Nonlinear ODEs Research Articles

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis.

Entropy generation analysis of microrotating Casson’s nanofluid with Darcy–Forchheimer porous media using a neural computing based on Levenberg–Marquardt algorithm

PurposeThe purpose of this paper is to showcase the utilization of the magnetohydrodynamics-microrotating Casson’s nanofluid flow model (MHD-MRCNFM) in examining the impact of an inclined magnetic field within a porous medium on a nonlinear stretching plate. This investigation is conducted by using neural networking techniques, specifically using neural networks-backpropagated with the Levenberg–Marquardt scheme (NN-BLMS).Design/methodology/approachThe initial nonlinear coupled PDEs system that represented the MRCNFM is transformed into an analogous nonlinear ODEs system by the adoption of similarity variables. The reference data set is created by varying important MHD-MRCNFM parameters using the renowned Lobatto IIIA solver. The numerical reference data are used in validation, testing and training sets to locate and analyze the estimated outcome of the created NN-LMA and its comparison with the corresponding reference solution. With mean squared error curves, error histogram analysis and a regression index, better performance is consistently demonstrated. Mu is a controller that controls the complete training process, and the NN-BLMS mainly concentrates on the higher precision of nonlinear systems.FindingsThe peculiar behavior of the appropriate physical parameters on nondimensional shapes is demonstrated and explored via sketches and tables. For escalating amounts of inclination angle and Brinkman number, a viable entropy profile is accomplished. The angular velocity curve grows as the rotation viscosity and surface condition factors rise. The dominance of friction-induced irreversibility is observed in the vicinity of the sheet, whereas in the farthest region, the situation is reversed with heat transfer playing a more significant role in causing irreversibilities.Originality/valueTo improve the efficiency of any thermodynamic system, it is essential to identify and track the sources of irreversible heat losses. Therefore, the authors analyze both flow phenomena and heat transport, with a particular focus on evaluating the generation of entropy within the system.

Magnetohydrodynamic Casson SWCNT-MWCNT/EG hybrid nanofluid flow over a stretching cylinder with Cattaneo-Christov double diffusion model

The present problem examines the flow features of a two-dimensional time-independent Casson hybrid nanofluid on a stretching cylinder with dual stratification and MHD effects. Heat and mass transfer effects are analyzed by utilizing the Cattaneo-Christov double diffusion scheme with entropy generation. Further, the energy losses due to viscous and Joulian dissipation are also considered in the problem. Due to the high thermal conductivity nature of the carbon nanotubes, which plays a crucial role in cooling systems, material sciences, electrical components and treatment of cancer cells, single and multi-walled carbon nanotube particles (SWCNT, MWCNT) are considered in the hybrid nanofluid. The carbon nanotubes are mixed with base fluid ethylene glycol (EG) to form the required hybrid nanofluid. Suitable transformations are adopted for transforming the non-linear PDEs to non-linear ODEs of the mathematical model. Then, the system of ODEs are resolved by utilizing bvp4c numerical approach in MATLAB software. The impact of the various flow pertinent parameters on the flow model are disclosed through graphs and tables and then explained elaborately. The important findings of the study reveal a reduction in the absolute value of coefficient of surface friction with the Casson and velocity slip parameters. The entropy generation rate (NG ) is diminished due to slip parameters, whereas the opposite effect is observed due to the SWCNT nanoparticle volume fraction. The thermal and mass relaxation parameters induce a significant amplification in the rates of heat and mass transport, respectively. Furthermore, it is observed that the heat and mass transfer rates of EG-based SWCNT – MWCNT hybrid nanofluid perform better than those of EG-based SWCNT nanofluid. The outcomes of the investigation exhibit an excellent agreement with the previously published results. The present analysis has applications in cancer therapy, microelectronics cooling, industrial heat generators, nuclear reactor maintenance, electrolytic condensers, and many more.

