On the non-existence of real-valued, analytical mass-density solutions corresponding to an expansion or compression of an ideal gas along the streamlines, by considering a steady, isentropic, 2D-flow through a Laval nozzle in orthogonal curvilinear coordinates in the Euclidean 2D-space

Abstract

Assuming that the streamlines are given by keeping constant one of the two orthogonal curvilinear coordinates in the Euclidean two-dimensional space, while considering a steady, two-dimensional, isentropic flow of an ideal gas through a convergent-divergent nozzle, and thus parallel to the curvilinear upper and lower walls of the nozzle, the theory of differential geometry together with the balance equations of physics was used to study the existence (or non-existence !) of real-valued, differentiable and integrable mass-density solutions to the problem, by means of analytical solutions, and corresponding to an expansion or compression of the gas along the streamlines. Thus, by initially assuming that the partial mass-density derivatives with respect to both curvilinear coordinates satisfy the integrability condition of Schwarz, the resulting system of four scalar partial differential equations led to an analytically derived quadratic equation for the determination of the ideal-gas mass density, based on generalised orthogonal curvilinear coordinates: Finally, the four orthogonal curvilinear coordinate systems, defined by the Killing two-tensors for the Euclidean two-dimensional space, were used, in order to examine whether these coordinate systems could satisfy the already mentioned generalised curvilinear-geometry equation as a quadratic equation, and the related requirements with regard to the partial mass-density derivatives, or not. Only real and nonzero, positive values for the mass density were considered, based on curvilinear streamlines of nonzero curvature.

Appendices

Appendix 1:

General relations, definitions and equations in order to solve the system of partial differential equations

In the following pages, the Einstein notation is used, with regard to indices. Superscripts are not denoting exponents. Exponents are always following parentheses or brackets.

The general equation for the mathematical operations with the del operator in curvilinear coordinates, where $\left( ... \right)$ in general denotes a tensor (a scalar, a vector, or a tensor of higher order), is

$$\begin{aligned} \nabla \otimes \left( ... \right) \equiv {\varvec{g}}^i \otimes \partial _i \left( ... \right) \end{aligned}$$

(41)

where the general symbol for any product, denoted by $\otimes$, stands either for the dot product $\left( \cdot \right)$, or the cross product $\left( \times \right)$, or for both the algebraic and the dyadic product ( ), and where

$$\begin{aligned} \partial _i \left( ... \right) \equiv \frac{ \partial }{ \partial z^i} \left( ... \right) . \end{aligned}$$

(42)

The contravariant base vectors, ${\varvec{g}}^i$, are defined in the following pages.

We consider a position vector in 2D curvilinear coordinates:

$$\begin{aligned} {\varvec{x}} = {\varvec{g}} \left( z^i \right) = x \left( z^1,z^2 \right) \varvec{\iota }_x + y \left( z^1,z^2 \right) \varvec{\iota }_y, \end{aligned}$$

(43)

where $\varvec{\iota }_x$ and $\varvec{\iota }_y$ are denoting the Euclidean base vectors as unit vectors in the Cartesian $x-$ and $y-$direction, respectively.

The corresponding covariant base vectors are defined by

$$\begin{aligned} {\varvec{g}}_i \equiv \frac{\partial {\varvec{g}}}{\partial z^i} . \end{aligned}$$

(44)

From the last definition follows for the 2D orthogonal curvilinear coordinates, that

$$\begin{aligned} {\varvec{g}}_i = \left( \partial _i x \right) \varvec{\iota }_x + \left( \partial _i y \right) \varvec{\iota }_y, \end{aligned}$$

(45)

where the covariant metric-tensor elements, $g_{ij}$, are defined as the scalar product of the covariant base vectors:

$$\begin{aligned} g_{ij} \equiv {\varvec{g}}_i \cdot {\varvec{g}}_j, \end{aligned}$$

(46)

and thus is $g_{ij} = g_{ji}$. In general is

$$\begin{aligned} g_{ik} g^{kj}= {\delta _i}^j, \end{aligned}$$

(47)

where $g^{kj}$ are the contravariant elements of the metric tensor: The elements of the Kronecker-delta, denoted by ${\delta _i}^j$, are according to their general definition equal to 1 for $i=j$, and equal to zero for $i \ne j$.

