Set Of Differential Equations Research Articles

On the augmentation of heat transfer with hybrid nanofluid containing microorganisms on flat plate under thermal radiation, using mixtures models

In this study, an investigation of a hybrid nanofluid flow across a stretched flat sheet is presented. On a flat, flexible surface, a water-based hybrid nanoliquid is propagating in a horizontal orientation, while a strong magnetic field is located along an upright direction to the flow. The flow is incompressible, magnetically influenced and electrically conducting. The aluminum – and graphene (Go) are dispersed in the working fluid. The main interest of this exploration corresponds to microorganism diffusivity with Brownian motion and thermophoresis diffusion for two different nanoparticles, which are modeled in a different set of equations. The flow and heat-mass transfer equations are constructed using Buongiorno's nanofluid model and hybrid nanofluid volumetric friction. The Runge–Kutta–Fehlberg procedure combined with the shooting procedure is employed to find the numerical solutions of governing equations. The results reveal that increasing magnetic force drags down the frictional coefficient. The highest energy transfer 19% was recorded with 1% of Al2O3 and 5% of Go. Higher Peclet number detracts the motile density. The faster heat transmission was observed by thermal radiation. The volumetric percentage of nanomaterials was influenced by both types of Brownian characteristics. The novel parameter [Formula: see text] influences the Sherwood and Nusselt numbers. When compared to Go, the temperature profile and the heat transmission rate for the second particle Al2O3 are higher. The Sherwood number declines with a rise in the thermophoresis parameter and rises with an improvement in the Lewis number and Brownian motion. Effect of [Formula: see text] and [Formula: see text] on mass transport rate are encouraging.

MHD boundary layer micropolar fluid flow over a stretching wedge surface: Thermophoresis and brownian motion effect

To investigate the consequence of thermophoresis and Brownian diffusion on convective boundary layer micropolar fluid flow over a stretching wedge-shaped surface. The effects of non-dimensional parameters namely coupling constant parameter (0.01 ≤ B1 ≤ 0.05), magnetic parameter (1.0 ≤ M ≤ 15.0), Grashof number (0.3 ≤ Gr ≤ 0.9), modified Grashof number (0.3 ≤ Gm ≤ 0.8), micropolar parameter (2.0 ≤ G2 ≤ 7.5), vortex viscosity constraint (0.02 ≤ G1 ≤ 0.2), Prandtl number (7.0 ≤ Pr ≤ 15.0), thermal radiation parameter (0.25 ≤ R ≤ 0.50), Brownian motion parameter (0.2 ≤ Nb ≤ 0.62), thermophoresis parameter (0.04 ≤ Nt ≤ 0.10), heat generation parameter (0.1 ≤ Q ≤ 0.5), Biot number (0.65 ≤ Bi ≤ 1.0), stretching parameter (0.2 ≤ A ≤ 0.5), Lewis number (3.0 ≤ Le ≤ 7.0), and chemical reaction parameter (0.2 ≤ K ≤ 0.7) on the steady MHD heat and mass transfer is investigated in the present study. The coupled non-linear partial differential equations are reduced into a set of non-linear ordinary differential equations employing similarity transformation. Furthermore, by using the Runge-Kutta method followed by the shooting technique, the transformed equations are solved. The main goal of this study is to investigate the numerical analysis of nanofluid flow within the boundary layer region with the effects of the microrotation parameter and velocity ratio parameter. The novelty of this paper is to propose a numerical method for solving third-order ordinary differential equations that include both linear and nonlinear terms. To understand the physical significance of this work, numerical analyses and tabular displays of the skin friction coefficient, Nusselt number, and Sherwood number are shown. The new approach of the present study contributes significantly to the understanding of numerical solutions to non-linear differential equations in fluid mechanics and micropolar fluid flow. Micropolar fluids are becoming even more of a focus due to the desire for engineering applications in various fields of medical, mechanical engineering, and chemical processing.

A semi-analytical rate-transient analysis model for fractured horizontal well in tight reservoirs under multiphase flow conditions

Abstract Rate-transient analysis (RTA) has been widely applied to extract reservoir/fracture properties using analytical and semi-analytical methods with simplifying assumptions. However, current RTA models may lead to misdiagnosis of flow regimes and incorrect estimates of reservoir/fracture information when complex fracture network, multiphase flow, and pressure-dependent properties occur in tight reservoirs simultaneously. A semi-analytical model is developed to account for multiphase flow, complex fracture network, and pressure-dependent properties. The technique uses the black oil formulation and butterfly model to determine three nonlinear partial differential equations (PDEs) that describe the flow of oil, gas, and water in the reservoir with complex fracture network. A modified Boltzmann variable considering the heterogeneity of complex fracture network is proposed to convert the fluid flow PDEs to a set of ordinary differential equations (ODEs) that can be solved through the Runge-Kutta method. A new rate-transient analysis workflow is also developed to improve flow regime identification (ID) and the accuracy for tight oil reservoirs with complex fracture network. It is applied to a synthetic case with equivalently modeled complex fracture network and multiphase flow. The estimated fracture properties are in excellent agreement with model inputs.

