Nonlinear Coupled Differential Equations Research Articles

Significance of Viscous Dissipation on Hydromagnetic Oscillatory Flow Affected by Nanoparticles Through a Boundary Layer Regime

In this article, the implications of viscous dissipation on buoyancy-induced heat and mass transfer of a nanofluid flow along a semi-infinite flat plate in an oscillating system is performed, taking into account the impacts of magnetohydrodynamics (MHD), chemical reaction, porous medium and heat sink. The consequences of various flow parameters on the velocity, temperature, and concentration distribution of the nanofluid, as well as the skin friction coefficient, Nusselt number, and Sherwood number has been solved semi-analytically. The basic governing equations have been determined by employing appropriate transformations techniques which result in high nonlinear coupled differential equations with physical conditions. Different types of nanoparticles are considered, and the salient features of all the embedded parameters on velocities, temperature and concentration fields have been displayed graphically and illustrated. It is concluded from the computational analysis that application of viscous dissipative parameter improves the temperature distributions, which consequently speed up the fluid velocity. This model has practical applications in the fields of biomedicine and resonance imaging. In addition, graphical comparison between Narayana and Venkateswarlu (2016) with the present work demonstrate an excellent relationship, thereby authenticating the precision and accuracy of this present work.

Non-similar analysis of micropolar magnetized nanofluid flow over a stretched surface

The study of micropolar nanofluids unveils intriguing applications, propelled by their exceptional heat transfer capabilities in comparison to conventional fluids. This investigation focuses on analyzing the behavior of magnetized micropolar nanofluid flow over a stretched surface, taking into account crucial factors such as viscous dissipation and heat source. The chosen base fluid is blood, with Copper [Formula: see text] nanoparticles serving as the selected material. Incorporating the single-phase (Tiwari-Das) model with boundary layer assumptions for micropolar nanofluid flow, we introduce the volume fraction of nanoparticles to assess heat transport. The governing system undergoes transformation into a set of dimensionless non-linear coupled differential equations through appropriate transformations. This transformation involves the utilization of a combination of the local non-similarity technique and bvp4c (MATLAB tool) to derive the system of nondimensional partial differential equations (PDEs) for micropolar nanofluid. Our systematic exploration delves into the consequences of nondimensional parameters on velocity, microrotation, and temperature profiles within the boundary layer, including the Eckert number, micropolar parameter, magnetic field parameter, heat source, Prandtl number, and microorganism parameter. Graphical representations vividly demonstrate that the velocity and temperature of micropolar nanofluid increase with the rise in material parameter values, while the microrotation profile decreases. Increasing the magnetic field parameter leads to a reduction in the velocity profile. Moreover, the micropolar temperature profile shows an increase with the rising Eckert number. Crucially, the research emphasizes that factors like the heat source and Eckert number play a role in decreasing the local Nusselt number. In contrast, an increase in the local Nusselt number is observed for material parameters. Furthermore, the skin friction coefficient decreases as micropolar parameter values increase, whereas an increase in the skin friction coefficient is noted for the magnetic field. The primary focus of this research lies in the development of suitable non-similar transformations for the investigated problem, aiming to yield authentic and efficient results. These results hold substantial promise to make meaningful contributions to future research on nanofluid flows.

Non-similar analysis of bio-convective micropolar nanofluid flow including gyrotactic microorganisms across a stretched geometry

This study explores the novel interaction of gyrotactic microorganisms within a two-dimensional bioconvective micropolar nanofluid flow along a stretched surface, employing the Tiwari-Das nanofluid model. Uniquely, blood is modeled as a micropolar fluid incorporating titanium dioxide ( Ti O 2 ) and silver ( Ag ) nanoparticles. The analysis incorporates influential factors such as viscous dissipation, magnetic field, and heat source effects. By converting the control system into dimensionless nonlinear coupled differential equations, a complete understanding of the system is attained through the application of the local non-similarity approach and bvp4c (MATLAB tool). Particularly, the study provides valuable insight into the influence of dimensionless control parameters on key profiles such as velocity, microrotation, temperature, concentration, and microbial profiles. The findings show that temperature, microrotation, and velocity profiles are all improved by raising the values of micropolar parameters. Furthermore, the microbe profile shows an inverse connection with the Lewis number and a positive association with the Peclet number, whereas the concentration profile declines with increasing Schmidt number values. Furthermore, the study provides a quantitative analysis through tables, revealing variations in the drag coefficient and Nusselt number. This innovative exploration sheds light on the unique behavior of micropolar bioconvective nanofluids, offering valuable contributions to the field of fluid dynamics and bioengineering.