The following products were used considering the Cartesian unit (base) vectors shown previously:

dot product:

$$\begin{aligned} \varvec{\iota }_i \cdot \varvec{\iota }_j = \delta _{ij} , \end{aligned}$$

(48)

cross product:

$$\begin{aligned} \varvec{\iota }_i \times \varvec{\iota }_j = \epsilon _{ijk} \varvec{\iota }_k , \end{aligned}$$

(49)

where $\delta _{ij}$ denotes also in this case the Kronecker-delta, and where $\epsilon _{ijk}$ is the Levi-Civita symbol, defined as $\epsilon _{ijk} \equiv 1$ for $i,j,k = 1, 2, 3$ and all even numbers of permutations of the indices i, j, k. Otherwise is $\epsilon _{ijk} \equiv -1$ for all odd numbers of permutations of the indices i, j, k. In any other case, when one of the indices appears at least two times, is $\epsilon _{ijk} \equiv 0$.

Finally, the dyadic product between $\varvec{\iota }_i$ and $\varvec{\iota }_j$ is simply denoted by $\varvec{\iota }_i \varvec{\iota }_j$.

Due to the already assumed orthogonality with regard to the 2D curvilinear coordinate lines, the 2D-metric tensor has the following covariant elements, written as a diagonal matrix:

$$\begin{aligned} \left( g_{ij} \right) = \begin{pmatrix} g_{11} &{} 0 \\ 0 &{} g_{22} \end{pmatrix}. \end{aligned}$$

(50)

Note that the orthogonality in general implies that $g_{12}=0$ and $g_{21}=0$.

Thus, the contravariant elements of the 2D-metric tensor, in general given by the inverse matrix, can in this 2D orthogonal case be written as

$$\begin{aligned} \left( g^{ij} \right) = \left( g_{ij} \right) ^{(-1)} = \begin{pmatrix} g^{11} &{} 0 \\ 0 &{} g^{22} \end{pmatrix}, \end{aligned}$$

(51)

where

$$\begin{aligned} g^{11} = \frac{1}{g_{11}}, \end{aligned}$$

(52)

and

$$\begin{aligned} g^{22} = \frac{1}{g_{22}}, \end{aligned}$$

(53)

are the inverse elements of the previously shown diagonal matrix.

Note that the contravariant base vectors are defined as

$$\begin{aligned} {\varvec{g}}^{i} = g^{ij} {\varvec{g}}_{j} . \end{aligned}$$

(54)

According to the already introduced terms and definitions, it follows that

$$\begin{aligned} g_{11} = \left( \partial _1 x \left( z^1, z^2 \right) \right) ^2 + \left( \partial _1 y \left( z^1, z^2 \right) \right) ^2 \ge 0, \end{aligned}$$

(55)

and

$$\begin{aligned} g_{22} = \left( \partial _2 x \left( z^1, z^2 \right) \right) ^2 + \left( \partial _2 y \left( z^1, z^2 \right) \right) ^2 \ge 0. \end{aligned}$$

(56)

The following equation is a requirement for any 2D orthogonal curvilinear coordinate system:

$$\begin{aligned} g_{12} = g_{21} = \left( \partial _1 x \left( z^1, z^2 \right) \right) \left( \partial _2 x \left( z^1, z^2 \right) \right) + \left( \partial _1 y \left( z^1, z^2 \right) \right) \left( \partial _2 y \left( z^1, z^2 \right) \right) = 0. \end{aligned}$$

(57)

The Christoffel symbols of second kind are in general calculated according to:

$$\begin{aligned} {{\Gamma ^k}_{ij}} = \frac{1}{2} g^{kl} \left( \partial _j g_{li} + \partial _i g_{lj} - \partial _l g_{ij} \right) , \end{aligned}$$

(58)

where in torsion-free spaces, as in this work, is ${{\Gamma ^k}_{ij}} = {{\Gamma ^k}_{ji}}$.

