Finite Element Solution Of Surface-Tension Driven Flows In Laser Surface-Melting

Lasers are very efficient in heating localized regions and hence they find a wide application in surface treatment processes. The surface of a material can be selectively modified to give superior wear and corrosion resistance. In laser surface-melting and welding problems, the high temperature gradient prevailing in the free surface induces a surface-tension gradient which is the dominant driving force for convection (known as thermo-capillary or Marangoni convection). It has been reported that the surface-tension driven convection plays a dominant role in determining the melt pool shape. In most of the earlier works on laser-melting and related problems, the finite difference method (FDM) has been used to solve the Navier Stokes equations [1]. Since the Reynolds number is quite high in these cases, upwinding has been used. Though upwinding gives physically realistic solutions even on a coarse grid, the results are inaccurate. McLay and Carey have solved the thermo-capillary flow in welding problems by an implicit finite element method [2]. They used the conventional Galerkin finite element method (FEM) which requires that the pressure be interpolated by one order lower than velocity (mixed interpolation). This restricts the choice of elements to certain higher order elements which need numerical integration for evaluation of element matrices. The implicit algorithm yields a system of nonlinear, unsymmetric equations which are not positive definite. Computations would be possible only with large mainframe computers.Sluzalec [3] has modeled the pulsed laser-melting problem by an explicit method (FEM). He has used the six-node triangular element with mixed interpolation. Since he has considered the buoyancy induced flow only, the velocity values are small. In the present work, an equal order explicit FEM is used to compute the thermo-capillary flow in the laser surface-melting problem. As this method permits equal order interpolation, there is no restriction in the choice of elements. Even linear elements such as the three-node triangular elements can be used. As the governing equations are solved in a sequential manner, the computer memory requirement is less. The finite element formulation is discussed in this paper along with typical numerical results.

Approximate solution of u'=-A2u using a relation between exp(-tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u(t) = -A(2)u. Using a representation of the semigroup exp(-A(2)t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w(t) = W-yy, with initial values v solving the initial value problem for v(y) = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2(nd) order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4(th) order equation for u to that of the 2(nd) order equation for v, followed by the solution of the heat equation in one space variable.

Activities in the $FeTiO_{3}-NiTiO_{3}$ Solid Solution from Alloy-Oxide Equilibria at 1273 K

Nine tie-lines between Fe-Ni alloys and FeTiO3-NiTiO3 solid solutions were determined at 1273 K. Samples were equilibrated in evacuated quartz ampoules for periods up to 10 days. Compositions of the alloy and oxide phases at equilibrium were determined by energy-dispersive x-ray spectroscopy. X-ray powder diffraction was used to confirm the results. Attainment of equilibrium was verified by the conventional tie-line rotation technique and by thermodynamic analysis of the results. The tie-lines are skewed toward the FeTiO3 corner. From the tie-line data and activities in the Fe-Ni alloy phase available in the literature, activities of FeTiO3 and NiTiO3 in the ilmenite solid solution were derived using the modified Gibbs-Duhem technique of Jacob and Jeffes [K.T. Jacob and J.H.E. Jeffes, An Improved Method for Calculating Activities from Distribution Equilibria, High Temp. High Press., 1972, 4, p 177-182]. The components of the oxide solid solution exhibit moderate positive deviations from Raoult's law. Within experimental error, excess Gibbs energy of mixing for the FeTiO3-NiTiO3 solid solution at 1273 K is a symmetric function of composition and can be represented as: Delta G(E) = 8590 (+/- 200) X-FeTiO3 X-NiTiO3 J/mol Full spectrum of tie-lines and oxygen potentials for the three-phase equilibrium involving Fe-Ni alloys, FeTiO3-NiTiO3 solid solutions, and TiO2 at 1273 K were computed using results obtained in this study and data available in the literature.

Iterative solution of a class of linear equations with application to reconstruction of three-dimensional object arrays

An iterative algorithm baaed on probabilistic estimation is described for obtaining the minimum-norm solution of a very large, consistent, linear system of equations AX = g where A is an (m times n) matrix with non-negative elements, x and g are respectively (n times 1) and (m times 1) vectors with positive components.

