Variational Finite Element Method Research Articles

Numerical study of thermodiffusion effects on boundary layer flow of nanofluids over a power law stretching sheet

This paper deals with the triple-diffusive boundary layer flow of nanofluid over a nonlinear stretching sheet. In this model, where binary nanofluid is used, the Brownian motion, thermophoresis, and cross-diffusion are classified as the main mechanisms, which are responsible for the enhancement of the convection features of the nanofluid. The boundary layer equations governed by the partial differential equations are transformed into a set of ordinary differential equations with the help of group theory transformations, which is introduced by Blasius (The boundary layers in fluids with little friction, 1950). The variational finite element method is used to solve these ordinary differential equations. We have examined the effects of different controlling parameters, namely the Brownian motion parameter, the thermophoresis parameter, modified Dufour number, nonlinear stretching parameter, Prandtl number, regular Lewis number, Dufour Lewis number, and nanofluid Lewis number on the flow field and heat transfer characteristics. The physics of the problem is well explored for the embedded material parameters through tables and graphs. The present study has many applications in coating and suspensions, movement of biological fluids, cooling of metallic plate, melt-spinning, heat exchangers technology, and oceanography.

Finite Element Solution of Unsteady Mixed Convection Flow of Micropolar Fluid over a Porous Shrinking Sheet

The objective of this investigation is to analyze the effect of unsteadiness on the mixed convection boundary layer flow of micropolar fluid over a permeable shrinking sheet in the presence of viscous dissipation. At the sheet a variable distribution of suction is assumed. The unsteadiness in the flow and temperature fields is caused by the time dependence of the shrinking velocity and surface temperature. With the aid of similarity transformations, the governing partial differential equations are transformed into a set of nonlinear ordinary differential equations, which are solved numerically, using variational finite element method. The influence of important physical parameters, namely, suction parameter, unsteadiness parameter, buoyancy parameter and Eckert number on the velocity, microrotation, and temperature functions is investigated and analyzed with the help of their graphical representations. Additionally skin friction and the rate of heat transfer have also been computed. Under special conditions, an exact solution for the flow velocity is compared with the numerical results obtained by finite element method. An excellent agreement is observed for the two sets of solutions. Furthermore, to verify the convergence of numerical results, calculations are conducted with increasing number of elements.

Numerical Solution of MHD Viscoelastic Nanofluid Flow over a Stretching Sheet with Partial Slip and Heat Source/Sink

We analyze the effect of velocity slip boundary condition on the flow and heat transfer of non-Newtonian nanofluid over a stretching sheet with a heat source/sink, under the action of a uniform magnetic field, orientated normally to the plate. The Brownian motion and thermophoresis effects are also considered. The boundary layer equations governed by the partial differential equations are transformed into a set of ordinary differential equations with the help of local similarity transformations. The differential equations are solved by the variational finite element method (FEM). We have examined the effects of different controlling parameters, namely, the Brownian motion parameter, the thermophoresis parameter, uniform magnetic field, viscoelastic parameter, Prandtl number, heat source/sink parameter, Lewis number, and the slip parameter on the flow field and heat transfer characteristics. Graphical display of the numerical examination is performed to illustrate the influence of various flow parameters on the velocity, temperature, concentration, and Nusselt and Sherwood numbers distributions. The present study has many applications in coating and suspensions, cooling of metallic plate, paper production, heat exchangers technology, and materials processing exploiting.

Boundary layer flow and heat transfer of viscoelastic nanofluids past a stretching sheet with partial slip conditions

The aim of the paper is to analyze the effect of velocity slip boundary condition on the flow and heat transfer of non-Newtonian nanofluid over a stretching sheet. The Brownian motion and thermophoresis effects are also considered. The boundary layer equations governed by the partial differential equations are transformed into a set of ordinary differential equations with the help of group theory transformations. The obtained ordinary differential equations are solved by variational finite element method (FEM). The effects of different controlling parameters, namely, the Brownian motion parameter, the thermophoresis parameter, viscoelastic parameter, Prandtl number, Lewis number and the slip parameter on the flow field and heat transfer characteristics are examined. The numerical results for the dimensionless velocity, temperature and nanoparticle volume fraction as well as the reduced Nusselt and Sherwood number have been presented graphically. The present study is of great interest in the fields of coatings and suspensions, cooling of metallic plates, oils and grease, paper production, coal water or coal–oil slurries, heat exchangers’ technology, and materials’ processing and exploiting.

