An explicit analytical solution for temperature rise in a half-space induced by ellipsoidal inclusions and its application for problems related to friction heating

Abstract

Green functions for temperature rise in a semi-infinite space containing an ellipsoidal inclusion are obtained in the present study. Explicit expression for disturbed temperature rise generated by eigen-temperature gradients uniformly distributed within a domain is derived. Further, the proposed analytical solution method is utilized to deal with temperature rise in heterogeneous half-space subjected to friction heating via applying the equivalent inclusion method (EIM), whose results are proven to be in good agreements with those of the benchmarks. Influences of heat load velocity, spatial orientation and aspect ratio of ellipsoidal inhomogeneity on temperature rise in a semi-infinite space are discussed. Finally, a model of semi-infinite medium with embedded dispersed ellipsoidal inhomogeneities of arbitrary spatial orientation is adopted to explore the application scope of the proposed solution method.     

Quadratic Nonlinear Optical Properties of N‐Aryl Stilbazolium Dyes

The new salts trans‐4′‐(dimethylamino)‐N‐R‐4‐stilbazolium hexafluorophosphate (R = methyl, Me 1 , phenyl, Ph 2 , 2,4‐dinitrophenyl, DNPh 3 , 2‐pyrimidyl, Pym 4 , Scheme 1) have been prepared. Their electronic absorption spectra show intense, visible intramolecular charge‐transfer bands, the energy (E_max) of which decreases in the order R = Me > Ph > DNPh > Pym. This trend arises from the steadily increasing electron deficiency of the pyridinium ring, a phenomenon also observed in cyclic voltammetric and ¹H nuclear magnetic resonance (NMR) data. Fluorescence‐free first hyperpolarizability β values of [ 1 – 4 ]PF₆ were measured by using femtosecond hyper‐Rayleigh scattering (HRS) with acetonitrile solutions and a 1300 nm laser, and static first hyperpolarizabilities β₀ were obtained by application of the two‐state model. The HRS results indicate that the N‐aryl chromophores in [ 2 – 4 ]PF₆ have considerably larger β₀ values than their N‐methyl counterpart in [ 1 ]PF₆, with a ca. 10‐fold increase in β₀ observed in moving from [ 1 ]PF₆ to [ 4 ]PF₆ (25 → 230 × 10^–30 esu). Stark (electroabsorption) spectroscopic studies in butyronitrile glasses at 77 K allowed the derivation of dipole moment changes Δμ₁₂ (10.9–14.8 D), which have been used to calculate β₀ according to the two‐state equation β₀ = 3Δμ₁₂(μ₁₂)²/2(E_max)² (μ₁₂ = transition dipole moment). With the exception of [ 1 ]PF₆, the Stark‐derived β₀ values are in reasonable agreement with those from HRS. However, the increase in β₀ in moving from [ 1 ]PF₆ to [ 4 ]PF₆ is only 2‐fold for the Stark data (90 → 185 × 10^–30 esu). The observed trend of increasing β₀ in the order [ 1 ]PF₆ < [ 3 ]PF₆ < [ 2 ]PF₆ < [ 4 ]PF₆ arises from a combination of decreasing E_max and increasing Δμ₁₂, with only a slight increase in μ₁₂ between [ 1 ]PF₆ and [ 4 ]PF₆. It is likely that the β₀ values for [ 3 ]PF₆ are lower than expected due to the steric effect of the ortho‐NO₂ group, which causes twisting of the DNPh ring out of the plane of the stilbazolium unit. A single crystal X‐ray structure shows that [ 2 ]PF₆ crystallizes in the space group Cc, with head‐to‐tail alignment and almost parallel stacking of the pseudo‐planar stilbazolium portions of the cations to form polar sheets within a polar bulk structure. [ 2 ]PF₆ is essentially isostructural with the related Schiff base salt trans‐4‐[(4‐dimethylaminophenyl)iminomethyl]‐N‐phenylpyridinium hexafluorophosphate ([ 8 ]PF₆). Second harmonic generation (SHG) studies on [ 2 ]PF₆ and [ 8 ]PF₆ using a 1907 nm laser and sieved powdered samples (53–63 μm) afforded efficiencies of 470 and 240 times that of urea, respectively. Under the same conditions, the well‐studied compound [ 1 ]p‐MeC₆H₄SO₃ gave an SHG efficiency of 550 times that of urea.     

Statistical circuit design with the use of a modified ellipsoidal technique

The ellipsoidal technique for design centering and feasible region approximation [1] is modified in this article. The feasible region is approximated by a hyperellipsoid determined by a sequence of decreasing volume, different shape, and center hyperellipsoids. The construction of the hyperplane cuts used for the generation of the sequence of the hyperellipsoids is modified to make use of the function values only. The hyperplane cuts are linearizations of the nonlinear constraints at proper boundary points. The choice of the nonlinear constraint to be linearized and the linearization point are optimized in order to obtain maximum reduction in the volume of the generated hyperellipsoid. This will speed up the convergence to the final hyperellipsoid. Practical examples to test the method are included and applications to centering, tolerancing, and yield estimation are also given. © 1997 John Wiley & Sons, Inc. Int J Microwave Millimeter-Wave CAE 7: 117–128, 1997.     

Stokes flow past an ellipsoid in a special ambient flow field: A marvelous cubic equation for a quadratic ambient field

A three-dimensional eigenspace appears in the analysis of an ellipsoid immersed in quadratic ambient viscous flow fields. To celebrate the mathematical achievements of Babatunde Ogunnaike, we focus on mathematical details of the eigenvalues and eigenfunctions of the double-layer operator that appears in the integral representation of velocity fields as a function of the shape parameters of the ellipsoid. Three special quadratic ambient fields that lack an ambient pressure gradient (toroidal fields) form a decoupled subsystem with eigenvalues associated with three real roots of a cubic equation. The discriminant of this cubic vanishes at discrete points in the parameter space of the ellipsoidal shape revealing repeated roots (and thus repeated eigenvalues) of the cubic. The discrete nature of these points is illustrated with explicit derivation of the discriminant's level set contours in the region near the repeated eigenvalues for the sphere.