Differential Equations Of Energy Research Articles

Evaluation of the effect of rotational speed and rheological nature on heat transfer of complex fluid between two cylinders

This paper presents numerical results for flow behavior between a cold inner cylinder and a hot outer cylinder. Both cyl-inders are placed horizontally. The space separating the two compartments is completely filled with a fluid of a complex rheological nature. In addition, the outer container is subjected to a constant and uniform rotational speed. The results of this work were obtained after solving the differential equations for momentum and energy. The parameters studied in this research are: the intensity of thermal buoyancy, the speed of rotation of the outer container and the rheological nature of the fluid. These elements are expressed mathematically by the following values: Richardson number (Ri = 0 and 1), Reyn-olds number (Re = 1 to 40), power-law number (n = 0.8, 1 and 1.4) and Prandtl number (Pr = 50). The results showed that the speed of rotation of the cylinder and the rheological nature of the fluids have an effective role in the process of heat transfer. For example, increasing the rotational speed of the enclosure and/or changing the nature of fluid from shear-thickening into shear-thinning fluid improves the thermal transfer rate.

Flow field and heat transfer characteristics in dimple pipe with different shape of dimples

In this work, a numerical study of a thermal performance of water flow inside a dimpled pipe. The effect of three types of dimples (circular, square and rhombus) studied in the numerical simulation. A commercial program called ANSYS was used to model the flow through a circular pipe .The three-dimensional governing differential equations of mass, momentum, and energy were used together with the (K − ε ) model to evaluate the impact of dimples on a turbulent flow and the velocity field. The study was carried out in the Reynolds number (Re) range (2500–15000). The research results demonstrate that the presence of a dimple on the pipe surface greatly increases the rate of heat transmission and the friction factor compared to a normal smooth pipe. Also, the numerical study demonstrated that the Nusselt number (Nu) in case of circular  dimples  at diameter  (4 , 6 and 8) mm was (22, 28 and 43) % greater than the smooth surface.  It is discovered that the improved pipe with circular dimples have a benefit for increased heat transmission efficiency compared with the square and rhombus dimples. Additionally , circular dimples have the ability to supply the lowest friction factor (f) when compared to other types of dimple. The pipe with circular dimples with D= 4mm , at Reynolds number 2500 provided the largest thermal performance criterion (PEC) value about 1.44.

Insight into the peristaltic motion through a tapered channel with Newton’s cooling subject to viscous dissipation, Lorentz force, and velocity slip

Peristalsis has gained significant attention due to its numerous applications in the medical field, engineering, and manufacturing industries. Therefore, the current work intends to look into the effects of variable liquid properties on the magnetohydrodynamics of peristaltic flow exhibited by viscous fluid through a tapered channel. The viscosity of the liquid differs over the thickness of the channel, and temperature-dependent thermal conductivity is considered. The constitutive relation for energy is formulated with the addition of viscous dissipation and heat generation/absorption. The assumption of velocity slip along with the convective boundary condition energizes the thermal system as well as the flow phenomena. The mathematical formulation is established on the grounds of low Reynolds number and long wavelength approximations. Perturbation solution were obtained for the resulting non-linear differential equations of momentum and energy for small values of variable viscosity and variable thermal conductivity. The effects of various relevant parameters on flow properties were investigated through graphical analysis. The results show that the maximum velocity does not occur in the middle of the tapered channel, but moves toward the upper wall with the increase in the variable viscosity difference between the walls. The application of viscosity is essential in many engineering and industrial processes.

