Set Of ODEs Research Articles

Sodium alginate-based unsteady nanofluid flow over a porous stretching surface with endothermic and exothermic reaction and bioconvection effects

The impact of nanofluid (NF) circulation under gyrotactic microorganisms and endo/exothermic chemical reactions across a permeable medium is the primary emphasis of this investigation. Sodium Alginate and Silicon dioxide are the base fluid and nanoparticle utilised here. The governing equations are reduced to a set of ODEs by employing the appropriate similarity variables. To get the numerical results, the shooting method and Runge–Kutta Fehlberg's fourth fifth order (RKF-45) are implemented. The numerical outcomes are conveyed graphically across numerous parameters. In addition, the engineering factors are analyzed and explained. It was found that the velocity profile decreased with the enhanced Casson and porous parameter values. An increment in temperature profile is experienced during exothermic reaction and is reduced for endothermic reaction and also the concentration profile got accelerated as the impact of chemical reaction parameter is upsurged. There is a significant growth in surface drag force from NF to base liquid of about 2%, rate of thermal and mass distribution of about 0.1% in context to various constraints.

Applications of partial differential equations to transient heat and mass transfer in MHD flow over a porous medium

The flow problem discussed in this research is an investigation of compressible viscous fluid in an unstable 2D MHD laminar driven vortex and stable boundary layer flow, passing a heated stretched sheet at varying speeds via a porous medium exposed to a magnetic field with viscous dissipation, thermal diffusion, thermal radiation, chemical reaction and Dufour effects. With the aid of similarity transformations, PDEs that come out in governing equations of this formulations are converted into non-linear sets of ODEs. Well-defined bvp4c technique used to numerically solve these converted equations. With use of various graphs, the consequences of different factors on the distributions of velocity, temperature and concentration were examined numerically and graphically. Physical parameters for this issue are also discussed.

A Double Legendre Polynomial Order N Benchmark Solution for the 1D Monoenergetic Neutron Transport Equation in Plane Geometry

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solution, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively.

Mathematical analysis of heat and mass transfer efficiency of bioconvective Casson nanofluid flow through conical gap among the rotating surfaces under the influences of thermal radiation and activation energy

In the current proceeding, the flow of incompressible non-Newtonian nanofluid called Casson nanofluid is considered. A conical gap occurred among the rotating disc and the cone filled with the fluid flow. Heat and mass transport through this nanofluid is done by the convection mode of heat transfer. The impacts of microorganisms, chemical processes, thermal radiation, minimal amount of energy, and magnetic field are also considered in the mathematical model of flow problem. The Casson nano-fluid governing equations are interpreted in cylindrical coordinates. By implementing proper similarity transformations, the modeling PDEs of energy, momentum, concentration, and microorganism density are transformed into non-linear ODEs. A set of non-linear ODEs that deals with the distributions of temperature, velocity, concentration, and motile microorganisms produced by this technique. MATLAB in-built ‘bvp4c‘ technique is utilized to solve these equations. Findings are displayed graphically and elaborated theoretically. The primary goal of this work is to examine the effects during the rotation of the disc and cone as well as the impacts of other variables on the rotation. The nano-fluid temperature and radial velocity are found to be negatively impacted by the rotation parameter whereas azimuthal velocity is positively impacted. The parametric values are taken as 0.1<Nb<0.7, 0.1<λ<0.4, 1.0<Pr<2.50.1<M<0.4, 1.0<Rb<4.0, 0.1<Nr<0.4, 0.1<Sc<3.0, 0.1<Nt<0.4 for the purpose of generating modified results. From published and current results noted that rotation parameter effects impacted the transfer rate of the nanofluid.

