Galerkin Method Of Weighted Residuals Research Articles

A numerical study of MHD nonlinear mixed convection flow over a nonlinear vertical stretching sheet with the buoyancy and suction/injection effects

The current research has a primary focus on characterizing buoyancy-driven nonlinear mixed convection. This phenomenon is studied within the context of an MHD Newtonian flow over a vertically stretching sheet with nonlinear characteristics. Flow patterns resembling those studied in this context have practical applications in various industrial sectors. Such flow patterns have practical applications in several industrial sectors, including the article and pulp industry, polymer manufacturing, electronic device cooling, solar energy collection, gas turbine plants, and nuclear power generation. The governing partial differential equations (PDEs) for nonlinear mixed convection are transformed into higher-order nonlinear ordinary differential equations (ODEs) through appropriate transformations, and these transformed ODEs are subsequently solved numerically. The software MATHEMATICA is used to solve the boundary layer flow equations, employing the semi-analytical Galerkin weighted residual method (GWRM). This approach yielded simultaneous solutions for both cases suction and injection. For both the suction and injection situations, graphical results, such as velocity, concentration, and temperature profiles are included. An augmentation in the nonlinear stretching parameter results in a reduction in velocity near the surface of the sheet. Conversely, away from the sheet, velocity increases. Moreover, an increase in nonlinear stretching parameters corresponds to higher values for both temperature and concentration. An increase in the suction/injection parameter causes a decrease in the velocity, temperature, and concentration fields. In the scenario of assisting flow, an increase in the thermal buoyancy parameter leads to higher velocity but lower temperature. Conversely, in the case of opposing flow, an increase in the thermal buoyancy parameter results in higher values for both velocity and temperature. Furthermore, a numerical examination is conducted to analyze the values of skin friction, Sherwood number, and Nusselt number.

The essentials of the finite element method to solve differential equations: an illustrative case in physics

Here we review the basic steps of the modern and accurate finite element method in its application to solve ordinary differential equations in physics. To exemplify this well-in-vogue numerical technique, we have chosen the second order Poisson-Boltzmann equation, which is a classic equation of colloid science. Aiming to formulate a viable, but didactic, implementation of the finite element technique, we have combined a linear basis of functions, the Galerkin weighted residuals method, the Swartz-Wendroff approximation and the Picard iteration algorithm. In summary, the finite element method transforms a differential equation into a simpler system of algebraic equations for the coefficients of the approximate solution in terms of a set of basis functions. We describe the full computational realisation of the finite element procedure and, also, we examine the corresponding Poisson-Boltzmann numerical predictions for various representative conditions.

Effect of Radiation on Casson Hybrid Nano-fluid Flow over an Inclined Surface Using Blasius Rayleigh-Stokes Variable: Application in Solar Aircraft

Solar energy is the most important heat source from the sun, with photovoltaic cells, solar power plates, photovoltaic lights, and solar pumping water being widely used. This study looks at solar energy analysis and a method for increasing the efficacy of solar aircraft by combining solar and nano-technological energy. To enrich the research on solar aircraft wings, the study is built on the investigation of heat transfer by employing a hybrid nano-fluid past inside the parabolic trough solar collector (PTSC). The thermal source is referred to as the solar radiative flow. The heat transfer efficiency of the wings was validated for different qualities such as porous medium, viscous dissipation, play heating, and thermal energy flow. The modelled energy and momentum equations were controlled by utilizing the Galerkin-weighted residual method (GWRM). This study used two types of nano-solid particles, copper (Cu) and zirconium dioxide (ZrO2), in ethylene glycol (EG) as the standard fluid. Various control parameters for velocity, temperature outlines, frictional factor, and Nusselt number were explained and shown in figures and tables. Also, analyses reveal that the thermal profile reduces with an increase in variable thermal conductivity parameters. This study will be of considerable economic value to marine engineers, mechanical engineers, physicists, chemical engineers, and others since its application will help them improve their operations. The findings revealed that the magnetic term is positively impacted by the Cu-ZrO2/EG hybrid nanofluid's thermal distribution.