Entropy analysis of electromagnetic trihybrid nanofluid flow with temperature-dependent viscosity in a Darcy-Forchheimer porous medium over a stretching sheet under convective conditions

This study introduces a novel theoretical model known as the trihybrid nanofluid (THNF), which demonstrates remarkable thermal conductivity properties to enhance heat transfer in liquids. The base liquid (H2O) was employed to immerse nanoparticles of Cu, Al2O3, and TiO2, thereby resulting in the creation of a THNF (Cu + Al2O3 + TiO2/H2O). This research focuses on the entropy generation of electromagnetic THNF flow in a Darcy-Forchheimer porous medium over a stretching sheet under convective conditions. The analysis considers the intricate interplay between various factors, including convection, radiation, heat source, chemical processes, and cross-diffusion effects. Complex partial differential equations for flow, energy, and concentration are transformed into nonlinear ordinary differential equations using similarity transformations. The Runge–Kutta–Fehlberg method with a shooting mechanism is used to solve these ODEs. The comprehensive graphical analysis visually exemplifies the impact of significant parameters on the profiles of velocity, temperature, and concentration, in conjunction with skin friction, Nusselt, and Sherwood numbers. The flow velocity decreased by 8.89 % in NFs with higher viscosity and Darcy-Forchheimer porous medium values, but it accelerated by 13.64 % in THNF with more convection. Electric fields boost THNF velocity and temperature by 20.94 % and 14.27 % respectively, while stronger magnetic fields reduce NF velocity by 15.34 % and elevate THNF temperature by 7.52 %. THNFs show the largest temperature increase of 18.67 % due to radiation, while it decreases by more than 7.51 % for NFs with an increasing Prandtl number. NP concentration drops by 10.19 % when Schmidt and chemical reactions increase. Elevating electric and magnetic fields can reduce heat transfer by 4.78 % and entropy generation by at least 6.45 % in NFs, while increasing the Bejan number by up to 18.83 % in THNFs. Radiation and diffusion increase the entropy generation by 6.71 % and decrease the Bejan number by 7.98 %.

Analysis of the Exothermic Reaction of Flame Ignition in the Combustion Chamber of a Gas Turbine Unit

This article analyses the exothermic reaction of flame ignition in the combustion chamber of a gas turbine unit, which is characteristic of combustion chambers operating on traditional hydrocarbon fuels. The combustion of gases as an explosive process in confined and semi-enclosed areas remains a poorly understood section of thermal physics. Without a detailed review of the physical and chemical processes taking place in the combustion chamber, it cannot be said whether the gas turbine unit will run sustainably. It is also important to know what combustion modes are in principle possible after a loss of stability in the combustion chamber in order to take action against this in advance. To describe flame ignition and quenching in the flow of the fuel–air mixture through a combustion chamber, a system of differential conservation equations of energy and reactive species supplemented with the equation of state is used. Nonstationary combustion processes in gas-turbine engines were studied, and flame ignition and blow-off were determined by the heat balance and by the continuity of chemical processes. Calculation methodologies for various operating modes of the combustion chamber of a gas turbine unit are developed and realized. The results of the calculations that were carried out are presented with graphical interpretation and with their analysis provided in sufficient detail. Based on this analysis, recommendations are then provided. From the graphs, it can be observed that the combustion chamber of a gas turbine unit reaches its maximum limit of stable operation at the optimum value of the reduced flow velocity in the openings of the air supply to the combustion and the mixing zones of the flame tube (λOC)opt = 0.22 when the fuel–air mixture is at maximum depletion, ensuring that combustion does not stop and flame failure does not occur. The topic of this article relates to the intensification of hydrocarbon fuel combustion and the technological improvement of combustion chambers in gas turbine units. This topic is of exceptional importance and relevance, emphasizing its significance. The purpose of this work is to develop and implement a methodology for calculating various modes of operation of the combustion chamber of a gas turbine unit.

