Experimental investigation of the electrokinetic flow in microchannels with internal electrodes

We investigate the electrokinetic flow in microchannels with internal electrodes. Experiments and numerical simulations are performed. The micro–particle–image velocimetry method is used to measure two–dimesional, two–component velocity fields over the complete height of the microchannel. Based on this measurements, the third velocity component, which cannot be measured directly, is calculated by an integration of the continuity equation. Due to the fact that microparticles, used for the μPIV are electrically non-neutral leads to the problem that these particles experience electrophoretic forces. That means that the particle movement appears to be a superposition of electroosmotic and electrophorectic effects. To verify the influence of electrophoretic effects on the microparticles, additional numerical calculations are made. (© 2012 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)     

Numerical exploration of a system of reaction–diffusion equations with internal and transient layers

A reaction pathway for a classical two-species reaction is considered with one reaction that is several orders of magnitudes faster than the other. To sustain the fast reaction, the transport and reaction effects must balance in such a way as to give an internal layer in space. For the steady-state problem, existing singular perturbation analysis rigorously proves the correct scaling of the internal layer. This work reports the results of exploratory numerical simulations that are designed to provide guidance for the analysis to be performed for the transient problem. The full model is comprised of a system of time-dependent reaction–diffusion equations coupled through the non-linear reaction terms with mixed Dirichlet and Neumann boundary conditions. In addition to internal layers in space, the time-dependent problem possesses an initial transient layer in time. To resolve both types of layers as accurately as possible, we design a finite element method with analytic evaluation of all integrals. This avoids all errors associated with the evaluation of the non-linearities and allows us to provide an analytic Jacobian matrix to the implicit time stepping method. The numerical results show that the method resolves the localized sharp gradients accurately and can predict the scaling of the internal layers for the time-dependent problem.     

Numerical simulation of internal flow fields of swirl coaxial injector in a hot environment

Based on the fractional volume of fluid (VOF), a pure Eulerian model for defining and capturing the gas/liquid interface is developed in this paper. This model can describe gas/liquid interface in high refinement, which is better than the original VOF methodology. To validate the proposed model and the algorithm, the computational code is employed to predict the flow performance in a cylindrical swirl injector under cold-flow condition, and the predicted results agree well with experimental measurements. Furthermore, the proposed model is used to simulate gas-liquid reacting flows inside a gas/liquid coaxial swirl injector operating in a hot environment. The turbulent combustion process is simulated with the k−ε−f−g model. The numerical simulation is carried out under actual operating condition of the coaxial injector. The injector performances, such as liquid film thickness, liquid film injection velocity, spray angle, pressure drop, are obtained based on the detailed information of the internal flow field. The predicted results also show that droplets are shed from the liquid film in the recess cup of the coaxial injector because of the large velocity gradient between the gas and liquid streams, and a burning area, which is characterized by high temperature, is present inside the injector.     

Numerical simulation and analysis of generalized difference method on triangular networks for electrical impedance tomography

Electrical impedance tomography is an inverse problem of elliptic differential equations. Numerical methods based on combining generalized difference method and Levenberg–Marquardt iteration on a planar domain are proposed. Positive semi-definiteness and existence of solution of the generalized difference scheme are proved. Element geometry matrix is introduced to shortcut calculation and standardize computer program. A series of numerical experiments verify the reliability of its mathematical model and the feasibility of the algorithm. A class of electrical current patterns is proposed to minimize the number of direct problems to be solved in each iteration. These methods have been applied successfully in practical simulation of electrical impedance tomography.     

Stationary internal layers in a reaction-advection-diffusion integro-differential system

A class of singularly perturbed nonlinear integro-differential problems with solutions involving internal transition layers (contrast structures) is considered. An asymptotic expansion of these solutions with respect to a small parameter is constructed, and the stability of stationary solutions to the associated integro-parabolic problems is investigated. The asymptotics are substantiated using the asymptotic method of differential inequalities, which is extended to the new class of problems. The method is based on well-known theorems about differential inequalities and on the idea of using formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers.     

Singularly perturbed problems with boundary and internal layers

This paper is an expanded version of the talk given by the authors at the International Conference “Differential Equations and Topology” dedicated to the centenary of the birth of L.S. Pontryagin. We present a brief survey and describe new ideas and methods of analysis in the asymptotic theory of solutions with internal layers, which is one of the topical fields of singular perturbation theory.     

