Studying the Accuracy and Applicability of the Finite Difference Scheme for Solving the Diffusion–Convection Problem at Large Grid Péclet Numbers

Journal of Applied Mechanics and Technical Physics - The work is devoted to studying a finite difference scheme for solving the diffusion–convection problem at large grid Péclet numbers....     

Rigorous upscaling of the reactive flow with finite kinetics and under dominant Péclet number

We consider the evolution of a reactive soluble substance introduced into the Poiseuille flow in a slit channel. The reactive transport happens in presence of dominant Péclet and Damköhler numbers. We suppose Péclet numbers corresponding to Taylor’s dispersion regime. The two main results of the paper are the following. First, using the anisotropic perturbation technique, we derive rigorously an effective model for the enhanced diffusion. It contains memory effects and contributions to the effective diffusion and effective advection velocity, due to the flow and chemistry reaction regime. Error estimates for the approximation of the physical solution by the upscaled one are presented in the energy norms. Presence of an initial time boundary layer allows only a global error estimate in L ² with respect to space and time. We use the Laplace’s transform in time to get optimal estimates. Second, we explicit the retardation and memory effects of the adsorption/desorption reactions on the dispersive characteristics and show their importance. The chemistry influences directly the characteristic diffusion width.     

Addendum to “Concentration and Lack of Observability of Waves in Highly Heterogeneous Media”

In [1] we introduced a class of 1?d wave equations with rapidly oscillating Hölder continuous coefficients for which the classical boundary observability property fails. We also established that these examples could be used to contradict Strichartz-type inequalities for the wave equation with low regularity coefficients. The object of this addendum is to further analyze this issue. As we will see, the argument in [1] only provides sharp counter-examples to the Strichartz estimates when the coefficient ρ belongs to L ^∞. We carefully analyze these counter-examples for Hölder continuous coefficients. We also give a new application of our construction which shows that some eigenfunction estimates for elliptic operators due to Sogge can fail when coefficients are not smooth enough.     

A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth

A third-order Lagrangian asymptotic solution is derived for gravity–capillary waves in water of finite depth. The explicit parametric solution gives the trajectory of a water particle and the wave kinematics for Lagrangian points above the mean water level, and in a water column. The water particle orbits and mass transport velocity as functions of the surface tension are obtained. Some remarkable trajectories may contain one or multiple sub-loops for steep waves and large surface tension. Overall, an increase in surface tension tends to increase the motions of surface particles including the relative horizontal distance travelled by a particle as well as the time-averaged drift velocity     

Finite amplitude oscillations of flanged laminas in viscous flows: Vortex–structure interactions for hydrodynamic damping control

In this paper, we study the problem of harmonic oscillations of a flanged lamina in a quiescent Newtonian incompressible viscous fluid. We conduct a comprehensive fluid–structure interaction investigation with the goal of assessing the effect of the presence of the flanges on the added mass and hydrodynamic damping experienced by the oscillating solid. We determine the complex nonlinear hydrodynamic function incorporating these effects via its real and imaginary parts, respectively, and its dependence on three nondimensional parameters that govern the flow evolution. We further investigate in detail the flow physics and the effects of nonlinearities on vortex shedding, convection, and diffusion in the vicinity of the oscillating structure. We find that the added mass effect is relatively independent of the oscillation amplitude and increases with the flange size. On the other hand, the hydrodynamic damping effect is remarkably affected by the interplay of geometry and dynamic parameters resulting into a peculiar non-monotonic behavior. We show the existence of a minimum in the hydrodynamic damping which can be attained via specific control of vortex–structure interaction dynamics and discuss its properties and significance from a physical perspective through analysis of the relevant flow fields. This novel finding has potential application for damping reduction in elastic systems where reduction of energy losses and increase of oscillation quality factor are desired.     

