Galerkin Finite Element Analysis Research Articles

Galerkin finite element analysis on radiative heat transfer in titanium dioxide-polyalphaolefin nanolubricant past a convergent/divergent channel with non-uniform heat source/sink effect

The current study inspects the radiative heat and mass transport of a polyalphaolefin (PAO) – based nanolubricant flow consisting of titanium dioxide (TiO2) nanoparticles in a convergent/divergent channel with thermophoresis, Brownian motion, and non-uniform heat source/sink effects. The mathematical modelling of the present problem results in a system of complex non-linear partial differential equations (PDEs). The Galerkin finite element method (GFEM) is adopted to solve complex equations and to obtain the solution for the field variables in convergent/divergent channels. The results revealed that as the values of Brownian motion get higher, the temperature dispersal in the channel quickens, whereas the concentration profile slows down. The rise in the value of the thermophoresis parameter increases the thermophoresis force, which causes liquid particles to be dispersed in the stream. The heat transmission rate increases with an increase in radiation parameter values.

Numerical evaluation of thermohydraulic parameters for diverse configurations of shell-and-tube heat exchanger

This research aims to optimize the internal design of shell-and-tube heat exchangers by investigating the thermal performance of various configurations. A comparative study of segmental baffles with plain and finned tube setups and helical baffles with different angular orientations is conducted using numerical simulations with the Galerkin finite element analysis. Key thermohydraulic parameters, such as shell-side pressure difference, heat transport coefficient, and heat transport coefficient per unit pressure drop, are assessed with shell-side mass flow rates. The outcomes direct higher shell-side oil mass flow rates correlate with increased pressure drop and enhanced heat transfer coefficients. Helical baffles exhibit superior performance to segmental baffles, offering a more significant heat transport coefficient per unit pressure drop. Within segmental baffles, finned tubes lead to higher pressure drops and heat transport coefficients than plain tubes while maintaining similar heat transport coefficients per unit pressure drop. Furthermore, lower helix angles (Φ = 10.81°) in helical baffles are associated with higher pressure drops and heat transport coefficients, whereas Φ = 25.52° achieves the highest heat transport coefficient per unit pressure drop.

Investigation of free convection in a wavy trapezoidal porous cavity with MWCNT- Fe3O4/Water hybrid nanofluid under MHD effects: Galerkin finite element analysis

Our study focuses on numerically investigating natural convection within a three-dimensional trapezoidal cavity featuring a corrugated hot bottom wall, with a particular emphasis on enhancing heat exchange using a water-based hybrid nanofluid. Employing a steady-state regime assumption, the study considers laminar and incompressible three-dimensional flow. The Darcy-Forchheimer model is incorporated to account for inertial advection effects within the porous layer. Predictions of the thermal and hydrodynamic behaviors of the system are achieved by solving dimensionless equations utilizing the Galerkin Finite Element Method (GFEM). Notably, a range of influential parameters, including the volume fraction of nanoparticles (φ), Darcy number (Da) spanning from 10−5 to 10−2, Hartman number (Ha) across the range of 0–100, φ within the range of 0–0.08, porosity (ε) ranging from 0.2 to 0.9, and Rayleigh numbers (Ra) varying from 103 to 106 was explored. The investigation also evaluates the geometrical impact of the enclosure by considering undulation numbers (N) of the warm bottom ranging from 1 to 4. Our results provide novel quantitative insights. Specifically, we observe an inverse relationship between undulation number (N), Hartman number (Ha), and porosity (ε) with large Rayleigh (Ra) flows, significantly influencing effective heat transfer. Notably, this influence diminishes at lower Ra flows. At the highest Ra, we find that increasing Da, ε, and φ enhances the Nusselt number (Nuavg) by 26 %, 17 %, and 23.5 %, respectively. Conversely, increasing Ha and N reduces Nuavg by 13 % and 40 %, respectively. These findings highlight the nuanced interplay of parameters and offer valuable quantitative data for optimizing heat transfer processes in similar systems.

