Математическое моделирование гидродинамических задач и их применение в многофазных системах / Mathematical modeling of hydrodynamic problems and their applications in multiphase systems тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Абунаб Ахмед Камал Ибрахим

• Абунаб Ахмед Камал Ибрахим
• кандидат наук
• 2024

Оглавление диссертации кандидат наук Абунаб Ахмед Камал Ибрахим

List of contents

Description of the dissertation work

Chapter 1. Mathematical concepts in hydrodynamics of multiphase systems

1.1 Introduction

1.2 Mathematical governing equations of fluid flow

1.3 Basic principles of mathematical modelling

1.3.1 Modeling cycle

1.4 Types of multiphase flow regimes

1.5 Physical phenomena addressed in the dissertation work

1.5.1 Gas hydrates decomposition

1.5.2 Granular material

1.5.3 Convection

1.6 Methodology and algorithms

1.6.1 The algorithm of Painlevé test

Chapter 2. Study of methane hydrates decomposition in a one-dimensional model in permafrost zones

2.1 Introduction

2.2 Theoretical conception of the physical problem

2. 3 Computational model and theoretical investigation

2.3.1 Three-phase porous media: a methane hydrate breakdown process

2.4 Mathematical model

2.5 Breakdown of MHs in porous medium between three-phase flow

2.6 Breakdown of MHs in a two-phase porous medium

2.7 Conclusion

Chapter 3. Hydrodynamic system interactions in fluidized granular medium of van der Waals

3.1 Introduction

3.2 Description of the physical problem

3.3 Mathematical model of granular hydrodynamics

3.3.1 The microscopic balance equations

3.3.2 Hydrodynamic Fourier's law

3.3.3 Equations of motion in a viscous and frictional model

3.3.4 Painlevé integrability technique and Backlund transformation

3.3.5 Painlevé analysis test algorithm for SVDWF with viscosity and friction

3.3.6 Backlund transformation of the SVDWF

3.3.7 Viscosity and friction dispersion properties of the model

3.3.8 Wave solution

3.4 Methods of solution

3.4.1 Tanh method for the SVDWF with viscosity and friction

3.4.2 Jacobi elliptic method for the SVDWF with viscosity and friction

3.5 The maximum growth rate

3.6 The phase plane technique for the SVDWF equation

3.7 Conclusion and outlook

Chapter 4. Analysis of multiple-soliton solutions for the Ginzburg Landau model of the (2 + 1)-dimensional coupled cubic-quintic complex in a convecting binary fluid

4.1 Introduction

4.2 Mathematical formulation

4.3 Linear stability analysis of the (2 + 1)-DCC-QCGLEs

4.4 Solution of traveling waves for the (2+1) -DCC-QCGLES

4.5 The simplest equation method

4.6 Application of the simplest equation method for (2+1)-DCC-QGLEs

4.6.1 Solutions of (2+1)-DCC-QGLEs using Bernoulli equation

4.6.2 Solutions of (2+1)-DCC-QGLEs using Riccati equation

4.6.3 Solutions of (2+1)-DCC-QGLEs using Burgers' equation

4.7 Painleve analysis

4.7.1 Painleve test of the (2+1)-DCC-QGLEs

4.7.2 Backlund transformation of the (2+1)-DCC-QGLEs and soliton solutions

4.8 Results and discussion

4.9 Conclusions

Thesis summary

Abbreviations

Bibliography

Введение диссертации (часть автореферата) на тему «Математическое моделирование гидродинамических задач и их применение в многофазных системах / Mathematical modeling of hydrodynamic problems and their applications in multiphase systems»

Description of the dissertation work

The thesis provides mathematical models of some hydrodynamic problems and their different applications in multiphase systems. In the first problem, the study highlights the thermophysical processes throughout the methane hydrates decomposition in porous media found in the sediments of the oceans. Gas capillary tubes are formed when methane hydrates decomposition in sediments with narrow pores, causing abrupt shifts in permeability. In the thesis, a new mathematical model for simulating the breakdown of methane hydrate in permafrost environments of porous medium is proposed. The model is based on fundamental thermodynamic concepts, including the thermal equilibrium equation in the hydrate stability zone and the conservation equations of liquid, gas, and energy. In addition, the proposed model was used to study methane hydrates breakdown processes in multiphase systems in porous media. These problems were then numerically solved using the finite difference technique. Algebraic equations for the nonlinear system were solved after the matrix for Jacobian was computed using the Newton-Raphson method.