Triple-diffusive magneto-nanofluid flow over a nonlinearly stretching sheet

Magnetic nanofluid constitutes a special class of nanofluids that exhibit both magnetic and fluid properties. Due to its extensive applications, researchers used the magneto-nanofluid to investigate the thermos-diffusion effects on boundary layer flows. The impact on the thermos-diffusion of magneto-nanofluid over a linear stretching sheet was first introduced by Awad et al., (boundary value problems, 1–13, 2013), then Goyal et al., (microfluidics and nano fluidics, 17(3), 591–604) examined the effects of Brownian motion, thermophoresis, and cross-diffusion on the enhancement of convection features of the power-law stretching sheet. In the present study, we extend Goyal’s finding of the thermophoresis effect by including a magnetic field. Consequently, now five partial differential equations with the magnetic field parameter need to be solved, which are converted to ordinary differential equations with similarity transformations. Then, the resulting nonlinear coupled differential equations are solved analytically using the method of Directly Defining the inverse Mapping with the help of Maple software. This is the first time this novel method is used to solve a system with four non-linear coupled differential equations. The analytical results of the study are compared with the available numerical results for a special case and found good agreement. This leads to the conclusion that this novel method is accurate and can be applied to solve not only higher order but also a system of several non-linear differential equations. Further, the skin friction is analyzed for various values of the magnetic parameter. It is observed that, the velocity profile decreases as the magnetic parameter increases. However, quite the opposite is true on the skin friction.

Bioconvection in a Porous Medium Saturated with a Casson Nanofluid

The bio-convective heat transfer of an incompressible, viscous, electrically conducting Casson nanofluid past a wedge has been analyzed. Furthermore, using a similarity variable, the governing flow equations are transformed to non-linear coupled differential equations corresponding to a two-point boundary value problem, which is solved numerically. A comparison of the solution technique is carried out with previous work and the results are found to be in good agreement. Numerical results for the coefficient of skin friction, Nusselt number, Sherwood number, micro-organism flux as well as the velocity, temperature, nanoparticles and micro-organisms concentration profiles are presented for different physical parameters. The analysis of the obtained results shows that the field of flow is significantly influenced by these parameters.

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

A novel numerical note on the enhanced thermal features of water-ethylene glycol mixture due to hybrid nanoparticles (MnZnFe2O4 – Ag) over a magnetized stretching surface

In this article, we presented a numerical solution of steady-state magnetohydrodynamic (MHD) flow of an incompressible, electrically conducting Casson hybrid nanofluid through a stretching surface under the incitement of viscous dissipation, heat generation and induced magnetism. A viscous mixture of water and ethylene glycol ( W : E G = 60 % : 40 % ) is considered as a base fluid. Enhanced thermal aspects of the working fluid due to suspended manganese zinc ferrite ( MnZnFe 2 O 4 ) and silver ( Ag ) nanoparticles are analyzed by successive over relaxation iterative method. A self-similar procedure is adopted to obtain dimensionless system of governing equations. Order reduction and finite differences discretization of nonlinear coupled differential equations lead to the approximate solution of proposed problem. Impacts of various dominant parameters on the hybridized Casson fluid are examined through the graphs sketched for the functions of velocity, temperature, and induced magnetism. Skin friction coefficient and thermal transfer characteristics at the stretching surface have been simultaneously observed via tabular data prepared for pure nanofluid ( Ag-WEG ) as well as for hybrid nanofluid ( MnZnFe 2 O 4 -Ag/WEG ). It is perceived that induced magnetic field tends to decline the Nusselt number while exhibits a significant growth in the surface drag. Moreover, a significant enhancement in temperature of working fluid has been observed due to Eckert number and heat generation parameter. This decisive role of enhanced thermal features will help to improve industrial growth, waste-water treatment, and automobiles engine efficiency.

Mixed convection of two layers with radiative electro-magnetohydrodynamics nanofluid flow in vertical enclosure

Mixed convection flow of two layers nanofluid in a vertical enclosure is studied. The channel consists of two regions. Region I is electrically conducting while Region II is electrically non-conducting. Region I is filled with base fluid water with copper oxides nanoparticles and Region II is filled with base fluid kerosene oil with iron oxides. The simultaneous effects of electro-magnetohydrodynamics and Grashof number are also taken into account. The governing flow problem consists of nonlinear coupled differential equations which is tackled using analytical technique. Analytical results have been obtained by the homotopy analysis method (HAM). The results for the leading parameters, such as the Hartmann numbers, Grashof numbers, ratio of viscosities, width ratio, volume fraction of nanoparticles, and the ratio of thermal conductivities for three different electric field scenarios under heat generation/absorption were examined. It is found that the effect of the negative electric load parameter assists the flow while the effect of the positive electric load parameter opposes the flow as compared to the case when the electric load parameter is zero. All outcomes for significant parameters on velocity and temperature are discussed graphically.