Heat and mass transfer correlations of supercritical n-decane considering the influence of pyrolysis reaction by simplifying internal heat and mass sources as pseudo-wall fluxes

The convective heat and mass transfer characteristics of the supercritical pressure (SCP) fuel in the regenerative cooling channel of hypersonic vehicles are significantly affected by pyrolysis reactions. In this study, we proposed a novel method to develop the Nusselt number (Nu) and Sherwood number (Sh) correlation of SCP fuel that considers the influence of pyrolysis reaction by simplifying internal heat and mass sources as pseudo-wall fluxes based on the differential global reaction (DGR) model of pyrolysis. The convective heat and mass transfer and pyrolysis characteristics of n-decane were investigated experimentally and numerically in a vertical downward-flow heated tube at pressures of 3-7 MPa. Theoretical analysis of the experimental conditions was conducted sequentially according to the partial differential equations (PDEs) of species continuity and energy, in which the internal source terms of heat and mass due to pyrolysis reactions were simplified and predicted using the DGR model of n-decane pyrolysis. A semi-empirical Sh correlation for predicting the radial distribution of fuel conversion was proposed associated with computational fluid dynamics (CFD) data by equivalently simplifying the internal mass source as pseudo-wall mass flux. A Nu correlation for convective heat transfer was proposed using the same methodology considering the influences of pyrolysis. The correction terms consider pyrolysis endothermicity and thermophysical property variation due to the conversion differences between the pipe wall and the bulk flow. Finally, the Nu correlation was compared to the experimental data, and the influence of heat transfer deterioration (HTD) due to pyrolysis coking was also analyzed quantitatively and empirically.

Computational method for energy transport of MHD nanofluids flow near non-aligned stagnation point with non-linear thermal radiation and interface slip

Haemodialysis, nuclear reactors, cooling and heating systems, and heat exchangers are glimpses of applications where thermal problems are considered as a fundamental for flow over aligned stagnation points with heat and mass transfer components. The purpose of this study is to present steady dual solutions for non-aligned stagnation point flow of nanofluid over a linearly contracting/expanding surface under temperature and velocity slips. The mathematical model is established using boundary layer assumption. The authors acknowledge from prior investigations and studies that the non-aligned stagnation point flow across a contracting/expanding surface is highly rare in the field of engineering and fluid dynamics research. In the backdrop of non-linear thermal radiation, the problem of oblique hydromagnetic stagnation point flow of an electrically conducting optically dense viscous incompressible nanofluid across a convectively heated stretching sheet is formulated. The effects of Brownian motion and thermophoresis are incorporated into the nanofluid model. With the use of appropriate similarity transformations, the governing nonlinear partial differential equations for momentum, energy, and nanoparticle concentration are converted into a set of nonlinear ordinary differential equations. Using the BVP4c approach, the modified equations are numerically solved. It is discovered that when the strength of the magnetic field increases, the non-alignment of the re-attachment point on the sheet surface becomes less pronounced. The thickness of the thermal boundary layer grows as the magnetic and Biot parameters expand. The extent of the skin friction factor ascends with increasing magnetic field and suction variable inputs. The heat transfer rate detract with radiative parameter.

Numerical Solution of Stagnation Point Flow of Non-Newtonian Fluid and Heat Transfer over a Porous Medium under Effect of Thermal Radiation and Magnetic Field

We study three key topics in this article, including Casson fluid flow at its stagnation point, heat transfer in the current magnetic field, and thermal radiation. Several equations related to coupled mass, momentum, and energy are controlled by the mathematical model. This is accomplished by using appropriate transformations to study dimensionless partial differential equations of motion, energy, and continuity. The Adam-Bashforth method is used to solve several nonlinear dimensionless partial differential equations. The analysis is based both on analytic and numerical representations of the impact of parameters which are relevant to the analysis, such as the Casson parameter β, porosity parameter K, magnetic field M, mixed convection parameter γ, and thermal radiation R. Finally, using the Adam-Bashforth approach, we are able to determine velocity profiles as well as temperature profiles.

