Set Of Nonlinear Differential Equations Research Articles

Particle migration in planar die-swell flows

We present a numerical study on particle migration in a planar extrudate flow of suspensions of non-Brownian hard spheres. The suspension is described as a Newtonian liquid with a concentration-dependent viscosity, and shear-induced particle migration is modelled according to the diffusive flux model. The fully coupled set of nonlinear differential equations governing the flow is solved with a stabilized finite element method together with the elliptic mesh generation method to compute the position of the free surface. We show that shear-induced particle migration inside the channel leads to a highly non-uniform particle concentration distribution under the free surface. It is found that particle migration dramatically changes the shape of the free surface when the suspension is compared to a Newtonian liquid with the same bulk properties. Remarkably, we observed extrudate expansion in the Newtonian and dilute suspension flows; in turn, at high concentrations, a die contraction appears. The model does not account for normal stress differences, and this result is a direct consequence of variations in the flow stress field caused by shear-induced particle migration.

An analytical treatment for MHD mixed convection boundary layer flow of Oldroyd-B fluid utilizing non-Fourier heat flux model

In current framework, an analytical treatment for mixed convection flow of an electrically conducting Oldroyd-B fluid adjacent to a vertical stretchable surface is provided. A non-Fourier heat flux approach is employed to formulate the energy balance relation. Using similarity approach, the governing equations are changed to a set of non-linear differential equations which are tackled by well-known analytical approach called homotopy analysis method (HAM). Appropriate range of auxiliary parameter is obtained by plotting the so called ℏ-curves. Velocity and temperature profiles are computed and elucidated in the existence of new physical mechanism, that is, thermal relaxation time in current research. The main implication of this research is that the relaxation and retardation times considerably alter the flow behavior near the surface. The results predict that heat penetration into the fluid reduces as the relaxation time of heat flux enlarges. Furthermore, the change in temperature gradient at the surface with increasing thermal relaxation time appears similar in magnitude at all considered Prandtl numbers. In assisting flow regime, we noticed a growth in velocity and temperature profiles for increasing strength of buoyancy force.

Analysis of heat and mass transfer with MHD and chemical reaction effects on viscoelastic fluid over a stretching sheet

This article addresses the mass and heat transfer analysis over an electrically conducting viscoelastic (Walters B′) fluid over a stretching surface in presence of transverse magnetic field. The impact of chemical reaction, as well as non-uniform heat source, are also taken into account. Similarity transformations are employed to model the equations. The governing equations comprises of momentum, energy, and concentration which are modified to a set of non-linear differential equations and then solved by applying confluent hypergeometric function known as “Kummer’s function”. The exact solution for heat equation is obtained for two cases i.e. (1) Prescribed surface temperature, (2) Prescribed wall heat flux. Physical behavior of all the sundry parameters are against concentration, temperature, and velocity profile are presented through graphs. The inclusion of magnetic field is counterproductive in diminishing the velocity distribution whereas reverse effect is encountered for concentration and temperature profiles.

System Dynamics Simulation Modelling of Managing a Maritime Shipping Organization

This paper is a result of the research team at the University of Split, Faculty of Maritime Studies in dynamics of conduct maritime transport organizations. Results of the paper are shown in paper [8], and wish of the authors is to present a model to wider audience. What is new in this paper are Subsystem of debts, Subsystem of credits, Subsystem of loans and Subsystem of commercial department. System dynamics simulation modelling is one of the most suitable and most successful scientific methods of modelling of dynamics of complex, non-linear, natural, technical and organizational systems. System dynamics models are basically continuous models, as the subjects are presented by a set of non-linear differential equations, i.e. “equations of state”. However, they are also discrete, as their basic time step, i.e. sampling “DT” is defined in full accordance with the Sampling Theorem (Shannon, Nyquist and Koteljnikov). Such selection of the basic time step of DT enables computer modelling of continuous simulation models in digital computer, which is very suitable for training students at maritime faculty, engineering and electro technical faculties, and for theoretical and practical training of engineers of ship processes, as it enables them, by applying system dynamics approach, to acquire new information and skills about the complex dynamics of performance of maritime systems and processes. In a system dynamics model a shipping company is an integrated part of the surroundings and cannot be observed separately, without mutual interference.