Finite-element simulation of mixed convection flow of micropolar fluid over a shrinking sheet with thermal radiation

An analysis is carried out to investigate the effect of thermal radiation on mixed convection flow of a msicropolar fluid over a shrinking sheet with prescribed surface heat flux. The velocity of the shrinking sheet and the surface heat flux are assumed to vary as a linear function of the distance from the origin. Using the boundary layer approximation and similarity transformations, the governing partial differential equations are transformed into a system of nonlinear coupled ordinary differential equations which are solved numerically by using a variational finite element method. The effects of suction, radiation, and buoyancy parameters on velocity, microrotation, and temperature functions are examined in detail. The skin-friction coefficient, local couple stress, and the local Nusselt number have also been computed. Under special conditions, an analytical solution is obtained only for the flow velocity, which is compared with the numerical results obtained by finite element method. An excellent agreement of the two sets of solutions is observed, which confirms the validity of the finite element method employed herein. Also, in order to check the convergence of numerical solutions, the calculations are executed by reducing the mesh size. The sensitivity of the solution as a function of suction through the permeable sheet has also been examined. The current study has important applications in industrial polymeric materials processing.

FEM Approach for Transient Heat Transfer in Human Eye

In this paper, a bio-heat transfer model of temperature distribution in human eye is discussed using appropriate boundary conditions for cornea and sclera. Variational finite element method with Crank-Nicolson scheme is used to calculate the transient temperature distribution in normal human eye. The temperature with and without the effect of blood perfusion and metabolism on retina is simulated and compared for various ambient temperatures, evaporation rates and lens thermal conductivities. The obtained results are compared with experimental results and past results found in literatures. The results show that the steady state corneal temperature is achieved in around 31 and 45 minute of exposure at ambient temperatures 10℃ and 50℃ respectively. Steady state eye temperature is achieved earlier at higher evaporation rate. Similar result is achieved for higher lens thermal conductivity and also for lower ambient temperature.

Newton iterative method for potential flow problems based on variational principle and finite element type solution

In this article, a new method is presented for simulation of incompressible flow field, which is called Newton iteration based on variational finite element method. Meanwhile, underlying theories and general steps in the solution procedure using this method have been described in detail. In order to validate this method, incompressible simulations on the flowfield around a circular cylinder were conducted. The achieved results were then compared with existing analytic solutions and good agreements have been obtained. In addition, the incompressible flow around an aerofoil was also calculated using both the proposed method and the conventional variational finite element method. It was found that the results obtained by the two methods were exactly the same; however, this proposed method can significantly improve the computational efficiency.

Preliminary study on the aerodynamic characteristics of an adaptive reconfigurable airfoil

A balloon-type airfoil, which has the classic airfoil shape but is full of air is designed in this paper. When the angle of attack changes, the pressure distribution on the airfoil surface will correspondingly be changed: the airfoil surface where the pressure is high will be pushed toward the inside of the airfoil resulting in the reduced curvature of this part of the airfoil and even a concave surface formed, but on the other hand, the curvature of the airfoil surface where the pressure is low will be increased. The initial shape of this adaptive reconfigurable airfoil used in this paper is NACA0012. As the forces acting on the airfoil surface should be in balance, a force balance equation is established at first. Then, the surface curvature of the airfoil as well as its shape can thereby be predicted. The flow field and the pressure distribution over the surface of the deformed airfoil can be simulated by use of the variational finite element method. In this paper, the pressure distribution over this adaptive airfoil as well as its corresponding shape under different incoming flows at angles of attack from −14° to 14° has been studied. At last, the XFoil software is used to calculate the coefficient of lift and drag and lift-to-drag ratio of the deformed airfoil. It has been found that this adaptive reconfigurable airfoil can automatically change its shape according to the pressure distribution over its surface and achieve better aerodynamic performance than NACA0012 airfoil in a wide range of application.

