where (xe, ye) are the coordinates of the experimental points, (x, y) are the coordinates of the calculated points.
The calculated points (x;, y() are functions of the parameters a1 = a2, a2 = H, a3 = R0:
xi - xi(t, a1 a2, a3),
yt - y,(h, ai, a2, a3). Let us expand (5) in a Taylor series in the vicinity of the
ki (a1, a2, a3)
xt = x (t , a°, a°, a°) + dXi Aa1 + dXt Aa2 + dXi Aa3,
yl = y,(ti, ai1, a°, a°) + ^ Aai + Aa2 + Aay
To find the minimum of functional (4), the conditions must be satisfied
Substituting (4) into (6) and differentiating, the system of equations (7) can be written as
Xei - xi - dx- Aai - dx- Aa2 - dx- Aa3)) +
+ | yei - y, -du Aai -f* Aa2 -f* Aa3))
oa1 oa2 oa3 jda1_
xei - x, - ^Dv1 -O*!.da2 -§xlAa3- +
yei-y, -yy. Yes, - Da1 - & Daz -
da1 da2 da3)da2j
dx. 5x- 5x- 15x-xei - xi --LD^ --LDa2 --LDaz - +
yei- yt -dR Da1 -M Da2 -^U- Da3 -
dxt dxt + dyt dyt =1 dak da, dak da,
I| (xei-xi)f + (yei - y, fi|, V da, da, 1
I I dxL dx± + dy_ dyj_
t dak da, dak da,
k = 1| i = 1k2k2.
I| (xei-x,)f* + (yei - Y,)f |,
I I dxj_ dxi + dy_ dy_
Dak da3 dak da3 k = 1V i = 1 k 3 k 3
I| (xei- x, + (yei - Y,) f
To solve the system of equations (8), it is necessary
dimo to calculate partial derivatives of the form
(6) , Where I = 1-^, k = 1-3. Since the analytical
the dependences (4) on the parameters a1 are unknown, the partial derivatives are determined numerically.
New values of ak (where k = 1-3) are calculated using the found values of Aak according to the formula
0 0 , . ak = ak + Ak
SOLUTION ALGORITHM
For the numerical solution of the system of equations (8), the following algorithm has been developed.
DIRECTOR etc.
Rice. Fig. 5. The shape of a water drop in the sessile drop method: 1 - experimental points; 2 - calculation using the optimization procedure.
1. Setting the initial approximation (a0, a0, a0) under the assumption that the shape of the drop is approximately described by an ellipse with semi-axes equal to the height of the drop and the maximum radius of the circle of the horizontal section.
2. Setting small deviations (Aab Aa2, Aa3).
3. Solving the system of equations (3) using the DIFSUB algorithm for given values (a0, a0, a0). Obtaining the 1st numerical solution. Determination of functional dependencies xn and yn using the algorithm for calculating the parameters of the cubic spline function SPLINE .
4. Solving the system of equations (3) using the DIFSUB algorithm for given values (a0 + Aa1, a0, a0). Obtaining the 2nd numerical solution. Determination of functional dependencies xi2 and yi2 using the SPLINE algorithm. Calculation of derivatives using the 1st and 2nd solutions
dx1 = Xg - xp dy1 = y2 - yn. da1 Aa1 da1 Aa1
5. Solving the system of equations (3) using the DIFSUB algorithm for given (a0, a0 +
Aa2, a0). Obtaining the 3rd numerical solution. Determining the functional dependencies of xi3 and yi3 using the SPLINE algorithm. Calculation of derivatives using the 1st and 3rd solutions
dX \u003d Xz - x / 1? g!± = Ya - Y/1. da2 Aa2 da2 Aa2
6. Solution of the system of equations (3) using
a3 + Aa3). Obtaining the 4th numerical solution. Determining the functional dependencies of xi4 and yi4 using the SPLINE algorithm. Calculation of derivatives using the 1st and 4th solutions
dX/ \u003d X / 4 - Xj 1 dyl \u003d Y / 4 - Y / 1.