Solving Neutral Delay Differential Equations Using Galerkin Weighted Residual Method Based on Successive Integration Technique and its Residual Error Correction

Objectives: The main objectives of this work are to solve Neutral Delay Differential Equations (NDDEs) using Galerkin weighted residual method based on successive integration technique and to obtain the Estimation of Error using Residual function. Methods: The Galerkin weighted residual method based on successive integration technique is proposed to obtain approximate solutions of the NDDEs. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomials, the Chebyshev polynomials, the Hermite polynomials, and the Fibonacci polynomials are considered. Findings: Numerical examples of linear and nonlinear NDDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. Novelty: From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems. Keywords: Galerkin Weighted Residual method, Polynomials, Hermite, Bernoulli, Chebyshev, Fibonacci, Successive integration technique, Neutral Delay Differential Equations

Bioconvection analysis of EMHD and dissipative Williamson nanofluid over a three dimensional Riga plate with Joule heating effect

ABSTRACT The current study investigates the weakly hydromagnetic and bioconvection nanofluid flow of Williamson fluid, which conveys gyrotactic microorganisms, over a three-dimensional Riga surface. The primary objective is to stabilize biological, mechanical, and thermal systems through the introduction of exponentially decaying rheology in both the momentum and energy equations, known as the electro-magneto-hydrodynamic actuator (EMHD). As such, the working fluid is assumed to be dissipative, with significant consideration given to the magnetic Reynolds number and a higher-order reaction rate. To simplify the phenomenon of suspended nanoparticles’ bioconvection, an appropriate similarity transformation is applied, converting the system of partial differential equations (PDEs) into systems of ordinary differential equations (ODEs). To analyze the governing flow parameters, the numerical approach, Galerkin Weighted Residual Method (GWRM), is employed. The results are presented through tables and graphs, providing valuable insights. The findings of the study highlight that Hartmann number improves the weak movement of the Williamson fluid, thermophoresis number positively affects all flow distributions. Moreover, the temperature field is influenced by Brownian motion, leading to inflation, while the concentration field experiences a decrease due to a lower number of fluid particles available for reaction. Furthermore, higher buoyancy forces indicate significant fluid movement, resulting in a reduction in the Williamson fluid chemical reaction rate.

Analysis of hydromagnetic Williamson fluid flow over an inclined stretching sheet with Hall current using Galerkin Weighted Residual Method

Analysis of hydromagnetic Williamson fluid flow over an inclined stretching sheet with Hall current using Galerkin Weighted Residual Method

Galerkin-Bernstein Approximations of the System of Time Dependent Nonlinear Parabolic PDEs

The purpose of the research is to find the numerical solutions to the system of time dependent nonlinear parabolic partial differential equations (PDEs) utilizing the Modified Galerkin Weighted Residual Method (MGWRM) with the help of modified Bernstein polynomials. An approximate solution of the system has been assumed in accordance with the modified Bernstein polynomials. Thereafter, the modified Galerkin method has been applied to the system of nonlinear parabolic PDEs and has transformed the model into a time dependent ordinary differential equations system. Then the system has been converted into the recurrence equations by employing backward difference approximation. However, the iterative calculation is performed by using the Picard Iterative method. A few renowned problems are then solved to test the applicability and efficiency of our proposed scheme. The numerical solutions at different time levels are then displayed numerically in tabular form and graphically by figures. The comparative study is presented along with L2 norm, and L∞ norm.

Thermal performance of radiative magnetohydrodynamic Oldroyd-B hybrid nanofluid with Cattaneo–Christov heat flux model: Solar-powered ship application

The demand for efficient applications of solar energy and nanomaterials has grown significantly in recent years due to industrial needs and the desire to minimize energy consumption and costs. However, the efficiency of photovoltaic cells and solar-powered ships is highly dependent on solar radiation. In this study, a model for the installation of a parabolic trough surface collector (PTSC) was developed to achieve high temperatures on solar-powered ships. The current study analyzed the thermal characterization function in solar-powered ships under the effect of heat radiation, magnetic field, and viscous dissipation by employing hybrid magnetite particles (Fe3O4) and silver (Ag). The Cattaneo–Christov model has been used to assess heat flux in the thermal boundary layer. The Galerkin weighted residual method (GWRM) has been employed to solve the ordinary differential equations using MATHEMATICA 11.3 software. The findings showed that the thermal efficiency of the hybrid nanofluid was significantly enhanced by 56.3% relative percentage. The motto of the current investigation is developing a model for the installation of PTSC on solar-powered ships under various provided assumptions.

