Dimensionless Problem Research Articles

Artificial neural network prediction on free convection in a triangular chamber with a v-shaped corrugated bottom wall filled with nanofluid

Artificial neural network (ANN) prediction on magnetohydrodynamic (MHD) free convection for complex systems using nanofluid plays a significant role in understanding and optimizing the system’s behaviour. The current research investigates how various governing parameters affect heat transport within a v-shaped corrugated bottom triangular chamber consisting of Al2O3-H2O nanofluid. The vertical wall is kept at a uniform low temperature, the inclined wall is adiabatic, and the v-shaped corrugated bottom wall is exposed to constant high temperature. The Galerkin weighted residual approach, based on the method of finite elements, is used to solve the dimensionless PDE problems with boundary constraints. The isotherms, streamlines, and average Nusselt numbers are displayed for various Hartmann numbers (0 ≤ Ha ≤ 100), Rayleigh numbers (103 ≤ Ra ≤ 106), and nanoparticles volume fractions (0.01 ≤ ϕ ≤ 0.05). It demonstrates how the average Nu is affected by Ha, Ra, and ϕ. Furthermore, it has been discovered that the average Nu increases with Ra and ϕ but decreases with Ha. However, the Levenberg-Marquardt back propagation method is employed for the artificial neural network (ANN) approach on 120 datasets to estimate the average Nu, demonstrating an excellent correlation with simulated data and an average R2 of 0.99689. MATLAB software is used to calculate the ANN predictions.

Dynamics of stratified-convected Eyring-Powell nanoliquid featuring chemically reactive species and Ohmic dissipation: Application of Levenberg-Marquardt artificial neural networks(ALM-ANNs)

Magnetic nanoliquids are pondered cutting-edge heat transference liquids, representing a modern generation because of their aptitude to function as smart liquids. The deployed magnetism externally exerts a readily manageable impact, escalating their adaptableness and practicality in various applications. In this analysis, electro magnetized Powell-Eyring rheological nanoliquid is modeled considering stretching cylinder based buoyancy-driven flow. Transport expressions include thermal radiation, Brownian diffusion, chemical reaction, thermophoresis, and thermal source effect. Prandtl theory of boundary-layer (BL) assists to develop the governing problems. Appropriate transformations are deployed to obtain the dimensionless problems which are then computed numerically through the novel Levenberg-Marquardt artificial neural networks(ALM-ANNs). It is worth to mention that the close agreement between the error analysis of the reference dataset and the proposed datasets roughly from E−10 to E−03, further justifies the approaches utilized in this study. Besides the excellent measures of performance in terms of MSE are achieved at level 5.77E−11, 4.41E−11, 2.09E−12, 2.16E−11, 1.03E−11, 1.65E−11, 2.00E−11, and 1.91E−11against 272, 416, 266, 391, 142, 366, 425, and 508 epochs for first instance of all eight various scenarios. The acquired outcomes are compared with alternate analytical (homotopy) scheme and good agreement is witnessed. Furthermore it is noticed that as the values of Pr,RandS1 increase in size, so does the EPFM temperature profile θ, and the evaluation of AE is between E-04 to E-10.

Linear and non-linear stability analysis of double-diffusive Hadley-Prats flow through horizontal porous layer with non-uniform thermal and solutal gradients

A scientific model is devised to understand the onset of double-diffusion Hadley-Prats flow through an infinite parallel permeable channel with non-uniform thermal, solutal gradients, and concentration based internal heat generation under the influence of convection conditions is considered. Darcy’s model is implemented in the context of a porous medium, which is characterized as isotropic and homogeneous. A linear and nonlinear stability analysis is conducted with the help of energy functional method and longitudinal roll perturbations are examined. The emerged dimensionless eigenvalue problem in the linear and nonlinear stability analysis is solved numerically by utilizing fourth order Runge-Kutta method (RK-4) method and shooting scheme. The critical values of wave and vertical thermal Rayleigh numbers are examined. A comprehensive analysis of solutions concerning the onset of convection mechanism is conducted. The produced results are discussed for the various values of emerged physical parameters. It is evident that, the presence of non-uniform thermal and solutal gradients and the internal heat generation mechanism causes the significant variations in the critical values of vertical thermal Rayleigh number in the channel regime.