Modal Discontinuous Galerkin Simulations of Richtmyer–Meshkov Instability at Backward-Triangular Bubbles: Insights and Analysis

This paper investigates the dynamics of Richtmyer–Meshkov instability (RMI) in shocked backward-triangular bubbles through numerical simulations. Two distinct gases, He and SF6, are used within the backward-triangular bubble, surrounded by N2 gas. Simulations are conducted at two distinct strengths of incident shock wave, including Ms=1.25 and 1.50. A third-order modal discontinuous Galerkin (DG) scheme is applied to simulate a physical conservation laws of two-component gas flows in compressible inviscid framework. Hierarchical Legendre modal polynomials are employed for spatial discretization in the DG platform. This scheme reduces the conservation laws into a semi-discrete set of ODEs in time, which is then solved using an explicit 3rd-order SSP Runge–Kutta scheme. The results reveal significant effects of bubble density and Mach numbers on the growth of RMI in the shocked backward-triangular bubble, a phenomenon not previously reported. These effects greatly influence flow patterns, leading to intricate wave formations, shock focusing, jet generation, and interface distortion. Additionally, a detailed analysis elucidates the mechanisms driving vorticity formation during the interaction process. The study also thoroughly examines these effects on the flow fields based on various integral quantities and interface characteristics.

Heat and mass transfer analysis for bi-dimensional bioconvective MHD nanofluid with varying thermal traits

ABSTRACT Bioconvection phenomena offer practical applications in biomicrosystems, fuel cells, biosensors, biocomputing, electronic thermal management, and renewable energy systems, showcasing their relevance and potential impact across various engineering sectors. Consequently, this study delves into the examination of a bi-dimensional magnetohydrodynamic nanofluid flow model encompassing the intricate dynamics of gyrotactic microorganisms, subject to the effects of suction and thermal radiation. The primary objective of the investigation is to optimize the thermal energy transfer efficiency of the fluid while concurrently minimizing associated costs, with the secondary aim of diminishing the frictional resistance at the fluid-solid interface. The basis for energy and momentum equations incorporating radiation using Rosseland’s approximation is established by the Buongiorno nanofluid model. The foundational PDEs along with their BCs, are reformulated into ODEs utilizing similarity variables, then transformed into a set of first-order ODEs ensuring compatibility with MATLAB’s bvp4c solver, which offers high convergence rates and accuracy due to its efficient algorithms and precision handling. Heat transfer rate boosts up by increasing suction and radiation parameters, while it dwindles due to an increase in the magnetic field. Enhancing suction and the magnetic field lowers skin friction. Microbial concentration surges by increasing the magnetic field, while suction minimizes it. Intensifying the thermophoresis parameter increases the concentration of nanoparticles.

Numerical simulation of Buongiorno's model on Maxwell nanofluid with heat and mass transfer using Arrhenius energy: a thermal engineering implementation

The thermal features of nanoparticles owing to progressive mechanisms are a fascinating phenomenon due to their applications in energy production, cooling procedures, heat transmission devices. Therefore, in the present study, the magnetohydrodynamic combined convection of Maxwell nanofluid and characteristics of heat transport in the presence of thermal radiation with a nonlinear relationship for modifications in the energy equation have been examined. Moreover, the features of activation energy in the presence of swimming microorganisms are considered. For motivation, the influence of bioconvection, magnetic field, and thermophoresis with convective boundary conditions are part of this investigation. The governing PDEs connected with momentum, energy, concentration, and density are converted into ODEs by using similarity functions. A transformed, dimensionless, nonlinear set of ODEs is tracked via a shooting scheme. The numerical results of prominent parameters have been analyzed in the form of graphs and tables using the computational software MATLAB. A significance improvement in the velocity profile is noted for the increasing value of Maxwell parameter. With rise of mixed convection parameter, both energy and volumetric friction field deteriorated. The determination of Biot number that is associated with the coefficient of heat transfer is more effective for growing the temperature and volumetric friction distribution. These conclusions may be appreciated in improving the efficiency of heat transfer strategies.