7. Calculation of the coefficients of system (8) and its solution using the algorithm for solving the system of linear equations SOLVE . Receipt (Aab Aa2, Aa3).
8. Calculation of new parameter values by formula (9)
algorithm DIFSUB
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KASHEZHEV A. Z., KUTUEV R. A., PONEZHEV M. Kh., SOZAEV V. A., KHASANOV A. I. - 2012
Ponomareva M.A., Yakutenok V.A. - 2011
The equation is also considered in two-dimensional and one-dimensional space. In two-dimensional space, the Laplace equation is written:
∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 (\displaystyle (\frac (\partial ^(2)u)(\partial x^(2)))+(\frac (\partial ^(2 )u)(\partial y^(2)))=0)Also in n-dimensional space. In this case, the sum is equal to zero n second derivatives.
Δ = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 + . . . (\displaystyle \Delta =(\frac (\partial ^(2))(\partial x^(2)))+(\frac (\partial ^(2))(\partial y^(2)))+ (\frac (\partial ^(2))(\partial z^(2)))+...)
1 r 2 ∂ ∂ r (r 2 ∂ f ∂ r) + 1 r 2 sin θ ∂ ∂ θ (sin θ ∂ f ∂ θ) + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2 = 0 ( \displaystyle (1 \over r^(2))(\partial \over \partial r)\left(r^(2)(\partial f \over \partial r)\right)+(1 \over r^( 2)\sin \theta )(\partial \over \partial \theta )\left(\sin \theta (\partial f \over \partial \theta )\right)+(1 \over r^(2)\sin ^(2)\theta )(\partial ^(2)f \over \partial \varphi ^(2))=0)
Singular points r = 0 , θ = 0 , θ = π (\displaystyle r=0,\theta =0,\theta =\pi ).
1 r ∂ ∂ r (r ∂ u ∂ r) + 1 r 2 ∂ 2 u ∂ φ 2 = 0 (\displaystyle (\frac (1)(r))(\frac (\partial )(\partial r)) \left(r(\frac (\partial u)(\partial r))\right)+(\frac (1)(r^(2)))(\frac (\partial ^(2)u)(\ partial \varphi ^(2)))=0)special point.
1 r ∂ ∂ r (r ∂ f ∂ r) + ∂ 2 f ∂ z 2 + 1 r 2 ∂ 2 f ∂ φ 2 = 0 (\displaystyle (1 \over r)(\partial \over \partial r)\ left(r(\partial f \over \partial r)\right)+(\partial ^(2)f \over \partial z^(2))+(1 \over r^(2))(\partial ^ (2)f \over \partial \varphi ^(2))=0)singular point r = 0 (\displaystyle r=0).
The Laplace equation arises in many physical problems of mechanics, heat conduction, electrostatics, hydraulics. The Laplace operator is of great importance in quantum physics, in particular in the Schrödinger equation.
Despite the fact that the Laplace equation is one of the simplest in mathematical physics, its solution faces difficulties. The numerical solution is especially difficult because of the irregularity of the functions and the presence of singularities.
where C 1 , C 2 (\displaystyle C_(1),C_(2)) are arbitrary constants.
The Laplace equation on a two-dimensional space is satisfied by analytic functions. Analytic functions are considered in the theory of functions of a complex variable, and the class of solutions of the Laplace equation can be reduced to a function of a complex variable.
The Laplace equation for two independent variables is formulated as follows
φ x x + φ y y = 0. (\displaystyle \varphi _(xx)+\varphi _(yy)=0.)If z = x + iy, and
f (z) = u (x , y) + i v (x , y) , (\displaystyle f(z)=u(x,y)+iv(x,y),)then the Cauchy-Riemann conditions are necessary and sufficient for the function f(z) was analytic:
∂ u ∂ x = ∂ v ∂ y , ∂ u ∂ y = − ∂ v ∂ x . (\displaystyle (\frac (\partial u)(\partial x))=(\frac (\partial v)(\partial y)),~(\frac (\partial u)(\partial y))=- (\frac (\partial v)(\partial x)).)Both the real and imaginary parts of analytic functions satisfy the Laplace equation. By differentiating the conditions