Galerkin finite element analysis of heat and mass transfer of Jeffrey, Maxwell and Oldroyd-B nanofluids in a vertical annulus with an induced magnetic field and a non–uniform heat source/sink

ABSTRACT The Maxwell, Jeffrey and Oldroyd-B nanofluid models are important for characterising fluid flow, heat and mass transfer in industrial chemical process systems and operations. Consequently, this study focuses on investigating thermophoresis, Brownian motion, a non-uniform heat source/sink and suction influence on Maxwell, Jeffrey and Oldroyd-B nanofluids towards a stretching cylindrical annular surface with an induced magnetic field (IMF). The non-linear ODEs are solved numerically by opting for GFEM for the transformed system. The obtained GFEM results are compared and are in exceptional agreement with existing results, in the limiting sense. The boundary layer of the velocity and IMF are reduced by the Lorentz force, and also in the vertical annulus the IMF is independent of magnetic parameter ( 1 ≤ β ≤ 5 ) and is reliant on the size of the radius ratio. For higher values of Nb ( 0.1 ≤ N b ≤ 0.5 ) the temperature distribution in the annulus accelerates, whereas the concentration profile decelerates. Additionally, heat and mass transfer characteristics are analysed in the annulus through ANN – BLMN. The important outcome of this study is that the Jeffrey model outperforms the Maxwell and Oldroyd-B nanofluid models in terms of heat and mass transfer rates.

Natural convection and entropy generation inside a square chamber divided by a corrugated porous partition

A thorough investigation of free convection and entropy generation occurring inside a differentially heated square chamber that is filled with water and divided by a water-saturated, corrugated porous partition is performed in this numerical study. The governing mathematical equations, that describe the flow and heat transfer phenomena for fluid and porous domains, are the Navier-Stokes (modified for the porous domain using the Darcy-Brinkman-Forchheimer model) and thermal energy equations. Those systems of equations are solved using the Galerkin finite element analysis. Parametric changes are carried out for different positions, thicknesses, amplitudes, and frequencies of corrugation of the porous partition, and the corresponding results are quantitatively presented in terms of average Nusselt number along the heated wall and total entropy generation of the entire chamber with increasing Rayleigh number (103 ≤ Ra ≤ 107). Corresponding variations are qualitatively visualized in terms of streamlines and isotherms for comparing the related parametric changes. Furthermore, a comparative analysis is included between the use of porous and solid partitions along with a chamber without any partition. Conclusive results show that using a porous rather than a solid partition can increase the average Nusselt number by 26.28% at Ra = 105 up to a maximum of 565% at Ra = 107. Similarly, lower thickness, higher frequency, and higher amplitude can increase the average Nusselt number by around 37.5%, 2.89%, and 1.17%, respectively.

Galerkin finite element analysis for buoyancy driven copper-water nanofluid flow and heat transfer through fins enclosed inside a horizontal annulus: Applications to thermal engineering

The hydrothermal significance of natural convection flow through the horizontal annulus reveal most important implication and those geometrical shapes are introduced in enormous industrial and engineering implementations related to superheaters, heat exchangers, hydrogen fuel cells, gas turbines, solar collectors, vehicle engines, and turbomachinery etc. With this motivation, the current advance study enlightens the natural convection heat transfer impacts of Cu−water nanofluid flow through horizontal annulus with inserting Fins at the inner cylinder. Different fins numbers (0, 2 and 4) are inserted at inner cylinder wall are heated. The outer cylinder kept at cooled temperature. The significance nanoparticles fraction (Φ=1%−5%), Rayleigh number (1000,10000and100000) and different fin numbers (0,2and4) on fluid flow, thermal field, and local Nusselt number investigated. The Galerkin finite element method has been employed to carry out the numerical solution for different values of flow controlling parameters. The impacts of prominent parameter on streamlines, isotherms and 2d graphs are analyzed. The analysis concluded that velocity and thermal outcomes upsurges for Rayleigh number. It is conclude that important thermal transfer increment can be observed due to the occurrence of solid nanoparticles and that is boosted by growing the solid particle fraction. The velocity is reduced by increasing the values of nanoparticle volume fraction. Additionally larger heat transfer for nanoparticles Volume fraction Φ=5% as compare to Φ=1%.

Galerkin finite element analysis for magnetized radiative-reactive Walters-B nanofluid with motile microorganisms on a Riga plate

In order to understand the characteristics of bio-convection and moving microorganisms in flows of magnetized Walters-B nano-liquid, we developed a model employing Riga plate with stretchy sheet. The Buongiorno phenomenon is likewise employed to describe nano-liquid motion in the Walters-B fluid. Expending correspondence transformations, the partial differential equation (PDE) control system has been transformed into an ordinary differential equation (ODE) control system. The COMSOL program is used to generate mathematical answers for non-linear equations by employing the Galerkin finite element strategy (G-FEM). Utilizing logical and graphical metrics, temperature, velocity, and microbe analysis are all studied. Various estimates of well-known physical features are taken into account while calculating nanoparticle concentrations. It is demonstrated that this model's computations directly relate the temperature field to the current Biot number and parameter of the Walters-B fluid. The temperature field is increased to increase the approximations of the current Biot number and parameter of the Walters-B fluid.

