Set Of Nonlinear Differential Equations Research Articles

Effects of Arrhenius Activation Energy on Thermally Radiant Williamson Nanofluid Flow Over a Permeable Stretching Sheet with Viscous Dissipation

This paper explores the role of viscous dissipation, Arrhenius activation energy, and thermal radiation of Williamson nanofluid flow over a permeable stretching sheet. The governing partial differential equations have been simplified through a transformation process, resulting in a set of non-linear differential equations. To find a solution for these equations, a numerical approach is employed, specifically the fourth order Runge-Kutta method. Additionally, a shooting technique is utilized to enhance the accuracy of the numerical solutions. Overall, the study involves reducing complex equations, solving them numerically, and refining the results through a combination of methods. The study investigates the impact of different physical parameters on key factors like velocity, temperature, skin friction coefficient, nano particle volume fraction, and rates of mass and heat transfer. This study exhibits that activation energy parameter enhances concentration profiles, whereas fitted rate constant shows opposite behavior. The activation energy into heat transfer model allows for the optimization of heat transfer systems utilizing Williamson nano fluids.

Effects of Arrhenius Activation Energy on Thermally Radiant Williamson Nanofluid Flow Over a Permeable Stretching Sheet with Viscous Dissipation

This paper explores the role of viscous dissipation, Arrhenius activation energy, and thermal radiation of Williamson nanofluid flow over a permeable stretching sheet. The governing partial differential equations have been simplified through a transformation process, resulting in a set of non-linear differential equations. To find a solution for these equations, a numerical approach is employed, specifically the fourth order Runge-Kutta method. Additionally, a shooting technique is utilized to enhance the accuracy of the numerical solutions. Overall, the study involves reducing complex equations, solving them numerically, and refining the results through a combination of methods. The study investigates the impact of different physical parameters on key factors like velocity, temperature, skin friction coefficient, nano particle volume fraction, and rates of mass and heat transfer. This study exhibits that activation energy parameter enhances concentration profiles, whereas fitted rate constant shows opposite behavior. The activation energy into heat transfer model allows for the optimization of heat transfer systems utilizing Williamson nano fluids.

Impact of the Stefan gusting on a bioconvective nanofluid with the various slips over a rotating disc and a substance-responsive species

This paper presents thorough computational and theoretical analyses of steady forced convective flow over a rotating disc submerged in a water-based nanofluid containing microorganisms. It delves into the examination of boundary layer flow characteristics of a viscous nanofluid, considering Stefan blowing effects and multiple slip conditions influenced by a magnetic field. Notably, the study accounts for novel aspects such as thermal radiation and both constructive and destructive chemical reactions. The movement of nanoparticles is elucidated based on thermophoresis and microscopic behaviors, while changes in volume fraction do not affect the thermo-physical properties of the nanofluid. To address the altered nonlinear set of differential equations, an effective numerical approach, the Keller-Box method, is implemented for critical and efficient solutions. These appropriate transformations are defined and applied. When compared to blowing suction, it shows a better enhancement in the rate of heat transfer, mass, and microorganisms. Some of the main observations are there is a decrease in wall skin friction in the directions of radial and tangential as magnetic field strength is increased. The evaluation of thermal boundary layer thickness and temperature is noted for the radiation parameter (Rd) improvement. The present analysis has applications in electromagnetic micro-pumps and nanomechanics. As to the applications in the science and engineering fields, technologies such as micro-electromechanical systems-based microfluidic devices and microfluidic-related technologies will be accepted.

Dynamics of vertical axis washing machines with uncertain unbalance

A washing machine is a household appliance that has an interesting and complex dynamical behavior, which can be well described by a set of nonlinear differential equations. When analyzing the dynamics of a washing machine, the steady state motion (periodic solution) is an important response to consider and can be evaluated as a solution of a periodic boundary-value problem. The unbalance generated by the unevenly distribution of clothes during centrifugation is highly random and, therefore, a stochastic model is necessary to take this characteristic into account. The novelty of this paper consists in the analysis of a washing machine dynamics considering the uncertainty in the unbalance. Therefore, a stochastic model is proposed for the dynamics of a washing machine. The steady state solutions are calculated using the shooting method combined with a sequential continuation to evaluate it across all the spin speeds of the machine. The probability distributions of the washing machine vibration at those different spin speeds are approximated using Monte Carlo simulations. The impact of the random unbalance in the vibration amplitude of the washing machine is also investigated and consists in a key input for the fatigue design of many components.

