Set Of Differential Equations Research Articles

Numerical Investigation of Thermophoresis and Activation Energy Effects on Maxwell Nano Fluid Over an Inclined Magnetic Field Applied to a Disk

Numerical model is conducted to investigate the behavior of an incompressible Maxwell nanofluid model flow on a convectively stretched surface, considering the effects of thermophoresis and an inclined magnetic field. The system, originally formulated as a set of partial differential equations, is transformed into a system of ordinary differential equations using similarity transformations. The resulting equations are solved using the Runge-Kutta-Fehlberg method in conjunction with the shooting technique. The obtained physical parameters from the derived system are presented and discussed through graphical representations. The numerical process is assessed by comparing the results with existing literature under various limiting scenarios, demonstrating a high level of proficiency. The key findings of this study indicate that the velocity field decreases as the fluid parameters increase, while the fluid temperature diminishes accordingly. Additionally, the heat transfer rate decreases with increasing fluid and thermophoresis parameters, but it increases with Biot and Prandtl numbers.

Propagation of magnetogasdynamics spherical shock wave in a gravitating gas with radiation heat flux

This article presents a mathematical model for characterizing the dynamic behavior of spherical shock waves in a self-gravitating, radiating ideal gas with the presence of an azimuthal magnetic field, emphasizing adiabatic conditions. The analysis assumes that the environment just ahead of the shock wave is stationary, and it accounts for variations in density, magnetic field, and fluid velocity within the disturbed medium just behind the shock front. Furthermore, the impact of thermal radiation within the context of an optically thin limit is incorporated into the energy equation of the governing system. Employing the Lie invariance method, the set of partial differential equations governing the flow within this medium is transformed into a system of nonlinear ordinary differential equations through the use of similarity variables. Four distinct cases of similarity solutions are derived by selecting different values for the arbitrary constants associated with the generators. Among these four cases, only two yield similarity solutions, one assuming a power-law shock path and the other an exponential-law shock path. In the case of a power-law shock path, the resulting set of nonlinear ordinary differential equations is numerically solved using the 4th-order Runge-Kutta method in MATLAB software. The article thoroughly discusses the influence of various parameters, including γ (adiabatic index of the gas), Ma−2 (Alfvén–Mach number), ϕ (ambient density exponent), and G0 (gravitational parameter), on the flow properties. The findings are presented graphically to provide a comprehensive understanding of the effects of these parameters.

Quantifying Reliability Indices of Garbage Data Collection IOT-based Sensor Systems using Markov Birth-death Process

The aim of this paper is to analyze the performance of a Garbage data collecting sensor network system (GDCSNS) through mathematical modelling and a reliability approach. The reliability assessment of such a system is essential to ensuring that it can collect data related to garbage at different locations consistently and accurately. After the determination of the reliability measures of the system, the next aim is to identify the weakest sensors of the system so that a timely maintenance strategy for the weakest sensor can be planned to avoid disruption in the collection of data from the sensor systems. In the considered system, three sensors have been installed at various location in the city that send the information to the center office (hub point) and then from the center office to the person who is responsible for collecting the garbage from the location and dumping it at some predefined places. These sensors collect data related to garbage level, weight, and other information and send it to computers at the city's central office. Markov modelling has been used to model the system. Based on the mathematical model, a state transition diagram and a set of Kolmogorov time-dependent differential equations have been obtained. The various state probabilities (explicit expressions) related to the performance of the system, namely, Reliability, Mean time to failure, have been obtained to understand the different maintenance policies that can be used. A sensitivity analysis has also been performed to determine the weakest sensor among the sensors.

An efficient numerical scheme for solving the melting transportation of energy with time dependent Carreau nanofluid

