Dimensional Equation Research Articles

New Painlevé Integrable (3+1)-Dimensional Combined pKP-BKP Equation: Lump and Multiple Soliton Solutions

We introduce a new form of the Painlevé integrable (3+1)-dimensional combined potential Kadomtsev--Petviashvili equation incorporating the B-type Kadomtsev–Petviashvili equation (pKP–BKP equation). We perform the Painlevé analysis to emphasize the complete integrability of this new (3+1)-dimensional combined integrable equation. We formally derive multiple soliton solutions via employing the simplified Hirota bilinear method. Moreover, a variety of lump solutions are determined. We also develop two new (3+1)-dimensional pKP–BKP equations via deleting some terms from the original form of the combined pKP–BKP equation. We emphasize the Painlevé integrability of the newly developed equations, where multiple soliton solutions and lump solutions are derived as well. The derived solutions for all examined models are all depicted through Maple software.

Unsteady magneto bioconvective Sutterby nanofluid flow: Influence of g-Jitter effect

This study is driven by the potential applications of a microgravity environment in various sectors and fields of engineering, especially in the manufacturing of semiconductors. An electrically conducting bioconvective Sutterby nanofluid flow was considered around an exponentially stretching surface through g-Jitter and nonlinear thermal radiation effects. The dimensional partial differential equations (PDEs) that govern the fluid flow system are changed into nondimensional PDEs through nonsimilar transformations. Once these PDEs have been linearized using Quasilinearization, the implicit finite difference scheme is used to solve them numerically. A MATLAB code was constructed for the numerical calculations and for analysing the results via graphs. These findings indicate that the greater values of amplitude ε1 result in a steep rise in the velocity and surface drag near the sheet's wall for assisting buoyancy. An enhancement of approximately 108 % in the drag coefficient is observed when ε1(the amplitude in g-Jitter) jumps to 1 from 0. The percentile increments in heat transport strength due to the rise of radiation Rd from 1 to 2 is 80 %, and due to the intensification of the temperature difference ratio φw from 1.5 to 2 is about 167 %. Elevating Peclet number Pe led to the depletion of the microbial boundary layer thickness. Oxytactic microbes' presence impacts the enhanced mass transport strength of nanoparticles. The current findings have been validated by comparing them to previously published results, and they have been shown to be in perfect agreement with those results.

A novel investigation of dynamical behavior to describe nonlinear wave motion in (3+1)-dimensions

This study focuses on the extended (3+1)-dimensional Sakovich equation, which is utilized for the description of nonlinear wave motion and manifests increased dispersion and nonlinear effects within the realm of nonlinear dynamics. The Lie symmetries are initially identified, followed by the construction of the Abelian algebra corresponding to these symmetries. Through the utilization of this Abelian algebra, the governing equation is transformed into an ordinary differential equation. Precise solutions are obtained using the unified technique, and the results are visually represented through 3D, 2D, and contour plots. Subsequently, the qualitative behavior of the equation is explored from multiple perspectives, encompassing bifurcation, chaos, and sensitivity analysis. Bifurcation is examined at critical points, while the dynamical system is subjected to an external force, leading to the detection of chaotic behavior employing diverse tools such as 3D and 2D phase portraits, time series, Poincaré maps, and multistability analysis. Furthermore, sensitivity analysis is conducted for various initial values, revealing the considerable sensitivity of the presented model, wherein even slight changes in the initial conditions result in significant variations. The reported outcomes are intriguing, showcasing the efficacy and applicability of the suggested methodologies for the assessment of soliton solutions and phase patterns in a diverse range of nonlinear models.

A CPC fractional model of the heat and mass transport mechanism in Carbon nanotubes with slip effects on velocity

A fractional technique is used to evaluate the temperature, mass, and velocity flow of single and double wall CNTs over a vertical plate. Slip boundary conditions and applied magnetic force are addressed. Human blood is used to examine how base fluid behaves. Applying the proper dimensionless variables results in the dimensionless formulation of initial and boundary conditions related to the governed dimensional concentration, momentum, and energy equations. The Laplace transform technique is used to resolve the dimensionless governing partial differential equations and get the solutions. The constant proportional Caputo (CPC) time-fractional derivative is a unique class of fractional model used in the simulation technique. The fundamental definitions are used to support the said model first. Using MWCNTs and SWCNTs in comparison to the flow characteristics, a thermal and mass study is given. The heat and mass transfer processes for single-walled carbon nanotubes (SWCNTs) have been shown to typically be progressive. The momentum profile decreases as the fractional variables rise. Multi-walled carbon nanotubes (MWCNTs) show more progressive velocity control as a result of the magnetic parameter. Graphs demonstrate the influence of embedding factors on the velocity, energy, and concentration profiles.

