Set Of Partial Differential Equations Research Articles

Extending a Physics-informed Machine-learning Network for Superresolution Studies of Rayleigh–Bénard Convection

Advancing our understanding of astrophysical turbulence is bottlenecked by the limited resolution of numerical simulations that may not fully sample scales in the inertial range. Machine-learning (ML) techniques have demonstrated promise in upscaling resolution in both image analysis and numerical simulations (i.e., superresolution). Here we employ and further develop a physics-constrained convolutional neural network ML model called “MeshFreeFlowNet” (MFFN) for superresolution studies of turbulent systems. The model is trained on both the simulation images and the evaluated partial differential equations (PDEs), making it sensitive to the underlying physics of a particular fluid system. We develop a framework for 2D turbulent Rayleigh–Bénard convection generated with the Dedalus code by modifying the MFFN architecture to include the full set of simulation PDEs and the boundary conditions. Our training set includes fully developed turbulence sampling Rayleigh numbers (Ra) of Ra = 106–1010. We evaluate the success of the learned simulations by comparing the power spectra of the direct Dedalus simulation to the predicted model output and compare both ground-truth and predicted power spectral inertial range scalings to theoretical predictions. We find that the updated network performs well at all Ra studied here in recovering large-scale information, including the inertial range slopes. The superresolution prediction is overly dissipative at smaller scales than that of the inertial range in all cases, but the smaller scales are better recovered in more turbulent than laminar regimes. This is likely because more turbulent systems have a rich variety of structures at many length scales compared to laminar flows.

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.

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.

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.

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 two‐stage reliable computational scheme for stochastic unsteady mixed convection flow of Casson nanofluid

AbstractResearchers can incorporate uncertainties in computational fluid dynamics (CFD) that go beyond the inaccuracies caused by numerical discretization thanks to stochastic simulations. This study confirms the validity of current stochastic modeling tools by providing examples of stochastic simulations in conjunction with numerical solutions for incompressible flows. A numerical technique for solving deterministic and stochastic models is developed in this work. Our approach employs the Euler‐Maruyama method for stochastic modeling, representing a stochastic version of the third‐order explicit‐implicit scheme. For the deterministic model, the scheme is third‐order accurate. The consistency and stability of the constructed scheme are provided in the mean square sense. The scheme is the predictor–corrector type that is built on two time levels. Moreover, a mathematical model of the Casson nanofluid flow with variable thermal conductivity is given with the effect of the chemical reaction. The appropriate transformations are used to condense the set of partial differential equations (PDEs) down to one that is dimensionless. The scheme is applied for the deterministic and stochastic models of dimensionless flow problems. The velocity profile's deterministic and stochastic behavior are shown using contour plots. Results show that growing values of the thermal mixed convection parameter enhance the velocity profile. This article presents the progress made in stochastic computational fluid dynamics (SCFD) and highlights the energy‐related aspects of our discoveries. Our computational approach and stochastic modeling techniques provide new insights into the energy properties of Casson nanofluid flow, specifically regarding the variability of thermal conductivity and chemical processes. Our objective is to clarify the complex interaction of these factors on energy dynamics. This article presents a contemporary summary of the latest SCFD advancements. Additionally, it highlights potential directions for future research and unresolved issues that require attention from the members of the field of computational mathematics.

A modification of explicit time integrator scheme for unsteady power-law nanofluid flow over the moving sheets

This paper introduces an exponential time integrator scheme for solving partial differential equations in time, specifically addressing the scalar time-dependent convection-diffusion equation. The proposed second-order accurate scheme is demonstrated to be stable. It is applied to analyze the heat and mass transfer mixed convective flow of power-law nanofluid over flat and oscillatory sheets. The governing equations are transformed into a dimensionless set of partial differential equations, with the continuity equation discretized using a first-order scheme. The proposed time integrator scheme is employed in the time direction, complemented by second-order central discretization in the space direction for the momentum, energy, and nanoparticle volume fraction equations. Quantitative results indicate intriguing trends, indicating that an increase in the Prandtl number and thermophoresis parameter leads to a decrease in the local Nusselt number. This modified time integrator is a valuable tool for exploring the dynamics of unsteady power-law nanofluid flow over moving sheets across various scenarios. Its versatility extends to the examination of unstable fluid flows. This work improves engineering and technological design and operation in nanofluid dynamics. Improving numerical simulations’ precision and computational efficiency deepens our comprehension of fundamental physics, yielding helpful information for enhancing systems that rely on nanofluids.