Significance of chemical reaction and viscous dissipation on magnetized-Marangoni convective flow of dusty Casson hybrid (AA7072+AA7075/sodium alginate) nanofluid with Soret and Dufour effects

ABSTRACT In the current study, mass and heat transfer flow of radiated Casson hybrid nanofluid with nanoparticles and dust deferment over sheet are explored. The Soret and Dufour effects, activation energy and microorganisms in the hybrid nanofluid model are described. Marangoni convection, a prominent occurrence in microgravity caused by surface tension gradients. It can be used for many different things, such as the creation of molten crystals, thin-film diffusion, the generation of vapour bubbles during nucleation, and semiconductor manufacturing. In sodium alginate, we wondered about the aluminium alloys AA7072 and AA7075. The aluminium alloys included in this study are specially made materials with improved heat transmission characteristics. aluminium and zinc are combined in AA7072 alloy in the ratios of 98 & 1, respectively, with the addition of silicon, ferrous metals, and copper. Similarly, AA7075 is a blend of aluminium, magnesium, zinc, and copper in the proportions of ~ 90, ~3, ~6, and ~ 1, respectively, with silicon ferrous and magnesium as additional metals. The controlling PDEs are transformed into nonlinear ODEs with the aid of a new set of similarity variables. These ODEs are then numerically solved using the RKF-45th method. The findings show that when the Marangoni convection parameter increases, the concentration, temperature, and microbiological characteristics drop while the velocity profile increases for both the dust and fluid phases.

Legendre wavelet integration operational matrix method for the numerical solution of unsteady MHD nanofluid flow between moving parallel plates

This study proposes a novel method based on Legendre wavelet integration operational matrix for solving a system of nonlinear ODE’s to model the two-phase, unsteady nanofluid flow and heat transfer between parallel plates in presence of magnetic field. To validate the applicability of proposed method, system of nonlinear BVP with exact solutions is employed as a test problem. Test problem results are compared with exact solution, and other numerical methods. The governing equations were then changed using similarity transformation into a set of nonlinear ODE’s. . Following this, the LWOMM is applied for the proposed problem. Findings demonstrated that, for a constant Reynolds number, skin friction coefficient value increases with rising Hartmann and squeezing numbers. Furthermore, it is demonstrated that Eckert number is a reducing function of the squeeze number and Nusselt number is an increasing function of Hartmann number.

MHD Maxwell dusty fluid in thermally stratified radiative flow with temperature-dependent thermal conductivity and Cattaneo-Christov model

It might be very important for the polymer processing industries to comprehend how Maxwell fluids behave on a stretched cylinder. Optimizing the extrusion and drawing processes can ensure the desired product qualities while avoiding faults. The objective of this study is heat transfer analysis on a Maxwell dusty fluid flow cylindrical surface with the Cattaneo-Christov concept. We immerse the cylinder in porous media, with a two-dimensional fluid regulating the flow. Our mathematical model further considers the effects of variable thermal conductivity, radiation, viscous and joule heating, magnetic field, thermal stratification, and slip velocity. Based on the presumptions, partial differential equations (PDE's) have been used to evolve the mathematical model. Using similarity transformations, the PDE's for heat and momentum for both phases are transformed into highly nonlinear ODE's.The numerical results have been obtained on these ordinary differential equations by using the RKF-45 method. This issue's main characteristic is that it examines the scenario's liquid and dust phases throughout. Results are given both visually and tabularly for the major parameters over a velocity, temperature, skin friction coefficient, and Nusselt number. When we compared our method to a previously published paper, we discovered a decent match. The findings, which were obtained for our system, show that the velocity and thermal gradient of both the phases of fluid and dust behave in an opposite trend in favor of rising Maxwell parameter, where the curvature parameter makes the rise in the same manner. Furthermore, the thermal transport profiles for both phases decline for the rising thermal time relaxation parameter.