Examining fluidized granular materials under the influence of viscosity and friction using the standard van der Waals model is the second problem. The mass, momentum, and energy equations of local densities are included in the system of macroscopic balance that is discussed. An analytical study of the non-linear partial differential equations governing the granular model investigated two types of collisions: elastic and inelastic collisions, using the hydrodynamic equations for granular matter motion. The integrability of the proposed model is analyzed by applying the Painlevé analysis. Besides, stability analysis of the standard van der Waals model in its ordinary differential equation form is demonstrated using phase portrait classifications. In conclusion, the model solutions under the impact of viscosity and friction are examined using two- and three-dimensional graphics, and qualitative agreements with earlier related research are demonstrated.

The third problem in this dissertation is to find the analytical solutions for some recent advances that have been made for Rayleigh-Bénard convection by applying the (2+1)-dimensional coupled cubic-quintic complex Ginzburg-Landau equations model for

slowly varying spatio-temporal amplitudes of the wave motion. In addition, novel traveling solitary wave solutions for the model equation are derived using a very useful method to investigate how complex physical coefficients affect the profiles of propagating waves. Furthermore, we introduce the Weiss-Tabor-Carnevale-Kruskal algorithm of the Painleve methodology to examine the integrability of the (2+1)-dimensional coupled cubic-quintic complex Ginzburg-Landau equations, and the truncated Painleve expansion is used to extract the Backlund transform, from which new solitary solutions can be acquired. The results also demonstrated good agreement with previous work and were more significant and accurate in the two and three dimensions of the proposed model. The synopsis that is being presented highlights the following aspects of the work: its relevance, its primary goals and tasks, its originality, the mathematical and numerical methodologies used, the arguments made for the defense, and the main conclusions of the dissertation. Pertinence of the thesis subject

Permafrost regions contain approximately as much carbon as conventional resource deposits combined. This carbon fills gaps in water molecules with a polyhedral network. Greater hydrocarbon reserves can be found in hydrate deposits than in conventional energy sources including petroleum, natural gas, and coal [1,2]. Because a sizable portion of methane is trapped inside a water crystal structure, methane hydrates is a solid clathrate chemical that resembles ice. Substantial methane hydrates deposits have been found approximately 1100 meters below sea level, under strata on Earth's ocean floors. Methane hydrates was formerly thought to be limited to the outermost parts of the Solar System, where water ice is prevalent, and temperatures are low. In oceans, methane gas and hydrogen-bonded water combine at high pressures and low temperatures to form methane hydrates. Studies have shown that this stability is essential to their development and breakdown processes. This demonstrates the importance of methane hydrates in the storage and transit of natural gas, particularly in regions with extreme environmental conditions. It is also important to remember that other gasses can be generated during this process in addition to methane, making methane hydrates a potential solution for energy storage and distribution challenges [3].

Recently, there has been a focus on the potential of hydrate-containing reservoirs as a major source of this valuable resource, which is natural gas that is extensively used. To extract natural gas from these reservoirs, various methods have been proposed, with depressurization being the most commonly used technique. The process entails lowering the reservoir's pressure to a point below the pressure at which hydrate dissociation occurs, causing the solid hydrates to convert back into gas and liquid phases. This approach has proven to be effective in releasing significant amounts of natural gas from these formations [4]. A number of methods and strategies, such as chemical injection, thermal stimulation, and carbon dioxide displacement, have been revealed for natural gas extraction from reservoirs with high hydrate content. Depressurization is still the accepted technique for obtaining natural gas from these special rocks, despite continuous research and technological developments [5].

On the other hand, the effects of friction and viscosity on the fluidized granular materials with the standard van der Waals normal model are examined. Consequently, a system of grains inside a box with a big aspect ratio on a 2D horizontal surface is taken into consideration. The horizontal direction of this system is assumed to be periodic. The top wall reflects grains, while at a special frequency and amplitude, the bottom one transports energy into the system through vertical sinusoidal vibrations. Because of the friction between the grains, horizontal momentum is conserved in elastic collisions with the wall. The granular temperature in the system standard framework is determined by taking the kinetic energy per particle into account, similar to that in molecular liquids [6].

Moreover, solitary wave propagations have been investigated in a variety of fields such as fluid dynamics, granular materials, physics of plasma, convective heat transfer, quantum mechanics, and solid-state. Much attention has been focused on the nonlinear partial differentia equations integrability that describes these physical models. It was found close relations between the Painleve property and the integrability of the latter partial differentia equations [7]. These relations can be obtained by applying some mathematical methods, for example, the Ablowitz-Ramani-Segur algorithm, the Weiss-Tabor-Carnevale method, and Weiss-Tabor-Carnevale with Kruskal Simplification method. Last decades, different methods for obtaining soliton solutions, kink, anti-kink

waves, and others of these nonlinear partial differentia equations have been introduced, developed, and extended.

