A Piezoelectric Finite Shell Element for the Geometrically Nonlinear Analysis of Sensors

A geometrically nonlinear finite element formulation to analyze piezoelectric shell structures is presented. The formulation is based on the mixed field variational functional of Hu–Washizu. Within this variational principle the independent fields are displacements, electric potential, strains, electric field, stresses and dielectric displacements. The mixed formulation allows an interpolation of the strains and the electric field through the shell thickness, which is an essential advantage when using a three dimensional material law. It is remarked that no simplification regarding the constitutive relation is assumed. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent resultant stress and resultant dielectric displacement fields. The shell structure is modeled by a reference surface with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electrical degree of freedom, which is the difference of the electric potential through the shell thickness. The developed mixed hybrid shell element fulfills the in–plane, bending and shear patch tests, which have been adopted for coupled field problems. A numerical investigation of a smart antenna demonstrates the applicability of the piezoelectric shell element under the consideration of geometrical nonlinearity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical study of flows driven by a rotating magnetic field in a square container

The influence of the geometry on the magnetically driven flow is studied by means of numerical simulations. Low–frequency, low–induction and low–interaction conditions are assumed. The rotating magnetic field (RMF) gives rise to a time–independent magnetic body force, computed via the electrical potential equation and Ohm's law and a time–dependent part that is neglected due to the low interaction parameter. Flow results of the cylindrical and square container are compared with respect to the magnetic body force, time–averaged velocity fields, first flow instabilities and Reynolds stress tensors. The dependency of the maximal velocity magnitude and the intensity of the magnetic induction is identical in axisymmetric and non–axisymmetric containers and in good agreement with Davidson's theory. However, significant differences are recognized, for instance, in the distribution of the Reynolds stress tensors. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effects of vortex breakdown bubbles on the mixing environment inside a base driven bioreactor

The bubble-type vortex breakdown inside a cylinder with flow driven by rotation of the base, has applications in mixing. We investigate this phenomena and its effect on the environment inside an open cylinder, with potential application as a tissue-engineering bioreactor, with tissue-scaffolds of two different geometries immersed in the fluid. Addition of scaffolds induces a blockage effect, hindering the flow in the central vortex core returning to the rotating base. This promotes early onset of vortex breakdown and alters the final shape of vortex breakdown bubbles. Placement of the scaffolds centrally on the cylinder axis yields almost identical levels and distributions of shear stress between the upper and lower surfaces of scaffolds. A change from a disk shaped to an ellipsoidal scaffold, of the same size, reduces the intensity of the maximum shear stresses at the scaffold surface by up to 50%. There is a range of Reynolds numbers where increasing Reynolds number, and hence possibly increasing mixing efficiency, leads to a decrease in the maximum levels of fluid forces at the scaffold surfaces. This is an important conclusion for scaffold based tissue engineering where improved mixing is sought, but often sacrificed in favor of minimizing fluid forces.     

微通道中电渗流及微混合的离子浓度效应

电渗流广泛应用于微流控芯片中的流体输运与混合.该文提出了一种离子浓度梯度对电渗流及微混合产生影响的变量模型,采用有限元分析方法对微通道中电渗流及微混合的离子浓度效应进行了数值模拟,分别讨论了zeta电势、介电常数等对微通道内流场和浓度场的影响规律,定量分析了微混合效率.结果表明,当zeta电势和介电常数随浓度变化时,微通道中流场分布不均匀,离子分布不对称.当溶液浓度趋近1 mol/L时,溶液基本无法进入微通道.微混合效率随溶液间浓度差的增大而减小,而且浓度差越大越能在较短距离内到达充分混合.     

A finite shell element with well balanced approximation functions for piezoelectric coupling

This contribution is concerned with a piezoelectric shell formulation. The present shell element has four nodes and bilinear interpolation functions. The nodal degrees of freedom are displacements, rotations and the electric potential on top and bottom of the shell. A 3D–material law is incorporated. In case of bending dominated problems incompatible approximation functions of the electrical and mechanical fields cause incorrect results. This effect occurs in standard element formulations, where the mechanical and electrical degrees of freedom are approximated with lowest order interpolation functions. In order to overcome this problem a mixed multi–field variational approach is introduced. It allows for approximations of the electric field and the strains independent of the bilinear interpolation functions. A quadratic approach for the shear strains and the electric field is proposed through the shell thickness. This leads to well balanced approximation functions regarding coupling of electrical and mechanical fields. A numerical example illustrates the more precise results in contrast to standard elements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometrical Effects on Nonlinear Electrodiffusion in Cell Physiology

We report here new electrical laws, derived from nonlinear electrodiffusion theory, about the effect of the local geometrical structure, such as curvature, on the electrical properties of a cell. We adopt the Poisson–Nernst–Planck equations for charge concentration and electric potential as a model of electrodiffusion. In the case at hand, the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson’s equation. We construct an asymptotic approximation for certain singular limits to the steady-state solution in a ball with an attached cusp-shaped funnel on its surface. As the number of charge increases, they concentrate at the end of cusp-shaped funnel. These results can be used in the design of nanopipettes and help to understand the local voltage changes inside dendrites and axons with heterogeneous local geometry.     