Exploration of Ostwald-de Waele non-Newtonian nanofluid subject to Lorentz force, and entropy optimization in a corrugated porous medium enclosure: Galerkin finite element analysis

In this work, we combine the rheological and nanofluidic properties of the fluid in a hermetically sealed chamber. The work was accomplished digitally using Galerkin finite element technique. The work aims to know the effect of these complex properties of the fluid on the quality of thermal activity of the type of free convection. The fluid of this study is composed of a complex fluid with viscosity added to a proportion of nanoparticles. As for the room, it has ripples on the walls and an elliptical obstacle in the middle. The thermal transfer studied here takes place between the hot elliptical obstacle and the cold walls of the room. based on this proposition, the issues studied here are: undulation number (N = 1, 2, 3, and 4); Power – law index (n = 0.8, 1, 1.2, and 1.4); Darcy number (Da = 10-2, 10-3, 10-4, and 10-5); Rayleigh number (Ra = 103, 104, 105, and 106); Hartmann number (Ha = 0, 25, 50, and 100); and the rotational angle of elliptic cylinder (γ = 0, 30, 60, and 90°). The finding shows that the Nusselt number (Nu), is augmented when these parameters increase: Da (for all values of Ra) and γ (large Ra). At the height’s Ra number (106) it was observed that increasing the Ha number and Da number reduced Nu by 18 % and 65 %, respectively. While Nu was enhanced by 129 % when increasing Da number at Ra = 106.

Galerkin finite element analysis for the augmentation in thermal transport of ternary-hybrid nanoparticles by engaging non-Fourier’s law

Boosting of thermal transportation is the demand of current era. Several techniques have been used to do so. One of an important way is the mixing of nanoparticles to boost thermal performance. Current investigation has been prepared to study the inclusion of tri hybrid nanoparticles in Prandtl fluid model past over a stretched heated sheet. Modelling of consider problem has been done due to consideration of movement in flow in Cartesian coordinates which results coupled partial differential equation system thermal transport in presented by considering generalized heat flux model and heat generation/absorption. The derived coupled complex partial differential equations (PDEs) system is simplified by engaging boundary layer theory. Such developed model is used in coolants regarding automobiles, dynamics in fuel and production of solar energy, fuel cells, optical chemical sensors, automotive parts, dental products, cancer therapy, electrical insulators and dental products. Handling of complex PDEs for the solution is a challenging task. Due to complexity in computational work these PDEs have been transformed into ordinary differential equations (ODEs) after applying similarity transformation afterwards converted ODEs have been approximated via finite element algorithm coded in MAPLE 18.0 symbolic computational package. Comparative study has been presented for the validity of code and authenticity of obtained result. It is observed that fluid velocity for tri-hybrid nanoparticles is higher than fluidic motion for pure fluid, nanofluid and hybrid nanomaterial.

GALERKIN FINITE ELEMENT ANALYSIS OF BUOYANCY-DRIVEN HEAT TRANSFER OF WATER-BASED CU-AL2O3 HYBRID NANOFLUID INSIDE A U-SHAPED ENCLOSURE WITH LOCALIZED HEAT

GALERKIN FINITE ELEMENT ANALYSIS OF BUOYANCY-DRIVEN HEAT TRANSFER OF WATER-BASED CU-AL2O3 HYBRID NANOFLUID INSIDE A U-SHAPED ENCLOSURE WITH LOCALIZED HEAT

Finite element method in thermal characterization and streamline flow analysis of electromagnetic silver-magnesium oxide nanofluid inside grooved enclosure

Employing magnetic effect, behavior of heat and fluidity of hybrid nanofluid based on the mixed water of magnesium oxide and silver is explored in the present work on the permeable and hollow ampoule. Inertia of penetrable layer is obtained by employing Darcy Forchheimer model especially and efficient thermos-physical model. Effective Galerkin Finite Element Analysis (GFEM) is executed to solve the governing equations of model. 10−6 error tolerance was used to examine the results. Impacts of isotherms and streamlines were marked by performing the parametrical analysis of important and effective constraints. Except the influencing constraints of the flow, there is a prominent effect of twisting ampoule structure on fluidity. This effect can be seen in the developed contours in the plots of streamlines. Heat diffusion is made away from the wall with the initial values of Rayleigh number (Ra) as well as Darcy number (Da), mainly from the hotter side of the wall that have two structures of cliff bar. Except the increment in Hartmann number (Ha) and low Rayleigh number (Ra), increment is observed in heat transfer rate with many constraints.