Investigating the Influence of Chemical Reaction on MHD-Casson Nanofluid Flow via a Porous Stretching Sheet with Suction/Injection

This study aims to investigate the influence of chemical reaction on the flow characteristics of a magnetohydrodynamic (MHD) Casson nanofluid over a porous stretching sheet with suction/injection. The nanofluid (NF) is comprised of a base fluid with suspended nanoparticles (NPs), and the Casson fluid (CF) model is employed to capture the non-Newtonian behavior. The governing partial differential equations (PDEs) for momentum, energy, and NP concentration are derived, incorporating the effects of magnetic field, viscous dissipation, chemical reaction, and porous medium. The resulting system of nonlinear ordinary differential equations (ODEs) is solved numerically using the Runge-Kutta-Fehlberg method together with the shooting process. The effects of various physical parameters, such as magnetic field strength, porous medium, CF parameters, NP volume fraction, suction/injection parameter, and chemical reaction parameter, on the flow characteristics are examined in detail. The results reveal that faster movement is linked to higher Grashof numbers and porous mediums, but weaker magnetic fields slow it down. The suction/injection affect velocity inversely. The Prandtl and Eckert numbers have opposite effects on temperature fields. The thermophoresis and Brownian motion parameters affect the opposite trends of the concentration and temperature distributions. The concentration reduces with chemical reaction parameters and the Lewis number. Heat transfer (HT) is enhanced for higher Brownian and thermophoresis. The findings of this study can have potential applications in various engineering fields, such as microfluidics, chemical processing, and thermal management systems, where precise control of fluid flow and heat transfer is essential.

Computation of stagnation coating flow of electro-conductive ternary Williamson hybrid mathrm{GO}-mathrm{AU}-{mathrm{Co}}_{3}{mathrm{O}}_{4}/mathrm{EO} nanofluid with a Cattaneo–Christov heat flux model and magnetic induction

Modern smart coating systems are increasingly exploiting functional materials which combine multiple features including rheology, electromagnetic properties and nanotechnological capabilities and provide a range of advantages in diverse operations including medical, energy and transport designs (aerospace, marine, automotive). The simulation of the industrial synthesis of these multi-faceted coatings (including stagnation flow deposition processes) requires advanced mathematical models which can address multiple effects simultaneously. Inspired by these requests, this study investigates the interconnected magnetohydrodynamic non-Newtonian movement and thermal transfer in the Hiemenz plane's stagnation flow. Additionally, it explores the application of a transverse static magnetic field to a ternary hybrid nanofluid coating through theoretical and numerical analysis. The base fluid (polymeric) considered is engine-oil (EO) doped with graphene left(GOright), gold left(Auright) and Cobalt oxide left(C{o}_{3}{O}_{4}right) nanoparticles. The model includes the integration of non-linear radiation, heat source, convective wall heating, and magnetic induction effects. For non-Newtonian characteristics, the Williamson model is utilized, while the Rosseland diffusion flux model is used for radiative transfer. Additionally, a non-Fourier Cattaneo–Christov heat flux model is utilized to include thermal relaxation effects. The governing partial differential conservation equations for mass, momentum, energy and magnetic induction are rendered into a system of coupled self-similar and non-linear ordinary differential equations (ODEs) with boundary restrictions using appropriate scaling transformations. The dimensionless boundary value problem that arises is solved using the bvp4c built-in function in MATLAB software, which employs the fourth-order Runge–Kutta (RK-4) method. An extensive examination is conducted to evaluate the impact of essential control parameters on the velocity f{^{prime}}left(zeta right), induced magnetic field stream function gradient g{^{prime}}left(zeta right) and temperature theta left(zeta right) is conducted. The relative performance of ternary, hybrid binary and unitary nanofluids for all transport characteristics is evaluated. The inclusion of verification of the MATLAB solutions with prior studies is incorporated. Fluid velocity is observed to be minimized for the ternary mathrm{GO}–mathrm{Au}–{mathrm{Co}}_{3}{mathrm{O}}_{4} nanofluid whereas the velocity is maximized for the unitary cobalt oxide left({mathrm{Co}}_{3}{mathrm{O}}_{4}right) nanofluid with increasing magnetic parameter (beta ). Temperatures are elevated with increment in thermal radiation parameter (Rd). Streamlines are strongly modified in local regions with greater viscoelasticity i.e. higher Weissenberg number (We). Dimensionless skin friction is significantly greater for the ternary hybrid GO–Au–C{o}_{3}{O}_{4}/EO nanofluid compared with binary hybrid or unitary nanofluid cases.