Additionally, numerous effective techniques for locating nonlinear wave solutions that are propagating in a variety of disciplines, including fluid, plasma, optics, and granular materials, have been developed in recent years. The nonlinear Schrodinger equation in the coupled form was solved using the energy-conservation deep-learning method to demonstrate the dynamic behaviors of vector solitons in birefringent fibers. Geng et al. [8] used the Hirota bilinear method to obtain the nondegenerate vector and binary soliton solutions of the two-coupled mixed derivative nonlinear Schrodinger equation. Solutions of various solitons equations were predicted for the nonlinear Schrodinger equation using a physical information neural network by Bo et al. [9] under pressure and temperature symmetric potentials. Finally, the traveling solitary wave solutions of the nonlinear differential equations were obtained approximately with the help of the simplest equation approach. Objectives of the dissertation work

The main goal of the research is to provide mathematical models of some hydrodynamic problems and their different applications in multiphase systems. The research objectives of the dissertation:

1. To examine the thermodynamic effects during methane hydrate decomposition in a three-phase flow by using numerical methods in a one-dimensional model.

2. To implement the numerical schemes in the form of robust algorithms and a software package for modeling in a one-dimensional fluid dynamic flow in a porous medium in the presence of hydrate-containing inclusions in permafrost regions.

3. To develop the new mathematical model that describes the phenomenon of fluidized granular materials with the standard van der Waals normal model under the influence of the effects of friction and viscosity in a two-phase medium in a horizontal box, the bottom wall vibrates, while the top represents the elastic and friction effects of the grains.

4. To examine the solutions for the traveling wave of the standard van der Waals normal model within fluidized granular materials by using the Backlund transformation, tanh function, and Jacobi elliptic function methods to study the phase separation phenomenon.

5. To determine the analytical solutions for some recent advances, which have been made for Rayleigh-Benard convection by applying the (2 + 1)-dimensional cubic-quintic complex Ginzburg-Landau equations model for slowly varying spatiotemporal amplitudes of the wave motion.

Research objects: The thesis work is primarily concerned with the formulation and development of mathematical frameworks for hydrodynamic problems in multi-phase systems using innovative algorithms and complex programs. Among these issues that have been studied are the processes for decomposition of methane hydrates in porous media found in permafrost areas through the formulation and development of a new mathematical model that simulates these processes and the study of the thermodynamic effects in the multiphase system. Also, in the study of fluidized granular materials with the standard van der Waals normal model under the influence of friction and viscosity in a two-phase medium in a horizontal box, the bottom wall vibrates while the top represents the elastic and friction effects of the grains. In the final part, the solitary wave solutions of the coupled cubic-quintic complex Ginzburg-Landau equations are examined. Scientific novelty of the dissertation work

1. In the dissertation, a new mathematical approach simulating the problem of decomposition of methane hydrates in porous media in permafrost zones was presented for a one-dimensional model. Additionally, complex codes and new numerical algorithms were used to solve the suggested mathematical framework. Furthermore, in two- and three-phase systems, the thermophysical characteristics during methane hydrate decomposition processes were investigated.

2. In the second part of the study a new mathematical model has been formulated to simulate the problem of fluidized granular materials with the standard van der Waals model under the effects of friction and viscosity are studied. The integrability of the new proposed model for this issue is analyzed by applying the Painleve analysis.

Moreover, the Backlund transformation was established using the Painlevé truncation expansion. New solutions for the traveling wave of the standard van der Waals model within fluidized granular materials were obtained by using the Backlund transformation, tanh function, and Jacobi elliptic function methods to study the phase separation phenomenon.

3. In order to examine the integrability of the (2+1)-dimensional cubic-quintic complex Ginzburg-Landau equations, the Weiss-Tabor-Carnevale-Kruskal algorithm of the Painlevé methodology is presented in the final issue of the dissertation work. Using the truncated Painlevé expansion, the Backlund transform was extracted to yield the new solitary solutions.

4. Also, the thesis addressed finding a new solution for some recent advances that have been made for Rayleigh-Bénard convection by applying the (2+1)- dimensional cubic-quintic complex Ginzburg-Landau equations model for slowly varying spatiotemporal amplitudes of the wave motion. In addition, novel traveling solitary wave solutions for the model equation were derived using a very useful method to investigate how complex physical coefficients affect the profiles of propagating waves.

