Mathematical and Computational Modeling of Catalytic Converter Using Navier–Stokes Equations in Curvilinear Coordinates

Abstract

This article discusses the problem of numerically solving the Navier–Stokes equations, the heat conduction equation, and the transport equation in the orthogonal coordinates of a free curve. Since the numerical solution domain is complex, the curvilinear mesh method was used. To do so, first, a boundary value problem was posed for the elliptic equation to automate the creation of orthogonal curved meshes. By numerically solving this problem, the program code for the curvilinear mesh generator was created. The motion of a liquid or gas through a porous medium was described by numerically solving the Navier–Stokes equations in freely curvilinear orthogonal coordinates. The transformation of the Navier–Stokes equation system, written in the stream function, vorticity variables, and cylindrical coordinates, into arbitrary curvilinear coordinates, was considered in detail by introducing metric coefficients. To solve these equations, the coefficients of which vary rapidly, a three-layer differential scheme was developed. The approximation, stability, and compactness of the differential scheme were previously studied. The considered problem was considered to be the mathematical model of a car catalytic converter, and computational experiments were conducted. Calculations were performed with the developed program code in different geometries of the computational domain and different values of grid size. The Reynolds number was changed from 100 to 10,000, and its effect on the size of the backflow in front of the porous medium was discussed. The software code, which is based on the differential equation of the Navier–Stokes equations written in the orthogonal coordinates of a curved line, and its calculation algorithm can be used for the mathematical and computer modeling of automobile catalytic converters and chemical reactors.

1. Introduction

The exhaust gases emitted by automobiles move in the lower layers of the atmosphere and negatively affect human respiratory health. Therefore, the automotive industry has been continuously improving the design of catalytic converters to reduce harmful emissions. However, in developing countries, many old vehicles remain in use, and most of them lack catalytic converters. To minimize the release of harmful substances into the environment, developing efficient models of catalytic converters is a highly relevant issue.

Additionally, the combustion chamber of an automobile engine acts as a chemical reactor, synthesizing toxic substances and expelling their residues into the atmosphere.

Catalytic converters have a curved geometry, a catalytic reaction with a temperature effect occurs in them, and a gaseous mixture is transported. To fully describe these processes mathematically, it is necessary to use the nonlinear Navier–Stokes equations.

Therefore, in this work, a method for numerically solving the Navier–Stokes equations in arbitrary curvilinear coordinates is presented through the mathematical modeling of automotive catalytic converters.

The Navier–Stokes equations were transformed from a cylindrical coordinate system to a free curvilinear coordinate system, and a three-step difference scheme and the method of orthogonal curvilinear grids were used for the resulting differential equations; the numerical solutions of the equations are found at the nodes of these grids. To describe the flow of a viscous fluid in a porous medium, the Navier–Stokes equation system is considered by using the stream function and vorticity variables. The stream function represents the velocity field structure, while vorticity describes the rotational motion of the fluid.

The following difficulties arise when converting the Navier–Stokes equations from cylindrical coordinates to free orthogonal curvilinear coordinates: during the conversion to new coordinates, the nonlinear terms of the equation become more complex; the system of equations becomes more complex due to the introduction of coordinate metric coefficients; the construction of the differential scheme of the transformed gradient, divergence, and Laplace differential operators is not widely considered in the known scientific literature; the generation of a curvilinear orthogonal grid and the exact boundary conditions are also very relevant. The problem of studying the approximation, stability, compactness, and accuracy of numerical solution methods is very relevant today.

The results of the research study presented in this article allow us to understand the distribution of flow in porous media and its dynamics.

There are several methods for numerically solving boundary value problems for the Navier–Stokes equations in regions with complex geometry.

An easy-to-use pseudo-domain method is presented in the works by Glowinski et al. [1,2,3,4,5]. This method allows the exact boundary conditions to be satisfied by using variational principles. The main goal of the method is to simplify the calculation by covering complex geometric domains with simple shapes. For solving mathematical physics problems in arbitrary domains numerically, the effectiveness of the fictitious domain method was demonstrated in the study by Smagulov et al. [6]. The peculiarity of this modification of the pseudo-region method is that it allows several important problems to be solved through a new formulation of the pressure and velocity at the boundary of the pseudo-region. In particular, the problem of the curved boundary of the region is solved, a boundary condition is imposed for pressure, and the construction of a homogeneous difference scheme arises.

This method is used in industry and in applied problems in various fields of science. The authors of [6] aimed to demonstrate the effectiveness and efficiency of the pseudo-domain method through the mathematical and computer modeling of complex geometric domains.

This method reduces computational complexity and allows for the easy solution of simple boundary value problems. However, the accuracy may be low for complex boundaries. Due to its high computational efficiency, this method is suitable for obtaining quick solutions.

The finite element method (FEM), widely used in computational practice, divides the area into various elements, including triangles, squares, hexagons, or in the three-dimensional case, tetrahedra, and the calculations are performed on each part. As the geometry becomes more complex, the mesh becomes more difficult, because the correct choice of element shapes and their coordination becomes more difficult.

The FEM is used in the automotive industry, especially in the design and improvement of catalytic converters to reduce emissions.

There are many works that have studied catalytic converters by using the FEM. One of them is Chuah et al. [7], which focuses on the numerical calculation of the efficiency of three-phase catalytic converters (TWCCs) in automobiles for reducing nitrogen oxides. The work created a software package, NEUTRALISATOR, based on the FEM. The method used in the work shows that it is a comprehensive tool for integrating the exhaust gas process in an automobile, its chemical reactions, and geometric parameters, which means that this method has a wide range of applications in solving engineering problems.

As the geometry becomes more complex, it becomes more difficult to create unstructured meshes, so the number of meshes in the FEM increases [8]. Therefore, due to the development of modern computer technology, the method of curvilinear meshes has become widely used, and the mathematical modeling of viscous incompressible fluid flow is often carried out based on the numerical solution of the system of Navier–Stokes equations in the stream function and vorticity variables [9].

Temirbekov [10], in his study, examined the Navier–Stokes equations for porous media in the Reynolds number range from 100 to 1000 and determined the effect of porous media on the dynamics of fluid flow.

Laurenzi et al. [11] solved Euler equations for gas dynamics in various cylindrical pipe configurations, along with an additional equation describing unburned gas fractions. Their approach evaluated the influence of pipe shapes on reducing exhaust gas residues, demonstrating that proper pipe geometry selection significantly enhances catalytic converter efficiency. Optimizing pipe design can lower harmful emissions, which is crucial for environmental protection.

As noted in the studies by Laurenzi et al. [11], the geometric shape of the catalytic converter plays an important role. Choosing the right shape of the converter increases its efficiency.

Settar et al. [12] conducted numerical studies on steam methane reforming (SMR) processes, focusing on catalytic layer configurations. They compared two designs: continuous catalytic layers and discrete catalytic layers separated by gaps. Their findings showed that using discrete layers prevented cold zones and improved methane conversion by 60%, achieved through an extended catalytic zone and continuous heat supply. This research study integrated equations of motion, energy, and gas dynamics, applying the Navier–Stokes equations to the gas mixture.

In the curvilinear grid method, to obtain numerical solutions, the equations must be transformed into curvilinear coordinates, resulting in a more complex form than the original equations. However, an advantage of this method is that it allows for a more detailed numerical solution without increasing the number of grid nodes. This means that while the grid size remains large in the physical domain, it becomes uniform in the computational domain. In the monograph by Godunov et al. [13], issues related to the solution of gas-dynamic problems in arbitrary curvilinear coordinates using moving grids are considered. Algorithms for constructing grids in the case of complex computational domains, systems of gas-dynamic equations in curvilinear coordinates, and difference calculation schemes adapted to specific problems and their features are described.

The curvilinear grid method is one of the most effective techniques for solving complex geometrical problems with high accuracy. Compared with other methods, it ensures precise boundary conditions, efficiently models complex regions, and is highly versatile for multidisciplinary studies. In the monograph in [14], adaptive curvilinear grids are constructed by using elliptic equations with tensor weight functions.

Danaev, Liseykin, and Yanenko’s work is devoted to the numerical solution of the modified Navier–Stokes equations for gas dynamics on a moving curved orthogonal grid [15].