Computer simulations of sodium formate solution in a mixing tank

Traditionally, solid–liquid mixing has always been regarded as an empirical technology with many aspects of mixing, dispersing and contacting were related to power draw. One important application of solid–liquid mixing is the preparation of brine from sodium formate. This material has been widely used as a drilling and completion fluid in challenging environments such as the Barents Sea. In this paper, large-eddy simulations of a turbulent flow in a solid–liquid baffled cylindrical mixing vessel with large number of solid particles are performed to obtain insight into the fundamental aspects of a mixing tank. The impeller-induced flow at the blade tip radius is modeled by using the dynamic-mesh Lagrangian method. The simulations are four-way coupled, which implies that both solid–liquid and solid–solid interactions are taken into account. By employing a soft particle approach the normal and tangential forces are calculated acting on a particle due to viscoelastic contacts with other neighboring particles. The results show that the granulated form of sodium formate may provide a mixture that allows faster and easier preparation of formate brine in a mixing tank. In addition it is found that exceeding a critical size for grains phenomena, such as caking, can be prevented. The obtained numerical results suggest that by choosing appropriate parameters a mixture can be produced that remains free-flowing no matter how long it is stored before use.     

The mixing of a viscous fluid in a layer between rotating eccentric cylinders

The motion of an incompressible viscous fluid in a thin layer between two circular cylinders, inserted into one another, with parallel axes is investigated. The cylinders rotate relative to one another about an axis parallel to the axes of the cylinders. The stream function of the unsteady plane-parallel flow that occurs is found by solving the boundary-value problem for the equations of hydrodynamic lubrication theory. The motion of the fluid particles is found from the solution of a non-autonomous time-periodic Hamiltonian system with a Hamiltonian equal to the stream function. The positions of fluid particles over time intervals that are a multiple of the period of rotation (Poincaré points) are calculated. The set of points is investigated using a Poincaré mapping on the phase flow. The observed transition to chaotic motion is related to the mixing of the fluid particles and is investigated both numerically and using a mapping, calculated with an accuracy up to the third power of the small eccentricity. The optimum mode of motion is observed when the area of the mixing (chaos) region reaches its highest value.     

Two-dimensional numerical simulation of a wave with a current past a circular cylinder. Part 1: Inline flow

This study considered a 2D numerical simulation with LES modeling of a sinusoidal oscillatory flow inline with an imposed uniform flow using primitive variables. The study is performed by a customized code particularly built for this simulation. This code allows an easy implementation to a customized combination of different boundary conditions and discretization schemes. The code is verified against well-recognized studies of a similar type to validate the calculated results. It is found that the drag and inertia coefficients of the wavy flow are reduced with the increased strength of the imposed uniform flow. A strong chaotic behavior was observed in the flow field structures. This behavior has a significant unpredictable influence on the resulting inline and transverse forces through the flow field pattern within the vicinity of the cylinder surface. This influence sometimes causes the exerted forces to increase when compared to the purely wavy flow. Based on this study, it is highly recommended not to uncouple wavy flows from uniform flows to simultaneously capture the combined interactions of such flow scenarios.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.     

Heat and mass transfer in MHD free convection from a moving permeable vertical surface by a perturbation technique

Numerical results are presented for heat and mass transfer effect on hydromagnetic flow of a moving permeable vertical surface. An analysis is performed to study the momentum, heat and mass transfer characteristics of MHD natural convection flow over a moving permeable surface. The surface is maintained at linear temperature and concentration variations. The non-linear coupled boundary layer equations were transformed and the resulting ordinary differential equations were solved by perturbation technique [Aziz A, Na TY. Perturbation methods in heat transfer. Berlin: Springer-Verlag; 1984. p. 1–184; Kennet Cramer R, Shih-I Pai. Magneto fluid dynamics for engineers and applied physicists 1973;166–7]. The solution is found to be dependent on several governing parameter, including the magnetic field strength parameter, Prandtl number, Schmidt number, buoyancy ratio and suction/blowing parameter, a parametric study of all the governing parameters is carried out and representative results are illustrated to reveal a typical tendency of the solutions. Numerical results for the dimensionless velocity profiles, the temperature profiles, the concentration profiles, the local friction coefficient and the local Nusselt number are presented for various combinations of parameters.     

Rectification,mixing, self-induced oscillations,measurement and control in asymmetrically grooved channels

Two separate constructions used in advanced microfluidics are combined to achieve controlled mixing and mass transport at maximum efficiency over minimal distance. One is the use of grooves to enhance mixing – an intensively investigated technique employed in electronic components cooling. So far, only grooves of ectangular cross–sections were used. The other construction builds on the well known effect of partial rectification in axially asymmetric channels and has been employed for valvless pumping. It is now shown that a cascade of axially asymmetric grooves retains and even improves the rectification efficiency of a single nozzle while offering the potential of simultaneous mixing enhancement by a factor of more than 2. The latter is achieved in a certain range of moderate Reynolds numbers characterized by self–induced oscillations at much higher frequency than that of flow actuation. Tuning the pressure drop provides precise control of the effective flow rate, up to suppression or reversion. The duration and intensity of mixing and shearing can thus be adjusted within a broad range and effected in very short channels without additional actuators. In the regime of self–induced oscillations, a few identical sensors with sufficient temporal resolution for temperature or concentration allow reliable determination of the flow rate as well as of the admixture composition of the transported fluid. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of constriction height on flow separation in a two-dimensional channel