Thermal analysis of MHD casson-sutterby fluid flow over exponential stretching curved sheet

The two-dimensional flow of Sutterby-Casson fluid flow over an exponentially curved sheet is considered. Incompressible and steady flow is taken into account. The magnetic and electric fields are ignored due to the small Reynolds number in the flow assumptions. Magnetic and chemical dipoles are considered at stretchable curved sheet. Heat generation and slip phenomena have been considered in the present analysis. The governing models of the above assumptions are reduced by implementing the boundary layer approximations and gained nonlinear coupled differential equations. The couple differential equation system is reduced in a dimensionless system (ordinary differential equations) using appropriate transformations. The dimensionless system is solved through a numerical scheme due to finding the results involving physical factors. The results are highlighted in graphs and as well as tables. Concentration achieved higher values by enlargement in the Brownian factor. The movement of particles becomes faster due to the enlargement of the Brownian factor, ultimately increment in concentration. Energy activation factor values enlarged which gained lesser values of Nusselt factor and mass transfer factor while friction factor remains constant. The reaction of chemical factor impact on mass transfer, friction factor, and heat transfer have been debated.

Exploring the dynamics of active swimmers microorganisms with electromagnetically conducting stretching through endothermic heat generation/assimilation flow: Observational and computational study

Enhancements brought about by motile gyrotactic microorganisms encompass improved mixing, efficient oxygen and nutrient transfer, environmental sensing, and reactions, as well as potential applications in bioremediation, contributing to the advancement of knowledge in fluid dynamics and biological locomotion. These microorganisms find utility in various fields such as biotechnology, environmental engineering, and fluid dynamics research, among others, playing a pivotal role in numerous ecological processes. The present study places its focus on the investigation of the effects associated with mass and heat transport in a fluid flow scenario featuring the presence of a heat source or sink. Specifically, the study aims to examine the impact of nanoparticles on Powell-Eyring fluid flow in the presence of a magnetic field (MHD) across a stretched sheet, taking into account factors such as activation energy, heat sources, and thermal radiation. Furthermore, the study explores the influence of motile gyrotactic microorganisms on various parameters. Recognizing the dearth of research in the realm of polymer extrusion processes, this study strives to bridge this research gap. Its primary objective is to develop a mathematical formulation employing a boundary layer approach, which encompasses the consideration of the interrelated effects of mass and heat transport in Eyring-Powell fluid flow across a stretched sheet in the presence of thermal radiation and chemical reactions from figure it shows that temperature profile rises as result of radiation parameter's increasing values (0.1<Rd<0.4). The impact of Lb is also showed with the help of graph. Greater diffusive mixing is implied by higher Lb for (0.1<Lb<0.4), which tends to distribute the microbes more equally throughout the fluid. This research study serves as an inspiration for the innovation of endogenous heat generation/consumption within the flow of motile organisms exhibiting gyro-taxis in the presence of magnetic flux density on a stretching sheet. To address the complexity of the problem, the renowned BVP4C package, a computational software tool in MATLAB, is utilized for solving a set of highly intricate and nonlinear coupled differential equations. The nonlinear nature of these equations poses significant challenges, necessitating the application of specialized numerical methods to ensure accurate and efficient solutions.

Computational modeling of Prandtl‐nanofluid flow using exponentially vertical surface in terms of chemical reaction

AbstractCurrent study examined the magnetohydrodynamic (MHD) Prandtl nanofluid of a thermal double‐diffusive flow through an exponentially vertical surface in association with heat generation, and thermophoresis effect. The novelty of this study is due to the analysis of Prandtl nanofluid model with Soret mechanism and chemically responding fluids. This suggested model is beneficial since it can significantly advance the domains of thermal and industrial engineering. The fluid flow phenomenon is characterized by nonlinear coupled differential equations involving two or more independent variables. A suitable numerical technique is used to handle the set of governing equations along with a stability and convergence analysis. According to recent study, the fluid velocity increases since all the parameters are set to higher levels. For the various parametric values, isotherms and streamlines have been explored. This suggested model is beneficial since it can significantly advance the domains of thermal and industrial engineering. For instance, thermal radiation is crucial in designing sophisticated energy‐transformed systems that operate at high temperatures. On the other hand, the phenomenon of Soret is useful in separating isotopes in chemical engineering. An important findings of the current investigations can be treated as, radiative heat encourages fluid temperature distribution since it is the measure of the electromagnetic element radiates from the fluid particle that convert it into thermal radiation. These studies have several applications in the manufacturing and biomedical fields, petrochemical industries, automobiles, medical sciences, and various production processes in industries.