Inertial Parameter Identification for Closed-Loop Mechanisms: Adaptation of Linear Regression for Coordinate Partitioning

Abstract This study investigates the use of linear-regression-based identification in rigid multibody system applications. A multibody system model, originally described with differential-algebraic equations (DAE), is transformed into a set of ordinary differential equations using coordinate partitioning. This allows the identification framework (where the system is described with ordinary differential equations) to be applied to rigid multibody systems described with nonminimal coordinates. The methodology is demonstrated via numerical and experimental validation on a slider–crank mechanism. The results show that the presented methodology is capable of accurately identifying the system's inertial parameters even with a short motion trajectory used for training. The presented linear-regression-based identification approach opens new opportunities to develop more accurate multibody models. The resulting updated multibody models can be considered especially useful for state-estimation and the control of multibody systems.

Numerical simulation for radiative hybrid nanofluid (TiO2+Fe3O4∕H2O) flow due to a non-uniform stretching sheet with variable permeability

Variable permeability plays a crucial role in various manufacturing and technical applications such as fixed-bed catalytic reactors, heat exchangers, and dying, among others. The primary focus of this paper is to investigate the impacts of variable permeability on hybrid nanofluid (HNF) flow due to nonlinear sheet stretching with thermal heat flux. The HNF is made up of a mixer of [Formula: see text] and [Formula: see text] nanoparticles and water serving as the base fluid. Using efficient similarity transformations, the flow-governing equations are converted into a set of ordinary differential equations, and the resulting system is computationally solved by using the MATLAB program (bvp4c). The impact of various physical variables on the fluid velocity and heat transfer characteristics is analyzed via graphs. It is found that as the permeability parameter rises, the velocity of the HNF diminishes while the temperature amplifies. The drag force coefficient declines with an intensification of the volume fraction of the [Formula: see text] nanoparticles. The HNF [Formula: see text] exhibits 0.45–0.75% increase in heat transfer rate when compared to a nanofluid [Formula: see text] for different values of heat source parameter. The current investigation is compared to the existing literature, revealing a good level of agreement.

Utilizing a third order difference scheme in numerical method of lines for solving the heat equation

Obtaining the numerical solution for partial differential equations (PDEs) is a fundamental task in computational mathematics and scientific computing. In this work, we propose a third order numerical method of lines (MOL) based on difference schemes for the efficient and accurate approximation of heat equation. The method combines the flexibility of MOL with the superior accuracy of higher order difference schemes. The proposed approach begins by discretizing the spatial domain, and then applies higher order difference schemes to approximate the spatial derivatives in the PDE. The MOL framework is utilized to transform the resultant set of ordinary differential equations to a system of algebraic equations, facilitating their solution through an appropriate numerical method of choice. We have utilized the Adams method for this purpose. We analyze the stability of the proposed method and establish conditions under which it yields accurate and reliable solutions. We present the theoretical foundations of the method and discuss its implementation. We also present numerical results that demonstrate the procedure’s precision and effectiveness. The results show that the method is able to achieve high accuracy with relatively few grid points, and the results are compared with those obtained by some existing numerical schemes accessible in the literature.

Structural and practical identifiability analysis in bioengineering: a beginner’s guide

Advancements in digital technology have brought modelling to the forefront in many disciplines from healthcare to architecture. Mathematical models, often represented using parametrised sets of ordinary differential equations, can be used to characterise different processes. To infer possible estimates for the unknown parameters, these models are usually calibrated using associated experimental data. Structural and practical identifiability analyses are a key component that should be assessed prior to parameter estimation. This is because identifiability analyses can provide insights as to whether or not a parameter can take on single, multiple, or even infinitely or countably many values which will ultimately have an impact on the reliability of the parameter estimates. Also, identifiability analyses can help to determine whether the data collected are sufficient or of good enough quality to truly estimate the parameters or if more data or even reparameterization of the model is necessary to proceed with the parameter estimation process. Thus, such analyses also provide an important role in terms of model design (structural identifiability analysis) and the collection of experimental data (practical identifiability analysis). Despite the popularity of using data to estimate the values of unknown parameters, structural and practical identifiability analyses of these models are often overlooked. Possible reasons for non-consideration of application of such analyses may be lack of awareness, accessibility, and usability issues, especially for more complicated models and methods of analysis. The aim of this study is to introduce and perform both structural and practical identifiability analyses in an accessible and informative manner via application to well established and commonly accepted bioengineering models. This will help to improve awareness of the importance of this stage of the modelling process and provide bioengineering researchers with an understanding of how to utilise the insights gained from such analyses in future model development.