Simulation Modelling of Managing a Maritime Shipping Organisation Using System Dynamics

This paper is a result of the research team at the University of Split, Faculty of Maritime Studies in dynamics of conduct maritime transport organizations. Results of the paper are shown in paper [8], and wish of the authors is to present a model to wider audience. What is new in this paper are Subsystem of debts, Subsystem of credits, Subsystem of loans and Subsystem of commercial department. System dynamics simulation modelling is one of the most suitable and most successful scientific methods of modelling of dynamics of complex, non-linear, natural, technical and organisational systems. System dynamics models are basically continuous models, as the subjects are presented by a set of non-linear differential equations, i.e. “equations of state”. However, they are also discrete, as their basic time step, i.e. sampling “DT” is defined in full accordance with the Sampling Theorem (Shannon, Nyquist and Koteljnikov). Such selection of the basic time step of DT enables computer modelling of continuous simulation models in digital computer, which is very suitable for training students at maritime faculty, engineering and electro technical faculties, and for theoretical and practical training of engineers of ship processes, as it enables them, by applying system dynamics approach, to acquire new information and skills about the complex dynamics of performance of maritime systems and processes. In a system dynamics model a shipping company is an integrated part of the surroundings and cannot be observed separately, without mutual interference.

System Dynamics Simulation Model of the Marine Diesel Engine Start up System

System dynamic simulating modelling is one of the most appropriate and successful scientific dynamics modelling methods of the complex, non-linear, natural, technical and organizational systems. Investigation of behavior dynamics of the ship/s propulsion system as one of the complex, dynamic, non-linear technical systems requires application of the most efficient modelling methods. Marine Diesel engine start up system, which it is consisted of ship/s piston compressor and its driving electric motor, i.e. in this case asynchronous motor, shall be presented in the System Dynamics modelling approach in POWERSIM simulation symbolic and language. System dynamic models are essentially continuous models since the realities have been demonstrated by a set of non-linear differential equation, i.e. “equations of conditions”, however they are at the same time discrete ones because their principle time interval calculation i.e. (discretisation) sampling “DT” is determined in full compliance with (Sampling Theorem) by Shannon and Koteljnikov.

Theory of atomic–molecular conversion under the conditions of Bose–Einstein condensation

A set of nonlinear differential equations that describe the dynamics of atomic–molecular conversion under the action of two arbitrarily shaped Raman pulses of resonance laser radiation has been derived in the mean-field approximation. The dynamics of the system subjected to Gaussian pulses has been studied in detail. It has been shown that the time evolution of the system depends on the initial conditions.

A Comparative Assessment of Simplified Models for Simulating Parametric Roll

The motion of a ship/offshore platform at sea is governed by a coupled set of nonlinear differential equations. In general, analytical solutions for such systems do not exist and recourse is taken to time-domain simulations to obtain numerical solutions. Each simulation is not only time consuming but also captures only a single realization of the many possible responses. In a design spiral when the concept design of a ship/platform is being iteratively changed, simulating multiple realizations for each interim design is impractical. An analytical approach is preferable as it provides the answer almost instantaneously and does not suffer from the drawback of requiring multiple realizations for statistical confidence. Analytical solutions only exist for simple systems, and hence, there is a need to simplify the nonlinear coupled differential equations into a simplified one degree-of-freedom (DOF) system. While simplified methods make the problem tenable, it is important to check that the system still reflects the dynamics of the complicated system. This paper systematically describes two of the popular simplified parametric roll models in the literature: Volterra GM and improved Grim effective wave (IGEW) roll models. A correction to the existing Volterra GM model described in current literature is proposed to more accurately capture the restoring forces. The simulated roll motion from each model is compared against a corresponding simulation from a nonlinear coupled time-domain simulation tool to check its veracity. Finally, the extent to which each of the models captures the nonlinear phenomenon accurately is discussed in detail.

Architecture of the Internet of Energy Network: An Application to Smart Grid Communications

Due to the global warming and energy crisis, the renewable distributed energy resources, such as wind turbines, are integrated into the grid. We model an AC microgrid with energy generating units, local loads, and electronic devices. Then, the set of non-linear differential equations are expressed as a state-space model. As the microgrid is located in the customer premises or remote areas, its condition needs to monitor in real-time. So, the smart sensor requires to deploy around the microgrid, and its sensing information transmits to the energy management system via the Internet as the sensing information is a massive amount of data. Combining the Internet of Things elements, such as sensors (Internet emended), and the Internet as a transmission medium will form the Internet of Energy, which is considered as a sign interest nowadays. Basically, the energy management center estimates the microgrid states to know the operating conditions of these foreseeable intermittent resources. For estimating the microgrid states, the H-infinity-based Mimi-max filter is proposed, which will no need to know the exact process and measurement noise statistics. Simulation results show that the proposed approach can well estimate the system states compared with the existing Kalman filter. As a result, this framework will assist to design a suitable microgrid framework and provides effective dynamic state estimations.