Stagnation point flow for the dynamics of thermal enhancement in nanofluid by a convective extending surface with suction/injection and heat source

We intend to analyze the influences of heat dissipation and heat generation on steady nanofluid stagnated flow along an extending surface along with thermal boundary layers. Water-based Cu and Al2O3 nanoparticles are considered. Moreover, the mechanism of injection/suction has been analyzed for such flows. Fundamental governing equations for the flow of an isotropic in-compressible thermally radiative single-phase nanofluid are defined by the set of nonlinear Partial Differential Equations (PDEs) which are transmuted to Ordinary Differential Equation (ODEs) and dealt with numerically by using Runge–Kutta (RK) fourth-order adopting shooting technique that gives approximate simulations. The numerical findings are illustrated as graphical and tabular representations after a careful parametric examination of determined parameters is carried out. The current outcomes specify that the velocity profiles decrease under the influence of magnetic strength while heat distribution positively changes under the effect of magnetic strength, heat source, radiative term and Biot number quantity. The temperature profiles behave negatively with the thermal Grashof number and Prandtl number. Moreover, the comparative simulations of the ongoing study with existing the literature for ensuing the obtained simulations have been worked out.

Efficient passive GDQLL scrutinization of an advanced steady EMHD mixed convective nanofluid flow problem via Wakif–Buongiorno approach and generalized transport laws

Owing to the practical importance of nanofluids and their adjustable thermal capabilities, this study intends to develop a robust generalized differential quadrature local linearization (GDQLL) algorithm for examining realistically the heat and mass aspects of electrically conducting nanofluids during their non-Darcian laminar motion nearby a convectively heating vertical surface of an active electromagnetic actuator (i.e., Riga plate). By invoking Wakif–Buongiorno model and Oberbeck–Boussinesq approximations along with other generalized transport laws (i.e., Cattaneo–Christov and non-Fick’s laws) and Grinberg’s concept, a set of gigantic partial differential equations is stated appropriately in the sense of the boundary layer approximations for describing exhaustively the present EMHD mixed convective nonhomogeneous flow under the passive control strategy of nanoparticles within the nanofluidic medium. Operationally, the dimensionless differential forms of the governing boundary equations are derived properly by introducing reasonable mathematical adjustments into the preliminary formulation. In this case, the differential complexity of the leading differential structure is reduced to a nonlinear coupled system of ordinary differential equations, whose discrete numerical solutions are computed perfectly via a well-structured GDQLL algorithm. As foremost outcomes, it is demonstrated that the nanofluid motion and its surface thermal enhancement rate can be reinforced significantly through the thermal strengthening in the convective heating and mixed convective process as well as via the electromagnetic improvement in the driven aspect of Lorentz’s forces.

Spectral-based optimal output-feedback boundary control of a cracking catalytic reactor PDE model

This research paper focuses on the design of a boundary optimal controller with output feedback for a tubular catalytic cracking reactor. The process is represented by a set of non-linear parabolic partial differential equations (PDEs). The approach involves utilising the linearised infinite-dimensional version of the model, taking advantage of the system generator being a Riesz spectral operator. A stabilising compensator is designed by leveraging the eigenvalues and eigenvectors of the system generator. This paper also includes numerical simulations to demonstrate the performance of the developed algorithm.

Numerical Investigation of Nanofluid as a Coolant in a Prismatic Battery for Thermal Management Systems

Abstract This study delves into the realm of numerical investigation of the heat transfer performance of nanofluids as coolants for prismatic batteries. Nanofluids are being employed in battery cooling systems to enhance overall thermal management and ensure the safe operation of batteries, particularly in situations involving high heat generation. In this study, different types of nanofluids were used along with a base fluid of ethylene glycol–water (EG–water 50%). The energy equations consider the effects of viscous dissipation and heat generation. The model generates a set of nonlinear partial differential equations, which can be transformed into ordinary differential equations (ODEs) using appropriate similarity variables. These ODEs are then solved numerically by employing the Runge–Kutta–Fehlberg method along with the shooting method to obtain solutions. The simulations in both 2D and 3D showcase the results for various parameters pertaining to thermal and velocity fields, heat transfer rate, and drag force. The findings reveal that heat generation leads to a staggering increase in temperature of 78.22%. However, using aluminum nanoparticles (NPs) as opposed to copper nanoparticles quickly reduced the battery’s maximum temperature by 9.31%. The exceptional heat generation strengths of CuO–EG and Al2O3–EG nanofluids also resulted in a significant increase in their heat transfer rates of around 40.42% and 42.13%, respectively. Additionally, the aluminum NPs exhibited a more rapid heat transfer rate of 4.06% when compared to the copper nanoparticles. This research contributes to the development of improved cooling strategies for prismatic battery applications, ultimately paving the way for enhanced battery performance, an extended lifespan, and improved safety in a wide range of industries and electric vehicles.

Physics informed neural networks reveal valid models for reactive diffusion of volatiles through paper

Predictive models for the transport of volatile organic compounds in paper need to consider the complex interplay of diffusion, adsorption, desorption, or chemical reactions. The relative importance of each of these processes is determined by the polarity of the volatile. Hence, it is challenging to pick a valid theoretical model that correctly predicts transport regardless of the polarity. Here, physics-informed neural networks (PINNs) assess which of five different models correctly describe transport of DMSO as polar and n-tetradecane as apolar model compound: (i) a pseudo first-order adsorption model for an irreversible sorption process, (ii) a first-order kinetics model allowing reversible sorption, (iii) a second-order model with a reversible process, and an effective diffusion model accounting for a constant (iv) and for a variable effective diffusivity (v). Each tested model is given as set of partial differential equations (PDE). Considering the model under testing and experimentally obtained spatially and temporally resolved concentration profiles through stacks of paper sheets, PINNs predict concentration of the volatiles and associated material constants such as sorption constants and effective diffusion coefficients by solving the inverse problem. Our PINNs revealed two models, pseudo first-order sorption and second-order reversible sorption, that correctly predict concentration profiles and polarity-driven differences in sorption times. While a PINN-based picking of valid transport models has important implications for the development of effective methods for controlling emission of volatiles from paper materials, PINNs represent a versatile mathematical tool to validate or refute the capability of PDE-based theoretical models to describe experimental data.