Furthermore, it can be shown that

$$\begin{aligned}{} & {} {{\Gamma ^1}_{11}} = \frac{1}{2 g_{11}} \partial _1 g_{11} \equiv \frac{1}{2} \partial _1 \left( \text {ln}g_{11} \right) , \end{aligned}$$

(59)

$$\begin{aligned}{} & {} {{\Gamma ^1}_{12}} = \frac{1}{2 g_{11}} \partial _2 g_{11} \equiv \frac{1}{2} \partial _2 \left( \text {ln}g_{11} \right) , \end{aligned}$$

(60)

$$\begin{aligned}{} & {} {{\Gamma ^1}_{21}} = {{\Gamma ^1}_{12}}, \end{aligned}$$

(61)

$$\begin{aligned}{} & {} {{\Gamma ^1}_{22}} = - \frac{1}{2 g_{11}} \partial _1 g_{22}, \end{aligned}$$

(62)

$$\begin{aligned}{} & {} {{\Gamma ^2}_{11}} = - \frac{1}{2 g_{22}} \partial _2 g_{11}, \end{aligned}$$

(63)

$$\begin{aligned}{} & {} {{\Gamma ^2}_{12}} = \frac{1}{2 g_{22}} \partial _1 g_{22} \equiv \frac{1}{2} \partial _1 \left( \text {ln}g_{22} \right) , \end{aligned}$$

(64)

$$\begin{aligned}{} & {} {{\Gamma ^2}_{21}} = {{\Gamma ^2}_{12}}, \end{aligned}$$

(65)

and

$$\begin{aligned} {{\Gamma ^2}_{22}} = \frac{1}{2 g_{22}} \partial _2 g_{22} \equiv \frac{1}{2} \partial _2 \left( \text {ln}g_{22} \right) . \end{aligned}$$

(66)

It can also be shown that any Christoffel symbol of second kind, ${{\Gamma ^k}_{ij}}$, is equal to zero when at least one of the indices i, j or k is equal to 3: This follows from the already introduced geometrical assumptions.

Thus is

$$\begin{aligned} {{\Gamma ^1}_{11}} + {{\Gamma ^2}_{12}} \equiv \frac{1}{2} \partial _1 \left( \text{ ln }g_{11} \right) + \frac{1}{2} \partial _1 \left( \text{ ln }g_{22} \right) \equiv \frac{1}{2} \partial _1 \left( \text{ ln } \left( g_{11} g_{22} \right) \right) \equiv \partial _1 \left( \text{ ln } \left( \sqrt{ g_{11} g_{22}} \right) \right) \end{aligned}$$

(67)

Substituting $g_{11} \equiv \lambda _1 G_{11}$ and $g_{22} \equiv \lambda _2 G_{22}$, where $\lambda _1$ and $\lambda _2$ are constant values, leads to:

$$\begin{aligned} {{\Gamma ^1}_{11}} + {{\Gamma ^2}_{12}} \equiv \partial _1 \left( \text {ln} \left( \sqrt{ G_{11} G_{22}} \right) \right) . \end{aligned}$$

(68)

Thus is

$$\begin{aligned} \int \left( {{\Gamma ^1}_{11}} + {{\Gamma ^2}_{12}} \right) \textrm{d} z^1 \equiv \text {ln} \left( \sqrt{ G_{11} G_{22}} \right) + D_2 \left( z^2 \right) , \end{aligned}$$

(69)

and

$$\begin{aligned} \text {exp} \left( - \int \left( {{\Gamma ^1}_{11}} + {{\Gamma ^2}_{12}} \right) \textrm{d} z^1 \right) \equiv \frac{\text {exp} \left( -D_2 \left( z^2 \right) \right) }{\sqrt{ G_{11} G_{22}}} . \end{aligned}$$

(70)

