Variational Finite Element Method Research Articles

Variable Viscosity Effects on Unsteady MHD an Axisymmetric Nanofluid Flow over a Stretching Surface with Thermo-Diffusion: FEM Approach

The present study investigated the unsteady magnetohydrodynamic (MHD) nanofluid flow over a radially nonlinear stretching sheet along with the viscosity dependent on temperature, convective boundary condition, thermo-diffusion, and the radiation effects. Moreover, the nanofluid’s viscous effects were considered dependent on temperature and the exponential Reynolds model was considered in this context. It was additionally assumed that a uniform suspension of nanoparticles is present in the base fluid. The Buongiorno model, which involves the thermophoresis and Brownian motion effects, was considered. For the sake of a solution, the variational finite element method was selected with coding in MATLAB and the numerical results were contrasted with the published articles. The influence of various physical parameters on the velocity, temperature, and concentration profiles are discussed by the aid of graphs and tables. It was detected that the nanofuid viscosity parameter declines the fluid flow velocity, while, for the temperature and the concentration profiles, it accomplished the reverse phenomenon.

Finite Element Simulation of Multi-Slip Effects on Unsteady MHD Bioconvective Micropolar Nanofluid Flow Over a Sheet with Solutal and Thermal Convective Boundary Conditions

In this article, the intention is to explore the flow of a magneto-hydrodynamic (MHD) bioconvective micro-polar Nanofluid restraining microorganism. The numerical solution of 2-D laminar bioconvective boundary layer flow of micro-polar nanofluids are presented. The phenomena of multi-slip, convective thermal and Solutal boundary conditions have been integrated. A system of non-linear partial differential equations are transformed into the system of coupled nonlinear ordinary differential equations by applying appropriate transformations, the transformed equations are then solved by applying the variational finite element method (FEM). The fascinating features of assorted velocity parameter, microrotation, temperature, microorganism compactness, solutal and nanoparticles concentration have been inspected. The rate of heat transfer, the skin friction coefficient, couple stress and Sherwood number for microorganisms have also been discussed graphically and numerically. The investigations illustrated that increase in material parameters causes a reduction in microorganism compactness, concentration and temperature. As a result of enhancement in the unsteadiness parameter, the fluid velocity, concentration of microorganisms and the temperature are observed to be declines. Energy and microorganism compactness profile affected by the improvement in the buoyancy ratio parameter. As the improvement in results of buoyancy ratio parameter effects on improvement in the energy and the microorganism compactness profile while the velocity profile is condensed. In the end, rationalized convergence of the finite element solution has been inspected; the computations are found out via depreciating the mesh size.

Finite Element Analysis of Thermo-Diffusion and Multi-Slip Effects on MHD Unsteady Flow of Casson Nano-Fluid over a Shrinking/Stretching Sheet with Radiation and Heat Source

In this article, we probe the multiple-slip effects on magnetohydrodynamic unsteady Casson nano-fluid flow over a penetrable stretching sheet, sheet entrenched in a porous medium with thermo-diffusion effect, and injection/suction in the presence of heat source. The flow is engendered due to the unsteady time-dependent stretching sheet retained inside the porous medium. The leading non-linear partial differential equations are transmuted in the system of coupled nonlinear ordinary differential equations by using appropriate transformations, then the transformed equations are solved by using the variational finite element method numerically. The velocity, temperature, solutal concentration, and nano-particles concentration, as well as the rate of heat transfer, the skin friction coefficient, and Sherwood number for solutal concentration, are presented for several physical parameters. Next, the effects of these various physical parameters are conferred with graphs and tables. The exact values of flow velocity, skin friction, and Nusselt number are compared with a numerical solution acquired with the finite element method (FEM), and also with numerical results accessible in literature. In the end, we rationalize the convergence of the finite element numerical solution, and the calculations are carried out by reducing the mesh size.

Finite Element Simulation of Multiple Slip Effects on MHD Unsteady Maxwell Nanofluid Flow over a Permeable Stretching Sheet with Radiation and Thermo-Diffusion in the Presence of Chemical Reaction

The aim of the present study is to investigate the multiple slip effects on magnetohydrodynamic unsteady Maxwell nanofluid flow over a permeable stretching sheet with thermal radiation and thermo-diffusion in the presence of chemical reaction. The governing nonlinear partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations with the aid of appropriate similarity variables, and the transformed equations are then solved numerically by using a variational finite element method. The effects of various physical parameters on the velocity, temperature, solutal concentration, and nanoparticle concentration profiles as well as on the skin friction coefficient, rate of heat transfer, and Sherwood number for solutal concentration are discussed by the aid of graphs and tables. An exact solution of flow velocity, skin friction coefficient, and Nusselt number is compared with the numerical solution obtained by FEM and also with numerical results available in the literature. A good agreement between the exact and numerical solution is observed. Also, to justify the convergence of the finite element numerical solution, the calculations are carried out by reducing the mesh size. The present investigation is relevant to high-temperature nanomaterial processing technology.