MODELING OF HUMAN HEAT TRANSFER IN URBAN BUILDINGS WITH CONSIDERING THE EFFECT OF HEAT ISLANDS

Computer modeling of heat transfer in the air volume of residential buildings was carried out taking into ac-count the effect of heat islands. The model includes directly the air volume, limited on both sides by the surfaces of asphalt and the wall of the building, as well as a model of the human body located at a given distance from the wall. In accordance with the climatic features of the territory of the Oryol region in the summer months with low cloudiness, the initial conditions of the computational problem are formed, which determine the modes of heat ex-change of heated surfaces with the air of residential buildings. For each of these modes, numerical modeling of natu-ral convective processes in the considered air volume was carried out. For this, differential equations of energy, mo-tion and continuity are solved for the system "complex of heated surfaces - air volume" for each combination of initial conditions, as well as various positions of the human body model relative to the heated wall of the building. It has been established that as a result of heat exchange in the air volume, a convective cell is formed, the shape and dimen-sions of which are determined by the temperature regime of building walls and asphalt. The influence of heat transfer processes on the surface of the human body on the parameters of this cell is revealed. The shape of this cell is estimat-ed under various initial conditions. Differences in heat fluxes on the surfaces of the human body facing the building and away from it are estab-lished. The dependences of the convective heat transfer of a person on the distance between him and the heated wall of the building are obtained. The quantitative values of the parameters of convective heat transfer of the body under various heat transfer modes, in particular, the temperature and speed of air movement in the upward flow, as well as the values of natural convective heat transfer, are determined.

Examination of bio convection with nanoparticles containing microorganisms under the influence of magnetism fields on vertical sheets by five-order Runge-Kutta method

In this paper, we analyzed vertical bio convection in nanofluids containing microorganisms. The novelty of this article is the numerical and analytical investigation of magnetic flow, radiation heat transfer, and viscous dissipation on bio convective fluid flow using the Five-order Runge-Kutta technique. Utilizing similitude parameters, determined ODE (ordinary differential equation) equations from partial differential equations for continuity, momentum, energy, and nanofluid concentration. Five-order Runge-Kutta was then used to solve the equations. The results show that it has a more significant influence on and then and. In addition, it exerts a force on neighboring particles, which causes them to shift from a hot zone to a great region. The density of microorganisms inside a part rises as it grows; when Le rises and Ha remains the same, x(ξ) falls, and When Ha rises, and Le remains the same, x(ξ) fall.

Effects of viscous dissipation, internal heat source/sink and Prandtl number on flow and heat transfer in a moving fluid over a moving flat surface with an applied magnetic field

The goal of this study is to see how heat sources and sinks, as well as viscous dissipation, affect natural convection magneto hydrodynamic flow of an incompressible moving fluid across a moving flat surface. Using partial differential equation similarity finding, additional momentum and energy equations were altered into non-linear ordinary differential equations. Keller box technique is a finite difference system that produces the desired results. The influences of movement, internal heat source/sink, Prandtl number, Eckert number and MHD parameter are clearly seen on the velocity and temperature outlines. The thermal boundary layer thickness is influenced by heat source/sink, Prandtl number and viscous dissipation.