In the work by Ladyzhenskaya, the stability and convergence of differential schemes for the Navier–Stokes equations in velocity–pressure variables were proven [16]. The methodological novelty of the article lies in the proposed three-level finite difference scheme for solving the system of equations and a non-iterative method for determining the velocity vortex at the domain boundary.

The scheme has been used by Vabishchevich for the Navier–Stokes equations in the stream function and vorticity (ψ, ω) variable [17].

When implementing the resulting differential schemes, the limitation on the time step was due only to the nonlinearity of the problem. That is, if we increase the time step, the calculation becomes unstable. In nonlinear problems, the dependencies between variables are complex, and large time steps lead to the loss of stability of the calculations and an increase in errors. Therefore, the limitation of the time step is necessary to ensure the accuracy and stability of the solution. New results on the numerical solution of the Navier–Stokes equations in complex geometric regions are presented in [10,18,19].

In this article, the Reynolds number was varied over a wide range (100 to 10,000), and a denser grid was used to analyze its impact on turbulence levels and flow structures. For high Reynolds numbers, the influence of flow resistance on flow structure and energy efficiency was extensively studied. These results reveal new opportunities for improving catalytic converter efficiency.

Our research is practically oriented toward the mathematical modeling of catalytic converter performance. For this purpose, a two-dimensional Navier–Stokes equation expressed in cylindrical coordinates was used as a mathematical model. The equations were formulated in terms of stream function and vorticity variables, with boundary conditions precisely applied to curved geometrical regions.

The considered problem has axial symmetry, so it is possible to study the properties of both the two-dimensional model and the three-dimensional problem.

The reason for analyzing the method of orthogonal curvilinear grids in the study is as follows: Since the computational domain is a unit square, the numerical calculations can be solved by refining the grid in the necessary part of the domain without increasing the grid size. In the physical domain, the grid step may be large, but in the finite element method, this is not possible because the areas of the triangles must be of the same size. The finite element method is highly dependent on the quality of the grid.

By further transforming the Navier–Stokes equations into curvilinear coordinates, the problem was solved numerically within a unit square, albeit with increased equation complexity.

Constructing a uniform finite difference scheme for such complex geometries is highly challenging. The solution involves transforming the Navier–Stokes equations into arbitrary curvilinear coordinates, where an orthogonal curvilinear grid was generated for numerical discretization. The method developed by Danaev [20] was used to construct this grid.

It has been proven by numerical experiments that the convergence rate of this method is less dependent on the change in the coefficients parameters.

Many methodological calculations were carried out, the results of which are presented in the form of isolines of the stream function with graphs for clarity.

Our model can be extended to study three-way catalytic converters (TWCCs) and their efficiency in reducing nitrogen oxide emissions. However, the following improvements are needed: the inclusion of chemical kinetics equations in the model, the inclusion of temperature effects, the inclusion of thermal dynamic equations, and the study of the effect of heat on the catalyst.

In this work, the model is considered for an incompressible fluid. If we adapt it to compressible gases, then we need to consider the gas-dynamic equations and use the gas equation of state; in the considered case, the velocity is not high, and the density does not change much, so this model can also be used for calculations.

2. Materials and Methods

2.1. Mathematical Model

To describe the unsteady flow in the smooth, curved regions of the catalytic converter boundary (Figure 1), the system of Navier–Stokes equations in cylindrical coordinates can be written in the following form [1]. Since the system of equations is a stream function and the speed of the winding are variables, the continuity equation is automatically satisfied.

$${\displaystyle \frac{\partial \omega }{\partial t}}+{\displaystyle \frac{\partial u\omega }{\partial x}}+{\displaystyle \frac{\partial \upsilon \omega }{\partial r}}={\displaystyle \frac{1}{Re}}\left(∆\omega +{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{\omega }{r}}\right)\right)-div\left({\displaystyle \frac{k_P}{r}}grad\psi \right)-{\displaystyle \frac{Gr}{{Re}^2}}{\displaystyle \frac{\partial \theta }{\partial x}},$$

(1)

$$div\left({\displaystyle \frac{1}{r}}grad\psi \right)=\omega ,$$

(2)

$${\displaystyle \frac{\partial {\phi }_i}{\partial t}}+{\displaystyle \frac{\partial u{\phi }_i}{\partial x}}+{\displaystyle \frac{\partial \upsilon {\phi }_i}{\partial r}}=div\left({\displaystyle \frac{D_i}{r}}grad{\phi }_i\right),$$

(3)

$${\displaystyle \frac{\partial \theta }{\partial t}}+{\displaystyle \frac{\partial u\theta }{\partial x}}+{\displaystyle \frac{\partial \upsilon \theta }{\partial r}}={\displaystyle \frac{1}{PePr}}\left({\displaystyle \frac{{\partial }^2\theta }{\partial x^2}}+{\displaystyle \frac{{\partial }^2\theta }{\partial r^2}}+{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{\theta }{r}}\right)\right)+{\displaystyle \frac{NuF}{PePr}}\left(T-\theta \right),$$

(4)

$${\displaystyle \frac{\partial T}{\partial t}}+{\displaystyle \frac{1}{PePr}}{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial }{\partial x}}\left(k_{T}r{\displaystyle \frac{\partial T}{\partial x}}\right)\right)+{\displaystyle \frac{\partial }{\partial r}}\left(k_{T}r{\displaystyle \frac{\partial T}{\partial r}}\right)={\displaystyle \frac{NuF}{PePr}}\left(T-\theta \right),$$

(5)

where

$$u={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial r}},\upsilon =-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial x}},\omega ={\displaystyle \frac{\partial u}{\partial r}}-{\displaystyle \frac{\partial \upsilon }{\partial x}},$$

(6)

$$k_P=\left\{\begin{array}{c}0,\mathrm{in}{\mathsf{\Omega }}_1\bigcup {\mathsf{\Omega }}_3\\ k_0\left(x,r\right),\mathrm{in}{\mathsf{\Omega }}_2\end{array}\right..$$

The physical quantities used in the model for Equations (1)–(6) are presented in

Table 1. Variables used in the calculation of the dimensionless parameter group.

Symbol	Description
$k_0\left(x,r\right)$	Conductivity coefficient of porous medium
$\psi$	Stream functions
$\omega$	Speed vortex
${\phi }_i$	Concentration of harmful substances
$\theta$	Exhaust gas temperature
$T$	Temperature of porous medium
$V$	Vorticity
$a$	Thermal conductivity coefficient
$\beta$	Thermal expansion coefficient
$g$	Acceleration due to gravity
$\Delta T$	Temperature difference scale
$\nu$	Kinematic viscosity coefficient
$k_T$	Thermal conductivity coefficient
$D_i=\left({CH}_4,{CO}_2,H_{2}O,H_2,N_2\right)$	Mass diffusivity of particles

, while the dimensionless parameters are listed in

Table 2. The mathematical model uses 6 dimensionless parameters.

Variables	Description
$Re={\displaystyle \frac{\mathrm{V}Dan}{\nu }}$	Reynolds number
$Pe={\displaystyle \frac{VDan}{a}}$	Péclet number
$\mathrm{F}\mathrm{r}={\displaystyle \frac{{\nu }^2}{\mathit{g}Dan}}$	Froude number
$Pr={\displaystyle \frac{Pe}{Re}}={\displaystyle \frac{\nu }{a}}$	Prandtl number
$Gr={\displaystyle \frac{{Re}^2}{Fr}}={\displaystyle \frac{\beta g{Dan}^3\Delta T}{{\nu }^2}}$	Grashof number
$Nu=0.664{Pr}^{1/3}\sqrt{Re}$	Nusselt number

.

The permeability coefficients of the porous medium $k_P$ and $F$ are defined as follows:

$$k_P(x,r)=\left\{\begin{array}{c}0,\mathrm{i}\mathrm{n}{\mathsf{\Omega }}_1\bigcup {\mathsf{\Omega }}_3,\\ {\displaystyle \frac{150{\left(1-\epsilon \right)}^2}{{\epsilon }^3}}{({\displaystyle \frac{Dan}{d_{\rho }}})}^2,\mathrm{i}\mathrm{n}{\mathsf{\Omega }}_2\end{array}\right.$$

$$F=\left\{\begin{array}{c}0,\mathrm{i}\mathrm{n}{\mathsf{\Omega }}_1\bigcup {\mathsf{\Omega }}_3,\\ {\displaystyle \frac{6(1-\epsilon )}{d_{\rho }}},\mathrm{i}\mathrm{n}{\mathsf{\Omega }}_2\end{array}\right.$$

(7)

where $\epsilon$ is the porosity coefficient, $Dan$ is the diameter of the apparatus, and $d_{\rho }$ is the diameter of the spheres in the porous medium.