An Experimental–Numerical Methodology for a Rapid Prototyped Application Combined with Finite Element Models in Vertebral Trabecular Bone

In the present study, a novel evaluation method involving rapid prototyped (RP) technology and finite element (FE) analysis was used to study the elastic mechanical characteristics of human vertebral trabecular bone. Three-dimensional (3D) geometries of the RP and FE models were obtained from the central area of vertebral bones of female cadavers, age 70 and 85. RP and FE models were generated from the same high-resolution micro-computed tomography (μCT) scan data. We utilized RP technology along with FE analysis based on μCT for high-resolution vertebral trabecular bone specimens. RP models were used to fabricate complex 3D objects of vertebral trabecular bone that were created in a fused deposition modeling machine. RP models of vertebral trabecular bone are advantageous, particularly considering the repetition, risks, and ethical issues involved in using real bone from cadaveric specimens. A cubic specimen with a side length of 6.5 mm or a cylindrical specimen with a 7 mm diameter and 5 mm length proved better than a universal cubic specimen with a side length of 4 mm for the evaluation of elastic mechanical characteristics of vertebral trabecular bones through experimental and simulated compression tests. The results from the experimental compression tests of RP models closely matched those predicted by the FE models, and thus provided substantive corroboration of all three approaches (experimental tests using RP models and simulated tests using FE models with ABS and trabecular bone material properties). The RP technique combined with FE analysis has potential for widespread biomechanical use, such as the fabrication of dummy human skeleton systems for the investigation of elastic mechanical characteristics of various bones.     

Positivity of temperature in the general Frémond model for shape memory alloys

In this paper, we consider a model introduced by M. Fr~mond to describe the martensitic phase transitions in shape memory alloys. In the derivation of his model, M. Fr'emond made the (physically reasonable) assumption that the state variable representing the absolute temperature is always positive. Although various results concerning existence and uniqueness of solutions to certain simplified versions of the governing field equations have been established in the past, it has been an open problem if the positivity of temperature can be recovered from the model. In our contribution, we give a rigorous proof that, under rather weak assumptions on the data of the system, any sufficiently smooth solution of the governing field equations has indeed the property that the absolute temperature variable attains positive values almost everywhere. The method of proof applies to all the simplified versions of the field equations that have been studied in the literature.Partially supported by DFG, SPP Anwendungsbezogene Optimierung und Steuerung, and by I.A.N. of C.N.R., Pavia — ItalyPartially supported by DFG, SPP Anwendungsbezogene Optimierung und Steuerung     

Local existence for Frémond’s model of damage in elastic materials

We consider a dissipative model recently proposed by M. Frémond to describe the evolution of damage in elastic materials. The corresponding PDEs system consists of an elliptic equation for the displacements with a degenerating elastic coefficient coupled with a variational dissipative inclusion governing the evolution of damage. We prove a local-in-time existence and uniqueness result for an associated initial and boundary value problem, namely considering the evolution in some subinterval where the damage is not complete. The existence result is obtained by a truncation technique combined with suitable a priori estimates. Finally, we give an analogous local-in-time existence and uniqueness result for the case in which we introduce viscosity into the relation for macroscopic displacements such that the macroscopic equilibrium equation is of parabolic type.Received: 31 July 2002, Accepted: 9 August 2003, Published online: 21 November 2003Correspondence to: E. Bonetti     

A model for universal spatial variations of temperature fluctuations in turbulent Rayleigh-Bénard convection

We propose a theoretical model for spatial variations of the temperature variance σ~2( z, r)( z is the distance from the sample bottom and r the radial coordinate) in turbulent Rayleigh-Bénard convection(RBC).Adapting the "attached-eddy" model of shear flow to the plumes of RBC, we derived an equation for σ~2 which is based on the universal scaling of the normalized RBC temperature spectra. This equation includes both logarithmic and power-law dependences on z/λ_(th), where λ_(th) is the thermal boundary layer thickness. The equation parameters depend on r and the Prandtl number Pr, but have only an extremely weak dependence on the Rayleigh number Ra Thus our model provides a near-universal equation for the temperature variance profile in turbulent RBC.     

N-Fold generalized Darboux transformation and semirational solutions for the Gerdjikov-Ivanov equation for the Alfvén waves in a plasma

Nonlinear Dynamics - Alfvén waves propagating parallel to the ambient magnetic field are modeled via the Gerdjikov-Ivanov equation. With respect to the transverse magnetic field perturbation...     