Tumor growth for remodeling process: A 2D approach

This paper aims to present a comprehensive framework for coupling tumor-bone remodeling processes in a 2-dimensional geometry. This is achieved by introducing a bio-inspired damage that represents the growing tumor, which subsequently affects the main populations involved in the remodeling process, namely, osteoclasts, osteoblasts, and bone tissue. The model is constructed using a set of differential equations based on the Komarova’s and Ayati’s models, modified to incorporate the bio-inspired damage that may result in tumor mass formation. Three distinct models were developed. The first two models are based on the Komarova’s governing equations, with one demonstrating an osteolytic behavior and the second one an osteoblastic model. The third model is a variation of Ayati’s model, where the bio-inspired damage is induced through the paracrine and autocrine parameters, exhibiting an osteolytic behavior. The obtained results are consistent with existing literature, leading us to believe that our in-silico experiments will serve as a cornerstone for paving the way towards targeted interventions and personalized treatment strategies, ultimately improving the quality of life for those affected by these conditions.

Elastoplastic solution for cylindrical cavity contraction in unsaturated soils

This paper develops an elastoplastic solution for cylindrical cavity contraction in unsaturated soils under constant suction conditions. The elastoplastic cavity contraction problem is formulated into a set of first-order ordinary differential equations by introducing a new auxiliary variable, which is solved as an initial value problem. The new solution is validated by comparison with numerical simulation results. Finally, parametric studies show that, as soil suction increases, the internal support pressure decreases faster with cavity contraction, the unloading-induced plastic zone becomes narrower, and the changes in effective stresses are smaller for a given tunnel convergence.

Entropy Generation and Thermal Radiation Impact on Magneto-Convective Flow of Heat-Generating Hybrid Nano-Liquid in a Non-Darcy Porous Medium with Non-Uniform Heat Flux

The principal objective of the study is to examine the impact of thermal radiation and entropy generation on the magnetohydrodynamic hybrid nano-fluid, Al2O3/H2O, flow in a Darcy–Forchheimer porous medium with variable heat flux when subjected to an electric field. Investigating the impact of thermal radiation and non-uniform heat flux on the hybrid nano-liquid magnetohydrodynamic flow in a non-Darcy porous environment produces novel and insightful findings. Thus, the goal of the current study is to investigate this. The non-linear governing equation can be viewed as a set of ordinary differential equations by applying the proper transformations. The resultant dimensionless model is numerically solved in Matlab using the bvp4c command. We obtain numerical results for the temperature and velocity distributions, skin friction, and local Nusselt number across a broad range of controlling parameters. We found a significant degree of agreement with other research that has been compared with the literature. The results show that an increase in the Reynolds and Brinckmann numbers corresponds to an increase in entropy production. Furthermore, a high electric field accelerates fluid velocity, whereas the unsteadiness parameter and the presence of a magnetic field slow it down. This study is beneficial to other researchers as well as technical applications in thermal science because it discusses the factors that lead to the working hybrid nano-liquid thermal enhancement.

Role of individual component failure in the performance of a 1-out-of-3 cold standby system: A Markov model approach

Abstract The current work assesses the availability and reliability of repairable 1-out-of-3 cold standby systems, undergoing switching failure, by the Markov models, where a finite set of differential equations govern the repairable system. A numerical approach based on KAOS programming in MATLAB is employed to solve the differential equations. Thus, numerical solutions for the time-dependent availability can then be obtained. The Laplace transform method is used for the reliability model to find the explicit form of the reliability function and the mean time to failure (MTTF) of the system. An availability estimation model for a 1-out-of-3 cold standby system is developed and the main electrical power network, local electricity generator, and domestic electricity generator are studied. Finally, the model is analyzed to illustrate the effect of both failures and repair rates as well as component capacity partitioning on system availability and MTTF.