Bejan number and entropy generation analysis of unsteady MHD non-Newtonian micropolar squeezed trihybrid nanofluid flow

Microscopic particles have incredible thermal conductivity making them essential in nanotechnology, electronics, materials science and heat exchangers. For suitable optimization of thermic engineering structure, thermal transportation has to increased but minimized the entropy generation. One of the basic reasons for the developing interest in tri hybrid nanofluids is its unique ability to increase thermal characteristics, that is very useful in numerous heat exchangers. Three distinct nanosized particles, namely silver, gold and diamond are considered with water as a base liquid. A set of dimensionless ODE’s are solved numerically by Bvp4c technique. The results of prominent parameters are elaborated and graphs are presented for velocity profile, entropy profile, Bejan and temperature profiles. Moreover, present result is compared with the literature and found a good agreement. It is observed that the Bejan number and entropy generation shows the opposite behavior for higher values of Casson and magnetic parameter. Further, it is noted that at the lower disk, squeezed Reynold number increases the skin friction coefficient and opposite trend noted against the upper disk. Temperature profile shows increasing trend for higher values of Eckert number. It is also noted that both velocities, temperature, entropy and Bejan number profiles for tri-hybrid nanofluid flow attains maximum values as compared to bi-hybrid and mono nanoparticles.

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.

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.

Significance of heat generation and thermophoretic particle deposition in Marangoni convective driven boundary layer flow of cross nanofluid with activation energy

The thermo-solutal Marangoni boundary layer flow has been studied because it has a number of real-world uses, such as drying silicon wafers, applying thin coats of paint or glue, employing adhesive in heat exchangers, and growing crystals in space. The physical phenomenon of Cross (FeSO4−(CMC−H2O(<0.4%)) nanofluid flow in a porous medium with thermophoretic particle deposition and Marangoni convective flow is explored. Non-Newtonian Cross nanofluid flow is explored in relation to the effect of activation energy. The process of thermophoretic particle deposition, which is important in both electrical engineering and aero-solution, is one of the most fundamental methods for transferring small particles across a heat gradient. Cross nanofluid containing FeSO4 as nanoparticle, and based fluid is (CMC- water). Using the appropriate similarity variables, the set of governing PDEs is transmuted into a set of ODEs. The RKF-45th method is used to numerically solve these simplified equations. The graphic examination of the concentration, velocity and thermal fields is developed for the embedded non-dimensional characteristics. By raising the Marangoni ratio parameter, the surface tension gradient gets stronger, causing the induced flows to get stronger through the liquid more effectively. The concentration and thermal profiles fall with an enhancement in Marangoni ratio parameter.

Optimizing entropy in mixed convective MHD dissipative nanofluid with cross-diffusion and nonlinear velocity slip

Entropy optimization is crucial for enhancing fluid dynamics and heat transfer, leading to improved efficiency, energy conservation, and environmental sustainability. Recognizing its significance, we examine the influence of radiation, Joule heating, heat sources, mixed convection, and nonlinear slip on the entropy-optimized flow of a dissipative nanofluid over a curved surface. To seamlessly integrate within the MATLAB bvp4c framework, we convert the set of multi-order ODEs, derived from fundamental PDEs via a similarity transformation approach, into a system of first-order ODEs. Graphs and analytical discussions are used to evaluate the parameters of the problem. The velocity within the boundary layer, along with entropy, experiences reduction due to the enhancement in both first- and second-order slip velocity parameters, as well as curvature parameters. The temperature plot improves with the increase in parameters such as the Soret number and Dufour number, as well as radiation. Conversely, the temperature plot reduces with the rise in suction values. The study indicates that a higher Brinkman number is associated with a lower Bejan number, while the first- and second-order velocity slip parameters have a contrasting effect, leading to an increase in the Bejan number.

Design of Bayesian stochastic networks for numerical treatment of Williamson fluid stretching flow model with mixed convected heat generation

In the presented article, a stochastic network paradigm through Bayesian Regularization backpropagation neural network (BRB-NN) is designed to interpret the dynamics of the Williamson fluid stretching flow model with mixed convected heat generation (WF-SFM). The governing nonlinear PDEs system of WF-SFM is reduced to a set of ODEs by incorporating appropriate transformations. The reference datasets for the anticipated BRB-NN approach are created with Adams solver for numerical solutions of WF-SFM by varying material variable W e , buoyancy factor λ , temperature characteristic parameter ε , thermal relaxation variable γ , heat source factor δ , and Prandtl numbers P r . The knacks of artificial intelligence (AI) based BRB-NN procedure is then employed on the generated dataset for WF-SFM. The bias training, biased testing, and validation of BRB-NN are conducted to compute the approximate numerical results for WF-SFM for sundry scenarios, and outputs are in good agreement in reference data that validate the worthy performance of the proposed BRB-NN which is further justified by results of absolute error, mean squared error, error histogram illustrations and regression measures. The viable performance in terms of mean square error (MSE) is achieved at levels ranging from E − 11 to E − 13 , consistently for all scenarios of WF-SFM. The accuracy and the justification of performance are effectively proven by the low level of MSE, to optimal regression metric index as well as distribution of instances error on histograms with negligible magnitude.

