Numerical studies on quasilinear and linear elliptic equations

We explore here the acceleration of convergence of iterative methods for the solution of a class of quasilinear and linear algebraic equations. The specific systems are the finite difference form of the Navier-Stokes equations and the energy equation for recirculating flows. The acceleration procedures considered are: the successive over relaxation scheme; several implicit methods; and a second-order procedure. A new implicit method—the alternating direction line iterative method—is proposed in this paper. The method combines the advantages of the line successive over relaxation and alternating direction implicit methods. The various methods are tested for their computational economy and accuracy on a typical recirculating flow situation. The numerical experiments show that the alternating direction line iterative method is the most economical method of solving the Navier-Stokes equations for all Reynolds numbers in the laminar regime. The usual ADI method is shown to be not so attractive for large Reynolds numbers because of the loss of diagonal dominance. This loss can however be restored by a suitable choice of the relaxation parameter, but at the cost of accuracy. The accuracy of the new procedure is comparable to that of the well-tested successive overrelaxation method and to the available results in the literature. The second-order procedure turns out to be the most efficient method for the solution of the linear energy equation.

Polarization Caging in Diffusion-Controlled Electron Transfer Reactions in Solution

In some bimolecular diffusion-controlled electron transfer (ET) reactions such as ion recombination (IR), both solvent polarization relaxation and the mutual diffusion of the reacting ion pair may determine the rate and even the yield of the reaction. However, a full treatment with these two reaction coordinates is a challenging task and has been left mostly unsolved. In this work, we address this problem by developing a dynamic theory by combining the ideas from ET reaction literature and barrierless chemical reactions. Two-dimensional coupled Smoluchowski equations are employed to compute the time evolution of joint probability distribution for the reactant (P-(1)(X,R,t)) and the product (p((2))(X,R,t)), where X, as is usual in ET reactions, describes the solvent polarization coordinate and R is the distance between the reacting ion pair. The reaction is described by a reaction line (sink) which is a function of X and R obtained by imposing a condition of equal energy on the initial and final states of a reacting ion pair. The resulting two-dimensional coupled equations of motion have been solved numerically using an alternate direction implicit (ADI) scheme (Peaceman and Rachford, J. Soc. Ind. Appl. Math. 1955, 3, 28). The results reveal interesting interplay between polarization relaxation and translational dynamics. The following new results have been obtained. (i) For solvents with slow longitudinal polarization relaxation, the escape probability decreases drastically as the polarization relaxation time increases. We attribute this to caging by polarization of the surrounding solvent, As expected, for the solvents having fast polarization relaxation, the escape probability is independent of the polarization relaxation time. (ii) In the slow relaxation limit, there is a significant dependence of escape probability and average rate on the initial solvent polarization, again displaying the effects of polarization caging. Escape probability increases, and the average rate decreases on increasing the initial polarization. Again, in the fast polarization relaxation limit, there is no effect of initial polarization on the escape probability and the average rate of IR. (iii) For normal and barrierless regions the dependence of escape probability and the rate of IR on initial polarization is stronger than in the inverted region. (iv) Because of the involvement of dynamics along R coordinate, the asymmetrical parabolic (that is, non-Marcus) energy gap dependence of the rate is observed.

b-Turn conformations in crystal structures of model peptides containing a,a-di-n-propylglycine (Dpg) and a,a-di-n-butyl-glycine (Dbg).

The crystal state conformations of three peptides containing the a,a-dialkylated residues, a,adi n-propylglycine (Dpg) and a,@-di-n-butylglycine (Dbg), have been established by x-ray diffraction. Boc-Ala-Dpg-Ala-OMe ( I ) and Boc-Ala-Dbg-Ala-OMe (III) adopt distorted type II @-turn conformations with Ala ( I ) and Dpg/Dbg (2) as the corner residues. In both peptides the conformational angles at the Dxg residue (I: 4 = 66.23 J/ = 19.3'; III: 4 = 66S0, J. = 21 .la)deviate appreciablyfrom ideal values for the i + 2 residue in a type II @-turn. In both peptides the observed(N. 0) distances between the Boc CO andAla(3) NHgroups are far too long (I:3.44 k; III: 3.63 k) for an intramolecular 4 + 1 hydrogen bond. Boc-Ala-Dpg-Ala-NHMe (II)crystallizes with two independent molecules in the asymmetric unit. Both molecules IIA and IIB adopt consecutive @-turn (type III-III in IIA and type III-I in IIB) or incipient 3,,,-helical structures, stabilized by two intramolecular 4 --t I hydrogen bonds. In all four molecules the bond angle N-C"-C' ( T ) at the Dxg residues are 2 1109 The observation of conformational angles in the helical region of 4,J/ space at these residues is consistent with theoretical predictions

Numerical estimation of hotspot temperatures in electrodes subjected to pulsed electron beam heating in vacuum

A numerical solution for the transient temperature distribution in a cylindrical disc heated on its top surface by a circular source is presented. A finite difference form of the governing equations is solved by the Alternating Direction Implicit (ADI) time marching scheme. This solution has direct applications in analyzing transient electron beam heating of target materials as encountered in the prebreakdown current enhancement and consequent breakdown in high voltage vacuum gaps stressed by alternating and pulsed voltages. The solution provides an estimate of the temperature for pulsed electron beam heating and the size of thermally activated microparticles originating from anode hot spots. The calculated results for a typical 45kV (a.c.) electron beam of radius 2.5 micron indicate that the temperature of such spots can reach melting point and could give rise to microparticles which could initiate breakdown.

An ethnobotanical survey of medicinal plants in the Eastern Himalayan zone of Arunachal Pradesh, India

Aim of the study: The medicinal plants are integral source of easily available remedy used in rural healthcare system. This study was conducted among three major ethnic groups namely the Nocte, the Nyishi and the Adi in the Eastern Himalayan region of Arunachal Pradesh to evaluate their comparative knowledge on medicinal plants. Materials and methods: The three remote districts of Arunachal Pradesh namely the Tirap, the Dibang Valley and the Papum Pare were surveyed through interviewing of randomly selected 237 participants using semi-structured questionnaire and regular field visits to selected districts. Results: We recorded the traditional use of 74 medicinal plants species belonging to 41 taxonomic plant families used for treating a total of 25 different diseases/ailments. The informant consensus factor (ICF) values demonstrated that local people tend to agree more with each other in terms of the plants used to treat malaria (0.71), jaundice (0.62), urological problems (0.56), dermatological disorders (0.45), pain (0.30), and respiratory disorder (0.33), and while the general health (0.15) and gastro-intestinal disorders category (0.28) were found low ICF values. Conclusion: Of the total 74 species recorded, the highest number of medicinal plants (36 species) was reported from the Adi of Lower Dibang Valley followed by the Nocte of the Tirap (25 species) and the Nyishi ethnic groups of Papum Pare districts (13 species). In the present study, we found that the men, elder people and illiterate ones had better knowledge on medicinal plants as compared to women, younger and literate people. Findings of this documentation study can be used as an ethnopharmacological basis for selecting plants for future phytochemical and pharmaceutical studies. (C) 2010 Elsevier Ireland Ltd. All rights reserved.

Exponential compact higher order scheme and their stability analysis for unsteady convection-diffusion equations

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Peclet number 0 <= Pe <= 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case.