Global nonlinear stability in the Bénard problem for a mixture near the bifurcation point

The nonlinear stability of the motionless state of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme, for stress-free and rigid-rigid boundary conditions and Schmidt numbers P_C greater than Prandtl numbers P_T, is studied in the region around the bifurcation point of linear instability. An improvement of the results in Mulone [11] is found for small values of p = P _C /P _T and P_T. For p sufficiently large the critical nonlinear Rayleigh number is very close to the linear one (with relative difference less than in the sea water case) Received December 12, 2002 / Published online April 23, 2003 RID="a" ID="a" e-mail: mbasurto@dmi.unict.it RID="b" ID="b" e-mail: lombardo@dmi.unict.it ID="Communicated by Brian Straugham, Durham"     

Bénard-Marangoni Instability of a Two-Layer System with Allowance for Variations in the Interfacial Internal Energy

The effect of variations of the internal surface energy due to local increments in the interfacial area on the conditions of onset of thermocapillary Marangoni instability in a two-layer system of reduced-viscosity fluids is studied. It is shown that in the linear approximation the effect considered leads to stabilization of the development of the monotonic instability mode.     

Bézier curves in the modification of boundary integral equations (BIE) for potential boundary-values problems

The paper presents a modification of the classical boundary integral equation method (BIEM) for two-dimensional potential boundary values problems. The proposed modification consists in describing the boundary geometry by means of Bézier curves. As a result of this analytical modification of the BIEM, a new parametric integral equation system (PIES) was obtained. The kernels of these equations include the geometry of the boundary. This new PIES is no longer defined on the boundary, as in the case of the BIEM, but on the straight line for any given domain. The solution of the new PIES does not require a boundary discretization since it can be reduced merely to an approximation of boundary functions. To solve this PIES a pseudospectral method has been proposed and the results obtained were compared with exact solutions.     

Justification of the Nonlinear Schrödinger Equation for the Evolution of Gravity Driven 2D Surface Water Waves in a Canal of Finite Depth

In 1968 V.E. Zakharov derived the Nonlinear Schrödinger equation for the two-dimensional water wave problem in the absence of surface tension, that is, for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. In this paper we give a rigorous proof that the wave packets in the two-dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the Nonlinear Schrödinger equation.     

A fourth order accurate finite volume scheme for numerical simulations of turbulent Rayleigh–Bénard convection in cylindrical containers

A finite volume scheme, which is based on fourth order accurate central differences in spatial directions and on a hybrid explicit/semi-implicit time stepping scheme, was developed to solve the incompressible Navier–Stokes and energy equations on cylindrical staggered grids. This includes a new fourth order accurate discretization of the velocity and temperature fields at the singularity of the cylindrical coordinate system and a new stability condition [J. Appl. Numer. Anal. Comput. Math. 1 (2004) 315–326]. The method was applied in direct numerical simulations of turbulent Rayleigh–Bénard convection for different Rayleigh numbers $\mathit{Ra}={10}^{\gamma }$, $\gamma =5\text{,}\dots \text{,}8$, in wide cylinders with the aspect ratios $a\equiv H/R=0.2$ and $a=0.4$ (where R denotes the radius and H – the height of the cylinder). To cite this article: O. Shishkina, C. Wagner, C. R. Mecanique 333 (2005).     

Homogenization limit for electrical conduction in biological tissues in the radio-frequency rangeLimite d'homogénéisation pour la conduction électrique dans les tissus biologiques dans le domaine des radiofréquences

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a finely mixed periodic medium. We show that the homogenization limit u₀ of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation $\text{?}\text{div}\text{(σ}{}_0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+A}{}^0\text{?}{}_{\mathrm{x}}\text{u}{}_0\text{+∫}{}_0{}^{\mathrm{t}}\text{A}{}^1\text{(t?τ)?}{}_{\mathrm{x}}\text{u}{}_0\text{(x,τ)}\text{d}\text{τ?}\text{F}\text{(x,t))=0}$ where σ₀>0 and the matrices A⁰, A¹ depend on geometric and material properties, while the vector function $\text{F}$ keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type. To cite this article: M. Amar et al., C. R. Mecanique 331 (2003).