Numerical analysis of the full MHD model with the Galerkin finite-element method

The steady-state, two-dimensional mixed-convection flow inside a square conduit subject to an external magnetic field is investigated here. The full MHD equations are used to model the set of governing partial differential equations for the problem under consideration. These equations along with appropriate boundary conditions are simulated by employing the variational finite-element method for the important parameters of the problem. The effects of these parameters on streamlines, temperature field, mid-section velocity and heat transfer rate are shown and discussed in detail. A comparison of the full MHD model with the low-Rm model is carried out. The grid independent test is also included to assure the convergence of the obtained solution. The analysis predicts that the intensity of flow circulations increases with increasing magnetic Reynolds number for any choice of the Prandtl number. Further, the average Nusselt number at both the right and bottom walls is an increasing function of Rm. Moreover, the low-Rm model predicts lower values of the average Nusselt number when compared to the result of the full MHD model.

ОЦІНКА ПРОЦЕСУ ЗАПОВНЕННЯ ГАЗОВОГО КОВПАКА ГЕОСОЛІТОННОГО РОДОВИЩА

Recently the efficiency of oil-gas deposits exploration has been closely connected with understanding of processes forming of such deposits. In forming, transferring and accumulation processes of hydrocarbons, the important role has mechanism of its vertical migrations. Among models in which such mechanisms were presented, geosoliton conception is well-known enough. Geosoliton conception is in well agreement with many of geologic and cosmogonic ideas. It unites principals of biogenic and abiogenic hypothesis of hydrocarbons forming, which come from mantle in composition of hot high-pressure fluids. The fluids come through the crack-pour faults in the central channel part of geosoliton. The abnormal high pressure of accumulated gases in the central part of geosoliton leads to their penetration in collector reservoirs. At that place deposit geosoliton system is forming. It consists of the gas caps, oil collars and reservoir water in the periphery. Purpose. Estimation of filling processes in the gas cap of geosoliton field. Methodology. To resolve the nonstationary piezoconductivity problem we use combined computer method consisting of variation finite element method and method of finite differences. Findings. As a result of the research, filling process of the gas cap essentially depends on filtration parameters of considering porous reservoir, its size and characteristic fiches of the filling origins. Filling gas cap velocity mainly determined by origins power and less depends on their sizes and quantity. In the subsequent stages of filling, the pressure gradually aligns in all reservoir places and then regularly increases. Originality. Presented method has some advantages in comparison with other similar methods. It enables calculating different reservoir filtration parameters and using manifold boundary conditions. Practical value. The obtained results allow making gas production process in gas-condensate reservoirs more effectively.

Finite element analysis on transient magnetohydrodynamic (MHD) free convective chemically reacting micropolar fluid flow past a vertical porous plate with Hall current and viscous dissipation

This article deals with boundary layer analysis of magnetohydrodynamics on an unsteady chemically reactive micropolar rotating fluid flow past a semi-infinite vertical plate with the impact of Hall current and viscous dissipation. The relevant equations on the fluid flow have been developed under Boussinesq approximation. The governing partial differential equations are transformed into a set of coupled partial differential equations using suitable dimensionless quantities. The resultant equations are solved numerically using variational finite element method. A parametric study illustrating the influence of different pertinent parameters is performed and the numerical results for translational velocity, microrotation velocity, temperature and concentration distributions near the boundary layers are discussed and presented graphically for the parametric variations. Finally, the skin-friction, wall couple stress, surface heat transfer and mass transfer rate dependency on the emerging thermo-physical parameters are also tabulated. The finite element code is benchmarked with the results reported in the literature to check the validity and accuracy under some limiting cases and excellent agreement with published solutions is achieved.

