Catastrophic instabilities and related results in a fluid of third grade

Thermodynamical considerations [1] have shown that the most general form for the stress constitutive relation of an incompressible fluid of grade three is T = ?p1 + μA₁ + α₁A₂ + α₂A²₁ + β(tr A²₁)A₁, where A₁ and A₂ are the first two Rivlin-Ericksen tensors. In addition, the material parameters μ, α₁, α₂ and α were shown in [1] to be restricted by certain inequalities (see (1.7), (1.8)). Here we show that the condition α₁ <0, which is compatible with the Clausius-Duhem inequality but not with the free energy being a minimum in equilibrium, leads to behavior which may not be physically acceptable. An explicit solution is presented for the second grade fluid, for which β =0 and μ, α₁ and α₂ are arbitrary, which demonstrates that if μ #62; 0 and α₁ < 0 then a rotating vortex system may increase indefinitely in amplitude.     

On Constitutive Equations of Cauchy Elastic Solids: All Kinds of Crystals and Quasicrystals

Our concern is with the problem of determining general reduced forms of constitutive equations of Cauchy elastic solids under all kinds of material symmetries, including well-known crystal classes and newly discovered quasicrystal classes. By means of Tschebysheff polynomials we present in unified forms simple irreducible representations for elastic constitutive equations under the infinitely many subgroup classes C _2m+1, C _2m+2, D _2m+1 and D _2m+2 for all m = 1, 2,... Moreover, we provide a simple representation for elastic constitutive equations under the most complicated point group, i.e. the icosahedral group. Each presented representation is expressed in terms of not more than nine polynomial tensor generators only. This revised version was published online in August 2006 with corrections to the Cover Date.     

Preparation of ternary Cr2O3–SiC–TiO2 composites for the photocatalytic production of hydrogen

Novel ternary Cr₂O₃–SiC–TiO₂ composites were synthesized by implanting Cr³⁺ into SiC–TiO₂ via sol–gel and impregnation approaches. The results from X-ray diffraction, scanning electron microscopy and transmission electron microscopy show that the Cr³⁺ species were doped onto the surface of the SiC–TiO₂ carrier. The diffuse reflectance ultraviolet–visible absorption spectra revealed that the absorption edges of the ternary Cr₂O₃–SiC–TiO₂ composites were gradually shifted red with increasing chromium content. The luminescence intensities of the composites decreased with increasing doped Cr³⁺ content due to the reduction in the number of recombination sites of electron–hole pairs. The ternary Cr₂O₃–SiC–TiO₂ composites showed high hydrogen-producing activities, which probably results from the formation of donor levels of the Cr³⁺ species in the forbidden band of the SiC semiconductor.     

Properties of an Al/(Al<Subscript>2</Subscript>O<Subscript>3</Subscript>+TiB<Subscript>2</Subscript>+ZrB<Subscript>2</Subscript>) hybrid composite manufactured by powder metallurgy and hot pressing

The properties and microstructure of an Al/(Al₂O₃ + TiB₂ + ZrB₂) hybrid composite made by using hot pressing of aluminum combined with different amounts of TiB₂, ZrB₂, and Al₂O₃ powders are studied. The mechanical properties of the composites are investigated on the basis of microhardness and compression tests. The results show that the microstructure of the composites is uniform and the particles are well distributed in the matrix.     

Entire Solutions with Merging Fronts to Reaction–Diffusion Equations

We deal with a reaction–diffusion equation u _t = u _xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ₁(x + c ₁ t) (c ₁ < 0) and ψ₂(x + c ₂ t) (c ₂ > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all . We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ₁(x + c ₁ t) and ψ₂(x + c ₂ t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c ₁, we show the existence of an entire solution which behaves as ψ₁( − x + c ₁ t) in and φ(x + ct) in for t≈ − ∞.     

Equivalence between Rank-One Convexity and Polyconvexity for Some Classes of Elastic Materials

Let W(F) = φ(λ ₁ ^s + λ ₂ ^s + λ ₃ ^s ) + ψ(λ ₁ ^r λ ₂ ^r + λ ₁ ^r λ ₃ ^r + λ ₂ ^r λ ₃ ^r ) + f(λ ₁ λ ₂ λ ₃) be a stored energy function. We prove that, for this function, rank-one convexity is equivalent to polyconvexity.under suitable assumptions on φ, ψ and f.     