Numerical assessment of irreversibility in radiated Sutterby nanofluid flow with activation energy and Darcy Forchheimer

Entropy generation is a concept that is primarily associated with thermodynamics and engineering, and it plays a crucial role in understanding and optimizing various processes and systems. Applications of entropy generation can be seen in turbo machinery, reactors, chillers, desert coolers, vehicle engines, air conditioners, heat transfer devices and combustion. Due to industrial applications entropy generation has gained attention of researchers. Owing such applications, current communication aims to model and analyzed the irreversibility in Sutterby nanoliquid flow by stretched cylinder. Momentum equation is reported by considering porosity, Darcy Forchheimer and magnetic field. While in energy equation radiation and Joule heating effects are accounted. Activation energy impact is accounted in the modeling of concentration equation. Thermodynamics second law is utilized for physical description of irreversibility analysis. Through similarity transformations dimensional equations representing flow are transformed to dimensionless ones. Numerical solution for ordinary system is obtained via Runge–Kutta-Fehlberg scheme in Mathematica platform through NDsolve code. Influence of prominent variables on velocity, entropy, temperature, Bejan number and concentration are graphically analyzed. Coefficient of skin friction, gradient of temperature and Sherwood number are numerically analyzed. The obtained results show that velocity field decreases through higher porosity and Forchheimer variables. Velocity and temperature curves shows an opposite trend versus magnetic parameter. A decay in concentration distribution is noticed through larger Schmidt number. Entropy generation amplifies against magnetic parameter and Brinkman number.

Diverse higher-order soliton solutions and novel hybrid behaviours of the $$(2+1)$$-dimensional KP-BBM equation

Diverse higher-order soliton solutions and novel hybrid behaviours of the $$(2+1)$$-dimensional KP-BBM equation

Dynamical analysis of multi-soliton and breather solutions on constant and periodic backgrounds for the (2+1)-dimensional Heisenberg ferromagnet equation

Dynamical analysis of multi-soliton and breather solutions on constant and periodic backgrounds for the (2+1)-dimensional Heisenberg ferromagnet equation

Formula omitted]-breather solutions, inelastic interactions of the lumps and resonant interactions of the breathers for a (2+1)-dimensional nonlinear evolution equation

Studies are conducted on a (2+1)-dimensional nonlinear evolution equation. We investigate the inelastic interactions of the two/three/four lumps based on the M-lump solutions, where M is a positive integer. We observe the different patterns of inelastic interactions of the lumps. We construct the H-breather solutions and carry out the asymptotic analyses for two/three-breather solutions via the symbolic computation, where H is a positive integer. We demonstrate that the resonant interactions occur when the phase shifts of the two/three breathers approach the infinity. We graphically show the resonant interactions of the two/three breathers.

Letter to the Editor: Singular-manifold view on a (3+1)-dimensional fourth-order nonlinear equation in a fluid via HFF 32, 1664 (2022)

Letter to the Editor: Singular-manifold view on a (3+1)-dimensional fourth-order nonlinear equation in a fluid via HFF 32, 1664 (2022)

Wellposedness for a (1+1)-dimensional wave equation with quasilinear boundary condition

We consider the linear wave equation on with initial conditions and a nonlinear Neumann boundary condition at x = 0. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where f is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at x = 0 have enough regularity to solve the initial value problem with their own initial data.

A diversity of patterns to new (3 + 1)-dimensional Hirota bilinear equation that models dynamics of waves in fluids

This article discusses the behavior of specific dispersive waves to new (3+1)-dimensional Hirota bilinear equation (3D-HBE). The 3D-HBE is used as a governing equation for the propagation of waves in fluid dynamics. The Hirota bilinear method (HBM) is successfully applied together with various test strategies for securing a class of results in the forms of lump-periodic, breather-type, and two-wave solutions. Solitons for nonlinear partial differential equations (NLPDEs) can be identified via the well-known mathematical methodology known as the Hirota method. However, this requires for bilinearization of nonlinear PDEs. The method employed provides a comprehensive explanation of NLPDEs by extracting and also generating innovative exact solutions by merging the outcomes of various procedures. To further illustrate the impact of the parameters, we also include a few numerical visualizations of the results. These findings validate the usefulness of the used method in improving the nonlinear dynamical behavior of selected systems. These results are used to illustrate the physical properties of lump solutions and the collision-related components of various nonlinear physical processes. The outcomes demonstrate the efficiency, rapidity, simplicity, and adaptability of the applied algorithm.