In this study, the diffusion model described by Equation (3) is formulated based on the Lewis number.

This model describes the distribution of harmful substance components. Therefore, the diffusion coefficient for the i component is expressed by using the Lewis number:

$$D_i={\displaystyle \frac{\lambda }{P{Le}_i{C}_{\rho }}}$$

(8)

where $P$—pressure; ${Le}_i$—Lewis number; $C_{\rho }$—specific heat capacity; and ${\lambda }_i$—diffusion coefficient.

The specific heat capacity and thermal conductivity of the mixture depend on the mass fractions of the components in both regions.

$$\lambda ={\sum }_{i=1}^n{(\phi }_i{\lambda }_i),n=1,\dots ,6$$

(9)

$$C_{\rho }={\sum }_{i=1}^n{(\phi }_i{\mathrm{C}}_{{\rho }_i}).$$

(10)

Since the mixture is assumed to be an ideal gas, its components follow Dalton’s law. Accordingly, the partial pressures of the mixture’s components can be expressed through their molar fractions in the mixture.

The equation of state for an ideal gas is used to calculate the density in the following form:

$$P{M}_{mix}=\rho R_{g}T$$

(11)

where $M_{mix}$—molecular mass of mixture; $T$—temperature; and $R_g$—Will Versailles gas constant.

$$M_{mix}={\displaystyle \frac{1}{\sum _{i=1}^n\left(\frac{{\phi }_i}{M_i}\right)}}$$

(12)

where $M_i$—molecular mass of component.

2.2. Boundary Conditions

For the completeness of the mathematical model presented above, the following boundary conditions are imposed:

$$\begin{array}{c}\mathrm{I}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}:x=0,0\le r\le r_0;\psi ={\psi }_0\left(r\right),\omega ={\omega }_0\left(r\right),{\displaystyle \frac{\partial \theta }{\partial x}}=0,\\ {\phi }_{{CH}_4},{\phi }_{H_{2}o},{\phi }_{CO},{\phi }_{H_2},{\phi }_{{NO}^{-}}\mathrm{values}\mathrm{are}\mathrm{given}.\end{array}$$

$$\mathrm{O}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}:x=L,0\le r\le f\left(L\right);{\displaystyle \frac{\partial \psi }{\partial x}}=0,{\displaystyle \frac{\partial \omega }{\partial x}}=0,{\displaystyle \frac{\partial \theta }{\partial x}}=0,{\displaystyle \frac{\partial {\psi }_i}{\partial x}}=0,$$

$$\begin{array}{c}\mathrm{A}\mathrm{t}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}:0\le x\le L,r\left(x\right)=f\left(x\right);{\displaystyle \frac{\partial \psi }{\partial n}}=0,\psi =const,\\ {\displaystyle \frac{\partial \theta }{\partial n}}=0,T=1,{\displaystyle \frac{\partial \theta }{\partial x}}=0,{\psi }_i=0,\end{array}$$

$$\begin{array}{c}\mathrm{O}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}:0\le \mathrm{x}\le \mathrm{L},r=0;\psi =0,{\displaystyle \frac{\partial \mathsf{\omega }}{\partial r}}=0,{\displaystyle \frac{\partial \mathsf{\theta }}{\partial r}}=0,\\ {\displaystyle \frac{\partial \mathrm{T}}{\partial r}}=0,{\displaystyle \frac{\partial {\psi }_i}{\partial r}}=0.\end{array}$$

(13)

2.3. Main Equations in New Coordinate System

The coordinate transformation is given by $\overrightarrow{x}:G\to X$, where the new local coordinates are related to the original coordinates as follows (however, in this article, for the sake of brevity, only Equations (1) and (2) are discussed in detail): $x=x_1=x_1\left(q_1,q_2\right),r=x_2=x_2\left(q_1,q_2\right)$.

When Equations (1)–(6) are transformed from the cylindrical coordinate system to a free orthogonal coordinate system, they take the following form:

$$\begin{array}{cc}\hfill J{\displaystyle \frac{\partial \omega }{\partial t}}+{\displaystyle \frac{\partial \left(u\omega \right)}{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_1}}& +{\displaystyle \frac{\partial \left(\upsilon \omega \right)}{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_2}}\hfill \\ & ={\displaystyle \frac{1}{Re}}\left[\left({\displaystyle \frac{\partial }{\partial q_m}}\left(J{g}^{km}{\displaystyle \frac{\partial \omega }{\partial q_k}}\right)\right)+\left({\displaystyle \frac{\partial }{\partial q_m}}\left(J{\displaystyle \frac{\omega }{r}}{\displaystyle \frac{\partial q_m}{\partial q_k}}\right)\right)\right]-{\displaystyle \frac{\partial }{\partial q_m}}\left(J{\displaystyle \frac{k}{r}}{g}^{km}{\displaystyle \frac{\partial \psi }{\partial q_k}}\right)-{\displaystyle \frac{Gr}{{Re}^2}}{\displaystyle \frac{\partial }{\partial q_m}}\left(J\theta {\displaystyle \frac{\partial q_m}{\partial x}}\right),\hfill \end{array}$$

(14)

$${\displaystyle \frac{\partial }{\partial q_m}}\left(J{\displaystyle \frac{1}{r}}{g}^{km}{\displaystyle \frac{\partial \psi }{\partial q_k}}\right)=J\omega ,$$

(15)

$$J{\displaystyle \frac{\partial {\phi }_i}{\partial t}}+{\displaystyle \frac{\partial \left(u{\phi }_i\right)}{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_1}}+{\displaystyle \frac{\partial \left(\upsilon {\phi }_i\right)}{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_2}}={\displaystyle \frac{\partial }{\partial q_m}}\left(J{g}^{km}{\displaystyle \frac{D_i}{r}}{\displaystyle \frac{\partial {\phi }_i}{\partial q_k}}\right),$$

(16)

$$J{\displaystyle \frac{\partial \theta }{\partial t}}+{\displaystyle \frac{\partial \left(u\theta \right)}{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_1}}+{\displaystyle \frac{\partial \left(\upsilon \theta \right)}{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_2}}={\displaystyle \frac{1}{PePr}}\left[{\displaystyle \frac{\partial }{\partial q_m}}\left(J{g}^{km}{\displaystyle \frac{\partial \theta }{\partial q_k}}\right)+{\displaystyle \frac{\partial }{\partial q_m}}\left(J{\displaystyle \frac{\theta }{r}}{\displaystyle \frac{\partial q_m}{\partial r}}\right)\right]+{\displaystyle \frac{NuF}{RePr}}(T-\theta ),$$

(17)

$$J{\displaystyle \frac{\partial T}{\partial t}}+{\displaystyle \frac{1}{PePr}}{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial q_m}}\left(J{g}^{km}{k}_{T}r{\displaystyle \frac{\partial T}{\partial q_k}}\right)={\displaystyle \frac{NuF}{PePr}}\left(T-\theta \right),$$

(18)

Here, the Jacobian determinant transformation is given by $J={\displaystyle \frac{\partial x_1}{\partial q_1}}{\displaystyle \frac{\partial x_2}{\partial q_2}}-{\displaystyle \frac{\partial x_1}{\partial q_2}}{\displaystyle \frac{\partial x_2}{\partial q_1}}$. This represents the Jacobian of the coordinate transformation. $g^{km}={\displaystyle \frac{\partial q_k}{\partial x_n}}.{\displaystyle \frac{\partial q_m}{\partial x_n}}$, with k, m, and n = 1, 2 being the contravariant tensor components and $g_{km}={\displaystyle \frac{\partial x_n}{\partial q_k}}.{\displaystyle \frac{\partial x_n}{\partial q_m}},$ with $k,m=\mathrm{1,2},$ being the covariant tensor components.