Effect of ion channel guiding on the saturation mechanism of single and two-stream free-electron lasers based upon a rectangular hybrid wiggler

The paper studies a simulation of a one-dimensional nonlinear free electron laser that includes a square hybrid wiggler, prebunched electron beam, and ion-channel guiding mechanism. The Lorentz equations of motion and the Maxwell equations were combined to derive a set of non-linear differential equations solved numerically within the framework of slowly varying amplitude and wavenumber. The relativistic electron beam is assumed to be cold, and self-fields of the electron beam and the slippage of the radiation wave are ignored. The appropriate parameters are chosen in such a way that the Raman regime is being considered, where there is high density and low energy. The effect of ion-channel densities on the saturation of the device is studied for both groups Ι and ΙΙ orbits. It is found that as the ion channel density increases, the range of saturation radiation for group Ι circuits increases, too. In contrast, the saturated radiation amplitude for group ΙΙ orbits is decreased with increasing the ion-channel density. Moreover, the results of the numerical calculations show that the saturation length for an FEL with a prebunched beam is considerably shorter than that of a uniform electron beam. A free electron laser (FEL) model called the two-stream FEL, which uses two relativistic electron beams of different energy, is also studied. Comparing the radiation amplitude between FEL and TSFEL shows that TSFEL is more efficient than FEL.

Energy model in Bianchi type-V spacetime via RIF integration techniques

This paper investigates the Noether versus Killing symmetries and their associated conservation laws of Bianchi type-V spacetimes. The study is carried out using the Maple RIFSIMP package, which converts a set of nonlinear differential equations into the reduced involutive form (RIF). This technique generates an RIF tree, where each branch is solved to obtain the explicit forms of Noether symmetries. Separate RIF trees are used to represent each group. We derive a general expression for the Noether symmetry connected to homothety, and then we investigate this Noether symmetry in each separate group. To explore whether dark energy is associated with Bianchi type-V spacetime, we examine the energy density of each spacetime obtained in association with the Ricci scalar. The symmetries of the larger groups, 11- and 17-dimensional, are presented separately in the tables.

Numerical analysis of radiative hybrid nanomaterials flow across a permeable curved surface with inertial and Joule heating characteristics

The water-based Cu and CoFe2O4 hybrid nano liquid flow across a permeable curved sheet under the consequences of inertial and Lorentz forces has been reported in this analysis. The Joule heating and Darcy Forchheimer effects on fluid flow have been also examined. In the presence of copper (Cu) and cobalt iron oxide (CoFe2O4) nanoparticles, the hybrid nano liquid is synthesized. Radiation and heat source features are additionally incorporated to perform thermodynamics analysis in detail. The second law of thermodynamics is employed in order to estimate the overall generation of entropy. The nonlinear system of PDEs (partial differential equations) is transformed into a dimensionally-free set of ODEs (ordinary differential equations) by employing a similarity framework. The Mathematica built in package ND Solve method is applied to compute the resulting set of nonlinear differential equations numerically. Along with the velocity, and temperature profiles, skin friction and Nusselt number are also computed. Figures and tables illustrate the effects of flow factors on important profiles. Evidently, the outcomes reveal that hybrid nanofluid (Cu + CoFe2O4+H2O) is more progressive than nanofluid (Cu + H2O) and base fluid (H2O) in thermal phenomena. Furthermore, the velocity profile is improved with the greater values of curvature parameter, while the inverse trend is observed against the magnetic parameters. Also, the velocity and energy distribution of hybrid nano-liquid flow boosts with the inclusion of Cu and CoFe2O4 nanoparticles into the base fluid. Velocity distribution diminishes with the increment of volume friction. For high values of inertial factor, skin friction improve while velocity and Nusselt number declines.

A MATHEMATICAL STUDY ON THE EFFECT OF MOLAR RATIOS OF THE REACTANTS FOR BIODIESEL PRODUCTION IN SC-CO2 MEDIUM

The present global situation is so alarming that the conventional energy resources are decaying day by day. This adverse situation encourages us to introduce unconventional and sustainable energy resources like biodiesel. The main source of biodiesel is the non-edible vegetable oil (Jatropha curcas oil) and it is produced by chemical catalytic method. But recently supercritical method has also been proposed as an alternative cost-effective method for its production. A good quality of biodiesel can be produced in a Supercritical-Carbon dioxide medium. The productivity of biodiesel preparation through supercritical method depends on molar ratio of the reactants (water and methanol). In this research article, a set of nonlinear differential equations has been formulated based on concentrations of Triglyceride, Diglyceride, Monoglyceride, Methanol, Free fatty acid, Water, Glycerol and Biodiesel. Our study is based on the effect of the molar ratios of the reactants and how we can get a cost-effective production of biodiesel. The validity of our model is attained by experimental results. The analytical results are also verified by our numerical findings