Thermal analysis of the water-based micropolar hybrid nanofluid flow comprising diamond and copper nanomaterials past an extending surface

This research demonstrates that hybrid nanofluids are superior to traditional fluids in terms of their capacity to transmit heat. An intensive investigation of hybrid nanofluids is needed to fill gaps in knowledge of heat transmission improvement methods. Therefore, scientists and researchers are combining different solid particles in various base fluids to boost the material's thermal conductivity. That's why the author has combined diamond and copper nanoparticles in a water solvent to examine the two-dimensional, steady, and incompressible magnetohydrodynamic (MHD) hybrid micropolar nanofluid flow over a permeable extending surface through the porous media. The Cattaneo-Christov heat and mass flux model is used to assess the heat and mass diffusion occurrences in temperature and concentration distributions. The similarity transformations are used to convert the PDEs along with their related boundary constraints into a set of nonlinear ODEs. A semi-analytical method known as HAM is used with the help of MATHEMATICA software to perform an analytical analysis on the set of nonlinear ODEs. The impacts of many important factors on the velocity, microrotation, thermal, and concentration functions are demonstrated through graphs and discussed in detail. The study results demonstrate that velocity distribution is a declining function of the magnetic field and porosity parameter. The magnetic field, Brownian motion, heat source, thermophoresis, and radiation parameters have positively influenced the thermal profile, while the thermal relaxation parameter has a negative influence on it. Nanoparticle volume fraction improves the Nusselt and Sherwood numbers. Heat transfer and skin friction are both amplified by the magnetic field. The skin-friction coefficient improves with an increase in the permeability parameter and the volume percentage of nanoparticles. Also, nanoparticles volume fraction enhances the mass transfer rate. Hybrid nanofluids have superior thermal characteristics over regular fluids.

Dynamics of magnetized viscous dissipative material of hybrid nanofluid with irregular thermal generation/absorption

The intriguing thermophysical properties of nanoparticles have considerable potential in thermal transportation for engineering applications. The inclusion of solid nanoparticles in the base fluid raises the fluid's thermal conductivity, which improves thermal transfer characteristics. Thus, an analysis is performed by assuming the Tiwari and Das model, for the flow and energy transportation aspects of hybrid nanoparticles in a shrinkable porous medium. The current study focuses on the thermal heat transfer performance on the dynamic of ethylene glycol-based titania and alumina nanoparticles. The flow-controlling system of PDEs is transformed into a system of non-linear ODEs by using similarity transformations, and the resulting equations are then solved numerically through the solver bvp4c in Matlab function with a limit of tolerance 10−6. The multiple branches of solutions are noted for distinct values of suction and shrinking parameters. The outcomes reveal that the inclusion of titania and alumina nanocomposites improves the thermophysical properties of the base fluid significantly. The findings indicate that hybrid nanofluids are more effective for increasing the thermal transport rate. Further, in the upper branch solution, the friction drags, and energy transportation rate raises for the higher influence of suction strength while it declines for the lower branch.

Radiation-influenced magnetohydrodynamic third-grade nanofluid flow around non-linearly stretched cylinder

Abstract This analysis considers the magnetized third-grade fluid stream and microorganisms over a non-linear stretchy cylinder. The radiation impacts are taken into consideration. The effects of the governing flow at the cylinder are represented in the form of PDEs employing boundary layer approximations. The system of the PDEs is further reduced in dimensionless form after applying the similarity transformations. The dimensionless system of non-linear ODEs is solved through the numerical technique bvp4c. The effects of radiation and magnetism on the third-grade liquid over a non-linear extending cylinder are highlighted in graphs and numerically in tabular form. The influence of fluid variables on the velocity curve, such as third-grade parameters, second-grade coefficients, and Reynolds number, is illustrated and explored. Suitable ranges for the parameters $( {1 \le \eta \le 10,\ 0.2 \le {{\alpha }_1} \le 0.5,\ 0 \le {{\alpha }_1} \le 1.5,\ 0.1 \le \beta \le 0.3,\ 0.1 \le \gamma \le 1.6,\ 0.05 \le M \le 0.15,\ 0.5 \le \delta \le 2.0,\ 0.7 \le Pr \le 1.3,\ 0.1 \le Rd \le 0.4,0.1 \le}$ ${e \le 0.4} )$ are chosen depending upon the convergence of the numerical method. The widths of the velocity and momentum boundary layers are revealed to be increasing functions of the curvature parameter. The temperature curve declines when boosting third-grade parameters, thermal stratification, and Hartmann number while boosting up for curvature and radiation parameters.