A machine learning approach for entropy due to natural convection flow of a nanofluid under the uniform inclined magnetic field

In this study, entropy generation of natural convection flow of Cu-water nanofluid under the influence of a uniform inclined magnetic field (MF) is firstly modeled by a machine learning approach using the data obtained from a numerical process. The two dimensional, time dependent dimensionless equations are numerically solved by using global radial basis function (RBF) method in spatial derivatives and the second order backward differentiation formula (BDF2) in time derivatives. In a set of combined dimensionless problem parameters, numerical simulations are performed. In the obtained data, the inputs are a set of combined dimnesionless problem parameters and the outputs are entropy indicators. Then, global and local entropy indicators are modeled by Neural Network (NN). In view of smaller mean squared error metric results, small number of layers or layer sizes in NN seems enough in smaller data while trilayer NN (TNN) is a better modeler in case of larger data.

Modeling of Average Nusselt Number by Machine Learning and Interpolation Techniques

Abstract In this study, an important heat transfer, fluid flow parameter, and average Nusselt number Nu¯ are statistically modeled by using the data obtained from a numerical process. The two-dimensional, time-dependent dimensionless equations of natural convection (NC) flow either in the absence or in the presence of a uniform inclined magnetic field (MF) are numerically solved by using global radial basis function (RBF) method in spatial derivatives and the second-order backward differentiation formula (BDF2) in time derivatives. Numerical simulations are performed in a set of combined dimensionless problem parameters. A dataset with inputs Rayleigh number Ra, Prandtl number Pr, and output Nu¯ in the absence of MF and a dataset with inputs Ra, Pr, Hartmann number Ha, inclination angle γ, and output Nu¯ in the presence of inclined uniform MF are saved. The obtained data are separated into train and test sets. Then, Nu¯ is first modeled by Neural Networks (NN). Second, interpolation is also examined. In terms of mean squared error (MSE) metric, NN outputs give the best goodness of fit results compared to curve fitting on test data. On the other side, it is shown that interpolation is also an alternative for modeling. This modeling issue enables one to get the desired result without making heavy numerical calculations many times.

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.

Mathematical modelling of a slow flameless combustion of a two-dimensional paper

We present a mathematical model for the slow combustion (smoldering) of a two-dimensional sheet of paper. We describe the evolution of the char region, and we investigate the effects of an orthogonal air flow on the shape of the combustion front. The mathematical formulation consists in a set of two nonlinear PDEs for the temperature and the oxygen concentrations coupled with one ODE for the cellulose concentration. The (dimensionless) problem is solved numerically by means of a spectral collocation scheme based on Chebyshev polynomials. Our results show that the Péclet and the Lewis number strongly influence the shape of the ignition front and that the advancement of the combustion front does not occur if advection and diffusion are neglected (zero Péclet and Lewis numbers). In particular we observe that the burning region and the ignition front are strongly influenced by the velocity of the airflow and by the mass and heat transport phenomena due to diffusion and advection. We shall see that the increasing of the ratio between the convective and diffusive characteristic times (Péclet number) and the decreasing of the ratio between the mass and heat diffusive characteristic times (Lewis number) have a “flattening effect” on the combustion front.

Optimization analysis of cold extrusion process parameters for valve screws

This study focuses on valve screws and establishes a cold extrusion simulation model using DEFORM-3D software to analyze the deformation characteristics and influencing factors during the extrusion process. Supported by the theory of rigid-plastic finite element, an orthogonal experimental design is conducted to establish a four-factor three-level simulation scheme with general fillet radius, die clearance, extrusion speed, and friction coefficient as the simulation variables, and maximum equivalent stress on the convex die, forming damage of the blank, and maximum wear depth on the concave die as the objective values. The prediction model based on BP neural network algorithm and orthogonal experimental data shows high accuracy. The optimization problem is simplified into a dimensionless numerical analysis problem using GC grey correlation analysis, and the optimal solutions within each variable range are determined using range analysis. The simulation results after optimization show a 20 5% reduction in maximum equivalent stress on the convex die compared to the original scheme, a 4.9% reduction in forming damage of the extruded part, and a 24.24% reduction in maximum wear depth on the concave die compared to before optimization. The actual verification confirms the good forming effect of valve screw extrusion, with well-formed contours and a uniform surface without wrinkles or defects. Finite element simulation provides valuable guidance for practical production.

Optimal Control of the Position of Moisture Transfer Sources in Porous Media

The paper proposes an approach for finding the optimal position of sources of known power for the quasi-linear Richards equation in a rectangular area. The Kirchhoff transformation is applied with the introduced scaling of coordinates and powers of submerged sources, which allows formulating a dimensionless problem. The task of this study is to find the position of submerged sources - such that the distribution of moisture at the final moment of time is close to the given values or the given target function.

