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Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics
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Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 22565

Special Issue Editor

Prof. Dr. Boqing Dong

E-Mail Website
Guest Editor
College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China
Interests: nonlinear partial differential equations; mathematical fluid dynamics

Special Issue Information

Dear Colleagues,

Most of the nonlinear problems in fluid dyanmics are governed by the nonlinear partial differential equations such as Navier–Stokes equations, Boussineqa equations, MHD equations, etc. The analysis and application of those nonlinear partial differential equations have attracted more and more attention over the past half century. The understanding of those important nonlinear partial differential equaitons is fascinating and challenging.

As a Special Issue of the international journal Mathematics, we invite investigators to contribute good quality and original research articles as well as review articles that will stimulate the continuing efforts to understand the related nonlinear partial differential equations in fluid dynamics. Potential topics include, but are not limited to: Navier–Stokes equations and Euler equations ; Magneto- hydrodynamic equations; Boussinesq equations; Kinetic equations in gas; Primitive equations and quasi-geostrophic equations in the ocean and atmosphere; and related nonlinear partial differential equations in applied sciences.

Prof. Dr. Boqing Dong
Guest Editor

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Keywords

  • nonlinear PDE
  • mathematical fluid dynamics
  • well-posedness and large time behaviors
  • related nonlinear analysis in applied sciences

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Published Papers (8 papers)