Pore-scale resolved simulation of quenching, acceleration, and transition to detonation of hydrogen explosions by metal foams

Pore-scale simulations of quenching, acceleration, and transition to detonation of gaseous hydrogen explosions by metal foams were performed. The model solves the three dimensional compressible reactive Navier-Stokes equations with detailed chemistry in a closed square channel with a metal foam placed inside. The structures of the foam were fully resolved using an immersed boundary method. Stoichiometric mixtures of hydrogen-oxygen and hydrogen-air were considered. An explosion in the channel was initiated with a high-pressure and high-temperature spark. The simulation results show that the explosion can be quenched, accelerate, or transition to detonation depending on the foam pores-per-inch (PPI). Hydrogen-air explosions were quenched with foams of ∼39 PPI or higher due to heat losses from the flame, which is consistent with experimental observations. The explosion was not quenched by foams with low PPI. Instead the low PPI foams exacerbated the explosion by causing flame acceleration. Worse, a 78 PPI foam caused a stoichiometric H2 and O2 mixture to transition to detonation. Hydraulic resistance due to products venting away from the flame surface was found to produce strong pressure gradients within the foam. This causes a self-pressurizing effect that grows as the flame propagates further into the foam. The pressure within the foam grows and steepens into a shock. The shock initiated the detonation by a Mach reflection, which is similar to deflagration-to-detonation transition mechanisms in traditional obstacle-laden channels. Statement of SignificanceMetal foam structures are often used as devices to inhibit flames in scenarios where explosion hazards are a concern. Direct numerical simulations were performed to increase understanding of how flames and explosions interact with metal foams. The simulation results show that high pores-per-inch (PPI) foams quench the explosion by removing heat from the flame. Lower PPI foams can have the opposite effect and exacerbate the explosion by promoting flame acceleration or even transition to detonation. Flames self-pressurize as they propagate through low PPI foams due to hydrodynamic friction and gas-dynamic effects from flowing product gases. In extreme cases, the pressure waves near the flame can steepen into a shock and initiate a detonation. These results indicate that improperly selected metal foam flame arresters can exacerbate the explosion.

On the decomposition mechanism of propanal: rate constants evaluation and kinetic simulations

The reactivity of aldehydes has been the subject of considerable interest in chemical kinetics, with propanal often chosen as the representative species. Despite its relevance, the reactivity of propanal is currently estimated from analogy and fitting of experimental data measured in limited temperature and pressure ranges, while the few literature theoretical studies have focused more on the exploration the potential energy surface (PES) than on the estimation of rate constants. The purpose of this work is to reinvestigate the propanal decomposition kinetics using the ab initio transition state theory based master equation approach with the intent of: (1) Determining accurate rate constants of key reaction channels; (2) Updating and validating an existing kinetic model by simulating available experimental data on propanal pyrolysis. It is found that propanal decomposition at the initial stages of pyrolysis occurs through four unimolecular barrierless reactions to form CHO + C2H5, CH2CHO + CH3, CH3CHCHO + H, and CH3CH2CO + H, and a termolecular pathway leading to the formation of C2H4 + CO + H2. High pressure rate constants were determined for each barrierless reaction channel using Variable Reaction Coordinate Transition State Theory and used to estimate phenomenological temperature and pressure dependent channel specific rate constants integrating the 1 dimensional master equation over the whole PES. The decomposition rate constants so determined are in agreement with the few available experimental data and significantly faster than previous literature estimates. The estimated kinetic parameters were finally implemented into the CRECK kinetic mechanism, leading to an improved agreement with shock tube pyrolysis data from the literature.

Finite element analysis of MHD naturally convective flow past an exponentially accelerated plate with viscous dissipation

The MHD finite element analysis naturally convective flow across an exponentially accelerating plate with viscous dissipation is investigated numerically. In this process dimensional partial differential equations are changed to dimensional less form. The Galerkin finite element method is emerged to solve non-dimensional partial differential equations. Obtained results are effectively depicted through the utilization of graphs, allowing for a comprehensive understanding of the impact that different physical parameters on primary velocity, transverse velocity, temperature, and concentration profiles. The study effectively differentiated between the present and prior findings, ultimately demonstrating a strong consensus of excellent agreement between the results. The key findings of this study are: The primary velocity increases on growing the values of Thermal Grashof number, Solutal Grashof number, Eckert number, and Dufour number and decreases on climb up the values of Magnetic parameter, Prandtl number, Schmidt number, and Hall parameter. The secondary velocity increases on raising the values of Thermal Grashof number and Solutal Grashof number and diminishes on enhancing the value of Hall parameter. The temperature increases on increasing the values Eckert number and Dufour number and decreases on improving the value of Prandtl number. The concentration decreases on increasing the value of Schmidt number.