The current work is categorized into the following formatBackground: Heat transfer controlling through therapy and blood flow has clinical importance in different levels of the human body. Biological faculties utilize blood in which iron oxide nanoparticles are incorporated for contrast agents and in the biomedical fields as they possess key and intrinsic characteristics, like low toxicity, capability of blood carrying nanoparticles, surface engineering and colloidal stability. This paper has a unique approach for analysis of blood flow containing nanoparticles with mathematical model of Carreau nanofluid. Furthermore, heat transport analysis has been provided by using the melting heat conditions and free convection.Carreau nanofluid for blood flow gives a set of partial differential equations (PDEs), which are handled by using the Keller Box scheme to obtain the numerical performances of the results. The obtained results are compared with Bvp4c, which is MATLAB built in program and found smooth agreement. The graphical plots and statistical tables have been provided for various dimensionless parameters to get the numerical simulations of the problem against velocity, temperature, and heat transfer Nusselt number. From the obtained results, the velocity and temperature of the fluid diminishes by the virtue of an amplification in unsteady parameter A. A positive variation in melting parameter M in the case of Prandtl number Pr and unsteady parameter A magnifies the heat transfer Nusselt number. This article shows the novelty in the sense that the effect of Tiwari and Das model accompanied with melting heat transfer phenomenon in the case of unsteady Carreau fluid has not investigated before in the available literature related to the Carreau fluid.

Uncertainty propagation in orbital dynamics via Galerkin projection of the Fokker-Planck Equation

The Fokker–Planck equation is a partial differential equation that describes how the probability density function of an object state varies, when subject to deterministic and random forces. The solution to this equation is crucial in many space applications, such as space debris trajectory tracking and prediction, guidance navigation and control under uncertainties, space situational awareness, and mission analysis and planning. However, no general closed-form solutions are known and several methods exist to tackle its solution. In this work, we use a known technique to transform this equation into a set of linear ordinary differential equations in the context of orbital dynamics. In particular, we show the advantages of the applied methodology, which allows to decouple the time and state-dependent components and to retain the entire shape of the probability density function through time, in the presence of both deterministic and stochastic dynamics. With this approach, the probability density function values at future times and for different initial conditions can be computed without added costs, provided that some time-independent integrals are solved offline. We showcase the efficacy and use of this method on some orbital dynamics example, by also leveraging the use of automatic differentiation for efficiently computing the involved derivatives.

Reliability and Performance Measurement of Safety-Critical Systems Based on Petri Nets: A Case Study of Nuclear Power Plant

Safety-critical systems (SCSs) mitigate the risk of catastrophic loss of assets and hence do have high dependability targets. Performance and reliability are the critical dependability attributes, particularly in control and safety systems, and hence essential to measure to ensure the dependability. Traditional methods either are not capable to capture the system dynamics or encounter state explosion problem. Also, the methods are not able to measure all critical performance attributes. This article proposes a novel approach to measure the performance and reliability of SCSs. Such systems contain multiple interconnecting processing nodes, the functional requirements of which are modeled using Petri net (PN). A set of ordinary differential equations (ODEs) is derived from the PN model that represents the state of the system. The ODE solution can be used to measure the critical performance attributes, such as latency time and throughput of the system. The proposed method can avoid the state explosion problem and also introduces new metrics of performance, along with their measurement: deadlock, liveness, stability, boundedness, and steady state. The proposed technique is applied to a case study of nuclear power plant. We obtained 99.887% and 99.939% accuracy of performance and reliability measurement, respectively, which proves the effectiveness of our approach.

Two-Dimensional Magnetotelluric Forward Modeling in Space-wavenumber Mixed Domain

In magnetotelluric exploration, if you want to do 2D or 3D numerical modelling, what you need in the computational and memory requirements are potentially enormous. In order to get efficient and precise forward simulation results, we propose a new 2D numerical modelling method in space-wavenumber mixed domain. This method uses the one-dimensional Fourier transform in the horizontal directions, so the two-dimensional partial differential equations in the spatial domain can be transformed into a set of independent one-dimensional differential equations with different wave numbers. Then we can use the finite different method to solve the one-dimensional equations. In this article, the COMMEMI-2D1 model was used to verify the accuracy of the algorithm. We compare our results with the traditional finite different method in TE mode, indicating that the algorithm improves computational efficiency.

Improved mean-field dynamical equations are able to detect the two-step relaxation in glassy dynamics at low temperatures

We study the stochastic relaxation dynamics of the Ising p-spin model on a random graph, which is a well-known model with glassy dynamics at low temperatures. We introduce and discuss a new closure scheme for the master equation governing the continuous-time relaxation of the system, which translates into a set of differential equations for the evolution of local probabilities. The solution to these dynamical mean-field equations describes the out-of-equilibrium dynamics at high temperatures very well, notwithstanding the key observation that the off-equilibrium probability measure contains higher-order interaction terms not present in the equilibrium measure. In the low-temperature regime, the solution to the dynamical mean-field equations shows the correct two-step relaxation (a typical feature of glassy dynamics), but with a too-short relaxation timescale. We propose a solution to this problem by identifying the range of energies where entropic barriers play a key role and defining a renormalized microscopic timescale for the dynamical mean-field solution. The final result perfectly matches the complex out-of-equilibrium dynamics computed through extensive Monte Carlo simulations.