Velocity and Thermal Slips Impact on the Williamson Fluid Flow above a Stretching Sheet in the Existence of Radiation and Inclined Magnetic Field

Research has been conducted on the study of the velocity and thermal slips' impact on the Williamson fluid flow above a stretching sheet in the existence of an inclined magnetic field and Radiation. By applying the proper similarity conversions, the governing equations (PDEs) are reduced to a set of non-linear ODEs, and a numerical solution is produced by using MATLAB in-built solver bvp4c package. The impacts of the dimensionless characteristics on the flow patterns are analyzed visually, and the values of the friction, Nusselt, and mass transfer quantities are tabulated to exemplify how the various physical factors have an influence. We noted that the velocity profile enhances the rising estimations of velocity slip and the temperature profile increases with Q increase.

Mechanisms of Diffusion thermo and Thermal diffusion on MHD Mixed Convection Flow of Casson fluid over a vertical cone with porous material in the presence of thermophoresis and a Brownian motion

In this present article, we analyzed the Effects of Diffusion thermo and Thermal Diffusion on magnetohydrodynamic (MHD) mixed convection flow for Casson nanofluid is deliberated a vertical cone with porous material. The modeled equations are transformed into a set of non-linear ODEs by employing similar transformable variables. These equations are then solved numerically using the shooting method, through the fourth-order Runge–Kutta integration procedure. Effects of some prominent physical parameters, such as diffusion thermo, Prandtl number, thermophoresis parameter, and magnetic parameter on the velocity, temperature, and concentration profiles are discussed graphically and numerically. Numerical calculations and graphs are used to illustrate the important features of the solution on fluid flow velocity, heat, and mass transfer characteristics under different quantities of parametric circumstances entering into the problem. Moreover, we computed the physical variables such as the coefficient of shear stress, rate of heat, and mass transfer. To establish the veracity of our present results, we compared them to previously published research and found substantial concordance.

Impact of Magnetohydrodynamics on Hyperbolic and Walters-B Non-Newtonian Fluids

The current study examines the impact of two non-Newtonian fluids hyperbolic tangent and Walters B fluids transferring heat and mass through an upward-facing, semi-infinite porous plate into the boundary layer. PDEs are used to determine how the flow analysis works. Using the proper similarity functions, the collection of PDEs is condensed into a set of dimensionless ODEs. The spectrum relaxation method (SRM) was used to solve the simplified equations. By initially decoupling the system of coupled equations, SRM solves differential equations repeatedly. The simultaneous flow of two non-Newtonian fluids with flow characteristics including radiation, Soret, viscous dissipation, Joule heating, magnetic field, etc. is elucidated in these communications, which are regarded as being one of a kind. The electrically conductive liquids' flow direction is subjected to a uniformly strong magnetic field.

A Kinetic Finite Volume Discretization of the Multidimensional PIDE Model for Gene Regulatory Networks

In this paper, a finite volume discretization scheme for partial integro-differential equations (PIDEs) describing the temporal evolution of protein distribution in gene regulatory networks is proposed. It is shown that the obtained set of ODEs can be formally represented as a compartmental kinetic system with a strongly connected reaction graph. This allows the application of the theory of nonnegative and compartmental systems for the qualitative analysis of the approximating dynamics. In this framework, it is straightforward to show the existence, uniqueness and stability of equilibria. Moreover, the computation of the stationary probability distribution can be traced back to the solution of linear equations. The discretization scheme is presented for one and multiple dimensional models separately. Illustrative computational examples show the precision of the approach, and good agreement with previous results in the literature.