Thermodynamic activities in the Pb(Zr1-XTiX)O-3 solid solution at 1373 K

Although Pb(Zr1-XTiX)O-3 solid solution is the cornerstone of the piezoelectric ceramics, there is no information in the literature on thermodynamic activities of the component phases in the solid solution. Using inter-crystalline ion exchange equilibria between Pb(Zr1-XTiX)O-3 solid solution with cubic perovskite structure and (Zr1-YTiY)O-2 solid solutions with monoclinic and tetragonal structures, activities of PbTiO3 and PbZrO3 in the perovskite solid solution have been derived at 1373 K using the modified Gibbs-Duhem integration technique of Jacob and Jeffes. Tie-lines from the cubic solid solution are skewed towards the ZrO2 corner. Activities in the zirconia-rich (Zr1-YTiY)02 solid solutions are taken from a recent emf study. The results for the perovskite solid solution at 1373 K can be represented by a sub-regular solution model: Delta G(E.M) (J mol(-1)) = X-PbTiO3 X-PbZrO3(5280X(PbTiO3) - 1980X(PbZrO3)) where Delta G(E.M) is the excess Gibbs energy of mixing of the cubic solid solution and Xi represents the mole fraction of component i. There is a significant positive deviation from ideality for PbTiO3-rich compositions and mild negative deviation near the PbZrO3 corner. The cubic solid solution is intrinsically stable against composition fluctuations at temperatures down to 840 K. The results contrast sharply with the recent calorimetric data on enthalpy of mixing which signal instability of the cubic perovskite solid solution. (C) 2007 Elsevier B.V. All rights reserved.

Solution structure of O-glycosylated C-terminal leucine zipper domain of human salivary mucin (MUC7)

Solution structures of a 23 residue glycopeptide II (KIS* RFLLYMKNLLNRIIDDMVEQ, where * denotes the glycan Gal-beta-(1-3)-alpha-GalNAc) and its deglycosylated counterpart I derived from the C-terminal leucine zipper domain of low molecular weight human salivary mucin (MUC7) were studied using CD, NMR spectroscopy and molecular modeling. The peptide I was synthesized using the Fmoc chemistry following the conventional procedure and the glycopeptide II was synthesized incorporating the O-glycosylated building block (N alpha-Fmoc-Ser-[Ac-4,-beta-D-Gal-(1,3)-Ac(2)alpha-D-GalN(3)]-OPfp) at the appropriate position in stepwise assembly of peptide chain. Solution structures of these glycosylated and nonglycosylated peptides were studied in water and in the presence of 50% of an organic cosolvent, trifluoroethanol (TFE) using circular dichroism (CD), and in 50% TFE using two-dimensional proton nuclear magnetic resonance (2D H-1 NMR) spectroscopy. CD spectra in aqueous medium indicate that the apopeptide I adapts, mostly, a beta-sheet conformation whereas the glycopeptide II assumes helical structure. This transition in the secondary structure, upon glycosylation, demonstrates that the carbohydrate moiety exerts significant effect on the peptide backbone conformation. However, in 50% TFE both the peptides show pronounced helical structure. Sequential and medium range NOEs, C alpha H chemical shift perturbations, (3)J(NH:C alpha H) couplings and deuterium exchange rates of the amide proton resonances in water containing 50% TFE indicate that the peptide I adapts alpha-helical structure from Ile2-Val21 and the glycopeptide II adapts alpha-helical structure from Ser3-Glu22. The observation of continuous stretch of helix in both the peptides as observed by both NMR and CD spectroscopy strongly suggests that the C-terminal domain of MUC7 with heptad repeats of leucines or methionine residues may be stabilized by dimeric leucine zipper motif. The results reported herein may be invaluable in understanding the aggregation (or dimerization) of MUC7 glycoprotein which would eventually have implications in determining its structure-function relationship.