The elements of the tensors have the following relation for the covariant tensor:

$$g_{11}.g_{22}=g_{12}^2+J^2$$

Since the coordinate system is orthogonal, the following relationship holds:

$$g_{12}=g_{21}=0,\mathrm{a}\mathrm{n}\mathrm{d}{J}^2=g_{11}.g_{22}.$$

Let us make the following transformations:

$$\begin{array}{c}J{g}^{11}={\displaystyle \frac{J^2\left({\left(\frac{\partial q_1}{\partial x_1}\right)}^2+{\left(\frac{\partial q_1}{\partial x_2}\right)}^2\right)}{J}}={\displaystyle \frac{{\left(\frac{\partial x_1}{\partial q_1}.\frac{\partial x_2}{\partial q_2}-\frac{\partial x_1}{\partial q_2}.\frac{\partial x_2}{\partial q_1}\right)}^2\left({\left(\frac{\partial q_1}{\partial x_1}\right)}^2+{\left(\frac{\partial q_1}{\partial x_2}\right)}^2\right)}{J}}=\\ {\displaystyle \frac{\left({\left(\frac{\partial x_1}{\partial q_1}\right)}^2{\left(\frac{\partial x_2}{\partial q_2}\right)}^2-2\frac{\partial x_1}{\partial q_1}.\frac{\partial x_2}{\partial q_2}.\frac{\partial x_1}{\partial q_2}.\frac{\partial x_2}{\partial q_1}+{\left(\frac{\partial x_1}{\partial q_2}\right)}^2{\left(\frac{\partial x_2}{\partial q_1}\right)}^2\right)\left({\left(\frac{\partial q_1}{\partial x_1}\right)}^2+{\left(\frac{\partial q_1}{\partial x_2}\right)}^2\right)}{J}}=\\ {\displaystyle \frac{\left({\left(\frac{\partial x_1}{\partial q_1}\right)}^2{\left(\frac{\partial x_2}{\partial q_2}\right)}^2{\left(\frac{\partial q_1}{\partial x_1}\right)}^2-2\frac{\partial x_1}{\partial q_1}.\frac{\partial x_2}{\partial q_2}.\frac{\partial x_1}{\partial q_2}.\frac{\partial x_2}{\partial q_1}{\left(\frac{\partial q_1}{\partial x_1}\right)}^2+{\left(\frac{\partial x_1}{\partial q_2}\right)}^2{\left(\frac{\partial x_2}{\partial q_1}\right)}^2{\left(\frac{\partial q_1}{\partial x_1}\right)}^2\right)}{J}}+\\ {\displaystyle \frac{\left({\left(\frac{\partial x_2}{\partial q_2}\right)}^2-2\frac{\partial x_2}{\partial q_2}.\frac{\partial x_1}{\partial q_2}.\frac{\partial x_2}{\partial x_1}+{\left(\frac{\partial x_1}{\partial q_2}\right)}^2{\left(\frac{\partial x_2}{\partial x_1}\right)}^2\right)+\left({\left(\frac{\partial x_1}{\partial x_2}\right)}^2{\left(\frac{\partial x_2}{\partial q_2}\right)}^2{\left(\frac{\partial q_1}{\partial x_1}\right)}^2-2\frac{\partial x_1}{\partial x_2}.\frac{\partial x_2}{\partial q_2}.\frac{\partial x_1}{\partial q_2}+{\left(\frac{\partial x_1}{\partial q_2}\right)}^2\right)}{J}}={\displaystyle \frac{({\left(\frac{\partial x_2}{\partial q_2}\right)}^2+{\left(\frac{\partial x_1}{\partial q_2}\right)}^2)}{J}}={\displaystyle \frac{g_{22}}{J}}={\displaystyle \frac{g_{22}}{\sqrt{g_{11}.g_{22}}}}=\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}},\end{array}$$

(19)

therefore,

$$J{g}^{22}={\displaystyle \frac{J^2\left({\left(\frac{\partial q_2}{\partial x_1}\right)}^2+{\left(\frac{\partial q_2}{\partial x_2}\right)}^2\right)}{J}}={\displaystyle \frac{\left({\left(\frac{\partial x_1}{\partial q_1}\right)}^2+{\left(\frac{\partial x_2}{\partial q_1}\right)}^2\right)}{J}}={\displaystyle \frac{g_{11}}{J}}={\displaystyle \frac{g_{11}}{\sqrt{g_{11}.g_{22}}}}=\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}.$$

(20)

To transform Equations (1) and (2) in a curvilinear cylindrical coordinate system into the local coordinate system, the following approach is applied:

$${\displaystyle \frac{\partial u}{\partial x_1}}={\displaystyle \frac{\partial u}{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial u}{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_1}},$$

(21)

$${\displaystyle \frac{\partial u}{\partial x_2}}={\displaystyle \frac{\partial u}{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_2}}+{\displaystyle \frac{\partial u}{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_2}}.$$

(22)

When taking the derivatives $x_1$ and $x_2$ in Equation (20), the following (Equation (22)) are obtained:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial x_1}{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial x_1}{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_1}}=1\\ {\displaystyle \frac{\partial x_2}{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial x_2}{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_1}}=0\end{array}\right.$$

(23)

By substituting $x_1$ and $x_2$ into Equation (21), the following (Equation (23)) are obtained:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial x_1}{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_2}}+{\displaystyle \frac{\partial x_1}{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_2}}=0\\ {\displaystyle \frac{\partial x_2}{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_2}}+{\displaystyle \frac{\partial x_2}{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_2}}=1\end{array}\right.$$

(24)

Now, we find ${\displaystyle \frac{\partial q_2}{\partial x_1}}$ and ${\displaystyle \frac{\partial q_1}{\partial x_1}}$ by using Formula (22); for ${\displaystyle \frac{\partial x_2}{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial x_2}{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_1}}=0$, from the equation ${\displaystyle \frac{\partial q_1}{\partial x_1}}$, we can transform as follows:

$${\displaystyle \frac{\partial q_1}{\partial x_1}}=-{\displaystyle \frac{\frac{\partial x_2}{\partial q_2}\frac{\partial q_2}{\partial x_1}}{\frac{\partial x_2}{\partial q_1}}}$$

Next, by putting ${\displaystyle \frac{\partial q_1}{\partial x_1}}$ into the equation ${\displaystyle \frac{\partial x_1}{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial x_1}{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_1}}=1$, the following formula will be obtained:

$$-{\displaystyle \frac{\partial x_1}{\partial q_1}}{\displaystyle \frac{\frac{\partial x_2}{\partial q_2}\left(\frac{\partial q_2}{\partial x_1}\right)}{\frac{\partial x_2}{\partial q_1}}}+{\displaystyle \frac{\partial x_1}{\partial q_2}}\left({\displaystyle \frac{\partial q_2}{\partial x_1}}\right)=1,$$

$${\displaystyle \frac{\partial q_2}{\partial x_1}}={\displaystyle \frac{\frac{\partial x_2}{\partial q_1}}{\left(\frac{\partial x_1}{\partial q_2}\frac{\partial x_2}{\partial q_1}-\frac{\partial x_1}{\partial q_1}\frac{\partial x_2}{\partial q_2}\right)}}=-{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial x_2}{\partial q_1}}.$$

(25)

where $J$ is the Jacobian, which represents the relationship between the new coordinate system and the original coordinate system. For the remaining partial derivatives, ${\displaystyle \frac{\partial q_i}{\partial x_j}}$, by using Equations (22) and (23), we obtain the following identities:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial q_2}{\partial x_1}}=-{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial x_2}{\partial q_1}}\\ {\displaystyle \frac{\partial q_1}{\partial x_1}}={\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial x_2}{\partial q_2}}\\ {\displaystyle \frac{\partial q_2}{\partial x_2}}={\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial x_1}{\partial q_1}}\\ {\displaystyle \frac{\partial q_1}{\partial x_2}}=-{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial x_1}{\partial q_2}}\end{array}\right.$$

(26)

Let us now express the nonlinear terms with respect to the new coordinates as follows:

$$\begin{array}{c}{\displaystyle \frac{\partial u\omega }{\partial x_1}}={\displaystyle \frac{\partial \left(u\omega \right)}{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_1}},\\ {\displaystyle \frac{\partial \upsilon \omega }{\partial x_2}}={\displaystyle \frac{\partial \left(\upsilon \omega \right)}{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_2}}.\end{array}$$