Mathematical modelling of HMT through porous stretching sheet using artificial neural network

The influence of heat radiation on HMT, MHD on a porous stretched sheet has been explained in this study. The governing PDEs are converted to ODEs via similarity transformations. The coefficient of skin friction, rate of HMT are calculated using the MATLAB software for various parameter values. The controlling boundary layer PDEs are turned into a system of coupled nonlinear ODEs via similarity techniques, which are numerically solved using the shooting technique in conjunction with the fourth order Runge Kutta method, and then ANN is applied to them. Validation of numerical results using ANN results. We discovered that using engineering standpoints, an ANN model may produce high-efficiency estimates for heat transfer rates. The achieved R squared values of 99% for predicting Skin friction, Nusselt number and Sherwood number coefficients highlight the remarkable effectiveness of neural network models in these predictions. This efficacy not only demonstrates the accuracy of the models but also results in a significant reduction in the computational time required compared to traditional numerical methods. Furthermore, when compared to alternative numerical approaches, the current ANN model stands out for its applicability to more complex mathematical models, because its efficiency in minimizing both time and processing capacity demands in solving such problems.

Numerical simulation of radiative flow with magnetized micropolar nanofluid along a curved stretched surface involving blood silica nanoparticles

Silicon nanoparticles can be used as contrast agents in medical imaging procedures such as computed tomography (CT), magnetic resonance imaging (MRI), and fluorescence imaging. They can aid in the early identification and diagnosis of illnesses by enhancing image sensitivity and resolution through adjustable optical and magnetic properties. Additionally, the knowledge gathered from this research might help create cooling systems for electronics and thermal management applications that are more effective. The focus of current research is on the steady two-dimensional radiative flow of blood with silicon dioxide particles along a curved, stretched surface using a magnetized micropolar nanofluid. Furthermore, the effect of radiation is mentioned. Formulating a mathematical representation of flow equations requires the use of variables in curvilinear form. The leading PDEs are converted into nonlinear ODEs using the non-similarity approach. A bvp4c based on a three-stage Lobatto approach is used to solve the modified nonlinear ODEs. Numerical results for skin factor, couple stress, and local Nusselt number are presented in tabular form, and velocity, microrotation, and temperature fields are presented in graphs. The micro-rotation and temperature field exhibits a declining tendency with rising material parameters, whereas the velocity profile shows the opposite pattern. In addition, the significance of various physical characteristics under various parametric assumptions for a curved, stretched channel is reviewed and emphasized.

A two-phase thin-film model for cell-induced gel contraction incorporating osmotic effects

We present a mathematical model of an experiment in which cells are cultured within a gel, which in turn floats freely within a liquid nutrient medium. Traction forces exerted by the cells on the gel cause it to contract over time, giving a measure of the strength of these forces. Building upon our previous work (Reoch et al. in J Math Biol 84(5):31, 2022), we exploit the fact that the gels used frequently have a thin geometry to obtain a reduced model for the behaviour of a thin, two-dimensional cell-seeded gel. We find that steady-state solutions of the reduced model require the cell density and volume fraction of polymer in the gel to be spatially uniform, while the gel height may vary spatially. If we further assume that all three of these variables are initially spatially uniform, this continues for all time and the thin film model can be further reduced to solving a single, non-linear ODE for gel height as a function of time. The thin film model is further investigated for both spatially-uniform and varying initial conditions, using a combination of analytical techniques and numerical simulations. We show that a number of qualitatively different behaviours are possible, depending on the composition of the gel (i.e., the chemical potentials) and the strength of the cell traction forces. However, unlike in the earlier one-dimensional model, we do not observe cases where the gel oscillates between swelling and contraction. For the case of initially uniform cell and gel density, our model predicts that the relative change in the gels’ height and length are equal, which justifies an assumption previously used in the work of Stevenson et al. (Biophys J 99(1):19–28, 2010). Conversely, however, even for non-uniform initial conditions, we do not observe cases where the length of the gel changes whilst its height remains constant, which have been reported in another model of osmotic swelling by Trinschek et al. (AIMS Mater Sci 3(3):1138–1159, 2016; Phys Rev Lett 119:078003, 2017).