Finally, considering the required partial differentiation in this paper, following equation was also used:

$$\begin{aligned} \partial _1 \left( \int \left( {{\Gamma ^1}_{11}} + {{\Gamma ^2}_{12}} \right) \textrm{d} z^1 \right) \equiv {{\Gamma ^1}_{11}} + {{\Gamma ^2}_{12}} \equiv \partial _1 \left( \text {ln} \left( \sqrt{ G_{11} G_{22}} \right) \right) \equiv \frac{1}{2} \partial _1 \left( \text {ln} \left( G_{11} G_{22} \right) \right) . \end{aligned}$$

(71)

Appendix 2:

Terms related to the mass-density roots according to the general quadratic equation

$$\begin{aligned}{} & {} f_1 \equiv \frac{(g_{22})^2}{\left( \lambda _1 \lambda _2 \right) ^2} \left( \frac{\kappa C}{(D)^2} \right) ^2 \left( f - \frac{1}{\kappa } \left( \partial _1 \left( \text {ln}G_{22} \right) \right) \left( \partial _2 \left( \text {ln} \left( (D)^{2 \kappa } C \right) \right) \right) \right) , \end{aligned}$$

(72)

$$\begin{aligned}{} & {} f \equiv \left( \partial _1 \left( \text {ln}G_{22} \right) \right) \left( \partial _2 \left( \text {ln} \frac{G_{22}}{G_{11}} \right) \right) - \partial _1 \partial _2 \left( \text {ln} \frac{G_{22}}{G_{11}} \right) , \end{aligned}$$

(73)

$$\begin{aligned}{} & {} f_2 \equiv \frac{2 \kappa C g_{22}}{\lambda _1 \lambda _2 (D)^2} \left( \frac{1}{2} \left( \partial _1 \partial _2\left( \text {ln}G_{22}\right) \right) + \left( \partial _1 \left( \text {ln}G_{22}\right) \right) \left( \partial _2 \left( \text {ln}G_{11}\right) \right) - \left( \partial _1 \partial _2\left( \text {ln}G_{11}\right) \right) \right) , \end{aligned}$$

(74)

$$\begin{aligned}{} & {} f_3 \equiv \left( \partial _1 \partial _2\left( \text {ln}G_{11}\right) \right) + \frac{\kappa -1}{2} \left( \partial _1 \left( \text {ln}G_{22}\right) \right) \left( \partial _2 \left( \text {ln}G_{11}\right) \right) , \end{aligned}$$

(75)

$$\begin{aligned}{} & {} g_{11} \equiv \lambda _1 G_{11}, \end{aligned}$$

(76)

and, finally

$$\begin{aligned} g_{22} \equiv \lambda _2 G_{22}. \end{aligned}$$

(77)

Appendix 3:

Dependence of other properties on the mass-density, based on the generalised orthogonal curvilinear geometry and on the boundary condition

Pressure:

$$\begin{aligned} p = \left( \rho \right) ^\kappa C. \end{aligned}$$

(78)

(Physical) Speed, based on normalised base vectors:

$$\begin{aligned} v = {\tilde{v}} \sqrt{g_{11}} = \sqrt{\lambda _1} \frac{D}{ \rho \sqrt{G_{22}}}. \end{aligned}$$

(79)

Absolute temperature (ideal gas):

$$\begin{aligned} T = \frac{p}{\rho R}. \end{aligned}$$

(80)

Speed of sound (ideal gas):

$$\begin{aligned} a = \sqrt{\left( \frac{\partial p}{\partial \rho } \right) _{S = \text {const.}}} = \sqrt{\kappa R T}. \end{aligned}$$

(81)

Mach number:

$$\begin{aligned} \text {Ma} = \frac{v}{a}. \end{aligned}$$

(82)

Appendix 4:

Constant mass flow rate through the nozzle in 2D orthogonal curvilinear coordinates, and the D -function

The mass flow rate is defined as

$$\begin{aligned} \frac{\textrm{d}m}{\textrm{d}t}= \int \rho {\varvec{v}} \cdot \textrm{d}{\varvec{A}}. \end{aligned}$$

(83)

Due to the dot product in the last equation, only the corresponding magnitudes of vector components which are parallel or antiparallel to each other shall be considered. Furthermore, the already mentioned assumption with regard to the velocity of the fluid should be taken into account. The magnitude of the velocity, $v \equiv |{\varvec{v}}|$, is given in Appendix 3.