Homotopy analysis of mixed convection flow of a magnetized viscoelastic nanofluid from a stretching surface in non-Darcy porous media with revised Fourier and Fickian approaches

This article addresses theoretically the mixed convection hydromagnetic flow of electrically conducting viscoelastic nanofluid from a vertical permeable stretching sheet in a non-Darcy porous medium with Cattaneo–Christov double-diffusion model to evaluate the heat transfer phenomena in steady boundary layer flow on a stretchable surface. In this regard, thermal and solutal hyperbolic wave relaxation effects are included for non-Fourier and non-Fickian models. Heat absorption is also included in the analysis. The Buongiorno two-component nanoscale model is adopted for simulating Brownian motion and thermophoretic body force effects. A non-Darcy drag force model is employed for the porous medium and the Reiner–Rivlin second-grade model for non-Newtonian characteristics. Via appropriate dimensionless similarity variables, the nonlinear dimensional partial differential conservation equations for momentum, energy and concentration with associated boundary conditions are rendered into a nonlinear dimensionless ordinary differential boundary value problem. The homotopy analysis method (HAM) is utilized to solve the boundary value problem and the impact of emerging parameters including thermal relaxation parameters, non-Newtonian material parameter, Darcy permeability parameter, porous inertial parameter and Hartmann magnetic number on the momentum, heat and mass transfer characteristics are visualized graphically and in tables. A 30th-order approximation for HAM is shown to produce sufficient accuracy for the velocity and temperature fields and a 40th-order estimates is adequate for the concentration field. It is observed that increasing the magnitude of thermal and solutal relaxation parameters has the opposite effect on the thermal and concentration distributions. The novelty of the work is the simultaneous inclusion of multiple effects (non-Fourier and non-Fickian thermal relaxation hyperbolic wave models, non-Darcy drag effects and heat generation/absorption) which are relevant to rheological nanomaterials processing and also the deployment of homotopy analysis as an alternative to conventional numerical methods such as finite differences, finite elements and MATLAB solvers. The study is relevant to the manufacture of electro-conductive polymers (ECPs) and smart (functional) magnetic nano-liquids.

MHD FLOW PAST OVER OSCILLATING VERTICAL PLATE THROUGH POROUS MEDIA IN THE PRESENCE OF HALL EFFECT AND CHEMICAL REACTION

MHD flow past over oscillating vertical plate with variable temperature and mass diffusion through a porous medium in the presence of Hall current effect and chemical reaction is studied here. The MHD fluid model contains partial differential equations of momentum, energy and diffusion equations. The Laplace-transform technique used to find the exact solution of equations involved in this paper. The resultant velocity profile is discussed with the help of graphs drawn for different parameters like, inclination of magnetic field, chemical reaction parameter, phase angle parameter and permeability parameter, and the numerical values Sherwood number have been also tabulated. It is noticed that the values obtained for velocity, concentration and temperature profiles are in concurrence with the actual flow of the unsteady MHD fluid.

A Numerical Assessment of Time-Dependent Magneto-Convective Thermal-Material Transfer over a Vertical Permeable Plate

The objective of this work is to investigate the influences of thermal radiation, heat generation, and buoyancy force on the time-dependent boundary layer (BL) flow across a vertical permeable plate. The fluid is unsteady, incompressible, viscous, and electrically insulating. The heat transfer mechanism happens due to free convection. The nondimensional partial differential equations of continuity, momentum, energy, and concentration are discussed using appropriate transformations. The impressions of thermal radiation and buoyancy forces are exposed in the energy and momentum equation, respectively. For numerical model, a set of nonlinear dimensionless partial differential equations can be solved using an explicit finite difference approach. The stability and convergence analyses are also established to complete the formulation of the model. The thermophysical effects of entering physical parameters on the flow, thermal, and material fields are analyzed. The variations in local and average skin friction, material, and heat transfer rates are also discussed for the physical interest. The analysis of the obtained findings is shown graphically, and relevant parameters pointedly prejudice the flow field. Studio Developer FORTRAN 6.2 and Tecplot 10.0 are applied to simulate the schematic model equations and graphical presentation numerically. The intensifying values of the magnetic field are affected decreasingly in the flow field. The temperature profiles decrease within the BL to increase the value of radiation parameters. The present study is on the consequences for petroleum engineering, agriculture engineering, extraction, purification processes, nuclear power plants, gas turbines, etc. To see the rationality of the present research, we compare these results and the results available in the literature with outstanding compatibility.