Numerical solution for mixed convection boundary layer flow of a nanofluid along an inclined plate embedded in a porous medium

The steady mixed convection boundary layer flow of an incompressible nanofluid along a plate inclined at an angle α in a porous medium is studied. The resulting nonlinear governing equations with associated boundary conditions are solved using an optimized, robust, extensively validated, variational finite-element method (FEM) and a finite-difference method (FDM) with a local non-similar transformation. The Nusselt number is found to decrease with increasing Brownian motion number (Nb) or thermophoresis number (Nt), whereas it increases with increasing angle α. In addition, the local Sherwood number is found to increase with a rise in Nt, whereas it is reduced with an increase in Nb and angle α. The effects of Lewis number, buoyancy ratio, and mixed convection parameter on temperature and concentration distributions are also examined in detail. The present study is of immediate interest in next-generation solar film collectors, heat-exchanger technology, material processing exploiting vertical and inclined surfaces, geothermal energy storage, and all those processes which are greatly affected by a heat-enhancement concept.

Studies of nonlinear deformation and stability of noncircular cylindrical shells in transverse bending

The variational finite element method in displacements is used to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with a noncircular contour of the cross-section. Quadrangle finite elements of shells of natural curvature are used. In the approximations of element displacements, the displacements of elements as solids are explicitly separated. The variational Lagrange principle is used to obtain a nonlinear system of algebraic equations for the unknown nodal finite elements. The system is solved by the method of successive loadings and by the Newton-Kantorovich linearization method. The linear system is solved by the Crout method. The critical loads are determined in the process of solving the nonlinear problem by using the Sylvester stability criterion. An algorithm and a computer program are developed to study the problem numerically. The nonlinear deformation and stability of shells with oval and elliptic cross-sections are investigated in a broad range of variation of the elongation and ellipticity parameters. The shell critical loads and buckling modes are determined. The influence of the deformation nonlinearity, elongation, and ellipticity of the shell on the critical loads is examined.

Stability of a cantilevered skew inhomogeneous plate in supersonic gas flow

This paper considers the vibrations of a skew inhomogeneous plate in gas flow. The plate is clamped in a certain section of one of its sides. Interaction of the flow with the plate is described using piston theory. The problem solution is based on the Hamilton’s variational principle and finite element method. The calculation results are compared with known data of theoretical studies and experiments. For the inhomogeneous plate, similarity parameters were established for the problem, which, in practically important cases, appears to be self-similar for one of the similarity parameters. This allows one to reduce the solution of this problem to the solution of an algebraic eigenvalue problem.

Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: A numerical study

Steady, laminar boundary fluid flow which results from the non-linear stretching of a flat surface in a nanofluid has been investigated numerically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The resulting non-linear governing equations with associated boundary conditions are solved using variational finite element method (FEM) with a local non-similar transformation. The influence of Brownian motion number ( Nb), thermophoresis number ( Nt), stretching parameter ( n) and Lewis number ( Le) on the temperature and nanoparticle concentration profiles are shown graphically. The impact of physical parameters on rate of heat transfer (− θ′(0)) and mass transfer (− ϕ′(0)) is shown in tabulated form. Some of results have also been compared with explicit finite difference method (FDM). Excellent validation of the present numerical results has been achieved with the earlier nonlinearly stretching sheet problem of Cortell [16] for local Nusselt number without taking the effect of Brownian motion and thermophoresis.

Numerical study of heat transfer enhancement in mixed convection flow along a vertical plate with heat source/sink utilizing nanofluids

Steady, mixed convection laminar boundary layer flow of incompressible nanofluid along the vertical plate with temperature dependent heat source/sink has been investigated numerically. The resulting non-linear governing equations (obtained with the Boussinesq approximation) are solved, using a robust, extensively validated, variational finite element method (FEM) for both spherical and cylindrical shaped nanoparticles with volume fraction ranging up to 4%, with associated boundary conditions and the effect of the parameters governing the problem are discussed. Different water-based nanofluids containing Cu, Ag, CuO, Al 2O 3, and TiO 2 are taken into consideration. The results show that the average Nusselt number is found to decrease for Ag, Cu, CuO, Al 2O 3, and TiO 2. The present study is of immediate interest in next-generation solar film collectors, heat exchangers technology, materials processing exploiting vertical surfaces, geothermal energy storage and all those processes which are highly affected with heat enhancement concept.

Mathematical estimation of cold effect in human dermal regions

In this paper we study cold related problems associated with human skin and subcutaneous tissues. We make use of the fact that the anatomy of the dermal layers in a human body plays a key role for the tolerance of thermal stresses like burn and cold injuries. The main response of the coldness is linked with the skin surface. There are several cold injuries associated with the dermal parts due to severe cold conditions disturbing the system. The response to cold and the hazards associated with the exposure are moderated by factors which influence heat production and heat loss including the severity and duration of cold stimuli and magnitude of the metabolic response. This model is an attempt to estimate the effect of ambient temperature on the internal parts of skin layers. We have addressed the issues associated with cold and defining more rigorous and comprehensive standard for modeling thermal injury in dermal parts. The model equation has been solved using variational finite element method.