Numerical and optimal thermal analysis of non-Newtonian fluids flow over a wedge by using response surface methodology (RSM): Sensitivity analysis

The numerical and optimal thermal analysis of Casson fluid flow over a wedge is investigated in this work. To perform numerical and sensitivity analysis we have convert third order non-linear PDEs into set of ODEs by using similarity transformations along with boundary conditions. These equations are simulated using numerical technique Keller Box method and then compared the values of skin friction coefficient and Nusselt number with those existing in literature and found excellent agreement. Then by using these numerical data we have performed sensitivity analysis by using Response Surface Methodology (RSM). ANOVA tables are generated and developed a correlations between input parameters and output responses. The coefficient of determination for output responses skinfriction coefficient and Nusselt number are 96.50% and 99.51% respectively which shows the best fitted correlations. Finally, we have performed sensitivity analysis which reflects that skinfriction coefficient is more sensitive to Falkner–Skan exponent (m) and there is no effect of Prandtl number (Pr) on skinfriction coefficient. It is also concluded that Nusselt number is sensitive to Prandtl number (Pr) and obtains its optimum value with variation in Prandtl number (Pr).

Numerical solution of entropy generation in nanofluid flow through a surface with thermal radiation applications

The main purpose of this research is to investigate a mathematical model of nanofluid passing through a stretching sheet having titanium dioxide (TiO2), copper oxide (CuO), and carbon nanotubes (CNTs) as nanoparticles, and sodium alginate is used as a base fluid. The present model can be very useful for numerous fields like Bioengineering, science, and technology. The main reason for using titanium dioxide, copper oxide, and carbon nanotubes nanoparticles as nanomaterials they have many applications in several areas. Moreover, impacts of non-linear thermal radiation and entropy generations are taken into account. By utilizing the similarity transformation technique given system of governing PDEs are transformed into a set of ODE. Then dimensionless equations are numerically tackled via the Keller box method a built-in function in the computer tool MATLAB. To support the present study, physical and numerical interpretations of involved parameters against velocity, temperature, and entropy generation profiles are also presented. From the analysis it is concluded that the combination of a magnetic field and suction reduces the rate of fluid mobility due to the synchronization of the magnetic and electric fields generated by the formation of the Lorentz force. In addition, boosting the thermal radiation parameter raises the present authors' heat transfer rate. The velocity profile drops when the Casson parameter is raised, but the heat and entropy production profiles grow. When the porosity parameter is increased, the velocity profile falls, and the heat and entropy production profiles rise. Additionally the temperature distribution is boosted up with increasing the values of nanoparticles volume fraction. The good agreement between current results and published data is observed. It is necessary to investigate the thermophysical characteristics of the magnetite nanofluid (titanium dioxide, copper oxide, and carbon nanotubes-sodium alginate nanofluid) under various situations. The MHD flow of nanofluid is significant because of its wide range of industrial uses, including flows over the tips of rockets, planes, submarines, and oil tankers. The novelty of this work is to scrutinize the MHD flow and heat transfer analysis of a nanofluid towards a stretched surface. The features of magnetite nanofluid (titanium dioxide, copper oxide, and carbon nanotubes-sodium alginate nanofluid) under entropy generation, thermal radiation and velocity slip are investigated. All of this contributes to the work's uniqueness and novel.

The Infinite State Representation of Fractional Order Differential Systems: A Survey - Part 2

Since the first fractional integration operator dedicated to simulation 25 years ago, the Infinite State representation has evolved into a general methodology for the analysis of fractional order system dynamics. Based on the frequency-distributed model of the fractional integrator, a fractional system is decomposed into an infinite dimensional set of ODEs, whose state variables are the frequency-distributed state variables of the fractional system. Thus, the Infinite State approach provides a conventional integer-order solution to the initialization problem, a new interpretation of the initial conditions of fractional derivatives, and a definition of fractional energy, preliminary to the analysis of system stability based on the Lyapunov approach. Its finite dimensional approximation enables the study of fractional transients thanks to numerical simulation and the discovery of hidden properties of fractional systems. Moreover, it is a helpful and necessary technique for the analysis of nonlinear and chaotic systems and represents a powerful characterization tool.