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Research

20 pages, 2876 KiB  
Article
Insight into Dynamic of Mono and Hybrid Nanofluids Subject to Binary Chemical Reaction, Activation Energy, and Magnetic Field through the Porous Surfaces
by Qadeer Raza, M. Zubair Akbar Qureshi, Behzad Ali Khan, Ahmed Kadhim Hussein, Bagh Ali, Nehad Ali Shah and Jae Dong Chung
Mathematics 2022, 10(16), 3013; https://doi.org/10.3390/math10163013 - 21 Aug 2022
Cited by 63 | Viewed by 2166
Abstract
The mathematical modeling of the activation energy and binary chemical reaction system with six distinct types of nanoparticles, along with the magnetohydrodynamic effect, is studied in this paper. Different types of hybrid nanofluids flowing over porous surfaces with heat and mass transfer aspects [...] Read more.
The mathematical modeling of the activation energy and binary chemical reaction system with six distinct types of nanoparticles, along with the magnetohydrodynamic effect, is studied in this paper. Different types of hybrid nanofluids flowing over porous surfaces with heat and mass transfer aspects are examined here. The empirical relations for nanoparticle materials associated with thermophysical properties are expressed as partial differential equations, which are then interpreted into ordinary differential expressions using appropriate variables. The initial shooting method converts the boundary condition into the initial condition with an appropriate guess and finally finds out an accurate numerical solution by using the Runge–Kutta method with numerical stability. Variations in nanoparticle volume fraction at the lower and upper walls of porous surfaces, as well as the heat transfer rate measurements, are computed using the controlling physical factors. The effects of the flow-related variables on the axial velocity, radial velocity, temperature, and concentration profile dispersion are also investigated. The Permeable Reynolds number is directly proportional to the regression parameter. The injection/suction phenomenon associated with the expanding/contracting cases, respectively, have been described with engineering parameters. The hybrid nanoparticle volume fraction (1–5%) has a significant effect on the thermal system and radial velocity. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics)
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17 pages, 861 KiB  
Article
Insight into Significance of Bioconvection on MHD Tangent Hyperbolic Nanofluid Flow of Irregular Thickness across a Slender Elastic Surface
by Muhammad Zeeshan Ashraf, Saif Ur Rehman, Saadia Farid, Ahmed Kadhim Hussein, Bagh Ali, Nehad Ali Shah and Wajaree Weera
Mathematics 2022, 10(15), 2592; https://doi.org/10.3390/math10152592 - 25 Jul 2022
Cited by 87 | Viewed by 2787
Abstract
This numerical investigation effectively establishes a unique computing exploration for steady magnetohydrodynamic convective streams of tangent hyperbolic nanofluid traveling across a nonlinearly elongating elastic surface with a variable thickness. In addition, the importance of an externally imposed magnetic field of tangent hyperbolic nanofluid [...] Read more.
This numerical investigation effectively establishes a unique computing exploration for steady magnetohydrodynamic convective streams of tangent hyperbolic nanofluid traveling across a nonlinearly elongating elastic surface with a variable thickness. In addition, the importance of an externally imposed magnetic field of tangent hyperbolic nanofluid is comprehensively analyzed by considering the substantial impact of thermal conductivity and thermal radiation consequences. The governing PDEs (partial differential equations) are transmuted into a nonlinear differential structure of coupled ODEs (ordinary differential equations) using a series of variable similarity transformations. Furthermore, these generated ODEs (ordinary differential equations) are numerically set using a novel revolutionary Runge-Kutta algorithm with a shooting approach constructed in a MATLAB script. In this regard, extensive comparison studies are carried out to validate the acquired numerical results. The interactions between the associated profiles and the relevant parameters are rationally explored and shown using graphs and tabular forms. The velocity distribution declined with improving Weissengberg number We and power-law index m, while the reverse performance can be observed for temperature. As enhancement in Brownian motion, Thermophoretic and radiation parameters significantly rise in temperature distribution. The use of many different technological and industrial systems, including nano-bioconvective systems, nano-droplet evaporation, nano-ink jet printing, and microbial fuel cells, would benefit this research study. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics)
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28 pages, 8405 KiB  
Article
Numerical Study of MHD Natural Convection inside a Cubical Cavity Loaded with Copper-Water Nanofluid by Using a Non-Homogeneous Dynamic Mathematical Model
by Mohamed Sannad, Ahmed Kadhim Hussein, Awatef Abidi, Raad Z. Homod, Uddhaba Biswal, Bagh Ali, Lioua Kolsi and Obai Younis
Mathematics 2022, 10(12), 2072; https://doi.org/10.3390/math10122072 - 15 Jun 2022
Cited by 16 | Viewed by 2386
Abstract