According to Fig. 2 and due to the generalised 2D orthogonal curvilinear coordinate system, the velocity is always perpendicular to any surface with $z^1 = \text {constant}$, and thus parallel to the normal vector of the element of area. Let the width of the 2D flow be denoted by b, perpendicular to the $(z^1, z^2)$-plane: The magnitude of the infinitesimal area element, $\textrm{d}A \equiv |\textrm{d}{\varvec{A}}|$ for $z^1 = \text {const.}$, is then equal to

$$\begin{aligned} \textrm{d}A = b \textrm{d}s, \end{aligned}$$

(84)

where $\textrm{d}s$ is the infinitesimal length for $z^1 = \text {constant.}$ According to differential geometry is

$$\begin{aligned} \textrm{d}s = \sqrt{g_{ij} \textrm{d}z^i \textrm{d}z^j}, \end{aligned}$$

(85)

and thus, in this case, where velocity is always perpendicular to any surface with $\textrm{d}z^1 = 0$, it follows that

$$\begin{aligned} \textrm{d}s = \sqrt{g_{22}} \textrm{d}z^2. \end{aligned}$$

(86)

According to the previously mentioned assumption regarding the velocity and the streamlines, and as already shown in Appendix 3, is

$$\begin{aligned} v = {\tilde{v}} \sqrt{g_{11}}. \end{aligned}$$

(87)

Putting all together, and based on the equations of Appendix 3, it follows that

$$\begin{aligned} \frac{\textrm{d}m}{\textrm{d}t}= b \sqrt{\lambda _1 \lambda _2} \int D \left( z^2 \right) \textrm{d}z^2, \end{aligned}$$

(88)

and thus

$$\begin{aligned} \frac{\textrm{d}m}{\textrm{d}t}= b \int \rho v \sqrt{g_{22}} \textrm{d}z^2. \end{aligned}$$

(89)

Note that D can also be written as:

$$\begin{aligned} D \left( z^2 \right) = \left( \rho v \sqrt{\frac{G_{22}}{\lambda _1}} \right) _{z^1 = z^1 \text { gas tank}}. \end{aligned}$$

(90)

Appendix 5:

General functions following from the mathematical requirements regarding the partial mass-density derivatives with respect to the curvilinear coordinates

$$\begin{aligned}{} & {} l_1 \left( z^1, z^2 \right) \equiv w \partial _1 \left( \frac{f_2}{f_1} \right) \left( \frac{f_2}{f_1} \frac{ \kappa C G_{22} }{2 \lambda _1 \left( D \right) ^2} + 1 \right) + \frac{ \kappa C G_{22} }{ \lambda _1 \left( D \right) ^2} w \partial _1 \left( \left( w \right) ^2 \right) + \left( \kappa + 1 \right) w \frac{f_2}{f_1} \Gamma ^2_{12} \end{aligned}$$

(91)

$$\begin{aligned}{} & {} l_2 \left( z^1, z^2 \right) \equiv \frac{ \kappa C G_{22} }{ \lambda _1 \left( D \right) ^2 } \left( \left( w \right) ^2 \partial _1 \left( \frac{f_2}{f_1} \right) + \frac{ \partial _1 \left( \left( w \right) ^2 \right) }{2} \frac{f_2}{f_1} \right) + \partial _1 \left( \left( w \right) ^2 \right) + 2 \left( \kappa + 1 \right) \Gamma ^2_{12} \left( w \right) ^2 \end{aligned}$$