Galerkin finite element analysis for peristaltic flow of micropolar fluid through porous soaked inclined tube independent of wavelength

In this novel numerical investigation, the application of well-renowned numerical technique known as Galerkin finite element method on full form of Navier-Stokes equations presented peristaltic flow of non-Newtonian fluid confined by a uniformly saturated porous medium. The rheological aspects of non-Newtonian material are discussed by considering micropolar fluid. The flow model consists of system of nonlinear partial differential equations with mixed boundary condition. The flow also experienced an externally applied magnetic field. The effects of inertial forces and the results independent of wavelength are obtained by dropping the presumptions of lubrication theory in modelling the governing equations. The numerical solution for formulated problem in terms of partial differential expressions is worked out via Galerkin finite technique in view of six nodal triangular elements. The enhancement in the inertial forces gives impressive pressure enhancement against wavelength while opposed the fluid flow in the vicinity of peristaltic walls of the tube but supported the fluid flow in the central region of the tube. The present results are also compared with the available results after applying lubrication theory and found in reliable agreement.

MHD darcy-forchheimer nanofluid flow and entropy optimization in an odd-shaped enclosure filled with a (MWCNT-Fe3O4/water) using galerkin finite element analysis

MHD nanoliquid convective flow in an odd-shaped cavity filled with a multi-walled carbon nanotube-iron (II, III) oxide (MWCNT-Fe3O4) hybrid nanofluid is reported. The side walls are adiabatic, and the internal and external borders of the cavity are isothermally kept at high and low temperatures of Th and Tc, respectively. The governing equations obtained with the Boussinesq approximation are solved using Galerkin Finite Element Method (GFEM). Impact of Darcy number (Da), Hartmann number (Ha), Rayleigh number (Ra), solid volume fraction (ϕ), and Heated-wall length effect are presented. Outputs are illustrated in forms of streamlines, isotherms, and Nusselt number. The impact of multiple parameters namely Rayleigh number, Darcy number, on entropy generation rate was analyzed and discussed in post-processing under laminar and turbulent flow regimes.

Galerkin finite element analysis of Darcy–Brinkman–Forchheimer natural convective flow in conical annular enclosure with discrete heat sources

This numerical study is intended for the analysis of thermal convection induced by buoyancy forces generated within a conical annular porous gap. The annulus was vertically positioned, it contains a discrete heat source and is filled with a Single-Walled Carbon Nanotubes-Water (SWCNT-H2O) nanoliquid exposed to a Lorentz force. To describe the porous medium in question, we have adopted the Darcy–Forchheimer model. Galerkin Finite Element Method (GFEM) has been used in this study to predict both thermal and hydrodynamic fields in the physical model. An extensive range of parameters are explored, i.e., the Rayleigh number (103 to 106), Hartman number Ha (1 to 100), and the volume fraction of nanoparticles (0 ≤ϕ≤0.08). For the purpose of exterminating the effects of heat source location on thermal and hydrodynamic fields, three locations (bottom, middle and upper) have been considered. Findings include current lines, isotherms, and Nusselt number evolution according to the previously stated variables. Predictably, our findings prove that heat transfer rate exhibits a decreasing function of Ha and an increasing function of Da and ϕ. Also, it was revealed that the convective regime is preponderant when the heat source was located in the bottom wall.

A novel case study of thermal and streamline analysis in a grooved enclosure filled with (Ag–MgO/Water) hybrid nanofluid: Galerkin FEM

Objective of the present work is to explore the fluidity and thermal performance of Silver and Magnesium oxide mixed water based hybrid nanofluid over the penetrable hollow ampoule under the magnetic influence. Effective thermophysical modelling has been adopted along with the Darcy Forchheimer model engaged exclusively for the inertia of the porosity layer. Governing model equations were solved with the efficient Galerkin Finite Element Analysis (GFEM) and the outcomes were scrutinized towards the error tolerance of 10−6. Parametrical study has been performed with crucial and influencing constraints to traces the behaviors of streamlines and isotherms. Added to the flow influencing constraints the meandering structure of the enclosure itself creates a significant impact on the fluidity which can be evident through the contours developed in the streamline plots. Initial values of the Rayleigh number (Ra) and the Darcy number (Da) make the thermal dissemination stays away from the walls particularly in the hooter wall side with two cliff bar structures. Thermal transference rates get notable assistance from most of the constraints except from the accumulative Hartmann number (Ha) and for the lower Rayleigh number (Ra).