Viscous Dissipation Effect on Magnetohydrodynamics Fluid Flow Over an Exponential Surface with the Influence of Thermal Radiation and Thermal Diffusion

This present investigation studies the effect of viscous dissipation in magnetohydrodynamics fluid flow over an exponential surface subject to the influence of thermal radiation and thermal diffusion. The coupled nonlinear guiding equations responsible for the flow, heat and mass transports presented as partial differential equations are revamped to the associated ordinary differential equation by application of the associated similarity variables and solved by Galerkin Weighted residual method (GWRM). The results of various parameters encountered are analyzed with graphs while the Sherwood number, Nussetl number, and local skin friction are computed and discussed. The study demonstrates, among other things, that the fluid has a strong thermal conductivity at low Prandtl numbers and that heat diffuses from the surface more quickly at low Prandtl numbers in comparison with the higher values.

Investigation of Mhd Non-Newtonian Nanofluid Stagnation-Point Flow with Variable Transport Properties and Multislip Effects: An Application to Solar Radiation

Heat and mass transfer performance of Casson nanofluid for both non-conducting (m=0), electrically conducting (m≠0) fluids with solar radiation effects in stagnation point flow is considered. In this model, entropy, irreversibility, and multi slip impacts over a shrinking, static, and stretching sheet are investigated. To minimize the energy used in the solar system, it is important to monitor the processes of heat and mass transfer in the solar radiation process. The slips boundary conditions acts as a closure of the fluid velocity, mass, and heat transfer differential equations. The equations obtained are solved numerically via Galerkin Weighted Residual Method (GWRM). In the limiting sense, the present results conform with the existing work. The Behaviors of the flow physical quantities, temperature, concentration, and velocity for distinct values of the applicable dimensionless numbers are demonstrated with tables and graphs. The results reveal that, for a theoretical account of thermal boundary layers, Prandtl number serves as a variable. Furthermore, higher values of variable thermal conductivity have a significant influence on the skin friction coefficient than the case of constant variable thermal conductivity even when the fluid viscosity is assumed to be variable. The structure of the new method can be applied to the development of oil production.

Computational study of Cattaneo–Christov heat flux on cylindrical surfaces using CNT hybrid nanofluids: A solar-powered ship implementation

This study examines the potential of using nanofluids in solar thermal energy systems. Nanofluids are known to exhibit high convection heat transfer coefficients, low specific heat, and density, making them ideal for improving the performance of solar thermal energy systems. However, this computational study investigates the application of Cattaneo-Christov heat flux on cylindrical surfaces using carbon nanotube (CNT) hybrid nanofluids, for use in a solar-powered ship. The work utilizes numerical simulations to analyze the heat transfer and fluid flow characteristics of the hybrid nanofluids. The research examines the use of single-walled and multi-walled carbon nanotubes (SWCNT and MWCNT) in engine oil (EO) as the working fluid. The Galerkin weighted residual method (GWRM) is utilized to solve the ordinary differential equations (ODEs) governing the system. The impact of various parameters, such as Cattaneo-Christov heat flux, solar thermal radiation, nonlinear stretching surface, slippery velocity, and porous media on the velocity equation, energy equation, and entropy generation are investigated and elaborated through detailed plots. The findings show that the MWCNT-SWCNT/EO hybrid nanofluid (HNF) exhibits maximum efficiency of around 2.4%, while the minimum efficiency is at 2.7%. This research provides valuable insights into the design and optimization of solar thermal systems for sustainable transportation.

Thermal analysis of ferromagnetic nanofluid flow in a channel over a dimpled cavity

The water-based nanofluid analysis of a ferromagnetic fluid flow in a channel has been conducted over a dimpled cavity. The horizontal flat walls are assumed to be adiabatic, whereas the dimpled part and the inner rectangular-finned-shaped obstacle are heated. The system of Partial Differential Equations (PDEs) along with boundary conditions have been solved via Galerkin Weighted Residual Finite Element Method. To predict the streamlines isotherms, local and average Nusselt numbers, all the simulations are performed for various Reynold and Hartmann numbers. High Reynolds numbers, results in significant improvement in heat transfer at the dimpled surface. It is believed that such investigation might be helpful to make a base for finding the experimental results for mixed convection of ferromagnetic nanofluid problem over a dimpled cavity.