Magnetically-driven nanofluid flow over a slippery-bended surface under thermal radiation and higher order chemical reaction

In this paper, nanofluid flow is considered on curved stretching surface under magnetic influence. Realistic velocity slip together with convective boundary condition is imported. The system is also blessed with radiation and higher order chemical reaction. Active and passive controls of nanoparticles are considered and under both boundary conditions the flow analysis is compared. Leading equations of the system is a set of partial differential equations which are transfigured by similarity variable into a set of highly nonlinear ordinary differential equations (ODEs). The system is solved by the Runge–Kutta fourth-order method (RK-4) with shooting technique. The simulation is done by MAPLE-2021 software. Outcomes are portrayed by several graphs and tables and comparison diagram for different conditions is also included. Velocity lines are compared for suction and injection effect but thermal and concentration profiles are compared under active and passive controls of nanoparticles. The velocity profile changed by 16.55% for higher magnetic profile and the mass transfer changed by 3.57% for actively controlled flow under velocity slip parameter. Chemical reaction parameter detained the concentration profile for both active and passive controls but gave lower magnitude for passively controlled flow.

Dynamics of Newtonian fluid through curved domain by considering unsteady effects

AbstractCross‐diffusion gradients, such as the Soret and Dufour effects, play a big role in the formation of binary alloys, the movement of oil and groundwater contaminants, and the separation of gas mixtures. Other applications where cross‐diffusion gradients are useful include: Temperature fluctuations cause matter diffusion, known as Soret effects. Concentration gradients drive heat diffusion, or the Dufour effect. This effect is named after its French discoverer. These findings could be applied to many engineering and industrial contexts and have many intriguing and potentially useful effects. Joule heating unites Soret and Dufour's work. The traditional nonlinear differential approach yields enough permutations. Convergent series can be used to solve temperature, velocity, and concentration problems. These changes can occur in temperature, velocity, or concentration. The drawings clarify all about the system's most important qualities and components. A comprehensive analysis of the Nusselt and Sherwood values is also done. After graphing the Nusselt and Sherwood values, they are analyzed. We are discussing computer science. This study found that Hartman number increases reduce one's perception of radial velocity. As Prandtl and Soret molecules increase, fluid temperature decreases. In this study, we employ numerical methods to solve the micropolar fluid flow problem on a stretched and curved disk. Our methods allow us to model fluid flow in three dimensions. We focus on micropolar fluid flow. Applying the necessary transformations to a set of partial differential equations simplifies it into a set of ordinary differential equations. The equations in both sets are transformed using similarities to convert one set of partial differential equations into another set of ordinary differential equations. The gunshot method and the Runge–Kutta algorithm can solve coupled equations numerically. The nondimensional radius of curvature can quantify and characterize many physical phenomena. Strain, microrotation velocity, and fluid velocity are examples. Due to the variances between curved and flat stretched sheets, the border layer strain cannot be neglected. Due to differences in stretching sheets,

Magnetized bioconvective thermal efficacy of ferromagnetic nanoparticles involving appliance of radiation and viscous dissipation

The research on nanotechnology is on peak because of its noteworthy applications including cure of cancer, industrialization of latest aircraft, enhanced heat transmission equipment, working as coolants for nuclear reactors and as coolants in radiators of vehicles. Here concept of microorganism is utilized to make suspension more stable, where nano sized particles are used with effect of bioconvection to solve the simultaneous equations having magnetic effects. Research proved that there are numerous applications of bioconvection in modern biotechnology and advanced bioengineering; so work on production of useful microorganism is on peak for microorganisms. Basic aim of recent research work is to explore a 2-D Sutterby ferrofluid flow having impacts of magnetic dipole. Moreover, important features of Brownian motion along with thermophoresis parameter are computed for Sutterby model having effects of magnetism termed as ferrofluid with consideration of important characteristics like convective conditions on boundary motile microorganisms, as well as thermal gradients all are under consideration. We transformed the set of non-linear partial differential equations in to set of ordinary differential equations after that we tackled it by use of well-known numerical method called bvp4c method. By using graphs, we expressed the impacts for physical parameters in detail. Results showed that temperature of ferrofluid increases with increase in parameter of thermophoresis as well as thermal effects whereas results in reduction of Prandtl number. The process of microorganism reduces for higher values of Peclet number.

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

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

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

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

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

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

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

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

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

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