Theoretical significance of the dissertation findings

The theoretical importance of the study focuses on the formulation of new mathematical models to describe physical phenomena that are addressed in the dissertation work. One such model that was developed during the period of study was a one-dimensional model to simulate the dissociation processes of methane hydrates in porous media, taking into account the occupancy of these porous media by (H2O, hydrate, and CH4) during three-phase flow. In addition, a new mathematical approach was proposed to describe the phenomenon of fluidized granular materials with the standard van der Waals normal model in a two-phase medium in a horizontal box. Moreover, the study highlighted the phenomenon of Rayleigh-Bénard thermal instability that usually occurs in a layer of binary fluid that has been heated from below. A new mathematical method and a developed mathematical model were proposed to describe this phenomenon according to mathematical concepts.

Practical significance of the dissertation findings

The proposed mathematical models that describe these physical phenomena depend on the new numerical and analytical techniques, and these models studied as part of the dissertation work are made publicly available. Besides, the finite difference technique was employed in Compaq Visual Fortran version 6.6, to examine the thermophysical properties of decomposition for methane hydrates in porous media during three-phase flow. On the other side, the effects of friction and viscosity on the fluidized granular materials with the standard van der Waals normal model are investigated. And the new solutions for the traveling wave of the standard van der Waals normal model within fluidized granular materials were obtained by applying the Backlund transformation, tanh function, and Jacobi elliptic function methods to study the phase separation phenomenon. In addition, novel traveling solitary wave solutions for the (2 + 1)-dimensional cubic-quintic complex Ginzburg-Landau equations were derived using a very useful method to investigate how complex physical coefficients affect the profiles of propagating waves. The practical value of the thesis results lies in obtaining different solutions to the mathematical models discussed in the dissertation work, which can be employed in many different applications such as optics, memory effects, pharmaceutical industries, chemical processes, condensed matter physics, and hydrodynamics.

Contribution of the author

All major results in the thesis were obtained by the author, either directly or through collaboration with staff at the Department of Modeling and Technologies for Oil Field Development.

[1] The applicant performed all the mathematical calculations of the proposed problems in hydrodynamic fluids in multiphase systems.

[2] The applicant and the co-working team carried out the analysis, investigation, processing of results, and interpretation and substantiation of the results obtained.

[3] The applicant proposed a good vision during the formulation and development of many of the mathematical problems addressed in the thesis, and the author was actively involved in every step of the process.

[4] The applicant wrote all original drafts of all manuscripts of the dissertation works and responded to all reviewer comments during publication processes.

Fundamental research techniques

Numerous kinds of numerical methods, such as the Newton-Raphson algorithm, finite difference method, and the technique of Gaussian elimination were employed to resolve the linear algebraic equation system at each time level. The computational algorithm code for the finite difference method was generated using Compaq Visual Fortran version 6.6, and these techniques were applied until a predetermined degree of solution reliability was achieved. Additionally, using Mathematica software, new solutions for the traveling wave of the standard van der Waals model within fluidized granular materials were obtained by applying the Backlund transformation, tanh function, and Jacobi elliptic function.

Scientific provisions submitted for defense

1. Development of a one-dimensional model to simulate the dissociation processes of methane hydrates in porous media in the permafrost zones. Additionally, complex codes and new numerical algorithms were used to solve the suggested mathematical framework. Furthermore, in two-and three-phase systems, the thermophysical characteristics during methane hydrate decomposition processes were investigated.

2. Implementation of the developed algorithms in the form of a software package that allows solving and investigating the phenomena of fluidized granular materials with the standard van der Waals model under the effects of friction and viscosity. Examine the integrability of the equations of the proposed model by applying the Painlevé analysis. Moreover, the Backlund transformation was established using the Painlevé truncation expansion. And the new solutions for the traveling wave of the standard van der Waals model within fluidized granular materials were obtained based on modified numerical techniques, such as the Backlund transformation, tanh function, and Jacobi elliptic function methods, to study the phase separation phenomenon.

3. Find novel traveling wave solutions for recent advances that have been made in the phenomenon of Rayleigh-Bénard convection by carrying out a series of numerical calculations on the proposed model by applying the (2+1)-dimensional coupled cubic-

quintic complex Ginzburg-Landau equations model for slowly varying spatiotemporal amplitudes of the wave motion. The novel traveling solitary wave solutions for the proposed model equation were derived based on the new simplest equation method.