A new explicit numerical scheme for enhancement of heat transfer in Sakiadis flow of micro polar fluid using electric field

This article suggests a fourth-order numerical approach for solving ordinary differential equations (ODEs) that are both linear and nonlinear. The suggested scheme is an explicit predictor-corrector scheme. For linear ODE, the proposed numerical scheme's stability area is discovered. The proposed strategy yields the same stability region as the traditional fourth-order Runge-Kutta method. In addition, partial differential equations (PDEs) are used to develop the mathematical model for the flow of non-Newtonian micro-polar fluid over the sheet and heat and mass transit using electric field effects. These PDEs are further transformed into dimensionless boundary value problems. Boundary value problems are resolved using the proposed shooting-based scheme. The findings show that increasing values of ion kinetic work and Joule heating parameters cause the temperature profile to climb. The results produced by the suggested strategy are compared to those discovered through earlier studies. The results of this study could serve as a starting point for future fluid-flow investigations in a secure industrial environment.

Micro-polar liquid flows with magnetohydrodynamic (MHD) mixed convection due to a nonlinear stretched sheet in a convective state

The topic of this study is the flow of a micro-polar liquid toward a nonlinear stretched surface using magnetohydrodynamics (MHDs). The analysis presents the effects of some important parameters on fluid flow. The mixed convection phenomena are used to examine the characteristics of heat transmission. By using the appropriate transformations, dimensional nonlinear equations can be transformed into dimensionless expressions. The governing dimensionless problems are solved using the modified Laplace decomposition approach by examining the effects of various boundary factors on flow and heat transfer.

The generalized Fourier’s and Fick laws effects on MHD free convection flows of Maxwell fluids by employing Caputo–Fabrizio time-fractional integral

The relevance of time-dependent magneto-free convection and its consequences for mass and energy transport are being increasingly understood in science. Unfortunately, very little is known about how the fractional generalized technique would affect a complete analysis of Maxwell fluid dynamics over a porous plate. Using the Caputo–Fabrizio time-fractional integral, the Fourier thermal flux law and the fractionally generalized Fick’s equation of mass flow are both generalized. Using the appropriate similarity transformations allows us to characterize the structured governing equations, which are nondimensionalized. The dimensionless energy, concentration, and velocity distribution problem is solved using the Laplace transform method. The graph demonstrates how physical and fractional parameters are affected. Fractional derivatives may be employed to accurately represent the rheology of such fluids. The Maxwell generalized fluid across an oscillating sheet was studied by Zheng et al. 3

Physical interference of magnetic dipole for retardation type nanofluid with bioconvection phenomenon

The bioconvection aspect for the rate type nanomaterial under dynamic of induced magnetic force has been numerically worked out. The Oldroyd-B nonlinear model is incorporated to inspect the interesting rheological dynamic of rate type classifications. Following nonlinear models, the relaxation and retardation features are observed. The numerical proposed data is fundamentally achieved via the shooting method. After developing the dimensionless problem expressions, the shooting numerical algorithm is followed for the computations. The physical onset of parameters is graphically listed with interesting applications. It is observed that a more strong induced magnetic field profile has been observed due to the presence of bio-convective Lewis number and Rayleigh number. An enriched profile of thermal phenomenon due to Grashof number is observed. The composed reflected outcomes present importance in thermal management systems, extrusion systems, plasma physics, chemical processes, nuclear systems, extrusion mechanism, biofuels, etc.

Artificial Neural Network Modeling for Predicting the Transient Thermal Distribution in a Stretching/Shrinking Longitudinal Fin

AbstractThis study emphasizes the aspects of heat transfer and transient thermal distribution through a rectangular fin profile when a stretching or shrinking mechanism is mounted on the surface of the fin. Furthermore, the effects of radiation, internal heat generation, and convection are all considered when developing the corresponding fin problem. The simulated time-dependent heat transfer equation is a partial differential equation that can be represented by dimensionless arrangement using appropriate nondimensional terms. The nonlinear dimensionless problem concerning the stretching/shrinking of a fin is numerically solved using the finite difference method (FDM), and the Levenberg–Marquardt method of backpropagation artificial neural network (LMM-BANN) has been used in this investigation. By varying the stretching/shrinking parameter, a set of data for the presented artificial neural network (ANN) is produced to discuss stretching and shrinking scenarios. The testing, training, and validation procedure of LMM-BANN, as well as correlation for verification of the validity of the proposed approach, establish the approximate solution to stretching/shrinking scenarios. The suggested model LMM-BANN is then validated using regression interpretation, mean square error, and histogram explorations. The ANN results and the procured numerical values agree well with the current numerical results.

