Galerkin Method Of Weighted Residuals Research Articles

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.

Analysis of Axisymmetric Vibration of Functionally-Graded Circular Nano-Plate Based on the Integral Form of the Strain Gradient Model

In this paper, it is aimed to analyze the linear vibrational behavior of functionally-graded (FG) size-dependent circular nano-plates using the integral form of the non-local strain gradient (NSG) model. The linear axisymmetric vibration of the circular FG nano-plates based on the non-local strain gradient (NSG) model is the focal point of this study. In this regard, the non-local elasticity theory (NET) and strain gradient (SG) models are used in conjunction with Hamilton's principle to obtain the governing equations. Discretization of the obtained governing equations is performed with the help of generalized differential quadrature rule (GDQR) and Galerkin weighted residual method (GWRM). The analysis is focused on the effect of non-local and material parameters, as well as the aspect ratio, heterogeneity index of FG material, different boundary conditions, and frequency number on the overall behavior of nano-plate. On using the Galerkin method, a system of linear differential equations is obtained and solved to determine the natural linear frequencies and mode shapes. The obtained results are then compared with the existing results in the literature. On using the proposed procedure in this paper, the dynamic behavior of nano-plate under different boundary conditions can be well described. In addition, the existing deficiencies in other non-local theories can be eliminated. The results of this investigation can be considered as a turning point in the improvement of theoretical results for achieving a better prediction of vibrational behavior in nanostructures.

Heat Transport of Casson Nanofluid Flow over a Melting Riga Plate Embedded in a Porous Medium

This article investigates the flow of Casson nanofluid induced by a stretching Riga plate in the presence of a porous medium. The implication of the Riga plate is to generate electromagnetohydrodynamic force which influences the fluid speed, and as well applicable in delaying boundary layer separation. The complexity of the equations governing the problem is reduced using similarity transformation. The resulting coupled nonlinear ordinary differential equations are solved by employing Chebyshev collocation scheme (CCS) and validated with Galerkin weighted residual method (GWRM). The influence of parameters, such as modified Hartmann number and melting parameter, on the nanofluid flow, heat, and mass transfer is considered. Some of the major findings include that modified Hartmann number tends to increase nanofluid flow. Also, increasing the value of melting parameter is in favor of both velocity and nanoparticle volume fraction profiles but diminishes temperature profile. The application of this work can be found in polymer synthesis, metallic processing, and electromagnetic crucible systems.

Estimation of unsteady hydromagnetic Williamson fluid flow in a radiative surface through numerical and artificial neural network modeling

In current investigation, a novel implementation of intelligent numerical computing solver based on multi-layer perceptron (MLP) feed-forward back-propagation artificial neural networks (ANN) with the Levenberg–Marquard algorithm is provided to interpret heat generation/absorption and radiation phenomenon in unsteady electrically conducting Williamson liquid flow along porous stretching surface. Heat phenomenon is investigated by taking convective boundary condition along with both velocity and thermal slip phenomena. The original nonlinear coupled PDEs representing the fluidic model are transformed to an analogous nonlinear ODEs system via incorporating appropriate transformations. A data set for proposed MLP-ANN is generated for various scenarios of fluidic model by variation of involved pertinent parameters via Galerkin weighted residual method (GWRM). In order to predict the (MLP) values, a multi-layer perceptron (MLP) artificial neural network (ANN) has been developed. There are 10 neurons in hidden layer of feed forward (FF) back propagation (BP) network model. The predictive performance of ANN model has been analyzed by comparing the results obtained from the ANN model using Levenberg-Marquard algorithm as the training algorithm with the target values. When the obtained Mean Square Error (MSE), Coefficient of Determination (R) and error rate values have been analyzed, it has been concluded that the ANN model can predict SFC and NN values with high accuracy. According to the findings of current analysis, ANN approach is accurate, effective and conveniently applicable for simulating the slip flow of Williamson fluid towards the stretching plate with heat generation/absorption. The obtained results showed that ANNs are an ideal tool that can be used to predict Skin Friction Coefficients and Nusselt Number values.

Thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect

Galerkin weighted residual method (GWRM) is applied and implemented to address the axial stability and bifurcation point of a functionally graded piezomagnetic structure containing flexomagneticity in a thermal environment. The continuum specimen involves an exponential mass distributed in a heterogeneous media with a constant square cross section. The physical neutral plane is investigated to postulate functionally graded material (FGM) close to reality. Mathematical formulations concern the Timoshenko shear deformation theory. Small scale and atomic interactions are shaped as maintained by the nonlocal strain gradient elasticity approach. Since there is no bifurcation point for FGMs, whenever both boundary conditions are rotational and the neutral surface does not match the mid-plane, the clamp configuration is examined only. The fourth-order ordinary differential stability equations will be converted into the sets of algebraic ones utilizing the GWRM whose accuracy was proved before. After that, by simply solving the achieved polynomial constitutive relation, the parametric study can be started due to various predominant and overriding factors. It was found that the flexomagneticity is further visible if the ferric nanobeam is constructed by FGM technology. In addition to this, shear deformations are also efficacious to make the FM detectable.

Influence of Chemical Reaction and Arrhenius Activation Energy on Hydromagnetic Non-Darcian Casson Nanofluid Flow with Second-Order Slip Condition

This article investigates the combined effect of second-order velocity slip, Arrhenius activation energy and binary chemical reaction on reactive Casson nanofluid flow in a non-Darcian porous medium. The governing equations of the problem were first non-dimensionalized and later reduced to ordinary nonlinear differential equations by adopting a similarity transformation. The emerging nonlinear boundary value problem was solved by using Galerkin weighted residual method (GWRM). The obtained results were compared with those found in the literature to validate our method. The impact of pertinent parameters on the velocity component, temperature distribution and concentration profile are presented using graphs and were discussed. The computational results show that an increase in second order slip parameter significantly results to an increase in the temperature as well as nanoparticle concentration profiles, while it reduces the velocity profile.

THE EFFECT OF ROTATION AND PULSATING THROUGHFLOW ON THE ONSET OF LONGITUDINAL CONVECTIVE ROLLS IN A POROUS MEDIUM SATURATED BY NANOFLUID

This study accentuates the combined effect of rotation and pulsating throughflow on the onset of longitudinal convec-tive rolls in a porous medium saturated by nanofluid. Pulsating throughflow modifies the basic profiles for temperature and the volumetric fraction of nanoparticle from linear to nonlinear with layer height, which in turn affects the stability significantly. To treat this problem, Buongiorno's two-phase mathematical model is used. Using the framework of linear stability theory and the frozen profile approach, the stability equations are derived and solved analytically using the Galerkin weighted residuals method. The effect of increasing the modified diffusive ratio NA and the nanoparticle Rayleigh-Darcy number RN, and decreasing the rotation parameter TD is to accelerate the onset of convection, while the Lewis number Le and porosity parameter φ have dual effect on the stability of the system in the presence of pulsating throughflow. Moreover, the frozen profile approach shows that the effect of pulsating throughflow in both directions is stabilizing. An increased amplitude of throughflow oscillations leads to increased stability by an amount that depends on frequency.

NUMERICAL SIMULATION OF ENTROPY GENERATION FOR CASSON FLUID FLOW THROUGH PERMEABLE WALLS AND CONVECTIVE HEATING WITH THERMAL RADIATION EFFECT

In this work, the influence of entropy generation analysis for an electrically conducting Casson fluid flow with convective boundary conditions has been numerically studied. The governing equations are analyzed numerically using weighted residual methods. Subsequently, the residuals were minimized using two different approaches of weighted residual method namely collocation weighted residual method (CWRM) and Galerkin weighted residual method (GWRM) and computed numerically using MATHEMATICAL software. The impacts of governing parameters on Casson flow velocity, temperature profile, local skin friction, and Nusselt number were analysed. The obtained solutions were used to determine the heat transfer irreversibility and bejan number of the model. The results of the computation show that the effect of thermophysical properties such as thermal radiation parameter, suction/injection parameter, magnetic field parameter, radiation parameter, and Eckert number has a significant influence on Skin friction coefficient (Cf) and local Nusselt number (Nu) when compared to the Newtonian fluid. The findings from this study are relevant to advances in viscoelasticity and enhanced oil recovery.