Finite Element Analysis of MHD Flow of Micropolar Fluid over a Shrinking Sheet with a Convective Surface Boundary Condition

This paper presents a numerical solution for the steady mixed convection magnetohydrodynamic (MHD) flow of an electrically conducting micropolar fluid over a porous shrinking sheet. The velocity of shrinking sheet and magnetic field are assumed to vary as power functions of the distance from the origin. A convective boundary condition is used rather than the customary conditions for temperature, i.e., constant surface temperature or constant heat flux. With the aid of similarity transformations, the governing partial differential equations are transformed into a system of nonlinear ordinary differential equations, which are solved numerically, using the variational finite element method (FEM). The influence of various emerging thermophysical parameters, namely suction parameter, convective heat transfer parameter, magnetic parameter and power index on velocity, microrotation and temperature functions is studied extensively and is shown graphically. Additionally the skin friction and rate of heat transfer, which provide an estimate of the surface shear stress and the rate of cooling of the surface, respectively, have also been computed for these parameters. Under the limiting case an analytical solution of the flow velocity is compared with the present numerical results. An excellent agreement between the two sets of solutions is observed. Also, in order to check the convergence of numerical solution, the calculations are carried out by reducing the mesh size. The present study finds applications in materials processing and demonstrates excellent stability and convergence characteristics for the variational FEM code.

NUMERICAL SIMULATION OF TIME-DEPENDENT NON-NEWTONIAN NANOPHARMACODYNAMIC TRANSPORT PHENOMENA IN A TAPERED OVERLAPPING STENOSED ARTERY

Nanofluids are becoming increasingly popular in novel hematological treatments and also advanced nanoscale biomedical devices. Motivated by recent developments in this area, a theoretical and numerical study is described for unsteady pulsatile flow, heat and mass transport through a tapered stenosed artery in the presence of nanoparticles. An appropriate geometric expression is employed to simulate the overlapping stenosed arterial segment. The Sisko non-Newtonian model is employed for hemodynamic rheology. Buongiorno’s formulation is employed to model nanoscale effects. The two-dimensional non-linear, coupled equations are simplified for the case of mild stenosis. An explicit forward time central space (FTCS) finite difference scheme is employed to obtain a numerical solution of these equations. Validation of the computations is achieved with another numerical method, namely the variational finite element method (FEM). The effects of various emerging rheological, nanoscale and thermofluid parameters on flow and heat/mass characteristics of blood are shown via several plots and discussed in detail. The circulating regions inside the flow field are also investigated through instantaneous patterns of streamlines. The work is relevant to nanopharmacological transport phenomena, a new and exciting area of modern medical fluid dynamics which integrates coupled diffusion, viscous flow and nanoscale drug delivery mechanisms.

Finite Element Analysis of Eyring–Powell Nano Fluid Over an Exponential Stretching Sheet

An analysis is made to study the two-dimensional boundary layer flow of Eyring–Powell nanofluid over an exponentially stretching sheet. The nanofluid model considered in the paper incorporates the effect of Brownian motion and thermophoresis. The condition for zero normal flux of nanoparticles at the stretching sheet is defined to impulse the particles away from surface. The governing partial differential equations are transformed into a set of coupled non-linear ordinary differential equations with appropriate similarity transformations. The transformed boundary layer equations are solved numerically, using variational finite element method. The flow features and the heat transfer characteristics and nanoparticle volume fraction are analyzed and discussed in detail for several sets of values of the governing flow parameters. Also the effects of the various flow parameters on the skin friction and the rate of heat transfer are obtained and illustrated graphically. A comparison of the present results with the existing published results shows excellent agreement, which confirms the validity of the present methodology.

Mathematical and numerical analysis of thermal distribution in cancerous tissues under the local heat therapy

The main purpose of this study is to investigate the thermal behavior of living tissues in the presence of spatial external heat source. An effort has been made to formulate the mathematical model to study the temperature distribution in in vivo tissues of the human body. The mathematical formulation is governed by bio-heat equation together with appropriate initial, boundary and interface conditions. The solution of the model was carried out using variational finite element method and computational simulations. The model describes the exchange of heat between the internal biological tissues and other surrounding media. The effect of external heat source under different conditions of atmospheric temperature and as a local hyperthermic method provides an important information to the temperature regulation in biological tissues under normal and malignant conditions. Thermal fluctuations at the targeted regions were obtained with respect to various time-dependent heating sources and scattering coefficients. The results obtained may be helpful for clinical purposes especially in the treatment of cancerous tumors through radiotherapy and other local hyperthermic approaches.