Quasilinear Hyperbolic Systems Applied to Describe the Magnetohydrodynamic Nanofluid Flow

This study examines the flow of hyperbolic nanofluid over a stretching sheet in three dimensions. The influence of velocity slip on the flow and heat transfer properties of a hyperbolic nanofluid has been investigated. The partial differential equations for nanoparticle solid concentration, energy, and motion were turned into ordinary differential equations. Nanoparticle mass fluxes at boundaries are assumed to be zero, unlike surface concentrations. The influence of the main parameters on flow characteristics, surface friction coefficients, and the Nusselt number has been visualized. The results suggest that Brownian motion has a negligible impact on the heat transfer rate. The ratio of the elastic force to the viscosity force was found to decrease the fluid velocity. The resulting thermophysical properties of nanofluids are in agreement with previous research. The present findings can be used to expand the potential for using nanofluids as a coolant in critical thermophysical and industrial installations.

Analysis of heat transfer and thin film flow of Au−Np over an unsteady radial stretching sheet

The analysis of heat transfer and thin film flow over an unsteady radial stretching surface is the primary focus of this work. For the two-dimensional motion of the pair stress nanofluid made of gold nanoparticles, a mathematical model is put forth. Nonlinear governing partial differential equations for momentum and energy are converted into a set of ordinary differential equations via similarity transformations. By using the BVP4c method, the resulting nonlinear differential equations were calculated. The relationship between the Nusselt number, temperature, skin friction, and fluid motion velocity is examined and graphically analyzed. This study suggests that thin film thickness increases for rising values of volume fraction whereas it dropped for increasing values of the magnetic parameter and unsteadiness parameters Maximum heat transfer rate and skin friction coefficient has been observed for blade and cylinder shaped nanoparticles and minimum for brick shape.

Energy–time optimal path planning in dynamic flows: Theory and schemes

We obtain, solve, and verify fundamental differential equations for energy–time path planning in dynamic flows. The equations govern the energy–time reachable sets, optimal paths, and optimal controls for autonomous vehicles navigating to any destination in known dynamic environments, minimizing both energy usage and travel time. Based on Hamilton–Jacobi theory for reachability and the level set method, the resulting methodology computes the Pareto optimal solutions to the multi-objective path planning problem, numerically solving the exact equations governing the evolution of reachability fronts and optimal paths in the augmented energy and physical-space domain. Our approach is applicable to path planning in various dynamic flow environments and energy types. We first validate the methodology through a benchmark case of crossing a steady jet for which we compare our results to semi-analytical optimal energy–time solutions. We then consider unsteady flow environments and solve for energy–time optimal missions in a quasi-geostrophic double-gyre flow field. Results show that our theory and schemes can provide all the energy–time optimal solutions and that these solutions can be strongly influenced by unsteady flow conditions.

Reliability of the bi-stable piezoelectric energy harvester under random excitation

The bi-stable piezoelectric energy harvester can produce higher voltage and efficiency than traditional energy harvester. In this manuscript, the reliability of the bi-stable piezoelectric energy harvester excited by Gaussian white noise is investigated. The equivalent uncoupled equation of system is deduced on the basis of the generalized harmonic transformation for the bi-stable piezoelectric energy harvester original coupling system. The averaged stochastic differential equation for system energy of the equivalent uncoupled system is established using the stochastic averaging method. The conditional reliability function and the mean first-passage time are derived through the backward Kolmogorov equation and the generalized Pontryagin equation associated with the averaged stochastic differential equation. The influences of noise intensity, electromechanical coupling coefficient, and time constant ratio on the conditional reliability function and mean first-passage time are discussed. The analytical results are highly consistent with the numerical results that are obtained by the Monte Carlo numerical simulation.