A note on the extensional viscosity of elastic liquids under strong flows

In this article a parametric study based on a balance between viscous drag and restoring Brownian forces is used in order to construct a nonlinear dumbbell model with a finite spring and a drag correction for a dilute polymer solution. The constitutive equations used are reasonable approximation for describing flows of very dilute polymer solutions such as those used in turbulent drag reduction. We investigate the response of an elastic liquid under extensional flows in order to explore the roles of a stress anisotropy and of elasticity in strong flows. It is found that for low Reynolds numbers, the extensional viscosity of a dilute polymer solution is governed by two parameters: a Deborah number representing the importance of the elasticity on the flow and the macromolecule extensibility that accounts for the viscous anisotropic effects caused by the macromolecule orientation. Two different asymptotic regimes are described.The first corresponds to an elastic limit in which the extensional viscosity is a function of the Deborah number and the particle volume fraction. The second is an anisotropic regime with the extensional viscosity independent of Deborah number but strongly dependent on macromolecule aspect ratio. The analysis may explain from a phenomenological point of view why few ppms of macromolecules of high molecule weight or a small volume fraction of long fibres produce important attenuation of the pressure drop in turbulent flows. On the basis of our analysis it is seen that the anisotropic limit of the extensional viscosity caused by extended polymers under strong flows should play a key role in the attenuation of flow instability and in the mechanism of drag reduction by polymer additives.     

Some remarks about the resolution of high velocity flows near low densities

High velocity flows which are exposed to strong rarefaction waves and creating low densities regions in it present difficulties and inaccuracies for numerical resolution. In particular, variables related to the internal energy are wrongly evaluated. Use of classical schemes solving the Euler equations in conservative variables introduces significant errors in the determination of temperature. We recommend to employ a non-conservative formulation of the energy equation. Results found to be more accurate in using the present internal energy formulation. In order to have the formulation available for both shock and strong rarefaction waves, we propose a hybrid formulation of conservative and non-conservative ones, depending on a shock indicator. The results are compared with exact solutions and show a significant improvement of the accuracy. The method is then extended to two-dimensional cases. Received 28 March 1997 / Accepted 18 June 1997     

The influence of organic sample solvents on the separation efficiency of basic compounds under strong cation exchange mode

This study investigated the influence of organic sample solvents on separation efficiency of basic compounds under strong cation exchange (SCX) mode. The mixtures of acidic aqueous solution and organic solvent such as acetonitrile, ethanol, methanol and dimethyl sulfoxide (DMSO) were tested as sample solvents. For later-eluting analytes, the increase of sample solvent elution strength was responsible for the decrease of separation efficiency. Thus, sample solvents with weak elution strength could provide high separation efficiencies. For earlier-eluting analytes, the retention of organic sample solvents was the main factor affecting separation efficiency. Weakly retained solvents could provide high separation efficiency. In addition, an optimized approach was proposed to reduce the effect of organic sample solvent, in which low ionic solvent was employed as initial mobile phase in the gradient. At last, the analysis of impurities in hydrophobic drug berberine was performed. The results showed that using acidic aqueous methanol as sample solvents could provide high separation efficiency and good resolution (R > 1.5).     

Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

Nouvelle approche pour la discrétisation de flux diffusifs en volumes finis à forte obliquité

We propose a new approach for the discretization of diffusive flux on a non orthogonal mesh based on the Deferred correction introduced by Peric. This new method is applied successfully to the solution of a Poisson problem in quadrangular domains meshed with very distorted control volumes. The interest of this approach lies in the fact that the precision of the used schemes is conserved despite meshes distortion level. The comparative study of the value of the spectral radius of iteration matrix enables us to explain why our Deferred correction approach is better than that of Peric, which diverges on highly skewed meshes. To cite this article: Y.M. Ahipo, Ph. Traore, C. R. Mecanique 334 (2006).     

On the strong convergence rate for positively associated random variables

For sequences of the positively associated random variables which are strictly weaker than the classical associated random ones introduced by Esary et al. (1967) [7], strong convergence rate is obtained, which reaches the available one for independent random variables in terms of Berstein type inequality. Further, we give the corresponding precise asymptotics with respect to the rate mentioned above, which extend and improve the relevant results in Fu (2009) [8].     

Blow-up criterion for compressible MHD equations

In this paper, we study the 3D compressible magnetohydrodynamic equations. We obtain a blow up criterion for the local strong solutions just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion (see J.T. Beal, T. Kato and A. Majda (1984) [1]) for the ideal incompressible flow. In addition, initial vacuum is allowed in our case.     

On the convergence rate of grad-div stabilized Taylor-Hood to Scott-Vogelius solutions for incompressible flow problems

It was recently proven in Case et al. (2010) [2] that, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to the Scott-Vogelius solution of that same problem. However, even though the analytical rate was only shown to be (where γ is the stabilization parameter), the computational results suggest the rate may be improvable to γ−1. We prove herein the analytical rate is indeed γ−1, and extend the result to other incompressible flow problems including Leray-α and MHD. Numerical results are given that verify the theory.     

Piecewise homotopy perturbation method for solving linear and nonlinear weakly singular VIE of second kind

The purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular Volterra integral equation of the second kind, especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem. For this purpose, we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like. Through modifying homotopy perturbation method, an algorithm is successfully established to solve the linear and nonlinear weakly singular Volterra integral equation of the second kind. And the convergence of the algorithm is proved strictly. Comparisons are made between our method and other methods, and the results reveal that the modified homotopy perturbation is effective.     

A sufficient descent LS conjugate gradient method for unconstrained optimization problems

In this paper, we make a modification to the Liu-Storey (LS) conjugate gradient method and propose a descent LS method. The method can generate sufficient descent directions for the objective function. This property is independent of the line search used. We prove that the modified LS method is globally convergent with the strong Wolfe line search. The numerical results show that the proposed descent LS method is efficient for the unconstrained problems in the CUTEr library.