Numerical Methods Coupled withRichardson Extrapolation for Computationof Transient Power Systems

The numerical solution of transient stability problems is a key element for electrical power system operation. The classical model for multi-machine systems is defined as a set of non-linear differential equations for the rotor speed and the generator angle for each electrical machine, this mathematical model is usually known as the swing equations. This paper presents how to use direct Richardson extrapolation of several orders for the numerical solution of the swing equations and compares it with other commonly used implicit and explicit solvers such as Runge-Kutta, trapezoidal, Shampine and Radau methods. A numerical study on a simple three machine system is used to illustrate the performance and implementation of algebraic Richardson extrapolation coupled to several solution methods. Normally, the order of accuracy of any numerical solution can be increased when Richardson Extrapolation is used. A numerical example is provided for an electrical grid consisting of three machines and nine buses undergoing a disturbance. It is shown that in this case Richardson extrapolation effectively increases the order of accuracy of the explicit methods making them competitive with the implicit methods.

Experimental Model Validation and Control of a Lactic Fermentation Process

In this work, the nonlinear dynamical model of a lactic fermentation process is widely analysed and control experiments are achieved. More precisely, a production of yogurt by Streptococcus termophilus and Lactobacillus bulgaricus in batch operation is taken into consideration. The process model is expressed by a set of nonlinear differential equations that describes the evolution of concentrations in the fermentation process. To validate the model, several simulations are performed in the Matlab programming and development environment. Furthermore, two experimental setups are used for batch fermentation experiments. From control point of view, the temperature and the pH are the basic dynamical factors that need monitoring and control in order to regulate the microbial growth and the lactic acid production. Different control architectures and tuning procedures are implemented. Specialized data acquisition and control software tools are used to perform the experiments. By using the features of these software tools, the time evolution of various process variables can be plotted and analysed. Several comparisons between the results obtained via simulation and with the two bioreactor setups are achieved.

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern.

Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.

Mechanical model of carbon dioxide vibrational spectrum

Classical dynamics methods have been used to study the nonlinear vibrations of a CO2 molecule. Consideration includes not only the anharmonicity valence angle, which enables one to explain the Fermi resonance, but also the physical nonlinearity of the force field (stiffness and softness of springs). In the farthest neighbor approximation (with regard to oxygen–oxygen interaction), a set of nonlinear differential equations in the Lagrangian form has been derived. Their analytical solution has been derived using the method of invariant normalization. The occurrence of a strange attractor has been discovered by numerical simulation. Recommendations for the selection of initial conditions are given that take into account the possibility of regular beatings that change into to chaotic beatings.

Thin Film Williamson Nanofluid Flow with Varying Viscosity and Thermal Conductivity on a Time-Dependent Stretching Sheet

This article describes the effect of thermal radiation on the thin film nanofluid flow of a Williamson fluid over an unsteady stretching surface with variable fluid properties. The basic governing equations of continuity, momentum, energy, and concentration are incorporated. The effect of thermal radiation and viscous dissipation terms are included in the energy equation. The energy and concentration fields are also coupled with the effect of Dufour and Soret. The transformations are used to reduce the unsteady equations of velocity, temperature and concentration in the set of nonlinear differential equations and these equations are tackled through the Homotopy Analysis Method (HAM). For the sake of comparison, numerical (ND-Solve Method) solutions are also obtained. Special attention has been given to the variable fluid properties’ effects on the flow of a Williamson nanofluid. Finally, the effect of non-dimensional physical parameters like thermal conductivity, Schmidt number, Williamson parameter, Brinkman number, radiation parameter, and Prandtl number has been thoroughly demonstrated and discussed.

The dynamics of nutrient, toxic phytoplankton, nontoxic phytoplankton and zooplankton model

<p>The objective of this paper is to study the dynamical behavior of an aquatic food web system. A mathematical model that includes nutrients, phytoplankton and zooplankton is proposed and analyzed. It is assumed that, the phytoplankton divided into two compartments namely toxic phytoplankton which produces a toxic substance as a defensive strategy against predation by zooplankton, and a nontoxic phytoplankton. All the feeding processes in this food web are formulating according to the Lotka-Volterra functional response. This model is represented mathematically by the set of nonlinear differential equations. The existence, uniqueness and boundedness of the solution of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. The occurrence of local bifurcation and Hopf bifurcation are investigated. Finally, numerical simulation is used to study the global dynamics of this model.</p>

Using numeric simulation in an online e-learning environment to teach functional physiological contexts

Background and objectivesMathematical models are suitable to simulate complex biological processes by a set of non-linear differential equations. These simulation models can be used as an e-learning tool in medical education. However, in many cases these mathematical systems have to be treated numerically which is computationally intensive. The aim of the study was to develop a system for numerical simulation to be used in an online e-learning environment. MethodsIn the software system the simulation is located on the server as a CGI application. The user (student) selects the boundary conditions for the simulation (e.g., properties of a simulated patient) on the browser. With these parameters the simulation on the server is started and the simulation result is re-transferred to the browser. ResultsWith this system two examples of e-learning units were realized. The first one uses a multi-compartment model of the glucose-insulin control loop for the simulation of the plasma glucose level after a simulated meal or during diabetes (including treatment by subcutaneous insulin application). The second one simulates the ion transport leading to the resting and action potential in nerves. The student can vary parameters systematically to explore the biological behavior of the system. ConclusionsThe described system is able to simulate complex biological processes and offers the possibility to use these models in an online e-learning environment. As far as the underlying principles can be described mathematically, this type of system can be applied to a broad spectrum of biomedical or natural scientific topics.