Limit Directions of Julia Sets of Entire Functions and Complex Differential Equations

The limiting directions of Julia sets of infinite order entire functions are studied by combining the theory of complex dynamic system and the theory of complex differential equations, in which the lower bound of the measure of limiting direction of Julia set of entire solutions of complex differential equations is obtained.

Revised set of equations describing the seismo-electromagnetic coupled wavefield in an ambient geomagnetic field

SUMMARY Temporal variations in the electromagnetic field generated by seismic waves are accounted for mainly by electrokinetic (EK) and motional induction (MI) effects. Currently, these effects are considered in terms of different sets of equations, even though both mechanisms are assumed in data interpretation. In principle, however, they should be described by a single set of equations, possibly including a new type of coupling effect. This inconsistency suggests that the deviation of conventional equations was somewhat incorrect, with some coupling mechanisms having been overlooked. Here, the derivation of the conventional equations was reconsidered, and necessary revisions were applied. The revised equations describe both EK and MI effects and include a previously unrecognized coupling effect (although its effect is negligible). The solution of the new equations is well approximated by a simple sum of EK and MI effects determined by conventional equations. In conclusion, it was confirmed that the conventional procedure for estimation of seismically induced EM variations in terms of the EK and MI effects is sufficiently accurate, in spite of the conventional governing equations missing some of coupling terms related to the EK and MI effects.

Linear-quadratic optimal control of second-order parabolic systems

This paper considers the linear-quadratic (LQ) optimal control problem for systems governed by a class of second-order parabolic partial differential equations (PDEs). This problem is significant in many areas of mathematics, control, engineering and so on, and thus has received great attention in the past decades. Different from the previous articles where the operator is applied to present the controller, the main contribution of this paper is to propose the discretisation-then-continuousization method, which is explicit and implementable. The solvability condition of the LQ optimal control problems is given based on a set of differential Riccati equations, and an explicit numerical calculation way of these equations and the design of the optimal controller are provided.

Entropy study of electromagnetohydrodynamic trihybrid nanofluid flow within non‐uniform peristaltic across microchannel

AbstractIn this work, an inspection of the entropy generation of ternary nanofluids inside an irregular peristaltic channel resulting from heat transfer and fluid viscosity in addition to the combined effects of electric and magnetic fields is conducted. The flow is supposed to be subject to a combination of external forces such as an electric field, a magnetic field, thermal radiation, and Joule heating due to the Lorentz force. This model is a representation of the heat transfer processes of heat exchangers and some types of solar cells. Variation in channel width is an effective factor that has not been addressed by many researchers before, these assumptions are simulated by a set of nonlinear partial differential equations due to the dual effect of electric and magnetic forces and under the influence of special boundary conditions. This set of equations has been converted to non‐dimensional equations and then solved analytically to study the behavior of velocities, heat transfer, and the rate of entropy generation inside the channel under the influence of important embedded parameters. Some notable results have been reached, such as a clear growth of the entropy generation system by using more than one type of nanoparticle as a result of the improvement that occurs in the thermal and magnetic conductivity of the fluid, in addition to the increase in viscosity. The irreversibility resulting from friction and the electric and magnetic fields clearly rises in the middle and diminishes at the walls compared to the irreversibility resulting from thermal transfer.

Eigenvalue-curve approach for fast assessment of rotor angle transient stability

Abstract The phase-out of nuclear and coal-based electricity production and their replacement by power-electronics based renewable energy sources, especially in Germany, leads to more and faster power system dynamics. Anyway, not all European countries plan to abandon bulk power generation, so there will still be a mix of inverter-based sources and synchronous machines in the future, justifying the necessity of rotor angle stability analysis. This task grows with the number of assets. It requires to numerically solve a large set of ordinary differential equations which gets more and more challenging to do in real-time during contingency analysis in power system operation. Therefore, this paper introduces a new approach to quickly estimate the transient stability of rotor angles without the need of solving differential equations. It is based on a linearized system description, some assumptions on the rotor angle displacement and uses the fault location and duration as parameters. It is proven, that this approach can be used to reduce the number of contingencies which need to be analyzed in detail during system operation and therefore helps transmission system operators to keep the power system stable and secure.