Validation of numerical and analytical results

The validity and reliability of the results and conclusions for all of the numerical results and traveling solitary wave solutions for the models of the study were ensured using different techniques, such as the finite difference method, tanh function, Jacobi elliptic function methods, and Backlund transformation, which is established using the Painleve truncation expansion. Moreover, the stability of the system and simulation analysis verify the dependability of the produced results; the outcomes correspond to agreements with previous models and include more accurate solutions. Besides, the results were discussed at international and all-Russian scientific conferences. The thesis results were published in peer-reviewed journals indexed by Scopus and Web of Science. Approbation of dissertation work

The dissertation's major findings were discussed and displayed at conferences in Russia and abroad, both orally and through conferences in Russia and abroad, both orally and through posters, and at seminars hosted by the "Department of Modeling and Technologies for Oil Field Development" in MIPT, Dolgoprudny, Russia.

1. Ahmed K. Abu-Nab. The 63rd All-Russian Scientific Conference of MIPT, 23-29 November 2020, MIPT, Dolgoprudny-Moscow region, Russia.

2. E. S. Selima, Ahmed K. Abu-Nab, A. M. Morad. VIII International Youth Scientific Conference Physics. Technologies. Innovations of the Phystech technical, 17-21 May 2021, Yekaterinburg, Russia.

3. Ahmed K. Abu-Nab, A. V. Koldoba, E. V. Koldoba, Y. A. Poveshchenko, V. O. Podryga, P. I. Rahimly, A. E. Bakeer. International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE-2022), 5-8 September 2022, Virtual, on-line.

Publications on the subject of work

Based on the different issues studied in the dissertation, four publications were published indexed in the WoS and Scopus databases.

1. Ahmed K. Abu-Nab, Alexander V. Koldoba, Elena V. Koldoba, Yury A. Poveshchenko, Viktoriia O. Podryga, Parvin I. Rahimly, Ahmed E. Bakeer. On the theory of methane hydrate decomposition in a one-dimensional model in porous sediments: numerical study. Mathematics, 11, 341 (2023). https ://doi. org/10.3390/math11020341

2. Ehab S. Selima, Ahmed K. Abu-Nab, Adel M. Morad. Dynamics of the (2+1)-dimensional coupled cubic-quintic complex Ginzburg-Landau model for convecting binary fluid: analytical investigations and multiple-soliton solutions. Mathematical Methods in the Applied Sciences, 1-25 (2023). doi.org/10.1002/mma.9454

3. Ahmed K. Abu-Nab, Alexander V. Koldoba, Elena V. Koldoba, Yury A. Poveshchenko, Viktoriia O. Podryga, Parvin I. Rahimly, Ahmed E. Bakeer. Numerical investigation of gas hydrate dissociation in the porous medium of multiphase flow. AIP Conference Proceedings, 2872, 060020 (2023). doi.org/10.1063/5.0163242

4. Adel M. Morad, Ehab S. Selima, Ahmed K. Abu-Nab. Bubbles interactions in fluidized granular medium for the van der Waals hydrodynamic regime. European Physical Journal Plus, 136, 306 (2021). doi.org/10.1140/epjp/s13360-021-01277-3

5. Ahmed K. Abu-Nab. The 63rd All-Russian Scientific Conference of MIPT, 23-29 November 2020, MIPT, Dolgoprudny-Moscow region, Russia.

6. E. S. Selima, Ahmed K. Abu-Nab, A. M. Morad. VIII International Youth Scientific Conference Physics. Technologies. Innovations of the Phystech technical, 17-21 May 2021, Yekaterinburg, Russia.

7. Ahmed K. Abu-Nab, A. V. Koldoba, E. V. Koldoba, Y. A. Poveshchenko, V. O. Podryga, P. I. Rahimly, A. E. Bakeer. International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE-2022), 5-8 September 2022. Virtual, on-line.

Structure and content of the thesis

The dissertation is organized as follows: an introduction, three chapters, and a conclusion that highlights the findings and accomplishments of the research. The goals of the study, its scientific uniqueness, practical significance, research methods, and defense provisions are all included in the introduction along with the significance of the work.