(27)

We obtain the velocity components of the stream function and their relation as follows:

$$u={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial x_2}},\upsilon =-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial x_1}}.$$

This is expressed in the following form by using local coordinates:

$$u={\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_2}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_2}}\right)={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_2}},$$

$$\upsilon =-{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_1}}\right)=-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_m}}{\displaystyle \frac{\partial q_m}{\partial x_1}}.$$

The further transformation of the nonlinear terms leads to the following result:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u\omega }{\partial x_1}}={\displaystyle \frac{\partial }{\partial q_m}}& \left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_n}}{\displaystyle \frac{\partial q_n}{\partial x_2}}\omega \right){\displaystyle \frac{\partial q_m}{\partial x_1}}={\displaystyle \frac{\partial }{\partial q_m}}\left[{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial \psi }{\partial q_1}}\left(-{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial x_1}{\partial q_2}}\right)+{\displaystyle \frac{\partial \psi }{\partial q_2}}\left({\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)\right)\omega \right]{\displaystyle \frac{\partial q_m}{\partial x_1}}\hfill \\ & ={\displaystyle \frac{\partial }{\partial q_1}}\left[{\displaystyle \frac{1}{r}}\left(-{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_1}{\partial q_2}}+{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)\omega \right]{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial }{\partial q_2}}\left[{\displaystyle \frac{1}{rJ}}.\left(-{\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_1}{\partial q_2}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)\omega \right]{\displaystyle \frac{\partial q_2}{\partial x_1}}\hfill \\ & ={\displaystyle \frac{\partial }{\partial q_1}}\left[{\displaystyle \frac{1}{rJ}}.\left(-{\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_1}{\partial q_2}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)\omega \right]{\displaystyle \frac{1}{J}}.{\displaystyle \frac{\partial x_2}{\partial q_2}}+{\displaystyle \frac{\partial }{\partial q_2}}\left[{\displaystyle \frac{1}{rJ}}.\left(-{\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_1}{\partial q_2}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)\omega \right]\left(-{\displaystyle \frac{1}{J}}.{\displaystyle \frac{\partial x_2}{\partial q_1}}\right)\hfill \\ & ={\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial }{\partial q_1}}\left[{\displaystyle \frac{1}{rJ}}\left(-{\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_1}{\partial q_2}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)\omega \right]{\displaystyle \frac{\partial x_2}{\partial q_2}}-{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial }{\partial q_2}}\left[{\displaystyle \frac{1}{rJ}}\left(-{\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_1}{\partial q_2}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)\omega \right]{\displaystyle \frac{\partial x_2}{\partial q_1}}\hfill \end{array}$$

(28)

Thus, the following nonlinear term is transformed as

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \upsilon \omega }{\partial x_2}}={\displaystyle \frac{\partial }{\partial x_2}}& \left(-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial x_1}}\omega \right)={\displaystyle \frac{\partial }{\partial q_m}}\left(-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_n}}{\displaystyle \frac{\partial q_n}{\partial x_1}}\omega \right){\displaystyle \frac{\partial q_m}{\partial x_2}}\hfill \\ & ={\displaystyle \frac{\partial }{\partial q_1}}\left[-{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_1}}\right)\omega \right]{\displaystyle \frac{\partial q_1}{\partial x_2}}+{\displaystyle \frac{\partial }{\partial q_2}}\left[-{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial q_1}{\partial x_1}}+{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial q_2}{\partial x_1}}\right)\omega \right]{\displaystyle \frac{\partial q_2}{\partial x_2}}\hfill \\ & ={\displaystyle \frac{\partial }{\partial q_1}}\left[-{\displaystyle \frac{1}{rJ}}.\left({\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_2}{\partial q_2}}-{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_2}{\partial q_1}}\right)\omega \right]\left(-{\displaystyle \frac{1}{J}}.{\displaystyle \frac{\partial x_1}{\partial q_2}}\right)\hfill \\ & +{\displaystyle \frac{\partial }{\partial q_2}}\left[-{\displaystyle \frac{1}{rJ}}.\left({\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_2}{\partial q_2}}-{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_2}{\partial q_1}}\right)\omega \right]\left({\displaystyle \frac{1}{J}}.{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)\hfill \\ & =-{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial }{\partial q_1}}\left[{\displaystyle \frac{1}{rJ}}\left({\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_2}{\partial q_1}}-{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_2}{\partial q_1}}\right)\omega \right]{\displaystyle \frac{\partial x_1}{\partial q_2}}-{\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial }{\partial q_2}}\left[{\displaystyle \frac{1}{rJ}}\left({\displaystyle \frac{\partial \psi }{\partial q_1}}{\displaystyle \frac{\partial x_2}{\partial q_2}}-{\displaystyle \frac{\partial \psi }{\partial q_2}}{\displaystyle \frac{\partial x_2}{\partial q_1}}\right)\omega \right]{\displaystyle \frac{\partial x_1}{\partial q_2}}\hfill \end{array}$$

(29)

The Laplace operator on the right-hand side of the equation in the new coordinates is written as

$$∆\omega ={\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial }{\partial q_m}}\left(J{g}^{km}{\displaystyle \frac{\partial \omega }{\partial q_k}}\right)$$

(30)

where $g^{km}$ is the metric tensor, where $m,k=1,2$.

$$div\left({\displaystyle \frac{k}{r}}grad\psi \right)={\displaystyle \frac{\partial }{\partial q_m}}\left(J{{\displaystyle \frac{k}{r}}g}^{km}{\displaystyle \frac{\partial \psi }{{\partial q}_k}}\right)$$

(31)

where $grad\psi \left({\displaystyle \frac{\partial \psi }{\partial x_1}},{\displaystyle \frac{\partial \psi }{\partial x_2}}\right)$ is a vector consisting of derivatives of the function $\psi$ with respect to variables $x_1$ and $x_2$.

The system of equations describing the flow of a viscous incompressible fluid, taking into account the temperature field through porous media, is written in the following form in an orthogonal curvilinear coordinate system:

$${\mathrm{a}}_{ij}={\displaystyle \frac{\partial x_i}{\partial q_j}},k,m=\mathrm{1,2},$$

$$\begin{array}{c}\sqrt{g_{11}{g}_{22}}{\displaystyle \frac{\partial \omega }{\partial t}}+{\displaystyle \frac{\partial }{\partial q_m}}\left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_n}}{\displaystyle \frac{\partial q_n}{\partial x_2}}\omega \right){\displaystyle \frac{\partial q_m}{\partial x_1}}-{\displaystyle \frac{\partial }{\partial q_m}}\left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_n}}{\displaystyle \frac{\partial q_n}{\partial x_1}}\omega \right){\displaystyle \frac{\partial q_m}{\partial x_2}}={\displaystyle \frac{1}{Re}}[{\displaystyle \frac{\partial }{\partial q_1}}\left(\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}{\displaystyle \frac{\partial \omega }{\partial q_1}}\right)+{\displaystyle \frac{\partial }{\partial q_2}}\left(\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}{\displaystyle \frac{\partial \omega }{\partial q_2}}\right)-{\displaystyle \frac{\partial }{\partial q_1}}\left({\displaystyle \frac{\omega }{r}}{\mathrm{a}}_{12}\right)+\\ {\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{\omega }{r}}{\mathrm{a}}_{11}\right)]-{\displaystyle \frac{\partial }{\partial q_1}}\left({\displaystyle \frac{k}{r}}\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}{\displaystyle \frac{\partial \psi }{\partial q_1}}\right)-{\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{k}{r}}\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}{\displaystyle \frac{\partial \psi }{\partial q_2}}\right)-{\displaystyle \frac{Gr}{R{e}^2}}\left({\displaystyle \frac{\partial \theta }{\partial q_1}}{\mathrm{a}}_{22}-{\displaystyle \frac{\partial \theta }{\partial q_2}}{\mathrm{a}}_{21}\right),\end{array}$$

(32)

$${\displaystyle \frac{\partial }{\partial q_1}}\left({\displaystyle \frac{1}{r}}\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}{\displaystyle \frac{\partial \psi }{\partial q_1}}\right)+{\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{1}{r}}\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}{\displaystyle \frac{\partial \psi }{\partial q_2}}\right)=\sqrt{g_{11}{g}_{22}}\omega ,$$