Multiple convected conditions in Williamson nanofluidic flow with variable thermal conductivity: Revised bioconvection model

The importance of bioconvection into the application of nanoparticles has had a significant impact on fundamental technological and industrial functions in recent years. These discussions are being carried forward by the bioconvection appearance on that flow by attracting nanoparticles into additional features on thermal radiation and activation energy. The investigation had been proposed on the appealing features of the mass and thermal convective boundary constraints. The high desirable subclass on rate-type fluid, especially Williamson fluid, was used to anticipate the absolute rheological parameters. The designated partial differential system was modified along with nonlinear ordinary differential equations applying enhance being related conversion, the designated partial differential system was modified along nonlinear ordinary differential equations. The governing equations are interpreted into ordinary differential equations, and in the following, shooting process was obeyed to build up the mathematical classification on the convert dimensionless flow problem. This value of physical restraint was displayed on the table and plots tests. This noted theoretical imitation could be highly successful in improving the solar energy systems and thermal extrusion processes. The outcomes reported that the rise in Weissenberg number and Hartmann number decreased the nanoparticles velocity distribution. The progressing temperature distribution was observed to develop radiation parameter and thermal conductivity parameter. That was further noted to modify bioconvection Lewis number and Peclet number that were reduced into motile microorganism distribution effectively.

Numerical Study of a Non-Linear Porous Sublimation Problem With Temperature-Dependent Thermal Conductivity and Concentration-Dependent Mass Diffusivity

Abstract Sublimation heat transfer occurs in a wide range of engineering processes, such as accelerated freeze drying (AFD), energy storage, and food technology. Particularly in the microwave AFD process, preservation of material with the least possible energy consumption is desirable. In connection with this, it is of interest to analyze the effect of temperature/concentration dependent heat/mass transfer properties. Given the limited literature available on sublimation, there is a general lack of physical understanding of this particular problem. The present work analyzes the nonlinear sublimation process driven by convective heat/mass transfer and evaporation of water vapor using the Legendre wavelet collocation method (LWCM). Results from the present work are shown to be in excellent agreement with the exact solution of the special case of a linear problem. Further, the present numerical technique shows good agreement with finite difference method in case of a completely nonlinear model. The model is used for a comprehensive investigation of the impact of the problem parameters, on the rate of sublimation. It is found that the sublimation rate increases with increasing values of β1 and decreasing values of β2. The impact of other dimensionless problem parameters such as Péclet numbers Pe1 and Pem, convection due to mass transfer of water vapor β, latent heat of sublimation l0 and Luikov number Lu on sublimation process is also discussed in detail. These observations offer a comprehensive theoretical and mathematical understanding of sublimation heat/mass transfer for improving the performance and efficiency of freeze-drying and related engineering processes.

Nanofluid containing motile gyrotactic microorganisms squeezed between parallel disks

Advanced nano and microtechnologies for nano/micro-electronic devices have made substantial advances in the past few years. These technologies are rapidly incorporating advanced fluid media such as nanofluids and biological microorganisms. Inspired by bio-nanofluid applications in medicine, biological systems and biotechnology in the present study, mathematical model is evolved for unsteady bio convective conducting nanofluid along with gyrotactic micro-organisms squeezed between parallel disks. The lower disk and upper disks are solids. The temperature field is improved by the methods of haphazard motion of nanoparticles and thermophoresis parameters. The nano-bio transfer model is written as a series of non-linear partial differential equations that are reduced into a set of ordinary differential equations using suitable transforms. The dimensionless problem is then numerically solved by RK-4th order scheme utilizing MATLAB bvp4c solver’s package to investigate the impact of the squeezing parameter, Hartman number, Brownian motion and thermophoresis parameter on motile microorganism velocity, temperature, nanoparticle concentration, and density. The friction factor, Nusselt number, Sherwood number and microorganism number distributions on Hartman number, thermophoresis and Brownian motion factors are also computed. The Brownian motion and the thermophoresis factors of nanoparticles cause an increment in temperature profiles for both suction and injection. The concentration and motile microorganism are both amplified for the Brownian parameter in the case of injection, whereas they are declined for suction and the opposite trend is observed for the thermophoresis parameter. The motile microorganism is deflated for both suction and injection with thermophoresis parameter. Suction and injection adversely affect the transfer properties at the disks. The resistive magnetic body force prevails in the core zone, resulting in a decrease in velocity. The heat generation in squeeze films with motile microorganisms can be successfully removed with magnetic nanoparticles which require a longer serviceability of the lubrication system, bio-medical systems and need for less maintenance and a longer lifespan approach. The finding is pertinent to novel bio-microsystems that combine nanofluid and the bioconvection phenomenon. The percentage increase in heat, mass, and microorganism transport rates is calculated.