Plastic variational principle based on the least work consumption principle

Plastic variational principles are foundation to solve the boundary-value problems of plastic mechanics with the variational method (or energy method) and finite element method. The most convenient way of establishing different kinds of variational principles is to set up the extreme principle related to the studied problem. Based on a general new extreme principle-the Least work consumption principle, the variational principles of the rigid-plastic and rigid-viscoplastic material were derived. In comparison with existing methods, the method in this paper is more clear and direct, and the physical meaning is clear-cut. This method can offer a new way for establishing other kinds of variational principles.

Aerodynamic design of airfoils based on variable-domain variational finite element method

Designing airfoils according to given pressure (or velocity) distribution is one kind of free boundary problems. Free boundary condition can be coupled with the flow governing equations by variable-domain variational calculus, which makes it possible to calculate simultaneously the flow field and the free boundary. An accurate deduction of the variable-domain variational principles is taken herein to design airfoils in compressible and incompressible flows. Furthermore, two grid types (H and O) are used in the calculation with better results for the O-type grid. It is shown that convergence is accelerated and good results can be obtained even if the initial guessed airfoil shape is a triangle, demonstrating the strong adaptability of this method.

A variational finite element method for source inversion for convective–diffusive transport

We consider the inverse problem of determining an arbitrary source in a time-dependent convective–diffusive transport equation, given a velocity field and pointwise measurements of the concentration. Applications that give rise to such problems include determination of groundwater or airborne pollutant sources from measurements of concentrations, and identification of sources of chemical or biological attacks. To address ill-posedness of the problem, we employ Tikhonov and total variation regularization. We present a variational formulation of the first-order optimality system, which includes the initial-boundary value state problem, the final-boundary value adjoint problem, and the space–time boundary value source problem. We discretize in the space–time volume using Galerkin finite elements. Several examples demonstrate the influence of the density of the sensor array, the effectiveness of total variation regularization for discontinuous sources, the invertibility of the source as the transport becomes increasingly convection-dominated, the ability of the space–time inversion formulation to track moving sources, and the optimal convergence rate of the finite element approximation.

Large variation finite element method for beams with stochastic stiffness

The behavior of beams with stochastic stiffness subjected to either deterministic or stochastic loading is studied via finite element method. The results are contrasted with exact solution to check the accuracy of the FEM for the case of large variations. It represents a generalization of the previous study in which the stiffness matrix was decomposed as a product of three matrices, two of which are numerical ones and the third matrix involves the uncertain stiffness analytically. To illustrate the proposed method, we evaluate the mean and the auto-correlation functions of the displacement of beams under various boundary conditions. Two statically determinate beams (clamped-free or simply-supported) and two statically indeterminate beams (clamped–simply-supported or clamped are both ends) are investigated in this study. The beams are subjected to a deterministic uniform pressure or a stochastic excitation.

Variational analysis for simulating free‐surface flows in a porous medium

A variational formulation has been developed to solve a parabolic partial differential equation describing free‐surface flows in a porous medium. The variational finite element method is used to obtain a discrete form of equations for a two‐dimensional domain. The matrix characteristics and the stability criteria have been investigated to develop a stable numerical algorithm for solving the governing equation. A computer programme has been written to solve a symmetric positive definite system obtained from the variational finite element analysis. The system of equations is solved using the conjugate gradient method. The solution generates time‐varying hydraulic heads in the subsurface. The interfacing free surface between the unsaturated and saturated zones in the variably saturated domain is located, based on the computed hydraulic heads. Example problems are investigated. The finite element solutions are compared with the exact solutions for the example problems. The numerical characteristics of the finite element solution method are also investigated using the example problems.

Nonlinear Deformation and Stability of Elliptic Cylindrical Shells under Torsion and Bending

The problem of nonlinear deformation and buckling of noncircular cylindrical shells under combined loading is solved by the variational finite-element method in the displacement formulation. A numerical algorithm for solving the problem is proposed. Stability of cylindrical shells with an elliptic cross-sectional contour under a combined action of torsion and bending is analyzed. The effect of cross-sectional ellipticity and nonlinear prebuckling deformation on the critical loads and buckling mode is studied.