Mathematical model of second-grade Casson fluid flow with Soret and Dufour impact over Riga sheet

We are examining a mathematical model of second-grade Casson fluid flow over a Riga surface. This analysis takes into account the effects of heat generation on a second-grade Casson fluid. Additionally, we have discussed the impacts of Soret and Dufour in the presence of thermal radiation. The study also includes an analysis of the influences of thermal and concentration slip. By considering these factors in the fluid flow, we have constructed a set of partial differential equations. Furthermore, these differential equations have been transformed into dimensionless ordinary differential equations. We have presented the influences of relevant parameters through tables and graphs. Velocity curve reveals declining behavior due to larger values of magnetic field factor. A high magnetic field coefficient means a strong magnetic field. As a result, the force acting on the charged particle increases, causing it to slow down or change direction. This change in force can cause the particle’s velocity to decrease.

Numerical treatment for double diffusion phenomenon with generalized heat flux effects in 3d nanomaterial flow over bi-directional stretched wall

Current study aims to numerically investigate the bi-directional flow of nanofluids in the presence of Cattaneo–Christov double diffusion for two heat sources, namely, prescribed surface temperature (PST) along with prescribed heat flux (PHF). The conventional mathematical model in partial differential equations (PDEs) of the system have been reduced into an equivalent set of ordinary differential equations (ODEs). The system dynamics in the form of approximate solutions of the ODEs is presented by Adams and Explicit Runge Kutta numerical solvers. For better understanding the influence of physical parameter of interest including Brownian movement parameter, thermophoresis parameter, concentration relaxation parameter, Schmidt number, Prandtl number, thermal relaxation parameter, temperature ratio parameter, stretching ratio parameter and Deborah number on temperature, as well as concentration profiles are presented exhaustively. Appropriateness of the scheme is ascertained through close resemblance of results for different state of the art counterparts with negligible error around 10–05 to 10–09.

Adaptive total energy shaping of a class of manipulators

AbstractPoint‐to‐point control of underactuated mechanical systems is a challenging problem, especially if some of the system's parameters are uncertain. A popular method to stabilize underactuated manipulators is total energy shaping, in which a new Hamiltonian function is assigned to the closed‐loop system while a set of partial differential equations (PDEs) should be solved. In this study, an adaptive total energy shaping controller based on the approach called interconnection and damping assignment passivity‐based control (IDA‐PBC) for some underactuated robots is designed. The proposed method can overcome uncertainties in the dynamical parameters and also the parameters of the input mapping matrix under the satisfaction of the PDEs. The stability of the desired pose is ensured by the Lyapunov approach via suitable design of adaptation laws without the requirement for the persistence of excitation condition or other similar conditions. The results are applied to a 3‐DOF underactuated manipulator and verified by simulations.

Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder

Abstract The ternary hybrid nanofluid leads to a significant enhancement in thermal performance applications like heat transfer in automotive engines, solar thermal energy storage, aerospace, and electronic cooling. The present study investigates the thermal characteristics of a ternary hybrid magnetized and radiated cross nanofluid comprising Al2O3, TiO2, and Ag nanoparticles in water subjected to combined convection flow around a vertical cylinder. Furthermore, innovative effects of the magnetic field, absorber surface of the cylinder, non-linear thermal radiations, and effective thermophysical characteristics of ternary nanofluid are taken, and a new model for heat transport is successfully achieved. The governing equations in the form of partial differential equations (PDEs) are obtained through Navier–Stokes and heat equations by applying current assumptions. The system of PDEs is converted into a set of ordinary differential equations (ODEs) via a similarity variable. The built-in code bvp4c in Matlab software further exercises the dimensionless ODE equations numerically. Adding multiple nanoparticles and the magnetic field effect enhances the heat transfer rate in the ternary hybrid cross nanofluid. The Weissenberg number reduces the velocity, the radiation parameter increases heat transport, and the increased volume friction of nanoparticles enhances thermal conductivity and rapid heat transport.

Degeneration phenomenon in linear ordinary differential equations

Abstract Given a linear constant coefficient ODE depending on a parameter, when this parameter approaches zero, the solution set converges to the solution set of the limit differential equation if the leading coefficient does not vanish. The situation is very subtle in the singular case, i.e., in the case when this coefficient becomes zero. The solution set then may even collapse completely. In this note, a formalism is developed in which the solution set of a linear constant coefficient ODE always depends continuously on the equation coefficients.