Free convective flow in a cubical cavity loaded with copper-water nanofluid was examined numerically by employing a non-homogeneous dynamic model, which is physically more realistic in representing nanofluids than homogenous ones. The cavity was introduced to a horizontal magnetic field from the left [...] Read more.
Free convective flow in a cubical cavity loaded with copper-water nanofluid was examined numerically by employing a non-homogeneous dynamic model, which is physically more realistic in representing nanofluids than homogenous ones. The cavity was introduced to a horizontal magnetic field from the left sidewall. Both the cavity’s vertical left and right sidewalls are preserved at an isothermal cold temperature (Tc). The cavity includes inside it four isothermal heating blocks in the middle of the top and bottom walls. The other cavity walls are assumed adiabatic. Simulations were performed for solid volume fraction ranging from (0 ≤ ϕ ≤ 0.06), Rayleigh number varied as (103 ≤ Ra ≤ 105), the Hartmann number varied as (0 ≤ Ha ≤ 60), and the diameter of nanoparticle varied as (10 nm ≤ dp ≤ 130 nm). It was found that at (dp = 10 nm), the average Nusselt number declines when Ha increases, whereas it increases as (Ra) and (ϕ) increase. Furthermore, the increasing impact of the magnetic field on the average Nusselt number is absent for (Ra = 103), and this can be seen for all values of (ϕ). However, when (dp) is considered variable, the average Nusselt number was directly proportional to (Ra) and (ϕ) and inversely proportional to (dp). Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics)
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31 pages, 7778 KiB  
Article
MHD Natural Convection and Radiation over a Flame in a Partially Heated Semicircular Cavity Filled with a Nanofluid
by Obai Younis, Milad Alizadeh, Ahmed Kadhim Hussein, Bagh Ali, Uddhaba Biswal and Emad Hasani Malekshah
Mathematics 2022, 10(8), 1347; https://doi.org/10.3390/math10081347 - 18 Apr 2022
Cited by 12 | Viewed by 2279
Abstract
The numerical analysis of MHD-free convective heat transfer and its interaction with the radiation over a heated flame inside a porous semicircular cavity loaded with SWCNTs–water nanofluid was explored for the very first time in the present work. The two circular arcs of [...] Read more.
The numerical analysis of MHD-free convective heat transfer and its interaction with the radiation over a heated flame inside a porous semicircular cavity loaded with SWCNTs–water nanofluid was explored for the very first time in the present work. The two circular arcs of the upper wall of the enclosure were preserved at a constant cold temperature, whereas the middle region of it was considered adiabatic. The midland region of the lower wall was heated partially, while other regions were also assumed adiabatic. An internal hot flame was included inside the cavity, while the cavity was exposed to a magnetic field. The results were illustrated for Hartmann number (0 ≤ Ha ≤ 100), Rayleigh number (104 ≤ Ra ≤ 106), heated region length (0.1 ≤ L ≤ 0.3), solid volumetric fraction (0 ≤ φ ≤ 0.04), Darcy number (10−3 ≤ Da ≤ 10−5) and radiation parameter (0 ≤ Rd ≤ 1). It was found that decreasing L is the best option for enhancing natural convection. Moreover, it was noted that (Nuout) is directly proportion to (Ra), (ϕ), (Rd) and (Da) increase. In contrast, it was in reverse proportion to (Ha). Furthermore, the results showed that augmentation of about (4%) and a decrement of (56.55%) are obtained on the average (Nu) on the heated length by increasing the radiation and the Hartmann number, respectively. Moreover, raising the radiation number from (0 to 1) causes an augmentation of about (73%) in the average (Nu) of the heated flame. Results also indicated that increasing the Hartmann number will cause a decrement of about (82.4%) of the maximum velocity profile in the vertical direction. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics)
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18 pages, 12932 KiB  
Article
Double Diffusive Natural Convection in a Square Cavity Filled with a Porous Media and a Power Law Fluid Separated by a Wavy Interface
by Lioua Kolsi, Shafqat Hussain, Kaouther Ghachem, Muhammad Jamal and Chemseddine Maatki
Mathematics 2022, 10(7), 1060; https://doi.org/10.3390/math10071060 - 25 Mar 2022
Cited by 11 | Viewed by 2571
Abstract
This study deals with the influence of a wavy interface separating two layers filled with power law fluid and porous media, respectively. The governing equations are solved using the Finite Element Method (FEM) and the numerical model is validated by comparing with experimental [...] Read more.
This study deals with the influence of a wavy interface separating two layers filled with power law fluid and porous media, respectively. The governing equations are solved using the Finite Element Method (FEM) and the numerical model is validated by comparing with experimental findings. The parameters governing the studied configuration are varied as: Rayleigh number (103 ≤ Ra ≤ 106), power law index (0.6 ≤ n ≤ 1.4), Darcy number (10−2 ≤ Da ≤ 10−6), buoyancy ratio (0.1 ≤ N ≤ 10) and Lewis number (1 ≤ Le ≤ 10). It is inferred that the temperature gradient increases by augmenting the Rayleigh number, as the flow is observed from the vertical to horizontal direction in both layers. Constant enhancement in the heat and mass transfer is also observed by enriching the buoyancy effect. Moreover, the average Nusselt and Sherwood numbers decline by increasing the width of the porous layer. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics)