(33)

$$\sqrt{g_{11}{g}_{22}}{\displaystyle \frac{\partial {\phi }_i}{\partial t}}+{\displaystyle \frac{\partial }{\partial q_m}}\left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_n}}{\displaystyle \frac{\partial q_n}{\partial x_2}}{\phi }_i\right){\displaystyle \frac{\partial q_m}{\partial x_1}}-{\displaystyle \frac{\partial }{\partial q_m}}\left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_n}}{\displaystyle \frac{\partial q_n}{\partial x_1}}{\phi }_i\right)={\displaystyle \frac{\partial }{\partial q_1}}\left({\displaystyle \frac{D_i}{r}}\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}{\displaystyle \frac{\partial {\phi }_i}{\partial q_1}}\right)+{\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{D_i}{r}}\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}{\displaystyle \frac{\partial {\phi }_i}{\partial q_2}}\right),$$

(34)

$$\begin{array}{cc}\hfill \sqrt{g_{11}{g}_{22}}{\displaystyle \frac{\partial \theta }{\partial t}}& +{\displaystyle \frac{\partial }{\partial q_m}}\left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_n}}{\displaystyle \frac{\partial q_n}{\partial x_2}}\theta \right){\displaystyle \frac{\partial q_m}{\partial x_1}}-{\displaystyle \frac{\partial }{\partial q_m}}\left({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \psi }{\partial q_n}}{\displaystyle \frac{\partial q_n}{\partial x_1}}\theta \right)\hfill \\ & ={\displaystyle \frac{1}{PePr}}\left[{\displaystyle \frac{\partial }{\partial q_1}}\left(\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}{\displaystyle \frac{\partial \theta }{\partial q_1}}\right)+{\displaystyle \frac{\partial }{\partial q_2}}\left(\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}{\displaystyle \frac{\partial \theta }{\partial q_2}}\right)-{\displaystyle \frac{\partial }{\partial q_1}}\left({\displaystyle \frac{\theta }{r}}{\mathrm{a}}_{12}\right)+{\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{\theta }{r}}{\mathrm{a}}_{11}\right)\right]+{\displaystyle \frac{N_{u}F}{PePr}}\left(T-\theta \right),\hfill \end{array}$$

(35)

$$\sqrt{g_{11}{g}_{22}}{\displaystyle \frac{\partial T}{\partial t}}+{\displaystyle \frac{1}{PePr}}{\displaystyle \frac{1}{r}}[{\displaystyle \frac{\partial }{\partial q_1}}\left(kr\left(\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}{\displaystyle \frac{\partial T}{\partial q_1}}\right)+{\displaystyle \frac{\partial }{\partial q_2}}\left(kr\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}{\displaystyle \frac{\partial T}{\partial q_2}}\right)\right]={\displaystyle \frac{Nu.F}{PePr}}\left(T-\theta \right).$$

(36)

2.4. Boundary Conditions in New Coordinate System

The boundary conditions in the new coordinate system were carefully defined to ensure accurate numerical simulations.

At the inlet, the boundary conditions are presented as $q_1=0,0\le q_2\le 1;$ $\psi ={\psi }_0\left(q_2\right),\omega ={\omega }_0\left(q_2\right)$, and ${\psi }_0,{\omega }_0$ are given functions.

On the axis of symmetry, the boundary conditions are presented as $0\le q_1\le 1,q_2=0;\omega =\psi =0$.

At the outlet, the boundary conditions are presented as $q_1=1,0\le q_2\le 1;$ ${\displaystyle \frac{\partial \psi }{\partial q_1}}=0,{\displaystyle \frac{\partial \omega }{\partial q_1}}=0.$

At the solid wall, the boundary conditions are presented as $0\le q_1\le 1,0\le q_2\le r\left(q_1\right);\psi =const,{\displaystyle \frac{\partial \psi }{\partial q_2}}=0.$ To calculate the velocity gradient on a solid wall, the Thom formula is used.

Equations (38) and (39) can be conveniently rewritten for further investigation as follows:

$$\sqrt{g_{11}{g}_{22}}{\displaystyle \frac{\partial \omega }{\partial t}}+\left(L_1+L_2\right)\omega =-L_3\psi ,$$

(37)

$${\displaystyle \frac{\partial }{\partial q_1}}\left({\displaystyle \frac{\mathrm{\Phi }}{r}}{\displaystyle \frac{\partial \psi }{\partial q_1}}\right)+{\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{1}{r\mathrm{\Phi }}}{\displaystyle \frac{\partial \psi }{\partial q_2}}\right)=\sqrt{g_{11}{g}_{22}}\omega ,$$

(38)

where

$$L_1\omega ={\displaystyle \frac{\partial }{\partial q_1}}\left[{\displaystyle \frac{1}{r}}({\displaystyle \frac{\partial \psi }{\partial q_2}}+{\displaystyle \frac{{\mathrm{a}}_{12}}{Re}})\omega \right]-{\displaystyle \frac{1}{Re}}{\displaystyle \frac{\partial }{\partial q_1}}\left(\mathrm{\Phi }{\displaystyle \frac{\partial \omega }{\partial q_1}}\right)+{\displaystyle \frac{Gr}{R{e}^2}}{\displaystyle \frac{\partial }{\partial q_1}}\left({\displaystyle \frac{\partial \theta }{\partial q_1}}\right),$$

$$L_2\omega =-{\displaystyle \frac{\partial }{\partial q_2}}\left[{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial \psi }{\partial q_1}}+{\displaystyle \frac{{\mathrm{a}}_{11}}{Re}}\right)\omega \right]-{\displaystyle \frac{1}{Re}}{\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{1}{\mathrm{\Phi }}}{\displaystyle \frac{\partial \omega }{\partial q_2}}\right)+{\displaystyle \frac{Gr}{R{e}^2}}{\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{\partial \theta }{\partial q_2}}\right),$$

$$L_3\psi =\left[{\displaystyle \frac{\partial }{\partial q_1}}\left({\displaystyle \frac{k}{r}}\mathrm{\Phi }{\displaystyle \frac{\partial \psi }{\partial q_1}}\right)+{\displaystyle \frac{\partial }{\partial q_2}}\left({\displaystyle \frac{k}{r\mathrm{\Phi }}}{\displaystyle \frac{\partial \psi }{\partial q_2}}\right)\right],$$

$$\mathrm{\Phi }=\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}},{\mathrm{a}}_{ij}={\displaystyle \frac{\partial x_i}{\partial q_j}},i,j=\mathrm{1,2}.$$

In an orthogonal coordinate system, the following conditions hold:

$$g_{12}{\displaystyle \frac{\partial \psi }{\partial q_1}}-g_{11}{\displaystyle \frac{\partial \psi }{\partial q_2}}=0,g_{12}=0,{\displaystyle \frac{\partial \psi }{\partial q_2}}=0$$

(39)

Considering the velocity vector $\overrightarrow{u}=(u,\upsilon )$, the derivatives of the stream function in the curvilinear coordinate system are expressed by the following relations:

$$u={\displaystyle \frac{1}{rI}}\left({\mathrm{a}}_{11}{\displaystyle \frac{\partial \psi }{\partial q_2}}-{\mathrm{a}}_{12}{\displaystyle \frac{\partial \psi }{\partial q_1}}\right),$$

$$\upsilon ={\displaystyle \frac{1}{rI}}\left({\mathrm{a}}_{22}{\displaystyle \frac{\partial \psi }{\partial q_1}}-{\mathrm{a}}_{21}{\displaystyle \frac{\partial \psi }{\partial q_2}}\right).$$

2.5. Description of Finite Difference Relations

In order to numerically solve the system of differential equations based on the Navier–Stokes equations, Equations (36) and (37) are considered by using the following iterative finite difference scheme [1]:

$$\begin{array}{c}\sqrt{g_{11}{g}_{22}}.{\displaystyle \frac{{\omega }_{ij}^{n+1/3}-{\omega }_{ij}^n}{\mathsf{\tau }}}+L_{1,h}{\omega }^{n+{\displaystyle \frac{1}{3}}}+L_{2,h}{\omega }^n=-L_{3,h}{\psi }^n,\\ \sqrt{g_{11}{g}_{22}}.{\displaystyle \frac{{\omega }_{ij}^{n+2/3}-{\omega }_{ij}^{n+1/3}}{\mathsf{\tau }}}+L_{1,h}{\omega }^{n+{\displaystyle \frac{1}{3}}}+L_{2,h}{\omega }^{n+2/3}=-L_{3,h}{\psi }^n,\end{array}$$