FEM study of blood gold electrically conducting micropolar nanofluid flow within a permeable channel

In this work, Hermite interpolation polynomials are utilized to develop a numerical model to analyze the influence of transport phenomena on blood gold electrically conducting micropolar nanofluid within a permeable channel with chemical reaction and thermal radiation. By considering blood as a micropolar fluid and applying Berman’s similarity transformation the resulting governing equations are transmuted to fourth-order nonlinear ODE which are solved utilizing FEM. The validity of the FEM code is established by comparing the present computations with existing work. Further, the influence of driving parameters are studied and graphically depicted. When utilizing a blood-gold micropolar nanofluid with a 1% volume fraction, it is observed that varying the Hartmann number from 0 to 15 leads to a decline in temperature profiles, resulting in a 2.51% reduction in heat transfer rate. Similarly, an increase in the chemical reaction parameter up to 10 causes species concentration profiles to decrease, leading to an 18.25% reduction in mass transfer rate compared to base fluid. Moreover, an increase in the volume fraction up to 5% results in an increase in coefficient of skin friction under the influence of the Hartmann number. These findings signify the critical role of Hartmann number in controlling blood velocity and chemical reaction parameter assists in lowering the species concentration levels. The results of the present investigation may help analyze flow models related to drug-delivery systems as gold nanoparticles are proficient drug-delivery and drug-carrying medium.

DJM solution of MHD flow of ternary hybrid nanofluid between nonparallel a porous media channels with velocity slip and radiation effects

This investigation on heat transfer and Jeffery Hamel ternary hybrid nanofluid flow across nonparallel a porous media channels while accounting for radiation, velocity slip, and mixture fluid ( H 2 O − C 2 H 6 O 2 ) effects is presented in this work. The nonlinear PDEs such as continuity, momentum and heat equations transformed to a system of nonlinear ODEs using similarity transformations then solved numerically and analytically, the analytical solution has been constructed using by Daftar Dar–Jafari method (DJM), the present results in particular cases are compared to results obtained by the HAM-based Mathematica package and by the Runge–Kutta Fehlberg fourth–fifth order (RKF-45) for validation. The effects of active parameters such as the nanofluid volume fraction, velocity slip parameter, Darcy number (Da), and thermal radiation parameter (Rd), are investigated on the velocity and temperature, skin friction, and Nusselt numbers. It is found that the velocity of ternary hybrid nanofluid upsurges for both convergent and divergent channel with the increment in the Hartman number (Ha). Also, results obtained reveal that the temperature profile experiences an upward shift with an increase in the Ha, and a downward shift with an increment of the Da. Furthermore, new modeling and the reliable analytic treatment via DJM approach for the ternary hybrid nanofluid is a sizable accomplishment of the current analysis.

ReMKiT1D - A framework for building reactive multi-fluid models of the tokamak scrape-off layer with coupled electron kinetics in 1D