We studied numerically the effect of the constriction height on viscous flow separation past a two-dimensional channel with locally symmetric constrictions. A numerically stable scheme in primitive variables (velocity and pressure) for the solution of two-dimensional incompressible time-dependent Navier–Stokes equations is employed using finite-difference approximation in staggered grid. The wall shear stresses at different heights of the constriction are computed and presented graphically. It is noticed that the maximum stress and the length of the recirculating region associated with two shear layers of the constriction increase with the increase of the area reduction of the constriction. The critical Reynolds number for symmetry breaking bifurcation for the 50%, 60% and 70% area reduction are obtained numerically. The flow field separates after the symmetry breaking bifurcation and the symmetry of the flow depends on the Reynolds number and the height of the constriction.     

Evaluating the transport in small-world and scale-free networks

We present a study of some properties of transport in small-world and scale-free networks. Particularly, we compare two types of transport: subject to friction (electrical case) and in the absence of friction (maximum flow). We found that in clustered networks based on the Watts–Strogatz (WS) model, for both transport types the small-world configurations exhibit the best trade-off between local and global levels. For non-clustered WS networks the local transport is independent of the rewiring parameter, while the transport improves globally. Moreover, we analyzed both transport types in scale-free networks considering tendencies in the assortative or disassortative mixing of nodes. We construct the distribution of the conductance G and flow F to evaluate the effects of the assortative (disassortative) mixing, finding that for scale-free networks, as we introduce different levels of the degree–degree correlations, the power-law decay in the conductances is altered, while for the flow, the power-law tail remains unchanged. In addition, we analyze the effect on the conductance and the flow of the minimum degree and the shortest path between the source and destination nodes, finding notable differences between these two types of transport.     

Theoretical investigation of the flow in a stenosis

The theoretical investigation of the flow in stenosis is presented. We use a new formulation of the incompressible Navier–Stokes equation in terms of an auxiliary field that differs from the velocity by a gauge transformation [1]. The gauge freedom allows us to formulate simple boundary conditions for the auxiliary field and the gauge field as well. The gauge field eliminates the pressure distribution in the Navier–Stokes equation. The numerical investigation of the creeping flow, depending on the geometrical parameters of the system, is performed. The influence of the pressure drop has been taken into account. An excellent agreement with the analytical results in frames of the film theory was observed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A hybrid model for electroosmotic flows in microchannels induced by internal electrodes

Internal electrodes, adjacent to insulating walls at defined zeta potential, lead to a non-continuous potential distribution at the wall. Hence, simplified treatment appears problematic due to the singularity of the electrical field strength. To avoid this difficulty, we develop a hybrid model, which solves the electrical problem, including a resolution of the EDL, while the flow problem is solved in the fluid bulk only. We apply this hybrid model to investigate the position of internal electrodes with regard to their influence onto the flow field, driven by electroosmosis in a modular rectangular microchannel. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation on the effect of trailing edge ejection on a turbine cascade

This paper presents a detailed experimental and numerical investigation for a turbine cascade with different trailing edge ejection. The numerical simulation is based on Three-Dimensional Navier–Stokes equations coupled with an effective ejection model, where a high resolution non-oscillatory scheme, LU-SGS implicit algorithm and Baldwin-Lomax turbulence model are employed. The experiments presented in this paper focused on a transonic turbine cascade performance with different ejection to validate the numerical simulation results. The results show that the blowing ratio has a small effect on the Mach number distribution and exit flow angle with two slot types. However the energy loss coefficient increases initially, and subsequently has a decrease tendency with the increasing of blowing ratio. The ejection from the symmetry slot blows away the vortex at the blade trailing edge and strengthens the mixing between the wake and main flow. The ejection from the pressure side cutback only clears up the vortex near the slot surface, and has small effect on the flow field near the trailing edge.     

MHD boundary-layer flow of a micropolar fluid past a wedge with constant wall heat flux

The steady laminar magnetohydrodynamic (MHD) boundary-layer flow past a wedge with constant surface heat flux immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated in this paper. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity variables, and then they are solved numerically by means of an implicit finite-difference scheme known as the Keller-box method. Numerical results show that micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics.     

Analysis of a FEM/BEM coupling method for transonic flow computations

A sensitive issue in numerical calculations for exterior flow problems, e.g.around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two-dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl-Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists.

     

Thin film flow over a non-linear stretching sheet in presence of uniform transverse magnetic field

A thin viscous liquid film flow is developed over a stretching sheet under different non-linear stretching velocities in presence of uniform transverse magnetic field. Evolution equation for the film thickness is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. It is observed that all types of stretching produces film thinning, but non-monotonic stretching produces faster thinning at small distance from the origin. Effect of the transverse magnetic field is to slow down the film thinning process. Observed flow behavior is explained physically.