Rayleigh-Benard convection during solidification of an eutectic solution cooled from the top

The interaction between laminar Rayleigh-Benard convection and directional solidification is studied for the case of an eutectic solution kept in a rectangular cavity cooled from the top. Experiments and numerical simulations are carried out using an NH4Cl-H2O solution as the model fluid. The flow is visualized using a sheet of laser light scattered by neutrally buoyant, hollow-glass spheres seeded in the fluid. The numerical modeling is performed using a pressure-based finite-volume method according to the SIMPLER algorithm. The present configuration enables us to visualize flow vortices in the presence of a continuously evolving solid/liquid interface. Clear visualization of the Rayleigh-Benard convective cells and their interaction with the solidification front are obtained. It is observed that the convective cells are characterized by zones of up-flow and down-flow, resulting in the development of a nonplanar interface. Because of the continuous advancement of the solid/liquid interface, the effective liquid height of the cavity keeps decreasing. Once the height of the fluid layer falls below a critical value, the convective cells become weaker and eventually die out, leading to the growth of a planar solidification front. Results of flow visualization and temperature measurement are compared with those from the numerical simulation, and a good agreement is found.

Ion exchange equilibria between (Mn,Co)O solid solution and (Mn,Co)Cr2O4 and (Mn,Co)Al2O4 spinel solid solutions at 1100oC

The compositions of the (Mn,Co)O solid solution with rock salt structure in equilibrium with (Mn,Co)Cr2O4 and (Mn,Co)Al2O4 spinel solid solutions have been determined by X-ray diffraction measurements at 1100° C and an oxygen partial pressure of 10–10 atm. The ion exchange equilibria are quantitatively analysed, using values for activities in the (Mn,Co)O solid solution available in the literature, in order to obtain activities in the spinel solid solutions. The MnAl2O4-CoAl2O4 solid solution exhibits negative deviations from Raoult's law, consistent with the estimated cation disorder in the solid solution, while the MnCr2O4-CoCr2O4 solid solution shows slightly positive deviations. The difference in the Gibbs free energy of formation of the two pure chromites and aluminates derived from the results of this study are in good agreement with recent results obtained from solid oxide galvanic cells and gas-equilibrium techniques.

Thermodynamic activities in the Pb(Zr1−XTiX)O3 solid solution at 1373K

Although Pb(Zr1−XTiX)O3 solid solution is the cornerstone of the piezoelectric ceramics, there is no information in the literature on thermodynamic activities of the component phases in the solid solution. Using inter-crystalline ion exchange equilibria between Pb(Zr1−XTiX)O3 solid solution with cubic perovskite structure and (Zr1−YTiY)O2 solid solutions with monoclinic and tetragonal structures, activities of PbTiO3 and PbZrO3 in the perovskite solid solution have been derived at 1373 K using the modified Gibbs–Duhem integration technique of Jacob and Jeffes. Tie-lines from the cubic solid solution are skewed towards the ZrO2 corner. Activities in the zirconia-rich (Zr1−YTiY)O2 solid solutions are taken from a recent emf study. The results for the perovskite solid solution at 1373 K can be represented by a sub-regular solution model:View the MathML sourcewhere ΔGE,M is the excess Gibbs energy of mixing of the cubic solid solution and Xi represents the mole fraction of component i. There is a significant positive deviation from ideality for PbTiO3-rich compositions and mild negative deviation near the PbZrO3 corner. The cubic solid solution is intrinsically stable against composition fluctuations at temperatures down to 840 K. The results contrast sharply with the recent calorimetric data on enthalpy of mixing which signal instability of the cubic perovskite solid solution.

Thermodynamics of Al(1-x)Ga(x)N solid solution: Inclination for phase separation and ordering

Although Al(1-x)Ga(x)N semiconductors are used in lighting, displays and high-power amplifiers, there is no experimental thermodynamic information on nitride solid solutions. Thermodynamic data are useful for assessing the intrinsic stability of the solid solution with respect to phase separation and extrinsic stability in relation to other phases such as metallic contacts. The activity of GaN in Al(1-x)Ga(x)N solid solution is determined at 1100 K using a solid-state electrochemical cell: Ga + Al(1-x)Ga(x)N/Fe, Ca(3)N(2)//CaF(2)//Ca(3)N(2), N(2) (0.1 MPa), Fe. The solid-state cell is based on single crystal CaF(2) as the electrolyte and Ca(3)N(2) as the auxiliary electrode to convert the nitrogen chemical potential established by the equilibrium between Ga and Al(1-x)Ga(x)N solid solution into an equivalent fluorine potential. Excess Gibbs free energy of mixing of the solid solution is computed from the results. Results suggest an unusual mixing behavior: a mild tendency for ordering at three discrete compositions (x = 0.25, 0.5 and 0.75) superimposed on predominantly positive deviation from ideality. The lattice parameters exhibit slight deviation from Vegard's law, with the a-parameter showing positive and the c-parameter negative deviation. Although the solid solution is stable in the full range of compositions at growth temperatures, thermodynamic instability is indicated at temperatures below 410 K in the composition range 0.26 <= x <= 0.5. At 355 K, two biphasic regions appear, with terminal solid solutions stable only for 0 <= x <= 0.26 and 0.66 <= x <= 1. The range of terminal solid solubility reduces with decreasing temperature. (C) 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Interdiffusion and solid solution strengthening in Ni-Co-Pt and Ni-Co-Fe ternary systems