Three-dimensional finite element model to study temperature distribution in peripheral layers of human limbs immediately after physical exercise

The physical exercise imposes challenges on the human thermoregulatory system, as heat exchange between the body and environment is substantially impaired, which can lead to decrease in performance and increased risk of heat illness. In view of above a three-dimensional finite element model is proposed to study the effect of different intensities of physical exercise on temperature distribution in peripheral regions of human limbs under moderate climatic conditions. Human limb is assumed to have a cylindrical cross-section. The peripheral region of the human limb is divided into three natural components, namely epidermis, dermis and subdermal tissues. The model incorporates the effect of important physiological parameters like blood mass flow rate, metabolic heat generation, and thermal conductivity of the tissues. Appropriate boundary conditions have been framed based on the physical conditions of the problem. The model is transformed into the discretized variational form and finite element method (FEM) has been employed to obtain the solution. The numerical results have been used to obtain the temperature profiles in the region immediately after exercise for an unsteady state case. The thermal information generated from the model can be useful for developing protocols for improving performance of sportsmen, military persons and labor-intensive workers.

Unsteady flow of a Maxwell nanofluid over a stretching surface in the presence of magnetohydrodynamic and thermal radiation effects

The problem of unsteady magnetohydrodynamic (MHD) boundary layer flow of a non-Newtonian Maxwell nanofluid over a stretching surface with thermal radiation is considered. The Maxwell model is used to characterize the non-Newtonian fluid behaviour. An appropriate similarity transformation is employed to transform the governing partial differential equations of mass, momentum, energy and nanoparticle concentration into ordinary differential equations. The coupled non-linear ordinary differential equations are solved by using the variational finite element method. The flow features and the heat transfer characteristics and nanoparticle volume fraction are analyzed and discussed in detail for several sets of values of the governing flow parameters. The results for the skin-friction coefficient, local Nusselt number and the local Sherwood number are presented in tables for various values of the flow controlling parameters.

MHD heat and mass transfer flow of a nanofluid over an inclined vertical porous plate with radiation and heat generation/absorption

MHD boundary layer flow, heat and mass transfer analysis of nanofluid over an inclined vertical plate saturated by porous medium with thermal radiation, magnetic field and heat generation/absorption is investigated in the present article. By using similarity variables the governing non-linear partial differential equations are transformed into ordinary differential equations and these equations together with associated boundary conditions are solved numerically by using versatile, extensively validated, variational Finite element method. The sway of key parameters, such as, Magnetic parameter (M), buoyancy ratio parameter (Nr), Radiation parameter (An), Heat source/sink parameter (Q), Brownian motion parameter (Nb), thermophoretic parameter (Nt) and Lewis number (Le) on hydrodynamic, thermal and concentration boundary layers are examined in detail and the results are shown graphically. Furthermore, the impact of these parameters on local skin friction coefficient (Cf), rate of heat transfer (Nux) and rate of mass transfer (Shx) is also investigated. The results are compared with the works published previously and found to be excellent agreement.

MHD NATURAL CONVECTION BOUNDARY LAYER FLOW OF NANOFLUID OVER A VERTICAL CONE WITH CHEMICAL REACTION AND SUCTION/INJECTION

In this article we have presented magnetohydrodynamic (MHD) natural convection boundary layer heat and mass transfer flow over a vertical cone embedded in porous medium filled with nanofluid under the enhanced boundary conditions in the presence of thermal radiation, chemical reaction, and suction/injection. The two slip effects, Brownian motion and thermophoresis, are considered in this problem. The governing nonlinear partial differential equations which represent the momentum, energy, and diffusion are transformed into ordinary differential equations with the help of suitable similarity transformations. The transformed conservation equations together with boundary conditions are then solved numerically by using a versatile, extensively validated, variational finite element method. The influences of key parameters such as magnetic field (M), buoyancy ratio parameter (Nr), Brownian motion parameter (Nb), thermophoresis parameter (Nt), thermal radiation (R), Lewis number (Le), suction (V0 > 0), or injection (V0 < 0) parameter and chemical reaction parameter (Cr) on velocity, temperature, and concentration evaluation in the boundary layer region are examined in detail. Furthermore, the effect of these parameters on the local skin-friction coefficient (Cf), local Nusselt number (Nux), and local Sherwood number (Shx) is also investigated. The results are compared with previously published work and found to be in admirable agreement.

