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.

Irreversibility Analysis for Eyring–Powell Nanoliquid Flow Past Magnetized Riga Device with Nonlinear Thermal Radiation

The report contained in this article is based on entropy generation for a reactive Eyring–Powell nanoliquid transfer past a porous vertical Riga device. In the developed model, the impacts of viscous dissipation, thermophoresis alongside nonlinear heat radiation and varying heat conductivity are modelled into the heat equation. The dimensionless transport equations are analytically tackled via Homotopy analysis method while the computational values of chosen parameters are compared with the Galerkin weighted residual method. Graphical information of the various parameters that emerged from the model are obtained and deliberated effectively. The consequences of this study are that the temperature field expands with thermophoresis, Brownian motion and temperature ratio parameters as the modified Hartmann number compels a rise in the velocity profile. The entropy generation rises with an uplift in fluid material term as well as Biot and Eckert numbers whereas Bejan number lessens with Darcy and Eckert parameters.

Application of exponential functions in weighted residuals method in structural mechanics. Part 3: infinite cylindrical shell under concentrated forces

Solution for cylindrical shell under concentrated force is a fundamental problem which allow to consider many other cases of loading and geometries. Existing solutions were based on simplified assumptions, and the ranges of accuracy of them still remains unknown. The common idea is the expansion of them into Fourier series with respect to circumferential coordinate. This reduces the problem to 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relation of 8 constant of integrations with boundary conditions are still beyond the possibilities of analytical treatment. In this paper we apply the decaying exponential functions in Galerkin-like version of weighted residual method to above-mentioned 8th order equation. So, we construct the sets of basic functions each to satisfy boundary conditions as well as axial and circumferential equilibrium equations. The latter gives interdependencies between the coefficients of circumferential and axial displacements with the radial ones. As to radial equilibrium, it is satisfied only approximately by minimizations of residuals. In similar way we developed technique for application of Navier like version of WRM. The results and peculiarities of WRM application are discussed in details for cos2j concentrated loading, which methodologically is the most complicated case, because it embraces the longest distance over the cylinder. The solution for it clearly exhibits two types of behaviors – long-wave and short-wave ones, the analytical technique of treatment of them was developed by first author elsewhere, and here was successfully compared. This example demonstrates the superior accuracy of two semi analytical WRM methods. It was shown that Navier method while being simpler in realization still requires much more (at least by two orders) terms than exponential functions.

Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method

The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate.

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.

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.

Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem

In this work, goal-oriented adjoint-based a posteriori error estimates are derived for a nonlinear phase-field discontinuity problem in which a scalar-valued displacement field interacts with a scalar-valued smoothed indicator function. The latter is subject to an irreversibility constraint, which is regularized using a simple penalization strategy. The main advancements in the current work are error identities, resulting estimators, and two-sided estimates employing the dual-weighted residual method, which address the influence of the phase-field regularization, penalization, and spatial discretization parameters. Some numerical tests accompany our derived estimates.

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.