Space-Time Finite Element Method for Fully Intrinsic Equations of Geometrically Exact Beam

In this paper, a space-time finite element method based on a Galerkin-weighted residual method is proposed to solve the nonlinear fully intrinsic equations of geometrically exact beam which are first-order partial differential equations about time and space. Therefore, it is natural to discretize it in time and space simultaneously. Considering the continuity and intrinsic boundary conditions in the spatial direction and the continuity and periodic boundary conditions in the time direction, the boundary value scheme of space-time finite element for solving the full intrinsic equations is derived. This method has been successfully applied to the static analysis and dynamic response solution of the fully intrinsic equations of nonlinear geometrically exact beam. The numerical results of several examples are compared with the analytical solution, existing algorithms, and literature to illustrate the applicability, accuracy and efficiency of this method.

Thermophoretic Movement Transport of Reactive Casson Nanofluid on Riga Plate Surface with Nonlinear Thermal Radiation and Uneven Heat Sink/Source

The flow of magnetized Casson nanofluid past a Riga surface is examined in this study. The model scrutinizes the impacts of nonlinear radiative, uneven heat sink/source, thermophoretic movement and chemical reaction. Additionally, model is considered water based nanofluid with nanoparticles CuO (Copper Oxide) and MgO (Magnesium Oxide). The flow mechanics transport through Riga plate is developed using coupled system of higher order equations have been mutated into ordinary order from partial order utilizing established similarity transform. These nonlinear equations are calculated by the usage of through the Galerkin weighted residual method (GWRM) along with Simpson’s one-third rule and procured results are visualized graphically. Verifications of attained numerical outcomes through available literature for limiting case are also presented and found in excellent agreement. Further, the results of engineering physical parameters were also highlighted. More so, the analysis show that an increment in the Casson term causes rise in the temperature profile for CuO and MgO nanofluid and also dominant behavior is noted in case of CuO nanofluid comparing with MgO nanofluid. Furthermore, higher values of volume fraction parameter rises the both velocity profiles f′ (η), g′ (η) for both MgO and CuO nanofluid cases.

Reliability investigation of exponentiated Weibull distribution using IPL through numerical and artificial neural network modeling

AbstractIn current investigation, a novel implementation of intelligent numerical computing solver depending on artificial neural networks (ANN) has been provided to interpret failure function (FF), reliability function (RF), hazard rate function (HRF), Mils ratio (MR), and mean time to failure (MTTF). This study investigates a reliability model centered on the exponentiated Weibull distribution (EWD) and the inverse power law (IPL) model employing the ANN model. The nonmonotonic failure rate can be modeled via this distribution. A data set for the proposed ANN has been generated for various scenarios of (Exponentiated Weibull Inverse Power Law Distribution) EWIPLD model by variation of involved pertinent parameters via the Galerkin weighted residual method (GWRM). Levenberg–Marquard training algorithm has been used in the multi‐layer perceptron (MLP) network model developed with 10 nodes in the hidden layer. The Coefficient of Determination (R) value for the ANN model has been obtained as 0.9999. The findings obtained, revealed that ANNs are an excellent technique that can be applied to predict reliability measures in conjunction with the right statistical model.

Entropy generation minimization on electromagnetohydrodynamic radiative Casson nanofluid flow over a melting Riga plate

AbstractMinimizing entropy generation is a technique that helps improve the effectiveness of real processes by studying the associated irreversibility of system performance of nanofluid. This study examines the entropy generation analysis of electromagnetohydrodynamic radiative Casson flow induced by a stretching Riga plate in a non‐Darcian porous medium under the influence of internal energy change, Arrhenius activation energy, chemical reaction, and melting heat transfer. The thermophysical features of the fluid are assumed constant in most of the literature. However, this current research bridges this gap by considering viscosity, conductivity, and diffusivity as temperature‐dependent variables. Also, the exponential decaying Grinberg term is used as a resistive force in this investigation due to the electromagnetic properties of the Riga plate in the momentum conservation equation. Some suitable dimensionless variables are introduced to remodel the transport equations into unitless ones and then solved numerically by employing Galerkin Weighted Residual Method. Analyses reveal that the Casson parameter declines the fluid velocity, while the existence of the melting parameter has the opposite effect. Also, this article includes some future recommendations.