Significance of improved heat conduction on unsteady axisymmetric flow of Maxwell fluid with cubic autocatalysis chemical reaction

This paper investigates the effect of chemical reactions on the flow of magnetized Maxwell fluid generated by an unsteady stretching surface. The thermal transport phenomenon is analyzed by using the Cattaneo-Christov theory. By applying the appropriate similarity transformations, the governing equations of motion turn into a set of nonlinear differential equations. For the velocity, temperature, and concentration fields, the resultant equations are then solved as series solutions using the homotopy analysis approach. Using graphical representations, the physical behavior of significant factors is examined in depth. The analysis reveals that higher Maxwell parameter values reduce the flow field while increasing energy transportation in the fluid flow. Further, it is noted that thermal distribution declines for the higher values of the thermal relaxation parameter. Additionally, the solutal distribution bootup for the increasing values of Schmidt number while it shows a decreasing trend for homogeneous and heterogeneous reactions strength. In order to verify our findings, a comparison to earlier research is also included.

Design of a Linear-Quadratic Regulator for the Control of an Oil-Heating Tank. Modeling and Simulation

In this paper, an oil-heating tank is presented, developing its respective modeling through the application of the laws of conservation of mass and energy. Then a set of nonlinear differential equations are obtained representative of the process. Subsequently, this system is linearized obtaining its linearized model. With this linear model, the design of two control systems is presented, based on a PI feedback output compensator and based on a linear-quadratic regulator, where the control objectives are the storage temperature and the fuel oil level. Finally, the process is simulated, verifying the feasibility of the operation with the designed control system

Mathematical modelling of heat and solutal rate with cross‐diffusion effect on the flow of nanofluid past a curved surface under the impact of thermal radiation and heat source: Sensitivity analysis

AbstractThis study analyses the effect of Brownian motion and thermophoresis on the flow of nanofluid along an impermeable curved surface. The Darcy‐Forchheimer drag vis‐à‐vis the radiating heat and the heat source enriches the flow phenomena. This drag force has several applications such as in (i) biomedical engineering for the flow of blood through curved arteries and veins, (ii) civil engineering for the flow of water through porous materials such as soil or rock, etc. The convective heat and solutal transport properties embedded in boundary conditions develop the heat transport phenomena. The dimensional governing equations are transformed into non‐dimensional form by using suitable substitution of transformed variables and stream function. Further, numerical practice is adopted to handle the set of nonlinear differential equations. The simulation of the optimized heat and solutal transfer rate for various factors is carried out using the ‘central composite design’ (CCD) associated with the ‘response surface methodology’ (RSM). The regression analysis and residual error are computed through the statistical approach of analysis of variance and finally, the sensitivity analysis is proposed for the various factors. However, the enhanced properties and flow behaviour for the diversified values of the physical parameters are presented and described briefly via the corroboration of the present methodology with the published work in particular cases. The notable outcomes are the axial velocity profile enriches for the increasing curvature constraints and the regression analysis is presented for the optimizing heat transfer rate using response surface methodology.

Numerical Investigation of Magnetohydrodynamic Flow of Reiner– Philippoff Nanofluid with Gyrotactic Microorganism Using Porous Medium

Nanoparticles facilitate the enrichment of heat transmission, which is crucial in many industrial and technical phenomena. The suspension of nanoparticles with microbes is another intriguing study area that is pertinent to biotechnology, health sciences, and medicinal applications. In the dispersion of nanoparticles, the conventional non-Newtonian fluid Reiner-Philippoff flows across a stretching sheet, which is examined in this article using numerical analysis. This study investigates the numerical investigation of Arrhenius reaction, heat radiation, and vicious variation variations on a Reiner-Philippoff nanofluid of MHD flow through a stretched sheet. Thus, for the current nanofluid, nanoparticles and bio-convection are highly crucial. The set of nonlinear differential equations is translated into Ordinary Differential Equations (ODEs) utilizing the requisite translation of similarities. These collected simple ODE are solved using the MATLAB computational tool bvp4c method. The graphical results for the velocity, concentration, motile microorganisms, and temperature profile are defined using the thermophoresis parameter and the Brownian motion respectively. Consider a tube containing gyrotactic microbes and a regular flow of nanofluid which is electrically conducted through a porous stretched sheet surface. This nonlinear differential problem is solved by a hybrid numerical solution method using fourth-order Runge-Kutta with shooting technique. The optimization method also performs well in terms of predicting outcomes accurately. As a result, the research applies the Bayesian Regularization Method (BRM) to improve the accuracy of the prediction results. Physical constraints are plotted against temperature, velocity, concentration, and microorganism profile trends and they are briefly described.