(40)

$$\begin{array}{c}\sqrt{g_{11}{g}_{22}}.{\displaystyle \frac{{\omega }_{ij}^{n+1}-{\omega }_{ij}^{n+2/3}}{\mathsf{\tau }}}=-L_{3,h}\left({\psi }^{n+1}-{\psi }^n\right),\\ {\left({\displaystyle \frac{1}{r}}\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}{\psi }_{q1}^{n+1}\right)}_{q1}+{\left({\displaystyle \frac{1}{r}}\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}{\psi }_{q2}^{n+1}\right)}_{q2}=\sqrt{g_{11}{g}_{22}}{\omega }_{i,j}^{n+1}.\end{array}$$

(41)

The system of Equations (39) and (40) describes the dynamics of the flow. To solve this problem, firstly, we calculate the values of ${\omega }_{ij}^{n+1/3}$ from the first equation, then calculate ${\omega }_{ij}^{n+1/3}$ from the first equation, and then calculate ${\omega }_{ij}^{n+2/3}$ from the second equation. Next, from Equation (40), we compute ${\psi }^{n+1}$. Finally, from the second equation of (40), we determine ${\omega }_{ij}^{n+1}$.

Thus, in solving Equations (39) and (40), the iterative process for $\omega$—speed vortex—continues until the following inequality is satisfied:

$${‖{\omega }^{n+1}-{\omega }^n‖}_{\mathrm{C}}\le \tau {\epsilon }_{eps}.$$

(42)

2.6. Method for Constructing Orthogonal Curvilinear Grid

To convert region $\mathsf{\Omega }$ into a parametric rectangle, we consider region $\mathsf{\Omega }$ as the sum of sub-regions of types ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2\bigcup {\mathsf{\Omega }}_3$. Figure 2a,b present orthogonal curvilinear grids with dimensions of 41 × 21, while Figure 3a,b display denser grids with dimensions of 61 × 41. Subfigures (a) illustrate the upper half of the catalyst geometry, whereas subfigures (b) correspond to the left half of the same domain.

To numerically construct an orthogonal curvilinear grid in the inner domain ${\mathsf{\Omega }}_1$, the differential equations from [21] are written as follows:

$$\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}{\displaystyle \frac{{\partial }^2{x}_1}{\partial q_1^2}}+{\displaystyle \frac{\partial }{\partial q_2}}\left(\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}{\displaystyle \frac{\partial x_1}{\partial q_2}}\right)=0,\mathsf{\iota }=\mathrm{1,2}$$

(43)

At the upper boundary of the region, the coordinate grid nodes are selected according to the numerical parameterization along the arc length of the curve [14]. To achieve this, the following differential equation is solved numerically as

$${\displaystyle \frac{\partial x_1}{\partial t}}={\displaystyle \frac{\partial }{\partial q_1}}\left(\mathrm{\Phi }{\displaystyle \frac{\partial x_1}{\partial q_1}}\right)$$

(44)

with the following boundary conditions:

$$x_1\left(0\right)=0,x_1\left(q_1^{*}\right)=x_1^{*}$$

where $\mathrm{\Phi }=\sqrt{1+{\left({\displaystyle \frac{\mathrm{d}\mathrm{f}\left(x_1\right)}{\mathrm{d}{x}_1}}\right)}^2}$ is a coefficient describing the length of the arc.

A non-uniform rectangular grid, corresponding to the curvilinear grid in the upper region ${\mathsf{\Omega }}_1$ is constructed for sub-region ${\mathsf{\Omega }}_2\bigcup {\mathsf{\Omega }}_3$.

For the remaining part of the contour of region ${\mathsf{\Omega }}_1$ the boundary conditions for Equation (42) are accepted as follows:

$$x_1=0,{\displaystyle \frac{\partial x_2}{\partial q_1}}=0,\mathrm{when}{q}_1=0,0\le q_2\le 1,$$

$${{\displaystyle \frac{\partial x_1}{\partial q_2}}=0,x}_2=0\mathrm{when}0\le q_1\le 1,q_2=0,$$

(45)

$$x_1=x_{1,L},{\displaystyle \frac{\partial x_2}{\partial q_1}}=0,\mathrm{when}{q}_1=1,0\le q_2\le 1.$$

According to the boundary conditions in (44), setting the normal derivatives to zero ensures the orthogonality of the curvilinear grid.

The numerical solution of Equation (42) for $\overrightarrow{x}:E\to \mathsf{\Omega }$, where $\overrightarrow{x}=(x_1,x_2)$, has fixed values at the upper boundary of ${\mathsf{\Omega }}_1$ and satisfies the boundary conditions in (44). The solution of Equations (42)–(44) is performed by using a finite difference scheme based on the stabilizing correction method [22].

$$\left(E-\tau L_{11}^h\right).\left(E-\tau L_{22}^h\right).{\displaystyle \frac{x_{\iota }^{n+1}-x_{\iota }^n}{\tau }}=L_{11}^h{x}_{\iota }^n+L_{22}^h{x}_{\iota }^n$$

(46)

where $L_{11}^h$ and $L_{22}^h$ are the finite difference analogs of the corresponding differential operators, as given by Equation (46).

$$L_{11}^h\left(x\right)={\displaystyle \frac{1}{h^2}}\left({\left(\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}\right)}_{i+1/2j}\left(x_{i+1j}-x_{ij}\right)-{\left(\sqrt{{\displaystyle \frac{g_{22}}{g_{11}}}}\right)}_{i-1/2j}\left(x_{ij}-x_{i-1j}\right)\right),$$

$$L_{22}^h\left(x\right)={\displaystyle \frac{1}{h^2}}{\left(\sqrt{{\displaystyle \frac{g_{11}}{g_{22}}}}\right)}_{ij}\left(x_{ij+1}-{2x}_{ij}+x_{ij-1}\right).$$

(47)

The boundary conditions in (44) are approximated with second-order accuracy and take the following form:

$$x_{1(i,l)}={\displaystyle \frac{4x_{l(i,2)}-x_{l(i,3)}}{3}},$$

$$x_{2(l,j)}={\displaystyle \frac{4x_{2(2,j)}-x_{2(3,j)}}{3}},$$

$$x_{2(N_1,j)}={\displaystyle \frac{4x_{2(N_{1-1},j)}-x_{2(N_{1-2},j)}}{3}}.$$

The derivatives included in $g_{11}$ and $g_{22}$ are everywhere approximated by central differences. Figure 2 and Figure 3 show the finite difference curvilinear grid constructed by using the described method. This grid is built within the domain characteristic of the boundary problem for the Navier–Stokes equations considered above.

The given domain is bounded at the top by the following lines:

$$x_2={\displaystyle \frac{b+a}{2}}+{\displaystyle \frac{b-a}{2}}thR\left(x_1-x_1^0\right)=f_1\left(x_1\right),0\le x_1\le x_1^{*},$$

$$x_2=f\left(x_1^{*}\right),x_1^{*}\le x_1\le x_{1L}.$$

(48)

where $a=0.2,b=0.5x_1^0=0.3,{x_1^{\mathrm{*}}=0.7,x}_{1L}=4,q_1^{\mathrm{*}}=0.35$. An uneven grid is formed on the segment $(x_1^0,x_{1L})$ by the following quadratic equation:

$$x_1\left(q_1\right)=\overline{a}{q}_1^2+\overline{b}{q}_1+\overline{c},q_1^{*}\le q_1\le 1$$

The coefficients $\overline{a},\overline{b},\overline{c}$ are determined from the following conditions:

$$x_1\left(1\right)=x_{1,L},x_1\left(q_1^{*}\right)=x_1^{*}.$$

These conditions are correspondence conditions of ${\mathsf{\Omega }}_1$ with a curvilinear lattice. In this case, the parameter $\tau$ was chosen equal to $\tau =0.05$.