Numerical simulation for peristaltic transport of radiative and dissipative MHD Prandtl nanofluid through the vertical asymmetric channel in the presence of double diffusion convection

This work's primary goal is to give a novel thorough investigation that examines the effects of the temperature- and concentration- dependent electrical conductivity on the peristaltic flow through the vertical asymmetric rotating channel filed with magnetic Prandtl nanofluid saturated in a porous medium. In addition to joule heating, viscous dissipation, and nonlinear thermal radiation, the fundamental characteristics of Brownian motion and thermophoresis processes are also considered when modelling the heat equation. Further, the influences of the double diffusion convection are examined, and the cases of biological scientific assumptions which is known as low Reynolds number and long wavelength are applied. The software Mathematica's built-in command (ND Solve function) is used to numerically solve the consequent nonlinear coupled differential equations system. The main findings showed that as the Soret and Dufour numbers increased, consequently increased the heat resistance, whereas the dissolvent concentrations and nanoparticle volume percentage had the reverse impact.

Investigation of radiative MHD Carreau nanofluid over an inclined stretching cylinder with activating energy, Wu’s slip, and convective heating

The influence of radiation, convective heating, second-order velocity slip condition, and activating energy on the magnetohydrodynamic (MHD) Carreau nanofluid and heat/mass transfer are depicted in this study. A mathematical model of the Carreau nanofluid is carried out with the effects of thermophoresis and Brownian motion. The governing equations are transformed into a system of nonlinear coupled differential equations. Solutions of the transformed equations are derived by the Matlab BVP4c solver. Efficiency of the solver is verified by the comparison with the previous literature. Moreover, the impacts of the dimensionless velocity, temperature, and nanoparticle concentration profile are elucidated graphically.

Matricials methods applied to the solution of the Ginzburg-Landau equations

In this contribution we propose two numerical methods for the solution of the system of two non-linear coupled differential Ginzburg-Landau equations. These proposals are based firstly on taking a matrix view considering the quasi-linear coupled system, as a second option, considering the computational molecule with its respective restriction of the values and eigen-vectors of the matrix. We clearly and concisely obtained the eigenvalues that lead us to an optimal spatio-temporal convergence solution of said system of equations. We compare the numerical convergence times obtained by these three methods with the known time found by applying the link variable method.

Hydromagnetic flow of two immiscible nanofluids under the combined effects of Ohmic and viscous dissipation between two parallel moving plates

The hydromagnetic flow of immiscible nanofluids between two parallel plates is investigated. Two regions, namely, (i) Region 1 is filled with base fluid kerosene oil with silver Ag nanoparticles, while (ii) Region 2 is filled with base fluid water with titanium dioxide TiO2 nanoparticles. Analytic solutions of nonlinear coupled differential equations have been obtained. The velocity and temperature profile results are discovered to flatten out when the magnetic field's inclination angle decreases. Impact of key parameters like inclination of the magnetic field, Hartmann number and load parameter with thermal conductivity ration have been illustrated graphically.

Sisko fluid modeling and numerical convective heat transport analysis over-stretching device with radiation and heat dissipation

We examine the time-independent boundary layer flow (BLF) and heat transmission of Sisko fluid subject to convective boundary conditions. The fluid flow is impacted through a nonlinear stretchable sheet in the existence of a magnetic field and porous medium. Thermal conductivity effect assumed on heat transfer. We also consider the impact of Joule heating and radiation in the present study. The model’s leading partial differential equations (PDEs) have reduced similarity transformation into nonlinear ordinary differential equations (ODEs). The resultant nonlinear coupled differential equations are tackled in MATLAB built-in solver (bvp4c). The repercussions of relevant parameters like material parameter ( A ) , the combined effect of the magnetic and porous medium parameter ( C ) , Sisko fluid power law index ( n ) , stretching parameter ( r ) , Prandtl number ( Pr ) , Joule heating parameter ( J ) , temperature parameter ( λ ) and radiation parameter ( N R ) on the temperature, velocity, skin friction coefficient, and Nusselt number are scrutinized via graphs. The results show that higher inputs of material parameter have a growing impact on velocity and a decreasing effect on temperature distribution. Increasing Sisko fluid has a reducing impact on velocity distribution. As witnessed from sketches, the shear stress distribution is decreased to increase the impression of the Sisko fluid, material, stretching, and combined parameters. Our computed outcomes are linked to existing outcomes and get the finest contract. The results benefit from comprehending the flow characteristics, flow behavior, and how to foresee it for individuals involved in designing high-temperature machinery in the industry.