Mechanically induced temperature oscillations in the coupled thermo-elasticity analysis of articulated dynamical systems

This article discusses a new approach for predicting and quantifying mechanically induced temperature oscillations in the coupled thermo-elasticity analysis of articulated mechanical systems (AMS). In this approach, the constrained equations of motion are solved simultaneously with discrete temperature equations obtained by converting heat partial-differential equation to a set of first-order ordinary differential equations. Dependence of the temperature gradients and their spatial derivatives on the position gradients, spinning motion, and curvatures is discussed. The approach captures dependence of the temperature-oscillation frequencies on the mechanical-displacement frequencies. The temperature field can be selected to ensure continuity of the temperature gradients at the nodal points. To generalize the AMS coupled thermo-elasticity formulation and capture the effect of the boundary and motion constraints (BMC) on the thermal expansion, the proposed method is based on integrating thermodynamics and Lagrange-D’Alembert principles. The absolute nodal coordinate formulation (ANCF) is used to describe continuum displacement and obtain accurate description of the reference-configuration geometry and change of this geometry due to deformations. A thermal-analysis large-displacement formulation is used to allow converting heat energy to kinetic energy, ensuring stress-free thermal expansion in case of unconstrained uniform thermal expansion. Cholesky heat coordinates are used to define an identity coefficient matrix for the efficient solution of the discretized heat equations. The approach presented is applicable to the two different forms of the heat equation used in the literature; one form is explicit function of the stresses while the other form does not depend explicitly on the stresses. Because of the need for using ANCF finite elements to achieve a higher degree of continuity in the coupled thermomechanical approach introduced in this article, the concept of the ANCF mesh topology is discussed.

Nanofluid Flow Past a Static Wedge with Velocity Slip Condition

Nanofluids are known for their exceptional thermophysical properties, especially their thermal conductivity, which make them ideal for a number of heat transfer applications. In this study, the mixed convection nanofluid flow past a static wedge with the presence of velocity slip condition is investigated. The problem is governed by a system of partial differential equations which then transformed into a set of nonlinear ordinary differential equations by using an appropriate similarity transformation. The transformed governing equations are then solved numerically by using MATLAB bvp4c solver. Numerical solutions obtained are presented graphically in the form of velocity and temperature profiles for different values of nanoparticles volume fraction, wedge angle and velocity slip. It is found that the increasing values of the wedge angle parameter causes the velocity of the fluid to increase whereas the temperature profile of the nanofluid decreases, additionally it observed that velocity of the fluid increase when velocity slip is increased. The findings of this study provide valuable insights into optimizing heat transfer processes in various engineering applications by leveraging the enhanced thermal properties of nanofluids and considering factors such as wedge angle and velocity slip

Computational study of magnetized and dual stratified effects on Non-Darcy Casson nanofluid flow: An activation energy analysis

The current research looks into magnetohydrodynamics Casson nanofluid flow and suction/injection implications for a nonlinearly stretched interface. To improve heat transport, joules of heat, radiation impacts and thermal stratification are used. A Darcy-Forchheimer porous media is used to conduct fluid flow. Chemical reactions involving energies of activation and solutal stratum are additionally considered. The Keller-box technique is used to solve the resulting non-linear set of ordinary differential equations. The impacts of a number of variables are examined employing diagrams of quantity, velocity, and heat. Increased assessments of porousness, Darcy-Forchheimer, and Casson fluid characteristics result in a decrease in velocity trends. For improved measurements of Lewis number and solutal fractionated factors, the intensity pattern reduces. The skin friction factor increases with increasing Casson parameter estimate. In the instance of the porous and nonlinear stretching parameters, the Nusselt number rises. For increasing levels of the Lewis number and Brownian motion factors, the Sherwood number drops. Furthermore, Nusselt number ratios are estimated with Prandtl numbers with regard to current information available, which shows remarkable consistency. For varied incorporated parameters, streamlines, heatlines, and masslines additionally displayed.