In this manuscript we present the recently developed flexible framework for building both fluid and electron kinetic models of the tokamak Scrape-Off Layer in 1D - ReMKiT1D (Reactive Multi-fluid and Kinetic Transport in 1D). The framework can handle systems of non-linear ODEs, various 1D PDEs arising in fluid modelling, as well as PDEs arising from the treatment of the electron kinetic equation. As such, the framework allows for flexibility in fluid models of the Scrape-Off Layer while allowing the easy addition of kinetic electron effects. We focus on presenting both the high-level design decisions that allow for model flexibility, as well as the most important implementation aspects. A significant number of verification and performance tests are presented, as well as a step-by-step walkthrough of a simple example for setting up models using the Python interface. Program summaryProgram title: ReMKiT1DCPC Library link to program files:https://doi.org/10.17632/j47ym66xzj.1Developer's repository link:https://github.com/ukaea/ReMKiT1D and https://github.com/ukaea/ReMKiT1D-PythonLicensing provisions: GPLv3Programming language: Fortran, PythonSupplementary material:https://doi.org/10.14468/fdq7-z869Nature of problem: The flexible generation and modification of 1D models pertaining to multi-fluid simulations of the tokamak Scrape-Off Layer (SOL) with electron kinetics and reaction support. This would then allow both for rapid iteration on reduced models as well as the evaluation of kinetic electron effects in equilibria and during transients, following the formalism previously developed for SOL-KiT [1]. The framework was not only envisioned as the successor to SOL-KiT, but a tool that would allow users to construct their own models coupled with electron kinetics capabilities.Solution method: The framework is written heavily utilizing Object-Oriented design principles, in particular using an extended version of the puppeteer pattern as presented by Rouson et al. [2], as well as the heavy use of the strategy pattern/dependency injection. The Fortran code is MPI parallel and utilizes the PETSc library for implicit time-stepping. MPI parallelization is extended to distribution function Legendre harmonics, allowing for improved strong scaling. Initialization of the Fortran framework is done using JSON configuration files generated by an accompanying Python interface, and data analysis is standardized using widely used data formats such as HDF5.Additional comments including restrictions and unusual features: The present manuscript focuses on the design and high-level implementation of the framework, as well as the demonstration of the workflow and various verification and performance benchmarking tests. Some details are avoided for the sake of brevity at various points, and these are meant to be available as part of the general code documentation or tutorials offered on the main repositories.

Quasilinearization variational iteration method for system of nonlinear ODEs

In this manuscript, we discuss a new technique for solving system of nonlinear differential equations, which is a modification of the variation iteration method (VIM) implemented using the quasilinearization method and Adomian’s polynomial. The quasilinearization variational iteration method (QVIM) is the name given to this proposed method. The proposed method’s convergence analysis in Banach space is also discussed here. Three application problems, including the Genesio-Tesi system, are considered to test the applicability of our approach. We also discuss the case study of the chaotic and non-chaotic solutions of the Genesio-Tesi system (GTS). The convergence behaviour of the method is studied for various values of parameter x. To assess the viability and efficacy of QVIM, we compare it to the existing well-known Adomian decomposition method. The results show that the proposed method is highly efficient and simple to implement.

Investigation of Melting Heat Transfer in Viscous Nanofluid Flow Including Micro-Organisms and Entropy Generation Due to an Inclined Exponentially Stretching Sheet

This study investigates melting heat transfer and entropy production in viscous nanofluid flow consisting of micro-organisms over an inclined exponentially stretching permeable sheet. The flow is considered via porous medium. Impacts of heat transport characteristics are invoked in the energy equation. In concentration equation we have included chemical reaction impact. The regulating PDEs are transformed into nonlinear ODEs in non-dimensional form using adequate similarity transformation relations. The analytical solution of the problem is obtained utilizing HAM. Various plots are drawn to exhibit impacts of the regulating parameters (Prandtl number, Porous medium parameter, Thermal Grashof number, Mass Grashof number, Micro-organism Grashof number, Thermophoresis parameter, Radiation parameter, Bio-convection Levis number, Brownian motion parameter, Chemical reaction parameter, Suction parameter, Peclet number, and Melting parameter) occurred in the problem on relevant fields (flow, temperature and concentration distribution) and entropy production and discussed. Further values of significant physical quantities skin friction coefficient, Nusselt number, Sherwood number, and motile microbes density computed using MATLAB based bvp4c function and HAM are displayed in tabular mode and found in excellent agreement. For validity of the results skin friction coefficient and Nusselt number values are compared to prior research, apparently good agreement is found. The effect of melting surface parameter is found to reduce the fluid flow and temperature field. Entropy production lessens with rising values of slip parameters but effects of radiation and porous medium parameters are found to upsurge it. It is also noticed that bioconvection Lewis number and Peclet number reduce the micro-organism density profile. Inclusion of entropy analysis is a novel feature of the study. The solution methodology also enriched the novelty of the investigation. The results of the study may be applied to improve the efficiency of thermal, fluid flow and energy systems. This study may also find applications in bio-nano-coolant systems and heat transfer devices.