The first chapter of this dissertation work is presented outlines the scientific literature, basic principles of mathematical modelling, fundamental ideas, and different physical phenomena of hydrodynamic problems in multi-phase systems. For instance, the Methane hydrates dissociation in porous media in the permafrost regions, the fluidized granular materials that are a collection of discrete, macroscopic, solid particles that interact to release energy, and the explanation of convection phenomena of the binary liquid with the related some physical concepts. Finally, methodology and algorithms introduced. The second chapter emphasizes the thermophysical processes for the disintegration of Methane hydrates in porous media are simulated and the numerical investigation of a one-dimensional model is highlighted. The methane gas and water molecules combine to form crystalline solid compounds known as methane hydrates, which are stable at high pressures and low temperatures. Thus, sea continental edges and permafrost regions can provide these favorable conditions for the creation of methane hydrates. Additionally, salinity and the amount of gas in the water influence the stability of methane hydrates. Three stages of flow (hydrate, H2O, CH4) are described in the proposed model, which comprises two components (H2O, CH4).

The third chapter the impact of friction and viscosity is examined in the analysis of fluidized granular materials using the standard van der Waals model. The mass, momentum, and energy equations of local densities are included in the system of macroscopic balance that is presented. Using the hydrodynamic equations for granular matter motion, analytical solutions of the non-linear partial differential equations governing the granular model are examined for two types of collisions: elastic and inelastic collisions. Furthermore, the Painlevé analysis was used to examine the suggested model's integrability.

The fourth chapter focuses on finding analytical solutions for Rayleigh-Benard convection by applying the (2 +1)- dimensional coupled cubic-quintic complex Ginzburg-Landau equations model for slowly varying spatio-temporal amplitudes of the wave motion. In addition, novel traveling solitary wave solutions for the model equation are derived using a very useful method to investigate how complex physical coefficients affect the profiles of propagating waves. And the obtained computational results indicate that the effects of the physical parameters of the considered equations can be demonstrated by utilizing 2D and 3D graphics for different values of these parameters.

Заключение диссертации по теме «Другие cпециальности», Абунаб Ахмед Камал Ибрахим

Thesis summary

The activity of this thesis focused on numerical and analytical studies of the different mathematical models for hydrodynamic problems and their applications in multiphase systems. The main results of the thesis can be summarized as follows.

❖ An implicit FDT was used to solve the systems of Eqs. (2.3)-(2.5) and (2.8) above using a moving in an ODM spatial grid established in the compressing solid sediments. The mathematical framework focuses on the investigation of MHs breakdown in an ODM. The solutions are advanced in finite time.

❖ The NRM is used to calculate the critical parameters in the spatial distribution over various time values, including pressure, temperature, and the rate at which pores saturate with hydrate or water, respectively.

❖ The solutions approach showed good numerical stability, accuracy, and consistency with the obtained results in Refs. [78,81].

❖ Grains are described as smooth rigid disks at a constant normal restitution coefficient, R, that characterizes the collisions. To achieve a qualitative comparison between the

proposed model and the previous models, we have solved the hydrodynamic model in 2D of the dense FGM.

❖ We consider the governing equations with the energy sink for dense quasi-elastic granular medium, which is similar to the compressible elastic hydrodynamic model. The Painlevé test is executed to check the integrability of the SVDWF equation and construct the BT using the truncation on the Painlevé expansion form. It is found that this equation is not integrable. In addition, the TWS of that equation are obtained using the BT, tanh-function, and Jacobi elliptic function methods.

❖ Also, the dispersion analysis for the linear wave theory of the SVDWF is derived to present the dispersive types. The relationship between phase and group velocities as well as the damping properties are shown in the figures. Finally, the stability classifications of the SVDWF model are illustrated using the vector fields of the phase plane method.

❖ The multiple-soliton and solitary wave solutions of these coupled forms of equations are extracted by applying the method of the simplest equation with Riccati, Bernoulli, and Burger's equations as the simplest equations. In addition, we use the Painlevé algorithm to verify the integrability of the (2 + 1)-DCC-QCGLEs and form the BT via the truncated Painlevé expansion. We find that these considered equations are not integrable according to the obtained compatibility conditions.

In the end, this dissertation's findings are consistent with those published in the literature that is currently accessible. Dependable data on mathematical modeling were covered in the dissertation for some hydrodynamic problems in multi-phase systems, such as MHs decomposition, fluidized granular materials with the SVDWM, and the phenomenon of convection in conjunction with thermal instability based on the complex Ginzburg-Landau (cGL) equation. The results have many more applications, such as crystallization, jamming, glass manufacturing technologies, optics, and plasma.

Список литературы диссертационного исследования кандидат наук Абунаб Ахмед Камал Ибрахим, 2024 год

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