Approximate symplectic approach for mistuned bladed disk dynamic problem

A blade disk system plays a crucial role in the energy conversion efficiency of turbomachinery. Generally, a bladed disk is designed as a tuned system, which implies that the dynamic behaviors of all the blades are identical. In practice, there are some differences between the blades, known as mistuning, that can result in localized vibration. Thus, in this study, an approximate symplectic method developed to reproduce the vibration localization in a mistuned bladed disk system, that can provide guidance for designing vibration control strategies for the system. Considering the morphological characteristics of the interface roughness and the dynamic/stickiness characteristics of the sliding joint surface, an improved dry friction model is established to describe the coupling effects between the blades. Considering the micro-dynamic behaviors of the multi-joint surfaces, the dynamic problem of the entire bladed disk system is formulated as a set of ordinary differential equations (ODEs). An approximate symplectic method is developed in the Hamiltonian framework to simulate the vibration of each blade in the mistuned bladed disk system. From the numerical results reported, the phenomenon of vibration localization in the mistuned bladed disk system is reproduced in the comparison with the results in the tuned bladed disk system. In addition, the influences of the initial positive pressure, contact angle, and roughness of the surfaces on the strength of the vibration localization are revealed in detail in the numerical simulations.

Research on nonlinear dynamic characteristics of high-speed gear in two-speed transmission system

The working performance and service life of the two-speed transmission system directly affects the performance and service life of helicopters and other equipment. One of the main tasks of the two-speed transmission system research is to improve its dynamic characteristics. For the two-speed transmission system in high-speed gear, a purely torsional nonlinear dynamic differential equation set considering the number of planetary gears, backlash, and clutch dynamic load is established by using the lumped parameter method, and the equations are dimensionless. Then the dimensionless differential equation set is solved by using the variable step-size fourth-order Runge–Kutta method, and the phase diagram and Poincare diagram of high-speed gear are obtained. By changing the dynamic friction coefficient of the friction clutch and the backlash of the gear pair, the influence of parameter change on the nonlinear dynamic characteristics of the system is analyzed. The results show that, with the increase of excitation frequency, the system has experienced single cycle, quasi-cycle, chaos, and double cycle, then changed from double cycle to chaotic motion, and then changed from chaotic motion to double cycle and single cycle motion in turn, and found the path to chaos. In the low-frequency band, reducing the friction coefficient of the friction clutch can reduce vibration amplitude; In the middle-frequency band, reducing the friction coefficient will make the system tend to unstable vibration. In the high-frequency band, it is a single-cycle movement, which is not affected by friction coefficient.

A novel approach to micropolar fluid flow between a non-porous disk and a porous disk with slip

This paper considers the problem of steady, laminar, incompressible, and two-dimensional micropolar fluid flow between two disks. The top disk is considered porous, while the lower one is not. The body forces and body couples were neglected, and the flow was assumed to be fully developed. The governing equations of the problem were reduced to a set of ordinary differential equations (ODEs) by Von-Karman's similarity transformations. Since the obtained governing ODEs have not been solved analytically according to the previous studies, in this article, the Modified Akbari-Ganji method (Modified AGM) and the hybrid analytical and numerical method (HAN-method) have been chosen to solve these equations analytically, which is one of the novelties of this study. However, most of this article's novelty is related to the physical results obtained from the analytical solution of these equations. The effects of various slip coefficients, Reynolds number, and micropolar parameters of vortex viscosity, spin gradient viscosity, and microinertia density on profiles of normal velocity, streamwise velocity, and microrotation. The validity of these two analytical solutions was proved by comparison with previously published results. The results of the two methods are almost the same in all cases, which can be seen as an indirect sign of the validity of the results of this study.

Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua

Non-prismatic slender continua are the prototypical models of many structural elements used in engineering applications, such as wind turbine blades and towers. Unfortunately, closed-form expressions for stresses and strains in such continua are much more difficult to find than in prismatic ones, e.g., the de Saint-Venant’s cylinder, for which some analytical solutions are known. Starting from a suitable mechanical model of a tapered slender continuum with one dimension much larger than the other tapered two, a variational principle is exploited to derive the field equations, i.e., the set of partial differential equations and boundary conditions that govern its state of stress and strain. The obtained equations can be solved in closed form only in a few cases. Paradigmatic examples in which analytical solutions are obtainable in terms of stresses, strains, or related mechanical quantities of interest in engineering applications are presented and discussed.