(92)

$$\begin{aligned}{} & {} j_1 \left( z^1, z^2 \right) \equiv - \frac{\partial _2 \left( \frac{f_2}{f_1} \right) }{2 \left( \kappa + 1 \right) } + \lambda _2 \frac{ \left( D \right) ^2}{ \kappa C} \frac{ \Gamma ^2_{11} }{G_{11}} - \frac{ \partial _2 C }{2 \kappa C} \frac{f_2}{f_1} \end{aligned}$$

(93)

$$\begin{aligned}{} & {} j_2 \left( z^1, z^2 \right) \equiv \frac{ \partial _2 \left( \left( w \right) ^2 \right) }{2 \left( \kappa + 1 \right) w} + \frac{ \partial _2 C }{ \kappa C } w \end{aligned}$$

(94)

Appendix 6:

Other equations and relations used in this work considering the four orthogonal coordinate systems, as defined by the Killing 2-tensors for the Euclidean 2D-space

The elliptic-hyperbolic curvilinear coordinates are defined as:

$$\begin{aligned}{} & {} x \left( z^1,z^2 \right) = \lambda _0 \text {sinh} \left( z^1 \right) \text {sin} \left( z^2 \right) , \end{aligned}$$

(95)

$$\begin{aligned}{} & {} y \left( z^1,z^2 \right) = \lambda _0 \text {cosh} \left( z^1 \right) \text {cos} \left( z^2 \right) . \end{aligned}$$

(96)

From the equations and definitions shown in Appendix 1, it follows that

$$\begin{aligned} g_{11} = g_{22} = \left( \lambda _0 \right) ^2 \left( \left( \text {cosh} \left( z^1 \right) \right) ^2 - \left( \text {cos} \left( z^2 \right) \right) ^2 \right) , \end{aligned}$$

(97)

which is equivalent to

$$\begin{aligned} g_{11} = g_{22} = \frac{\left( \lambda _0 \right) ^2}{2} \left( \text {cosh} \left( 2z^1 \right) - \text {cos} \left( 2z^2 \right) \right) . \end{aligned}$$

(98)

According to the already introduced definitions is

$$\begin{aligned} \lambda _1 = \lambda _2 = \frac{\left( \lambda _0 \right) ^2}{2} \end{aligned}$$

(99)

and

$$\begin{aligned} G_{11} = G_{22} = \text {cosh} \left( 2z^1 \right) - \text {cos} \left( 2z^2 \right) . \end{aligned}$$

(100)

Moreover, the orthogonality of the curvilinear coordinate lines is confirmed, due to

$$\begin{aligned} g_{12} = g_{21} = 0. \end{aligned}$$

(101)

According to the equations shown in Appendix 2, in this case is $f=0$, and thus

$$\begin{aligned}{} & {} f_1 = -\frac{8}{\kappa \left( \lambda _0 \right) ^4} \left( \frac{\kappa C}{ \left( D \right) ^2 } \right) ^2 \text {sinh}\left( 2z^1 \right) \left( \text {cosh} \left( 2z^1 \right) - \text {cos} \left( 2z^2 \right) \right) \partial _2 \left( \text {ln} \left( (D)^{2 \kappa } C \right) \right) , \end{aligned}$$

(102)

$$\begin{aligned}{} & {} f_2 = \frac{ 24 \kappa C \text {sinh} \left( 2z^1 \right) \text {sin} \left( 2z^2 \right) }{ \left( \lambda _0 \right) ^2 \left( D \right) ^2 \left( \text {cosh} \left( 2z^1 \right) - \text {cos} \left( 2z^2 \right) \right) } , \end{aligned}$$

(103)

and finally:

$$\begin{aligned} f_3 = \frac{ 2 \left( \kappa - 3 \right) \text {sinh}\left( 2z^1 \right) \text {sin}\left( 2z^2 \right) }{ \left( \text {cosh} \left( 2z^1 \right) - \text {cos} \left( 2z^2 \right) \right) ^2} . \end{aligned}$$

(104)

polar coordinates (circular segments as curvilinear streamlines):

$$\begin{aligned}{} & {} x \left( z^1,z^2 \right) = z^2 \text {cos} \left( z^1 \right) \end{aligned}$$