Influence of nonlinear thermal radiation and Magnetic field on upper-convected Maxwell fluid flow due to a convectively heated stretching sheet in the presence of dust particles

A numerical investigation of two-dimensional MHD boundary layer flow and thermal characteristics of an electrically conducting dusty non-Newtonian fluid over a convectively heated stretching sheet has been considered. The effects of nonlinear thermal radiation, heat source or sink and viscous dissipation are also taken into the account. The Rosseland approximation is used to model the nonlinear thermal radiation. Suitable similarity transformations are used to transform the flow governing equations into a set of nonlinear differential equations of one independent variable. The Shooting method is adopted to solve transformed equations. The effects of various material parameters on the flow and heat transfer in terms of velocity and temperature distributions are drawn in the form of graphs and are briefly discussed. The numerical computations for the Nusselt number and skin friction drag are also carried out for the emerging parameters of interest in the problem. The obtained numerical results show the good agreement with the existing one for limiting case.

Mixed convective flow of Maxwell nanofluid past a porous vertical stretched surface – An optimal solution

Present investigation is devoted to examine the mixed convective flow of Maxwell nanofluid with Soret and Dufour effects through a porous medium. Effects of variable temperature and concentration over a linearly permeable stretched surface are also taken into account. An optimal solution is obtained for the highly nonlinear set of differential equations using BVPh 2.0 Mathematica package. Graphs of different emerging pertinent parameters against velocity, temperature and concentration distributions are plotted and discussed accordingly. Numerically tabulated values of local Nusselt and Sherwood numbers are also part of this investigation. It is witnessed that concentration field is decreasing and increasing function of Brownian motion and thermophoretic parameters respectively. Further, opposite behavior of Soret number on temperature and concentration distributions is seen.

On the plasma-based growth of ‘flowing’ graphene sheets at atmospheric pressure conditions

A theoretical and experimental study on atmospheric pressure microwave plasma-based assembly of free standing graphene sheets is presented. The synthesis method is based on introducing a carbon-containing precursor (C2H5OH) through a microwave (2.45 GHz) argon plasma environment, where decomposition of ethanol molecules takes place and carbon atoms and molecules are created and then converted into solid carbon nuclei in the ‘colder’ nucleation zones. A theoretical model previously developed has been further updated and refined to map the particle and thermal fluxes in the plasma reactor. Considering the nucleation process as a delicate interplay between thermodynamic and kinetic factors, the model is based on a set of non-linear differential equations describing plasma thermodynamics and chemical kinetics. The model predictions were validated by experimental results. Optical emission spectroscopy was applied to detect the plasma emission related to carbon species from the ‘hot’ plasma zone. Raman spectroscopy, scanning electron microscopy (SEM), and x-ray photoelectron spectroscopy (XPS) techniques have been applied to analyze the synthesized nanostructures. The microstructural features of the solid carbon nuclei collected from the colder zones of plasma reactor vary according to their location. A part of the solid carbon was deposited on the discharge tube wall. The solid assembled from the main stream, which was gradually withdrawn from the hot plasma region in the outlet plasma stream directed to a filter, was composed by ‘flowing’ graphene sheets. The influence of additional hydrogen, Ar flow rate and microwave power on the concentration of obtained stable species and carbon−dicarbon was evaluated. The ratio of sp3/sp2 carbons in graphene sheets is presented. A correlation between changes in C2 and C number densities and sp3/sp2 ratio was found.

Heat Transfer Analysis of Third-Grade Fluid Flow Between Parallel Plates: Analytical Solutions

Heat transfer analysis for the plane Couette, and, Poiseuille flows of a third grade fluid is presented. Both the problems are modeled accordingly, and later, the sets of nonlinear differential equations are solved by using Variation of Parameter Method (VPM). Exact solution of the first problem is obtained in just two iterations which shows the effectiveness of VPM. For the second problem, i.e. Poiseuille flow, a numerical solution is also sought for the sake of comparison. Analytical and numerical results show an excellent agreement that is presented with the help of a table. The effects of physical parameters on velocity and temperature profiles are demonstrated graphically. Comprehensive discussions have also been made to interpret the results obtained.