Galerkin finite element analysis of magneto two-phase nanofluid flowing in double wavy enclosure comprehending an adiabatic rotating cylinder

In this work, the finite element method is employed to simulate heat transfer and irreversibilities in a mixed convection two-phase flow through a wavy enclosure filled with water–alumina nanoliquid and contains a rotating solid cylinder in the presence of a uniform magnetic field. Impact of the variations of undulations number (0 ≤ N ≤ 5), Ra (103 ≤ Ra ≤ 106), Ha (0 ≤ Ha ≤ 100), and angular rotational velocity (− 500 ≤ Ω ≤ 500) were presented. Isotherms distribution, streamlines and isentropic lines are displayed. The governing equations are verified by using the Galerkin Finite Element Method (GFEM). The Nusselt numbers are calculated and displayed graphically for several parametric studies. The computational calculations were carried out using Buongiorno's non-homogeneous model. To illustrate the studied problem, a thorough discussion of the findings was conducted. The results show the enhacement of the maximum value of the flow function and the heat transfer process by increasing the value of Rayleigh number. Furthermore the irreversibility is primarily governed by the heat transfer component and the increment of the waviness of the active surfaces or the cylinder rotational velocity or hartmann number will suppress the fluid motion and hinders the heat transfer process.

Galerkin finite element analysis of thermal aspects of Fe3O4-MWCNT/water hybrid nanofluid filled in wavy enclosure with uniform magnetic field effect

In this contribution, a numerical investigation was made for heat transfer and steady magneto-hydrodynamic natural convection in a fined cold wavy-walled porous enclosure with a hot elliptic inner cylinder occupied by hybrid Fe3O4-MWCNT /water nanofluid. The enclosure is also placed in a uniform magnetic field. The governing equations are verified by using the Galerkin Finite Element Method (GFEM). Our numerical conclusions are expressed in terms of distributions of isotherms, Nusselt number, and streamlines, which are important, control parameters for the heat convection, and flow in the enclosure. The findings elucidate that the Nuavg rises with the growth of Ra, porosity ratio, and Da ratio increase, though it reduces with the growth of Hartmann number.

Galerkin finite element analysis of magneto-hydrodynamic natural convection of Cu-water nanoliquid in a baffled U-shaped enclosure

In this paper, single-phase homogeneous nanofluid model is proposed to investigate the natural convection of magneto-hydrodynamic (MHD) flow of Newtonian Cu–H2O nanoliquid in a baffled U-shaped enclosure. The Brinkman model and Wasp model are considered to measure the effective dynamic viscosity and effective thermal conductivity of the nanoliquid correspondingly. Nanoliquid's effective properties such as specific heat, density and thermal expansion coefficient are modeled using mixture theory. The complicated PDS (partial differential system) is treated for numeric solutions via the Galerkin finite element method. The pertinent parameters Hartmann number (1 ≤ Ha ≤ 60), Rayleigh number (103 ≤ Ra ≤ 106) and nanoparticles volume fraction (0% ≤ ϕ ≤ 4%) are taken for the parametric analysis, and it is conducted via streamlines and isotherms. Excellent agreement between numerical results and open literature. It is ascertained that heat transfer rate enhances with Rayleigh number Ra and volume fraction ϕ, however it is diminished for larger Hartmann number Ha.

A Selection of Benchmark Problems in Solid Mechanics and Applied Mathematics

In this contribution we provide benchmark problems in the field of computational solid mechanics. In detail, we address classical fields as elasticity, incompressibility, material interfaces, thin structures and plasticity at finite deformations. For this we describe explicit setups of the benchmarks and introduce the numerical schemes. For the computations the various participating groups use different (mixed) Galerkin finite element and isogeometric analysis formulations. Some programming codes are available open-source. The output is measured in terms of carefully designed quantities of interest that allow for a comparison of other models, discretizations, and implementations. Furthermore, computational robustness is shown in terms of mesh refinement studies. This paper presents benchmarks, which were developed within the Priority Programme of the German Research Foundation ‘SPP 1748 Reliable Simulation Techniques in Solid Mechanics—Development of Non-Standard Discretisation Methods, Mechanical and Mathematical Analysis’.

Effect of Thermo Diffusion on Mass And Heat Transfer Flow on Convective Viscous Electrically Conducting Fluid Through a Porous Medium Bounded by a Semi-Infinite Vertical Plate with Variable Electrically Conductivity Diffusion Thermo Chemical Reaction

In this paper we investigate thermochemical diffusion reaction and effects of thermal diffusion on the mass and heat transfer flux Although the effects of thermal diffusion are vast, they can be transmitted through a very wide medium.. By employing Galerkin-finite element analysis the equations solved with three nodded line segments