Chemically reactive magnetized tangent hyperbolic bidirectional nanofluid flow subject to interaction of gyrotactic moment of microorganisms and nanoparticles

The exploration of nanofluids have many applications in oils, microelectronics, compact heat exchangers, cooling of an infinite metallic plate, in power transportation, mineral water, plastic and rubber sheet manufacturing nuclear reactor design, electronics, solar energy, making it an identical influential scientific field. The inspiration of activation energy occurring within a chemical reaction has been extensively useful in manufacturing of oil reservation, biodiesel, geothermal accomplishment, oil, base liquid procedure, food manufacturing, oil amalgamate, sewage systems, along with a significant renewable energy source. We have studied the effect of MHD on steady 3D tangent hyperbolic nano fluidic flows through an extended surface containing gyrotactic microorganisms by considering new mass flux conditions in this research article. With the help of Rossland's approximation, we determine solar radiation and the activation energy describing the chemical reaction. By using suitable transformation, the mathematical modeling of the determinate flow can provide the set of non-linear differential equations. With the help of different parameters, we obtain a graphical interpretation of the motile density, local Nusselt number and Sherwood number. As an outcome of the stretching parameter and Prandtl number the temperature of the tangent hyperbolic liquid declines. By increasing the thermophoresis and magnetic parameter the nano fluid concentration initiates to ascend directly. The microorganism profile is found to be increasing for improving values of power law index parameter, while an opposite behavior can be observed for bio-convective Lewis number.

Dynamics of unsteady axisymmetric of Oldroyd-B material with homogeneous-heterogeneous reactions subject to Cattaneo-Christov heat transfer

In several industrial and engineering applications, the Catteno-Christov heat flux plays a major role in the flow and heat transport of non-Newtonian fluids. Therefore, the goal of this study is to investigate the characteristics of homogeneous-heterogeneous reactions in the axisymmetric flow of Oldroyd-B materials using heat conduction analysis. The heat transfer phenomenon is examined from the non-Fourier heat flux model perspective. The governing flow field equations are developed using the rheological formulation of the Oldroyd-B fluid model, which is transformed into a set of nonlinear differential equations by utilizing the proper similarity conversions. The series solutions for the velocity, thermal, and solutal distribution are derived. The physical behaviors of relevant parameters are discussed in detail. The findings reveal that for the curvature parameter, the fluid velocity improves near the cylinder surface and no motion occurs away from the surface, while this predicts the dual behavior on temperature and concentration distributions. Further, fluid relaxation and retardation time exhibit opposite behavior on fluid velocity. Additionally, the results demonstrate that the homogeneous response parameter exhibits contradicting behavior on the concentration field while the thermal relaxation parameter lowers the temperature field.

The Earth's rotational modes revisited

In the conventional treatment of the Earth's rotational dynamics using the Earth's angular momentum description (AMD), it is customary to assume that the velocity/displacement of a mass element in the liquid core (LC) has a displacement as well as an explicit rigid rotation component in addition to the uniform (solid-body) rotation. This makes for a very complex set of non-linear differential equations in the treatment of the dynamics of this body. In this work, I will use a simple three-layer Earth model with a rigid mantle (MT), a rigid inner core (IC), and an incompressible and homogeneous LC to show that in the alternative linearised dynamics of this body, it is redundant to assign a rigid rotation component to the motion. Next, I will use an approximation commonly used in dealing with the Earth's rotational dynamics, and further assume that the MT rotates uniformly, to show that the linearised equations yield identical analytical results to those in the literature for the periods of the inner-core wobble (ICW) and the free inner-core nutation (FICN).

Analytical Models of Intra- and Extratumoral Cell Interactions at Avascular Stage of Growth in the Presence of Targeted Chemotherapy.