LTNE thermoconvective instability in Newtonian rotating layer under magnetic field utilizing nanoparticles

In the present framework, the simultaneous effect of rotation, magnetic field, heat source and local thermal non-equilibrium (LTNE) is investigated on a water-based nanofluid layer. A mathematical model depicting LTNE behavior between base fluid and nanoparticles phases along with slip mechanisms, i.e., Brownian motion and thermophoresis is taken into consideration. Both linear and weakly nonlinear analyses have been performed for a given dynamical system. The normal mode technique and Galerkin weighted residual method (GWRM) are employed to find the stationary mode of convection for free–free (FF), rigid–rigid (RR) and rigid–free (RF) boundary conditions in linear stability. In case of nonlinear stability, the minimal truncated Fourier series is implemented and the resulting system of equations is solved using MATLAB built-in ode solver to visualize the streamlines, heatlines and quantify the heat and mass transport. The magnetic field, rotation, interface heat transfer and thermal diffusivity ratio parameters delay the onset of convection in the nanofluid layer, while the modified thermal capacity ratio and heat source advance it. Thus, heat transfer enhancement has been reported with the increment of both modified thermal capacity ratio and heat generation, whereas the opposite trend is noticed for other parameters in nanofluid layer. For Ra > Rac, the isotherms for both fluid and particle phases are flattened near the wall which depicts the increase in the convection in the nanofluid layer. The present investigation of Rayleigh–Benard convection in nanofluids may be useful for providing parameter ranges where the desired instability can be made to manifest depending upon the need for an engineering application.

On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution.

Among various magneto-elastic phenomena, flexomagnetic (FM) coupling can be defined as a dependence between strain gradient and magnetic polarization and, contrariwise, elastic strain and magnetic field gradient. This feature is a higher-order one than piezomagnetic, which is the magnetic response to strain. At the nanoscale, where large strain gradients are expected, the FM effect is significant and could be even dominant. In this article, we develop a model of a simultaneously coupled piezomagnetic–flexomagnetic nanosized Euler–Bernoulli beam and solve the corresponding problems. In order to evaluate the FM on the nanoscale, the well-known nonlocal model of strain gradient (NSGT) is implemented, by which the nanosize beam can be transferred into a continuum framework. To access the equations of nonlinear bending, we use the variational formulation. Converting the nonlinear system of differential equations into algebraic ones makes the solution simpler. This is performed by the Galerkin weighted residual method (GWRM) for three conditions of ends, that is to say clamp, free, and pinned (simply supported). Then, the system of nonlinear algebraic equations is solved on the basis of the Newton–Raphson iteration technique (NRT) which brings about numerical values of nonlinear deflections. We discovered that the FM effect causes the reduction in deflections in the piezo-flexomagnetic nanobeam.

THERMAL INSTABILITY OF OLDROYDIAN VISCO-ELASTIC NANOFLUID IN A POROUS MEDIUM FOR MORE REALISTIC BOUNDARY CONDITIONS

The thermal instability of Oldroydian visco-elastic nanofluid in a porous medium with more realistic boundary conditions is investigated. The Brinkman model, used for the porous medium, incorporates the effect of Brownian diffusion and thermophoresis. The eigen value problem is solved by employing the Galerkin weighted residuals method. The influence of the Lewis number, Brinkman-Darcy number, nanoparticle Rayleigh number, modified diffusivity ratio, and porosity parameter on the stationary convection are studied analytically. The Lewis number, modified diffusivity ratio, and nanoparticles Rayleigh number destabilize, and the Brinkman-Darcy number and porosity parameter stabilize the stationary convection.

Semi-analytical solutions for different non-linear models of dual phase lag equation in living tissues

In this study, the heat transfer in a multilayer heated living tissue is investigated. A non-linear dual phase lag (DPL) heat conduction model is presented assuming temperature-dependent blood perfusion rate, heat conductivity and metabolic heat generation. Semi -analytical solutions for the non-linear model are obtained employing the Galerkin weighted residuals method. The effects of the variable thermophysical properties on the temperature distribution within the tissue are investigated through comparison of the results with the corresponding results for the constant thermophysical properties. The results show that the temperature dependence of the blood perfusion rate has a significant effect on the temperature distribution in the different layers of the tissue. As far as the tumor region is considered, 0.43 °C increase in its temperature is observed in comparison with that obtained assuming the constant blood perfusion rate at t = 1800 s. Moreover, increasing the temperature dependence of the thermal conductivity leads to the reduction in the temperature of the tissue except in the muscle layers. Considering the temperature-dependent metabolic heat generation increases the temperature of all of the layers compared to the corresponding temperatures obtained assuming the constant metabolic heat generation.