Magnetized heat transfer visualization through computational modeling of third-grade fluid via exponentially stretching cylinder

The steady MHD boundary-layer axis-symmetric flow of a third-grade fluid passing through an exponentially expanded cylinder in the vicinity of a magnetic field is investigated in this study. The problem is mathematically modeled. Suitable similarity transformations are carried out to convert the partial differential equations into nonlinear ordinary differential equations. The Runge–Kutta fourth-order shooting technique is used to solve the transmuted system of nonlinear ODEs. Graphical representations of numerical findings are used to examine the effects of various physical factors on the velocity and temperature profiles. The influence of fluid variables on the velocity curve, such as third-grade parameters, second-grade parameters, and Reynolds number, is illustrated and explored. The skin-friction coefficient expression is computed and given. The widths of the velocity and momentum boundary layers are revealed to be increasing functions of the curvature parameter. It is found that the third-grade fluid has a higher velocity profile than Newtonian and second-grade fluids. Also, the stretched cylinders cause a more progressive shift in heat and mass pattern for flow than flat plates do.

Analysis of resonant/nonresonant vibrations of simply-supported Kirchhoff nanoplates under in–plane magnetic field based on a strongly coupled two-mode model

The paper aims to study the nonlinear vibrations of simply supported square and rectangular nanoplates in the 2D magnetic field. The investigated model is developed in the framework of the von Kármán nonlinear theory, whereas nonlocal effects are taken into consideration due to the Eringen nonlocal theory of elasticity. PDEs governing system dynamics include the stress function. Both an in-plane magnetic field caused by the Lorentz force yielded by Maxwell's equations, as well as a transverse harmonical excitation are taken into account. The obtained results are based on the Bubnov–Galerkin approach and the two-mode deflection approximation. The latter reduces the problem of infinite dimensions to the system of coupled nonlinear ODEs, which is investigated by the multiple scale method (MSM). The employed MSM in the framework of Mathematica symbolic computations yielded the analytical approximate solutions, which were validated via the numerical Adams method. Both nonresonant and external/internal resonances exhibited by the studied nanoplates are thoroughly analysed. Numerous novel nonlinear phenomena are detected and analysed, with emphasis put on their physical interpretation.

Thermal Marangoni convection in two-phase quadratic convective flow of dusty MHD trihybrid nanofluid with non-linear heat source

The current study examines the effect of heat generation on thermal Marangoni convective boundary layer flow of dusty trihybrid nanofluid across a flat surface with thermal radiation and non-linear mixed convection. We consider, two phase dusty liquid model with non-linear heat generation. The fluctuation of surface tension gradients leads to the discovery of Marangoni convection. It can be used for growing crystals, drying silicon wafers, stabilising soap films, and wielding. The main objective of this study is to ascertain the trihybrid nanofluid thermal mobility. A trihybrid nanofluid consisting of magnesium oxide (Mgo), titanium oxide (TiO2), silver (Ag) and water as the base fluid is used. This model can improve the efficiency and dependability of thermal systems in a variety of applications by helping to optimize heat transfer processes in materials processing, electronics cooling, and the creation of cutting-edge cooling technologies in energy systems. The existing PDEs are converted into nonlinear ODEs via similarity variables. The nonlinear ODEs are then solved numerically using shooting technique (RKF-45th approach). When the Marangoni convection parameter rises, higher surface tension gradients lead to stronger induced flows and more efficient heat transfer inside the liquid. As the temperature profiles of the dust and fluid phases drop, the distribution of these characteristics in the liquid becomes more homogeneous.