MHD flow and heat transfer of Casson nanofluid over a wedge

Heat transfer characteristics of a steady, two dimensional, magnetohydrodynamic boundary layer flow of Casson nanofluid over a wedge have been investigated by incorporating the effects of Brownian motion and thermophoresis. The flux of volume fraction of nanoparticles is taken zero on the boundary. Using suitable similarity transformation, the governing partial differential equations for the modelling of boundary layer flow are reduced to ordinary differential equations. The resulting coupled non-linear ordinary differential equations are successfully solved numerically with the help of variational finite element method. To validate the present analysis the numerical results are compared with previously published work available in the literature and they found to be in good agreement. The effects of the flow controlling parameters on velocity, temperature and nanoparticle volume fraction profiles are investigated. The skin-friction co-efficient, Nusselt number and Sherwood number presented for several set of values of the physical parameters and salient features are analyzed.

Boundary layer flow and heat transfer of a non-Newtonian nanofluid over a non-linearly stretching sheet

Purpose The purpose of this paper is to study the boundary layer flow and heat transfer of a power-law non-Newtonian nanofluid over a non-linearly stretching sheet. Design/methodology/approach The governing equations describing the problem are transformed into a nonlinear ordinary differential equations by suitable similarity transformations. The resulting equations for this investigation are solved numerically by using the variational finite element method. Findings It was found that the local Nusselt number increases by increasing the Prandtl number, stretching sheet parameter and decreases by increasing the power-law index, thermophoresis parameter and Lewis number. Increases in the stretching sheet parameter, Prandtl number and thermophoresis parameter decrease the local Sherwood number values. The effects of Brownian motion and Lewis number lead to increases in the local Sherwood number values. Originality/value The work is relatively original as very little work has been reported on non-Newtonian nanofluids.

MHD boundary-layer flow of a non-Newtonian nanofluid past a stretching sheet with a heat source/sink

The goal of the present paper is to examine the magnetohydrodynamic effects on the boundary layer flow of the Jeffrey fluid model for a non-Newtonian nanofluid past a stretching sheet with considering the effects of a heat source/sink. The governing partial differential equations are reduced to a set of coupled nonlinear ordinary differential equations by using suitable similarity transformations. These equations are then solved by the variational finite element method. The profiles of the velocity, temperature, and nanoparticle volume fraction are presented graphically, and the values of the Nusselt and Sherwood numbers are tabulated. The present results are compared with previously published works and are found to be in good agreement with them.

Unsteady two-layered blood flow through a \(w\)-shaped stenosed artery using the generalized Oldroyd-B fluid model

A theoretical study of an unsteady two-layered blood flow through a stenosed artery is presented in this article. The geometry of a rigid stenosed artery is assumed to be \(w\)-shaped. The flow regime is assumed to be laminar, unsteady and uni-directional. The characteristics of blood are modelled by the generalized Oldroyd-B non-Newtonian fluid model in the core region and a Newtonian fluid model in the periphery region. The governing partial differential equations are derived for each region by using mass and momentum conservation equations. In order to facilitate numerical solutions, the derived differential equations are nondimensionalized. A well-tested explicit finite-difference method (FDM) which is forward in time and central in space is employed for the solution of a nonlinear initial boundary value problem corresponding to each region. Validation of the FDM computations is achieved with a variational finite element method algorithm. The influences of the emerging geometric and rheological parameters on axial velocity, resistance impedance and wall shear stress are displayed graphically. The instantaneous patterns of streamlines are also presented to illustrate the global behaviour of the blood flow. The simulations are relevant to haemodynamics of small blood vessels and capillary transport, wherein rheological effects are dominant. doi:10.1017/S1446181116000134

TEMPERATURE DISTRIBUTION AND THERMAL DAMAGE OF PERIPHERAL TISSUE IN HUMAN LIMBS DURING HEAT STRESS: A MATHEMATICAL MODEL

The physiological processes taking place in human body are disturbed by the abnormal changes in climate. The changes in environmental temperature are not effective only to compete with thermal stability of the system but also in the development of thermal injuries at the skin surfaces. Therefore, it is of great importance to estimate the temperature distribution and thermal damage in human peripherals at extreme temperatures. In this paper, the epidermis, dermis and subcutaneous tissue were modeled as uniform elements with distinct thermal properties. The bioheat equation with appropriate boundary conditions has been used to estimate the temperature profiles at the nodal points of the skin and subcutaneous tissue with variable ambient heat and metabolic activities. The model has been solved by variational finite element method and the results of the changes in temperature distribution of the body and the damage to the exposed living tissues has been interpreted graphically in relation with various atmospheric temperatures and rate of metabolic heat generation. By involving the metabolic heat generation term in bioheat equation and using the finite element approach the results obtained in this paper are more reasonable and appropriate than the results developed by Moritz and Henriques, Diller and Hayes, and Jiang et al.