A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

The solitary wave solutions of the stochastic Heisenberg ferromagnetic spin chain equation using two different analytical methods

Here, we consider the stochastic (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation which is forced by the multiplicative Brownian motion in the Stratonovich sense. We utilize the (G′/G)-expansion method and the mapping method to attain the analytical solutions of the stochastic (2 + 1)-dimensional Heisenberg ferromagnetic chain equation. Various types of analytical stochastic solutions, such as the hyperbolic, elliptic, and trigonometric functions, have been obtained. Physicists can utilize these solutions to understand a variety of important physical phenomena because the magnetic soliton has been categorized as one of the interesting groups of nonlinear excitations representing spin dynamics in the semiclassical continuum Heisenberg systems. Moreover, we employ MATLAB tools to plot 3D and 2D graphs for some obtained solutions to address the influence of Brownian motion on these solutions.

Solving 2D damped Kuramoto-Sivashinsky with multiple relaxation time lattice Boltzmann method

Lattice Boltzmann method (LBM) is a powerful fluid flow solver. Using this method to solve other PDEs might be a difficult task. The first challenge is to find a suitable local equilibrium distribution function (EDF) capable of recovering the desired PDE. The next difficulty arises from the explicit nature of LBM. The conditional stability of the LBM algorithm affects the numerical solution accuracy. Damped Kuramoto–Sivashinsky (DKS) equation is a fourth–order PDE that recently challenged many numerical methods' abilities. This equation is highly sensitive to a parameter that causes three states of solutions in a small interval of freedom. In this paper, we challenged LBM to solve the two–dimensional DKS equation by finding EDF using the Chapman–Enskog analysis up to the fourth–order. Also, the von Neumann analysis and a simple genetic algorithm are applied to find reliable values for the free parameters. Furthermore, a modification on image-based ghost node method is proposed for implementation of the boundary conditions in the complex geometries.

Symmetries and Conservation Laws for a Class of Fourth-Order Reaction–Diffusion–Advection Equations

We have studied a class of (1+1)-dimensional equations that models phenomena with heterogeneous diffusion, advection, and reaction. We have analyzed these fourth-order partial differential equations within the framework of group methods. In this class, the diffusion coefficient is constant, while the coefficients of advection and the reaction term are assumed to depend on the unknown density u(t,x). We have identified the Lie symmetries extending the Principal Algebra along with all the conservation laws corresponding to the different forms of the coefficients, and have derived several brief applications.

Dynamics of complexiton, Y-type soliton and interaction solutions to the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation in liquid with gas bubbles

This paper focuses on some novel exact solutions of the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation (KSE). Based on the linear superposition principle (LSP) and weight algorithm (WA), we construct the complexiton solutions by introducing the pairs of the conjugate parameters. Besides, the Y-type soliton solutions are extracted by imposing the resonant conditions on the N-soliton solutions. Finally, two different types of the interaction solutions (ISs) are also explored by exerting the test function method and symbolic computation. The graphical representations of the corresponding solutions are presented to reveal the dynamical characteristics. To the best of our knowledge, the attained outcomes of this article are brand new and have never been reported in other literature, and can enable us make sense of the physical behaviors of the (3 + 1)-dimensional KSE better.

CFD study of a horizontal axis wind turbine NREL Phase II

The present work aims to study the aerodynamic characteristics of the NREL phase II (generated only with S809 profile along the span for an untwisted case) rotor that is a horizontal axis downwind wind turbine rotor and which is assumed to stand isolated in the space. The two dimensional steady-incompressible flow Reynolds average NavierStokes equations, are solved by using the commercial CFD package Ansys Fluent. The 2D computations are first performed on S809 airfoil in order to define the most suitable model to be used; the turbulence closure model has been chosen among four possible candidates (standard , Spalart-Allmaras, based on comparison of pressure coefficient for the different configurations with experimental results. Secondly, through a three dimensional study we tried to simulate the experiment for wind speed velocities of 7.2, 10.56, 12.85, 16.3, and 9.18m/s. Results of pressure and torque for considered wind turbine rotor have been directly compared to the available experimental data. The comparisons show that CFD results along with the turbulence model used can predict the span-wise loading of the wind turbine rotor with reasonable agreement. The work presented here is the first stage of project that aims at giving a better understanding of the main influence of the rotational effect on boundary layer separation, and identify the stalled configuration in order to control this latter in future work.