Diffusion couple experiments are conducted in Co-Ni-Pt system at 1200 degrees C and in Co-Ni-Fe system at 1150 degrees C, by coupling binary alloys with the third element. Uphill diffusion is observed for both Co and Ni in Pt rich corner of the Co-Ni-Pt system, whereas in the Co-Ni-Fe system, it is observed for Co. Main and cross interdiffusion coefficients are calculated at the composition of intersection of two independent diffusion profiles. In both the systems, the main interdiffusion coefficients are positive over the whole composition range and the cross interdiffusion coefficients show both positive and negative values at different regions. Hardness measured by performing the nanoindentations on diffusion couples of both the systems shows the higher values at intermediate compositions.

Interdiffusion and solid solution strengthening in Ni-Co-Pt and Ni-Co-Fe ternary systems

Diffusion couple experiments are conducted in Co-Ni-Pt system at 1200 degrees C and in Co-Ni-Fe system at 1150 degrees C, by coupling binary alloys with the third element. Uphill diffusion is observed for both Co and Ni in Pt rich corner of the Co-Ni-Pt system, whereas in the Co-Ni-Fe system, it is observed for Co. Main and cross interdiffusion coefficients are calculated at the composition of intersection of two independent diffusion profiles. In both the systems, the main interdiffusion coefficients are positive over the whole composition range and the cross interdiffusion coefficients show both positive and negative values at different regions. Hardness measured by performing the nanoindentations on diffusion couples of both the systems shows the higher values at intermediate compositions.

Exact N-wave solutions of generalized Burgers equations

Exact N-wave solutions for the generalized Burgers equation u(t) + u(n)u(x) + (j/2t + alpha) u + (beta + gamma/x) u(n+1) = delta/2u(xx),where j, alpha, beta, and gamma are nonnegative constants and n is a positive integer, are obtained. These solutions are asymptotic to the (linear) old-age solution for large time and extend the validity of the latter so as to cover the entire time regime starting where the originally sharp shock has become sufficiently thick and the viscous effects are felt in the entire N wave.

Evidence of clustering in an aqueous electrolyte solution: a small-angle X-ray scattering study

The work reported hen was motivated by a desire to verify the existence of structure - specifically MP-rich clusters induced by sodium bromide (NaBr) in the ternary liquid mixture 3-methylpyridine (Mf) + water(W) + NaBr. We present small-angle X-ray scattering (SAXS) measurements in this mixture. These measurements were obtained at room temperature (similar to 298 K) in the one-phase region (below the relevant lower consolute points, T(L)s) at different values of X (i.e., X = 0.02 - 0.17), where X is the weight fraction of NaBr in the mixture. Cluster-size distribution, estimated on the assumption that the clusters are spherical, shows systematic behaviour in that the peak of the distribution shifts rewards larger values of cluster radius as X increases. The largest spatial extent of the clusters (similar to 4.5 nm) is seen at X = 0.17. Data analysis assuming arbitrary shapes and sizes of clusters gives a limiting value of cluster size (- 4.5 nm) that is not very sensitive to X. It is suggested that the cluster size determined may not be the same as the usual critical-point fluctuations far removed from the critical point (T-L). The influence of the additional length scale due to clustering is discussed from the standpoint of crossover from Ising to mean-field critical behaviour, when moving away from the T-L.

An Approximate Solution Of One-Dimensional Piston Problem

A new theory of shock dynamics has been developed in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth or decay of shock strength for accelerating or decelerating piston starting with a nonzero piston velocity. The results show good agreement with those obtained by Harten's high resolution TVD scheme.