In this study, we propose a set of nonlinear differential equations to model the dynamic growth of avascular stage tumors, considering nutrient supply from underlying tissue, innate immune response, contact inhibition of cell migration, and interactions with a chemotherapeutic agent. The model has been validated against available experimental data from the literature for tumor growth. We assume that the size of the modeled tumor is already detectable, and it represents all clinically observed existent cell populations; initial conditions are selected accordingly. Numerical results indicate that the tumor size and regression significantly depend on the strength of the host immune system. The effect of chemotherapy is investigated, not only within the malignancy, but also in terms of the responding immune cells and healthy tissue in the vicinity of a tumor.

Numerical solution for MHD flow and heat transfer of Maxwell fluid over a stretching sheet

In this article, the numerical solutions for the flow of heat transfer for an incompressible Maxwell fluid on a stretching sheet channel are presented in this study. By applying appropriate transformations, the system of governing partial differential equations is transformed into a system of ordinary differential equations. A successive linearization method (SLM) is used to describe and solve the resulting nonlinear equations numerically using MATLAB software. The main goal of this paper is to compare the results of solving the velocity and temperature equations in the presence of β1 changes through SLM for introducing it as a precise and appropriate method for solving nonlinear differential equations. Tables with the numerical results are created for comparison. This contrast is important because it shows how precisely the successive linearization method can resolve a set of nonlinear differential equations. Non-Newtonian parameters on the flow field, like mixed convection, Hartman, Deborah, and Prandtl numbers, are explored and illustrated graphically. Apart from that, a great deal of agreement has been seen between the current results and the published data that have been evaluated and compared in a limited way.

Massive particle pair production and oscillation in Friedman Universe: its effect on inflation

We study the classical Friedman equations for the time-varying cosmological term tilde{Lambda } and Hubble function H, together with quantised field equations for the production of massive Mgg H particles, namely, the tilde{Lambda }CDM scenario of dark energy and matter interactions. Classical slow components {mathcal O}(H^{-1}) are separated from quantum fast components {mathcal O}(M^{-1}). The former obeys the Friedman equations, and the latter obeys a set of nonlinear differential equations. Numerically solving equations for quantum fast components, we find the production and oscillation of massive particle-antiparticle pairs in microscopic time scale {mathcal O}(M^{-1}). Their density and pressure averages over microscopic time do not vanish. It implies the formation of a massive pair plasma state in macroscopic time scale {mathcal O}(H^{-1}), whose effective density and pressure contribute to the Friedman equations. Considering the inflation driven by the time-varying cosmological term and slowed down by the massive pair plasma state, we obtain the relation of spectral index and tensor-to-scalar ratio in agreement with recent observations. We discuss the singularity-free pre-inflation, the CMB large-scale anomaly, and dark-matter density perturbations imprinting on power spectra.

Modelling of a piezoelectric beam with a full-bridge rectifier under arbitrary excitation: experimental validation

Abstract This paper presents a set of nonlinear differential equations to model a piezoelectric energy harvesting (PEH) system with a full-bridge waveform rectifier (FWR) under arbitrary base excitations. The PEH comprises a piezoelectric element modeled as a current source with a capacitor in parallel, which are connected with an inductor and a resistor. The inductor is proposed to smooth the current generated by the piezoelectric element under rapid mechanical variations and improve the convergence of the set of differential equations. The equations are obtained using piecewise linear modelling for the diodes. The main advantage of this piecewise linear modelling is considering different bias points to represent the nonlinear characteristics of real diodes. Numerical simulations are employed to obtain the optimum inductor value through a comparison with an analytical result, validated with experimental tests. A real case of random acceleration in a bike is applied to the PEH-FWR to evaluate the performance of the proposed equations. They are validated with experiments, a LTspice formulation and a numerical previous one. The proposed formulation can estimate the output DC voltage and energy for a large range of excitation frequencies, including resonant and nonresonant conditions and arbitrary or harmonically externally excited PEHs with FWR.

Similarity Investigation of Thermosolutal Mixed Convection in a Saturated Porous Medium

This contribution investigates the thermosolutal convection phenomenon around a thin wall, vertically immersed in a fluid saturating a porous medium of non-uniform permeability, taking into account the thermal conditions of the wall which is exposed to a fluid suction/injection. The conservation equations with boundary conditions have been transformed by the similarity method into a set of nonlinear differential equations. The resulting equations are numerically resolved using a fifth-order Runge-Kutta scheme coupled with the shooting technique. A graphical and physical interpretation of the found results as a function of the control parameters was performed. It is noticed that the heat transfer rate and the mass transfer rate at wall, are intensified for free convection and for significant permeability.