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17 pages, 5730 KiB  
Article
Generalized Exp-Function Method to Find Closed Form Solutions of Nonlinear Dispersive Modified Benjamin–Bona–Mahony Equation Defined by Seismic Sea Waves
by Muhammad Shakeel, Attaullah, Essam Roshdy El-Zahar, Nehad Ali Shah and Jae Dong Chung
Mathematics 2022, 10(7), 1026; https://doi.org/10.3390/math10071026 - 23 Mar 2022
Cited by 31 | Viewed by 2287
Abstract
Using the new generalized exp-function method, we were able to derive significant novel closed form solutions to the nonlinear dispersive modified Benjamin–Bona–Mahony (DMBBM) equation. The general framework of the new generalized exp-function method has been given. Many novel closed form solutions have been [...] Read more.
Using the new generalized exp-function method, we were able to derive significant novel closed form solutions to the nonlinear dispersive modified Benjamin–Bona–Mahony (DMBBM) equation. The general framework of the new generalized exp-function method has been given. Many novel closed form solutions have been obtained in the form of hyperbolic, trigonometric, and rational function solutions. Using the computer application Wolfram Mathematica 10, we plotted 2D, 3D, and contour surfaces of closed form solutions found in this work. In the form of a table, the acquired results are compared to the known solutions in the existing literature. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics)
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14 pages, 98376 KiB  
Article
Mixed Convection inside a Duct with an Open Trapezoidal Cavity Equipped with Two Discrete Heat Sources and Moving Walls
by Fateh Mebarek-Oudina, Hanane Laouira, Ahmed Kadhim Hussein, Mohamed Omri, Aissa Abderrahmane, Lioua Kolsi and Uddhaba Biswal
Mathematics 2022, 10(6), 929; https://doi.org/10.3390/math10060929 - 14 Mar 2022
Cited by 27 | Viewed by 2197
Abstract
The current research presents a numerical investigation of the mixed convection inside a horizontal rectangular duct combined with an open trapezoidal cavity. The region in the bottom wall of the cavity is heated by using two discrete heat sources. The cold airflow enters [...] Read more.
The current research presents a numerical investigation of the mixed convection inside a horizontal rectangular duct combined with an open trapezoidal cavity. The region in the bottom wall of the cavity is heated by using two discrete heat sources. The cold airflow enters the duct horizontally at a fixed velocity and a constant temperature. All the other walls of the duct and the cavity are adiabatic. Throughout this study, four various cases were investigated depending on the driven walls. The effects of the Richardson number and Reynolds number ratio are studied under various cases related to the lid-driven sidewalls. The results are presented in terms of the flow and thermal fields and the average Nusselt number. The yielded data show that the average Nusselt number rises as the Richardson number and Reynolds number ratio increases. Furthermore, the Reynolds number ratio and the movement of the cavity sidewall(s) have a significant effect on the velocity and temperature contours. By the end of the study, it is shown that the maximum rates of heat transfer are related to Case 1 where the left sidewall moves downward and heater 2, which is placed near the left sidewall. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics)
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18 pages, 40157 KiB  
Article
MHD Hybrid Nanofluid Mixed Convection Heat Transfer and Entropy Generation in a 3-D Triangular Porous Cavity with Zigzag Wall and Rotating Cylinder
by Aissa Abderrahmane, Naef A. A. Qasem, Obai Younis, Riadh Marzouki, Abed Mourad, Nehad Ali Shah and Jae Dong Chung
Mathematics 2022, 10(5), 769; https://doi.org/10.3390/math10050769 - 28 Feb 2022
Cited by 71 | Viewed by 3904
Abstract
The purpose of this work was to conduct a numerical examination of mixed convective heat transfer in a three-dimensional triangular enclosure with a revolving circular cylinder in the cavity’s center. Numerical simulations of the hybrid Fe3O4/MWCNT-water nanofluid are performed [...] Read more.
The purpose of this work was to conduct a numerical examination of mixed convective heat transfer in a three-dimensional triangular enclosure with a revolving circular cylinder in the cavity’s center. Numerical simulations of the hybrid Fe3O4/MWCNT-water nanofluid are performed using the finite element approach (FEM). The simulation is carried out for a range of parameter values, including the Darcy number (between 10−5 and 10−2), the Hartmann number (between 0 and 100), the angular speed of the rotation (between −500 and 1000), and the number of zigzags. The stream function, isotherms, and isentropic contours illustrate the impact of many parameters on motion, heat transfer, and entropy formation. The findings indicate that for enhancing the heat transfer rates of hybrid nanofluid in a three-dimensional triangular porous cavity fitted with a rotating cylinder and subjected to a magnetic field, Darcy number > 10−3, Hartmann number < 0, one zigzag on the hot surface, and rotation speed >500 in flow direction are recommended. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics)
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