(105)

$$\begin{aligned}{} & {} y \left( z^1,z^2 \right) = z^2 \text {sin} \left( z^1 \right) \end{aligned}$$

(106)

$$\begin{aligned}{} & {} g_{11} = \left( z^2 \right) ^2 \end{aligned}$$

(107)

$$\begin{aligned}{} & {} g_{22} = 1 \end{aligned}$$

(108)

and according to the already introduced definitions:

$$\begin{aligned}{} & {} \lambda _1 = 1 \end{aligned}$$

(109)

$$\begin{aligned}{} & {} \lambda _2 = 1 \end{aligned}$$

(110)

$$\begin{aligned}{} & {} G_{11} = \left( z^2 \right) ^2 \end{aligned}$$

(111)

$$\begin{aligned}{} & {} G_{22} = 1 \end{aligned}$$

(112)

$$\begin{aligned}{} & {} g_{12} = g_{21} = 0 \end{aligned}$$

(113)

$$\begin{aligned}{} & {} f_1 = 0 \end{aligned}$$

(114)

$$\begin{aligned}{} & {} f_2 = 0 \end{aligned}$$

(115)

$$\begin{aligned}{} & {} f_3 = 0 \end{aligned}$$

(116)

(2D) parabolic coordinates:

$$\begin{aligned}{} & {} x \left( z^1,z^2 \right) = z^1 z^2 \end{aligned}$$

(117)

$$\begin{aligned}{} & {} y \left( z^1,z^2 \right) = \frac{1}{2} \left( \left( z^1 \right) ^2 - \left( z^2 \right) ^2 \right) \end{aligned}$$

(118)

$$\begin{aligned}{} & {} g_{11} = g_{22} = \left( z^1 \right) ^2 + \left( z^2 \right) ^2 \end{aligned}$$

(119)

and according to the already introduced definitions:

$$\begin{aligned}{} & {} \lambda _1 = \lambda _2 = 1 \end{aligned}$$

(120)

$$\begin{aligned}{} & {} G_{11} = G_{22} = \left( z^1 \right) ^2 + \left( z^2 \right) ^2 \end{aligned}$$

(121)

$$\begin{aligned}{} & {} g_{12} = g_{21} = 0 \end{aligned}$$

(122)

$$\begin{aligned}{} & {} f_1 = - 2 \kappa \left( \frac{C}{ \left( D \right) ^2} \right) ^2 \partial _2 \left( \text {ln} \left( \left( D \right) ^{2 \kappa } C \right) \right) z^1 \left( \left( z^1 \right) ^2 + \left( z^2 \right) ^2 \right) \end{aligned}$$

(123)

$$\begin{aligned}{} & {} f_2 = \frac{12 \kappa C}{ \left( D \right) ^2 } \frac{z^1 z^2 }{ \left( z^1 \right) ^2 + \left( z^2 \right) ^2 } \end{aligned}$$

(124)

$$\begin{aligned}{} & {} f_3 = 2 \left( \kappa - 3 \right) \frac{ z^1 z^2 }{ \left( \left( z^1 \right) ^2 + \left( z^2 \right) ^2 \right) ^2 } \end{aligned}$$

(125)

Cartesian coordinates:

$$\begin{aligned}{} & {} x \left( z^1,z^2 \right) = z^1 \end{aligned}$$

(126)

$$\begin{aligned}{} & {} y \left( z^1,z^2 \right) = z^2 \end{aligned}$$

(127)

$$\begin{aligned}{} & {} g_{11} = g_{22} = 1 \end{aligned}$$

(128)

and according to the already introduced definitions:

$$\begin{aligned}{} & {} \lambda _1 = \lambda _2 = 1 \end{aligned}$$

(129)

$$\begin{aligned}{} & {} G_{11} = G_{22} = 1 \end{aligned}$$

(130)

$$\begin{aligned}{} & {} g_{12} = g_{21} = 0 \end{aligned}$$

(131)

$$\begin{aligned}{} & {} f_1 = f_2 = f_3 = 0 \end{aligned}$$

(132)