Painlevé analysis, Painlevé–Bäcklund, multiple regular and singular kink solutions of dynamical thermopherotic equation drafting wrinkle propagation

The thermophoretic motion (TM) system with a variable heat transmission factor, based on the Korteweg-de Vries (KdV) equation, is used to model soliton-like thermophoresis of creases in graphene sheets. Painlevé test is employed to discover that the equation is Painlevé integrable. Then an auto-Bäcklund transformation using the truncated Painlevé expansion is obtained. Concerning the additional variables, the auto-Bäcklund transformations convert the nonlinear model to a set of linear partial differential equations. Finally, various explicit precise solutions based on the acquired auto-Bäcklund transformations are investigated and the researched solutions are illustrated in 3D, 2D and contour plots. Furthermore, the Cole-Hopf transformation is used in conjunction with Hirota’s bilinear technique to get multiple regular and singular kink solutions.

Transformer oil‐based Casson ternary hybrid nanofluid flow configured by a porous rotating disk with hall current

AbstractThe present work examines the 3D, incompressible, magnetohydrodynamic, mixed convective, and Darcy‐Forchheimer Casson /transformer oil ternary hybrid nanofluid flow configured by a porous rotatory disk. This research work takes into account the influences of hall current, thermophoresis, heat absorption/generation, Brownian motion, joule heating, chemical reaction, thermal radiation, activation energy, and velocity slip condition. The administrative partial differential equations (PDEs) that expound the problem are modified into a set of ordinary differential equations (ODEs) by imposing a similarity conversion, which is further solved numerically by adapting the worthy bvp4c scheme. The velocity, thermal, and mass profiles are delineated through graphs, and the features of skin friction coefficient, heat, and mass transmission are also explicated thoroughly via tables. One significant observation of this research is that the Casson parameter weakens transversal velocity whereas a reverse tendency is noticed against the radial velocity. Our obtained outputs affirm that Casson ternary hybrid nanofluid has respectively 10.93 % and 8.86% higher heat deliverance rate when compared to Casson nanofluid and Casson hybrid nanofluid. Besides, the Casson ternary hybrid nanofluid has respectively 33.68 % and 27.52% higher mass deliverance rate when compared to the Casson nanofluid and Casson hybrid nanofluid. Also, the sturdy values of the Casson parameter lessen the skin friction coefficient. The inferences derived from our research may advantageously be used in rotating types of machinery, medical equipment, gas turbine rotators, etc.

Illustration of thermal radiation on the mixed convection of Williamson nanofluid over an inclined cylindrical surface

This study emphasizes the two-dimensional incompressible flow of Williamson nanofluid caused by an inclined, infinite, expanding/contracting cylinder that is inclined at an angle with the horizontal axis. The focus is on the behavior of non-Newtonian fluid on the flow, heat, and mass transfer during the process. The properties of the fluid are modeled using a set of partial differential equations in association with the characterizing parameters obtained during the presentation of non-dimensional form. The problem is solved numerically using the shooting-based Runge–Kutta fourth-order technique. The results are shown graphically for the behavior of the several contributing parameters. The effect of various pertaining parameters is analyzed henceforth. The physical quantities of interest are also discussed in the tables. The study provides useful insight into the behavior of the fluid, i.e. suction/injection shows opposite behavior on the fluid velocity profiles where suction decelerates the profile and injection enhances it significantly.

A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium

Abstract The analysis of the fluid flow with the energy transfer across a stretching sheet has several applications in manufacturing developments such as wire drawing, hot rolling, metal extrusion, continuous casting, paper production, and glass fiber fabrication. The current examination presents the hybrid nanofluid flow past a convectively heated permeable sheet. The ferrous oxide (Fe3O4) and Gold (Au) nanoparticles have been dispersed in the blood. The significances of thermal radiation, inclined magnetic field, and space-dependent heat source have been observed in this work. The modeled equations are presented in the form of partial differential equations and reformed into the set of ordinary differential equations (ODEs) by using the similarity substitution. The Matlab built-in package (bvp4c) is employed to resolve the transform nonlinear set of ODEs. The significance of flow constraints versus the velocity and temperature profiles is demonstrated in the form of Figures and Tables. The numerical outcomes for the physical interest quantities are presented in tables. It has been perceived from the results that raising the angle of inclination from 0° to 90° reduces both the velocity and energy profile. The escalating values of Eckert number, constant heat source, and space-dependent heat source factor accelerate the temperature profile. The velocity and temperature distributions are very effective in the cases of hybrid nanofluid (Au–Fe3O4/blood) when compared to nanofluid (Au/blood). The skin friction and rate of heat transfer are very effective in the cases of hybrid nanofluid (Au–Fe3O4/blood) when compared to nanofluid (Au/blood).