The Navier–Stokes equation system in curvilinear coordinates, modeling the catalytic converter described above, was numerically computed. The program code was written in Python 3.9. A Ryzen 9 9950X computer, with 32 GB RAM, 2 TB SSD, and RTX 4080 Super was used to perform high-performance computations. Calculations were carried out over a wide range of problem parameters. The calculation results for different values of Reynolds number are shown.

3. Results and Discussion

As can be seen from

Table 3. Flow characteristics based on Reynolds number (Re).

Re	${\mathit{x}}_1$	${\mathit{x}}_2$
100	0.5716559337482368	0.3218785880697778
500	1.065695666358492	0.37084370176015174
1000	1.065695666358492	0.37084370176015174
2000	1.3733211011294242	0.37492454616578524
5000	1.0660258676562395	0.3965031311383946
10,000	1.0660258676562395	0.3965031311383946

and

Table 4. Continuation of Table 3.

${\mathit{\psi }}_{\mathit{m}\mathit{a}\mathit{x}}$	Number of Iterations	Time
$2.23\times {10}^{-2}$	2070	273 s
$2.91\times {10}^{-2}$	4207	514 s
$3.25\times {10}^{-2}$	6287	667.0934 s
$3.45\times {10}^{-2}$	9423	903.8450 s
$3.55\times {10}^{-2}$	9531	981.1325 s
$3.54\times {10}^{-2}$	15,000	1371.7946 s

, the value of ${\psi }_{max}$ increases with the increase in Re. All the results in the table are calculated for different values of Re in the range ε = 0.2–0.4. The value of the current function at the solid boundary increases by $0.2\times {10}^{-2}$, while in the elastic region, the value of the current function increases as shown in the table.

The influence of the mesh on the accuracy was analyzed by conducting computational experiments. The program code written in Python was run on 31 × 21, 41 × 21 and 61 × 41 meshes. As a result of the calculations, it was found that with an increase in the Re number, ${\psi }_{max}$ also increases.

The program code does not cause any changes in the mesh size and calculation, because everything is executed automatically. The mesh size was calculated automatically. In the refinements, 41 × 21 and 61 × 41 times were used, the accuracy of the calculation was satisfactory, and in both cases, the number of iterations and calculations were different. The results remained exactly the same.

The difference between the values of the two different current functions between the meshes is less than 0.78%, the coordinates $x_1$ and $x_2$ change, the number of iterations on the mesh increases, but the stability is maintained. The difference in the values of the vorticity $\omega$ at the corner points does not exceed 1%. In addition, despite the increase in mesh density, the current function and the velocity curve remain unchanged. No artificial oscillations were observed, which confirms the reliability of the method.

For the following calculations, we consider $\epsilon \left(r\right)=\left\{\begin{array}{c}0.4at0\le r\le 0.2,\\ {\displaystyle \frac{100r^2+59}{210}}at0.2\le r\le 0.5,\end{array}\right.$ for the upstream flow of the catalyst in the as-received condition.

In Figure 4 above, the isolines of the stream function at $Re=100$ do not have reverse flow zones. When the Reynolds number increases to 500, in Figure 5, a reverse flow zone appears in front of the porous medium.

Figure 6 and Figure 7 below show the changes in the isolines of the current function when $Re=2000$ and $Re=5000$. As can be seen in Figure 6, the area of the coil increases with the increase in $Re$ as a result of the resistance of the porous medium. In the coil medium, $Re=2000$ and ${\mathsf{\psi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$=$3.2\times {10}^{-2}$, and for $Re=5000$, ${\mathsf{\psi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$=$3.4\times {10}^{-2}$.

Above, Figure 4 and Figure 7 show the flow of fluid in a curved pipe with only a diffuser section, the length of which is 4 m. The main advantage of the curved grid method is that it is possible to calculate the flow in such a long pipe, since the computational domain is a unit square consisting of a uniform grid. The following studies, Figure 8 and Figure 9, are devoted to the numerical modeling of a pipe with a diffuser and a confuser. In these numerical calculations, the length of the pipe is 2 m. In the diffuser and confuser sections, the frequency of the grids is increased, since the velocity gradient changes rapidly in the vertical regions, and above it, there is a porous medium in the middle of the pipe.

The complexity of this flow is quite high, because the geometry of the computational domain is complex. The flow of liquid is in the upper section of the catalyst.

From Figure 8a,b, we observe significant differences in the computational results for $Re=100$ and $Re=1000$.

In these calculations, it was necessary to reduce the time step parameter, specifically by a factor of 10 compared with the previous simulations, setting $\tau =5\times {10}^{-3}$. Without such a reduction in the time step parameter, the computation would become unstable.

In the Figure 9a,b, the stream function contour lines are shown for $Re=2000$, and $Re=5000$.

For $Re=2000,$ the maximum stream function value in the vortex region is ${\psi }_{max}$ = $3.2\times {10}^{-2}$, while for $Re=5000$, it is ${\psi }_{max}$ = $3.4\times {10}^{-2}$.

The system of Navier–Stokes equations used for modeling the catalytic converter is based on physical laws.

The transition from cylindrical coordinates to any arbitrary curvilinear orthogonal coordinates has been rigorously proven through strict mathematical expressions.

Well-established methods were applied to construct an orthogonal grid in a curvilinear domain. The difference scheme for equations in arbitrary curvilinear orthogonal coordinates was implemented by using a three-level algorithm.

The stability and convergence of this scheme are well known. A computational code was developed based on this difference scheme, and numerous methodological calculations were performed over a wide range of parameters.

We increased the Reynolds number from 100 to 10,000, and the rise in Reynolds number led to an increase in the turbulence level of the flow as follows:

1.

For Reynolds numbers $Re\le 1000$, the flow remains laminar, characterized by smooth, orderly motion with minimal mixing between fluid layers.

2.

For Reynolds numbers $2000\le Re\le 5000$, the flow becomes more complex, transitioning from laminar to transitional flow with the development of turbulence. The flow exhibits some chaotic behavior, but large-scale structures are still somewhat organized.

3.

For Reynolds numbers $Re\ge 8000$, the flow enters a turbulent regime, where the effects of nonlinear convective terms become more pronounced. This is marked by highly irregular fluid motion, eddies, and strong mixing within the flow.

The computational experiments confirmed the validity of the theoretically proven assumptions. The numerical scheme is stable, and its solution converges correctly. The obtained results accurately describe the physical process of the research object.

4. Conclusions

The method developed in the article was used to numerically solve the problem of the movement of a viscous incompressible fluid in a porous medium in a curved pipe with axisymmetric geometry.

The value of the vortex on solid walls was calculated according to the Tom formula, since the value of the vortex on a solid wall is unknown.

The results of the calculations showed that the applied differential scheme is stable: as the Reynolds number increases, the volume of the vortex zones increases, and the number of iterations increases.

To increase the efficiency of the flow in the catalytic converter, it is necessary to reduce the volume of the reverse flow zones. An increase in the Reynolds number leads to an increase in the flow vortex zone. The developed program code allowed us to change the geometry of the pipe and conduct a calculation experiment.

With the help of modern computers, it is possible to conduct computational experiments for various geometries by using our program code. These computational experiments are financially much cheaper than natural experiments in the atmosphere. As a result, the environmental efficiency of catalytic converters was significantly improved.

In this article, the mathematical model was compared with experimental data. The experimental results show that the particles near the walls of the catalytic converter degrade more quickly.

The effectiveness of the numerical method used for solving the mathematical model was demonstrated by solving the Roach test problem. The computational accuracy is high, and the computation time is low.

According to the simulation results, the most effective catalyst geometry is the one we have considered, which allows one to reduce the turbulence in the diffuser. For example, the limiting case is the Vitoshinsky profile.

The model can be improved by incorporating the effects of thermodynamics and chemical reactions, which are essential to assessing the overall efficiency of the catalyst. To achieve this, this article discusses a macroscopic model, and the effects of thermodynamics and chemical reactions can be determined by adding additional macroscopic models. These models would help account for temperature distribution, chemical reaction rates, and other large-scale phenomena that influence the system. The model in this article can be applied to define the geometry of the catalyst and model the processes occurring inside it. This would allow for the simulation of the catalyst’s internal processes, including heat and mass transfer, as well as the chemical reactions taking place, providing a more comprehensive understanding of the converter’s behavior and performance.

This model can be used not only in the automotive industry, but also in the following areas: industrial gas purification, gas reactors in chemical production, and environmental protection systems.