Influence of magnetization, variable viscosity and thermal conductivity on Von Karman swirling flow of H2O-FE3O4 and H2O-Mn-ZNFe2O4 ferromagnetic nanofluids from a spinning DISK: Smart spin coating simulation

Motivated by smart nano-ferromagnetic spin coating applications, A theoretical study is described for steady swirling Von Karman thermo-magnetic water-based flowing nanoliquids containing ferromagnetic nanoparticles from a rotating disk in Darcian permeable media. The governing mass, momentum and temperature equations are converted into nonlinear-coupled ordinary derivative momentum and energy equations via appropriate similarity transformations, resulting boundary value ordinary differential problem is solved by a Galerkin weighted residual method (GWRM) along with Simpson’s one-third rule. Verification of the GWRM solutions is achieved with numerical shooting quadrature (MAPLE) and very good correlation is demonstrated. Ferromagnetic Fe3O4 nanofluid is observed to achieve superior thermal conductivity enhancement relative to ferromagnetic Mn-ZnFe2O4 nanofluid. Increasing magnetic field intensity δ substantially modifies the viscosity and produces a consistent retardation in both axial and radial velocity whereas it weakly enhances the tangential velocity field. With greater ferromagnetic interaction number β axial velocity is enhanced strongly, and radial velocity is also boosted.

Thermosolutal LTNE Porous Mixed Convection Under the Influence of the Soret Effect

Abstract The onset of thermosolutal convection in a fluid-saturated porous medium in the presence of a horizontal pressure gradient and the Soret effect under the two-temperature model of local thermal nonequilibrium (LTNE) is investigated. The Darcy model with the time-dependent velocity term in the momentum equation is employed together with the Oberbeck–Boussinesq approximation. The eighth-order eigenvalue differential equation obtained by employing linear stability analysis is solved using Galerkin's method of weighted residuals (GMWR) and also analytically. The onset of convection instills through oscillatory rather than stationary mode, and the critical stability parameters for the same are determined. The horizontal pressure gradient reinforces together with the solute concentration gradient and the scaled interphase heat transfer coefficient in evidencing mixed behavior on the criterion for the onset of oscillatory convection. Besides, the similarities and differences between the results of thermal and thermosolutal LTNE porous mixed convection are discerned.

Effects of Heat Generation/Absorption on Magnetohydrodynamics Flow Over a Vertical Plate with Convective Boundary Condition

Heat generation effect in a steady two-dimensional magnetohydrodynamics (MHD) flow over a moving vertical plate with a medium porosity has been studied. By similarity transformation variables, the coupled non-linear ordinary differential equations describing the model are obtained. The resulting equation is then solved, using Galerkin Weighted Residual Method (GWRM), where the effect of heat generation, Magnetic Parameter as well as other physical parameters encountered were examined and discussed. Some of the major findings were that increase in heat generation and convective heat parameter enhances the plate surface temperature as well as temperature field which allows the thermal effect to penetrate deeper into the quiescent fluid.

Unsteady flow analysis of Maxwell fluid with temperature dependent variable properties and quadratic thermo-solutal convection influence

The current work analyzed the surface effect on the motion of Maxwell fluid with variable transport properties (viscosity, diffusivity, thermal conductivity, and electrical field) in porous, magnetized, radiative, and nonuniform (quadratic) convection processes. In this study, the flow examination modeled the motion of Maxwell fluid under MHD and variable properties influence. The flow field solutions were obtained numerically via the Spectral Collocation Approach (SCM) and justified with Galerkin Weighted Residual Method (GWRM). It was shown that the flat surface dominates the flow fields, a rise in variable viscosity diminished the fluid momentum, variable thermal conductivity and diffusivity enhance the consumption of more fluid particles, the velocity field is promoted to a hike in nonlinear Boussinesq approximation numbers, while sheet variable thickness number is aimed at downsizing the flow distributions. Moreover, the current analysis is applicable in metal spinning, polymer extrusion, machine design, architecture, aluminum aircraft skin, and structural steel beam where the extrudate materials are stretched into a sheet.