Effect of an external force on the by-pass transition mechanism in internal flows electrical double layer effect in mini and micro channels

The effect of the electric double layer (EDL) on the bypass transition mechanism is explored through direct numerical simulations. The electrokinetic effects destabilize rapidly the flow when the local disturbance and/or the Reynolds number are respectively strong and large enough to overcome the transient growth regime. It is found that a weak perturbation quickly leads to the transition through bypass and non-linear interactions under the EDL effect, while an order of magnitude larger disturbance is incapable to destabilize the macro-scale flow. The EDL develops some new transitional wall structures during the bypass process. It is concluded that electrostatic effects can be efficiently used to enhance the mixing and heat transfer in microchannels, providing that the diffuse layer is large enough.     

Element-based index reduction in electrical circuit simulation

The numerical simulation of very large scale integrated circuits is an important tool in the development of new industrial circuits. In the course of the last years, this topic has received increasing attention. Common modeling approaches for circuits lead to differential-algebraic systems (DAEs). In circuit simulation, these DAEs are known to have index 2, given some topological properties of the network. This higher index leads to several undesirable effects in the numerical solution of the DAEs. Recent approaches try to lower the index of DAEs to improve the numerical behaviour. These methods usually involve costly algebraic transformations of the equations. Especially, for large scale circuit equations, these transformations become too costly to be efficient. We will present methods that change the topology of the network itself, while replacing certain elements in oder to obtain a network that leads to a DAE of index 1, while not altering the analytical solution of the DAE. This procedure can be performed prior to the actual numerical simulation. The decreasing of the index usually leads to significantly improved numerical behaviour. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and stability of solutions with periodically moving weak internal layers

We consider the periodic parabolic differential equation under the assumption that ε is a small positive parameter and that the degenerate equation f(u,x,t,0)=0 has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution up(x,t,ε) which is T-periodic in t, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as ε→0.     

Boundary and internal layers in the reaction-diffusion problem with a nonlocal inhibitor

A nonlinear parabolic integral problem arising in dynamic simulation of processes in activator-inhibitor systems is considered. Based on the asymptotic theory of such problems previously developed by the authors, the existence of solutions with boundary and internal layers is proved and their asymptotic behavior is found.     

Numerical valuation of discrete double barrier options

In the present paper we explore the problem for pricing discrete barrier options utilizing the Black-Scholes model for the random movement of the asset price. We postulate the problem as a path integral calculation by choosing approach that is similar to the quadrature method. Thus, the problem is reduced to the estimation of a multi-dimensional integral whose dimension corresponds to the number of the monitoring dates.We propose a fast and accurate numerical algorithm for its valuation. Our results for pricing discretely monitored one and double barrier options are in agreement with those obtained by other numerical and analytical methods in Finance and literature. A desired level of accuracy is very fast achieved for values of the underlying asset close to the strike price or the barriers.The method has a simple computer implementation and it permits observing the entire life of the option.     

Numerical simulation of flows in sinusoidally-walled channels

Incompressible viscous flow in a channel with sinusoidally–walled boundaries is investigated numerically. The ultimate aim of the investigation is the computation of the linear stability with respect to two– and three–dimensional perturbations of the basic two–dimensional flow. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of nonlinear electro-elastostatics

The numerical simulation of nonlinear electro-elastostatics is addressed in this work. Based on the governing equations of the electric field and the governing equations of non-linear elastostatics, a variational formulation is formulated and the finite element method is employed to solve the non-linear electro-mechanical coupling problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of Richtmyer-Meshkov instability

The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2 and density ratio 1:20 (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock with the material interface, and effect of initial perturbation modes on R-M instability are investigated numerically. It is noted that the shock refraction is a main physical mechanism of the initial phase changing of the material surface. The multiple interactions of the reflected shock from the origin with the interface and the R-M instability near the material interface are the reason for formation of the spike-bubble structures. Different viscosities lead to different spike-bubble structure characteristics. The vortex pairing phenomenon is found in the initial double mode simulation. The mode interaction is the main factor of small structures production near the interface.

     

Numerical simulation of Richtmyer-Meshkov instability

The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2 and density ratio 1:20 (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock     

Numerical investigation of nanoindentation of viscoelastic polymer layers and parameters re-identification

In the present contribution, we focus on the numerical investigation of nanoindentation of viscoelastic polymer layers using the finite element code ABAQUS®. The time-dependent behaviour of the viscoelastic layer is modeled with Boltzman hereditary integration based on Prony series. The normal contact between the rigid indenter and the deformable layer are treated numerically as a unilateral non-penetration constraint problem. The computer-based numerical optimisation routine is treated by employing the Matrix Laboratory MATLAB® combined with ABAQUS®. The stochastic strategy is used based on principles of biological evolution, i. e. an evolution strategy. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)