Novel Analysis of Two Kinds Hybrid Models in Ferro Martial Inserting Variable Lorentz Force Past a Heated Disk: An Implementation of Finite Element Method

In this article, the rheology of Ferro-fluid over an axisymmetric heated disc with a variable magnetic field by considering the dispersion of hybrid nanoparticles is considered. The flow is assumed to be produced by the stretching of a rotating heated disc. The contribution of variable thermophysical properties is taken to explore the momentum, mass and thermal transportation. The concept of boundary layer mechanism is engaged to reduce the complex problem into a simpler one in the form of coupled partial differential equations system. The complex coupled PDEs are converted into highly nonlinear coupled ordinary differential equations system (ODEs) and the resulting nonlinear flow problem is handled numerically. The solution is obtained via finite element procedure (FEP) and convergence is established by conducting the grid-independent survey. The solution of converted dimensionless problem containing fluid velocity, temperature and concentration field is plotted against numerous involved emerging parameters and their impact is noted. From the obtained solution, it is monitored that higher values of magnetic parameter retard the fluid flow and escalating values of Eckert number results in to enhance temperature profile. Ferro-fluid flow and heat energy for the case of the Yamada Ota hybrid model are higher than for the case of the Hamilton Crosser hybrid model. Developing a model is applicable to the printing process, electronic devices, temperature measurements, engineering process and food-making process. The amount of mass species is reduced vs. incline impacts of chemical reaction and Schmidt parameter.

Investigation on the role of tri-nanoparticles for non-Fourier thermal enhancement by MHD non-Newtonian rheological fluid composed charged particles

Heat transfer enhancement in partially ionized Williamson fluid using tri-nanoparticles has not been studied yet. This article provides a comparative analysis of the enhancement of heat and mass transfer in Williamson fluids for mono, hybrid, and tri-nanoparticles. This is a theoretical study based on mathematical modelling using governing laws of heat transfer in a fluid medium (here fluid medium is Williamson fluid). The system of governing PDEs is transformed via a change of variables suggested under the similarity principle. Finite element method (FEM) is implemented to solve the dimensionless problems. Computer code is tested and convergent meshed free solutions are computed. Aluminium oxide, titanium oxide, and silicon oxide are taken as tri-nanoparticles whereas ethylene glycol is considered plasma and is subjected to the applied magnetic field of non-uniform intensity. An optimized improvement in thermal performance is noticed for the cause of tri-nanoparticles. The wall of heat transfer rate for mono, hybrid, and tri-nanofluids was computed and their numerical values are compared. It is noted that the wall heat transfer rate by tri-nanofluid is maximum in respect of wall heat transfer rates by mono and hybrid nanofluids. Wall shear stress exerted by tri-nanofluid is noted to have a minimum value in comparison with the wall shear stress exerted by mono and hybrid nanofluids. Hall and ion slip effects on the flow of tri-nanofluid are stronger than those on the flow of mono and hybrid nanofluids.

Explicit finite difference simulations for accelerating cross diffusion flow over a perpendicular shield with parabolic motion

The heat transfer phenomenon subject to thermos-diffusion effects convey important applications in the heating processes, extrusion systems, chemical processes and various engineering systems. The objective of current work is to observe the contribution of Soret and Dufour effects in oscillating shield for cross diffusion flow. The perpendicular shield with oscillating motion induced the flow. The magnitude of oscillations is assumed to be small so that laminar flow due to oscillating shield has been resulted. The motivations for addressing the thermos-diffusion phenomenon due to oscillating of shield are due to applications in oscillatory pumps, moving surface, metal detectors, power systems etc. The dimensionless problem is obtained via introducing the appropriate set of variables. The numerical outcomes are suggested by using the most interesting explicit finite difference scheme. The physical illustration for flow parameters is presented. Moreover, the aspect of physical quantities involving the flow are graphically reported.