The exploration of thin liquid film flow under the influences of various physical parameters by a new analytical innovation

AbstractA new analytical innovation is utilized to explore the thin liquid film flow of magnetohydrodynamic (MHD) fluid over an unsteady porous stretching surface. Here magnetic field, thermal radiation, and variable viscosity are taken. The self‐similarity variables have been used to transform the modeled partial differential equations into a set of non‐linear coupled differential equations. These non‐linear differential equations for the velocity and temperature profiles have been tackled through an innovation containing homotopy, auxiliary functions, and convergence control parameters. This method is called the second configuration of the Optimal Homotopy Asymptotic Method and is denoted by (OHAM‐2). Galerkin's process is chosen for the optimizations of parameters. The achieved coupled equations are also solved numerically (ND‐Solve Method), and the obtained outcomes are compared with the results elaborated by the suggested algorithm. Here the utmost attention is on the rapid convergence of the proposed algorithm and the consequences of various tangible variables on the velocity and temperature profiles. This straightforward algorithm consists of a few steps but gives better outcomes. The technique can be applied to solve partial differential equations and their systems forming in various disciplines. This technique can also solve Integro differential equations. The residuals achieved by the proposed method for the velocity and temperature fields are shown graphically. The errors and residuals are also taught in Table 1 and Table 2. The effects of parameters are inculcated in Table 3 and Table 4. The results are validated with the published work as shown in Table 5. Nomenclature is indicated in the text below.

The role of thermal noise on key parameters governing the improvement of tokamak fusion reactor efficiency using aneutronic fuel

This article examines the time evolution of energy, particle density, synchrotron radiation loss, plasma frequency, electron particle density, and the relationship between these parameters and blackbody radiation in the presence of thermal noise for the tokamak fusion reactor with aneutronic fuel. To achieve this goal, a suitable thermal noise term is added to the set of energy and particle balance equations. These nonlinear point kinetic coupled differential equations are simultaneously solved by numerical methods. The results of our calculations related to fusion power, particle density, fusion energy density, and quality factor in cases: (i) without and (ii) with noise show that in the conditions where thermal noise is considered, the mentioned parameters and quality factor are increased and in fact system performance improves. Also, our numerical results related to synchrotron radiation loss, plasma frequency, electron particle density, and energy intensity in cases (i) and (ii) show that in the conditions when blackbody radiation occurs (3.75 and 7.225 s respectively for (i) and (ii) cases), synchrotron radiation reaches the maximum values of and for (i) and (ii) cases, respectively.

Application of the successive over relaxation method for analyzing the dusty flow over a surface subject to convective boundary condition

Heat transport features of a dusty fluid flow within a resistive porous medium are numerically interpreted under the magnetohydrodynamic environment. Fluid and dust phase characteristics are overlooked against the preeminent parameters. The main concern, in this work, is to analyze how much the resistive porous media affects the dusty flow in the presence of Lorentz force. The use of convection boundary conditions will aid attain the desired heat transfer rate in engineering applications. The nonlinear partial differential equations governing the fluid and dust phase are transmuted into a system of coupled and highly nonlinear ordinary differential equations which are first discretized using central differences, and then tackled numerically using the successive over-relaxation (SOR) method. This technique creates an effective way to solve complex dynamical problems comprising highly nonlinear-coupled differential equations. The flow features of the concerned problem are explored for different parameters such as the magnetic field, the porosity of the medium, the relative density of the two phases, ratio of free stream velocity, specific heat capacity ratio of the fluid to the dust phase as well as Prandtl number. Our results may be helpful to understand the dusty airflow during storms, mud particles in canals, ponds, and rivers.

Magnetic fluid flow and heat transfer due to a uniform source and vorticity

Abstract The present work investigates the ferrohydrodynamic flow and heat transfer due to a uniform source and irrotational vortex under the influence of a stationary magnetic field. A uniform source generates only a two-dimensional flow. However, in the presence of the vorticity with a uniform source, the flow becomes three-dimensional. The governing equations are expressed as a system of nonlinear coupled differential equations. The transformed differential equations are solved using the finite element approach for both the two-dimensional and three-dimensional flow models. With variations in the strength of the source parameter, Reynolds number, and ferromagnetic interaction numbers, the behavior of two-dimensional and three-dimensional flow is investigated. In three-dimensional flow, the influence of swirling effects on the velocity and temperature profiles are weak as compared to two-dimensional flow. The main role of the three-dimensional vortex flow of ferrofluid is to generate rotational viscosity, and it is not possible in the case of two-dimensional flow case.