Implementation of non-probabilistic methods for stability analysis of nonlocal beam with structural uncertainties

In this study, a non-probabilistic approach-based Navier’s method (NM) and Galerkin weighted residual method (GWRM) in terms of double parametric form have been proposed to investigate the buckling behavior of Euler–Bernoulli nonlocal beam under the framework of Eringen’s nonlocal elasticity theory, considering the structural parameters as imprecise or uncertain. The uncertainties in Young’s modulus and diameter of the beam are modeled in terms of triangular fuzzy numbers. The critical buckling loads are calculated for hinged–hinged, clamped–hinged, and clamped–clamped boundary conditions, and these results are compared with the deterministic model in special cases, demonstrating robust agreement. Further, a random sampling technique-based method, namely Monte Carlo simulation technique (MCST), has been implemented to compute the critical buckling loads of uncertain systems. Also, the critical buckling loads obtained from the uncertain model in terms of lower bound and upper bound by the non-probabilistic methods, viz. NM and GWRM, are again verified with the MCST with their time periods, demonstrating the efficacy, accuracy, and effectiveness of the proposed uncertain model. A comparative study is also carried out among the non-probabilistic methods and MCST to demonstrate the effectiveness of methods with respect to time. Additionally, a parametric study has been performed to display the propagation of uncertainties into the nonlocal system in the form of critical buckling loads.

Numerical Solutions of Black-Scholes Model by Du Fort-Frankel FDM and Galerkin WRM

The main objective of this paper is to find the approximate solutions of the Black-Scholes (BS) model by two numerical techniques, namely, Du Fort-Frankel finite difference method (DF3DM), and Galerkin weighted residual method (GWRM) for both (call and put) type of European options. Since both DF3DM and GWRM are the most familiar numerical techniques for solving partial differential equations (PDE) of parabolic type, we estimate options prices by using these techniques. For this, we first convert the Black-Scholes model into a modified parabolic PDE, more specifically, in DF3DM, the first temporal vector is calculated by the Crank-Nicolson method using the initial boundary conditions and then the option price is evaluated. On the other hand, in GWRM, we use piecewise modified Legendre polynomials as the basis functions of GWRM which satisfy the homogeneous form of the boundary conditions. We may observe that the results obtained by the present methods converge fast to the exact solutions. In some cases, the present methods give more accurate results than the earlier results obtained by the adomian decomposition method [14]. Finally, all approximate solutions are shown by the graphical and tabular representations.

Instability of Vertical Throughflows in Porous Media under the Action of a Magnetic Field

The instability of a vertical fluid motion (throughflow) in a binary mixture saturating a horizontal porous layer, uniformly heated from below, uniformly salted from below by one salt and permeated by an imposed uniform magnetic field H , normal to the layer, is analyzed. By employing the order-1 Galerkin weighted residuals method, the critical Rayleigh numbers for the onset of steady or oscillatory instability, have been determined.

3D fluid-solid full coupling numerical simulation of soil deformation induced by shield tunnelling

For twin tunnels, succeeding tunnelling may cause damage to preceding tunnel in close proximity during excavation. Therefore it is important to predict the soil deformation around the tunnels to ensure safety during shield construction. However, the effect of groundwater is usually ignored and soil parameters are considered constant in previous study. This study has been undertaken using the Biot’s theory and rheological theory. And the parameters of soil are considered to be variable. On that basis, a 3D fluid-solid full coupling mathematical model of soil deformation induced by shield tunnelling is established. Furthermore, the equations are discretized by the Galerkin Weighted Residual Method (GWRM). The analysis of finite mathematical model is developed by FORTRAN95. The mathematical model was proved by comparing the calculated and measured settlement from Chengdu Metro Line 4. And then the soil deformation and the pore water pressure around the tunnels were predicted. The results show that the error between measured values and calculated values of soil deformation is relatively small.