Functionalized TiO2@ZrO2@Y2O3 ：Eu3＋ core-multishell microspheres and their photoluminescence properties

TiO2@ZrO2@Y2O3 ：Eu3＋ composite particles with a core-multishell structure were synthesized through the combination of a layer-by-layer （LBL） self-assembly method and a sol-gel process. The obtained sam- ples were characterized with scanning electron microscopy （SEM）, transmission electron microscopy （TEM）, X-ray photoelectron spectroscopy （XPS）, X-ray diffraction （XRD）, and fluorescence spectropho- tometry. The results showed that the composite particles had a core-multishell structure, spherical morphology, and a narrow size distribution. The presence of a ZrO2 layer on the TiO2 core can effec- tively prevent the reaction between the TiO2 core and a Y203 shell; the temperature for the reaction between the TiO2 core and the Y203 shell in the TiO2@ZrO2@Y2O3 ：Eu core-multishell phosphor can be elevated by 300 ℃ compared to that for TiO2@ZrO2：Eu. Upon excitation of the core-multishell particles in the ultraviolet （254 nm）, the Eu3＋ ion in the Y2O3 ：Eu3＋ shell shows its characteristic red emission （611 nm, 5D0→7F2）, and the photoluminescence （PL） intensity of the phosphor with the core-multishell structure was obviously greater than that of the core-shell TiO2@Y2O3 ：Eu phosphor.     

On the fundamental linear fractional order differential equation

This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t₀s)^2m + 2z(t₀s)^m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t₀)^2m\fracd^2mx(t)dt^2m + 2z(t₀)^m\fracd^mx(t)dt^m + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.     

Nonhomogeneous Dirichlet Navier—Stokes Problems in Low Regularity Lp Sobolev Spaces

The time-dependent Navier--Stokes problem on an interior or exterior smooth domain, with nonhomogeneous Dirichlet boundary condition, is treated in anisotropic L_p Sobolev spaces (1 < p < ￥ \infty ) of Bessel-potential type H^s+2,s/2+1_p H^{s+2,s/2+1}_p or Besov type B^s+2,s/2+1_p B^{s+2,s/2+1}_p by use of a reformulation of the linearized problem to a parabolic pseudodifferential boundary value problem. Earlier studies required s > \frac 1p - 1 s > {\frac 1p} - 1 ; the present work extends the solvability to spaces with s > \frac 1p - 2 s > {\frac 1p} - 2 for zero initial data (s > - 2 s > - 2 if f = 0), s > \frac 2p - 2 s > {\frac 2p} - 2 for nonzero initial data, with s,p subject to other conditions stemming from the nonlinearity.     

Explicit solitary-wave solutions to generalized Pochhammer-Chree equations

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Multi-ion intercalated Ti3C2Tx MXene and the mutual modulation within interlayer

Intercalation of ions between the adjacent MXene layers can change the interlayer environment and influence the electrochemical ion storage capacity. In order to understand the effect of multi-ions confined by the MXene layers on the performance of electrochemical energy storage, Co²⁺, Mn²⁺ and Ni²⁺ intercalated into Ti₃C₂T_x MXene which already pre-intercalated Al³⁺ are obtained by spontaneous static action. Based on the monitor of (002) crystal orientation, intercalated multi-ions can regulate and control the interlayer environment of MXenes via stress, which induces lattice shrinkage occurring in the c axis. Limited by ion storage mechanism-performance, the multi-ion occupies the interspace of MXene and affects the electrochemical performance. This work would offer guidance to understand the relationship among the multi-ion and MXene by two-dimensional (2D) layered materials.     

Burning of single carbon particle approached by activation energy asymptotics

The method of activation energy asymptotics is used to treat the combustion of a single carbon particle in quiescent gas mixture with high temperature. Both heterogeneous reactions 2C+O₂ 2CO, C+CO₂ 2COand homogeneous reaction 2CO+O₂ 2CO₂ are considered. It is shown that the burning of the particle principally is carried out during a diffusion-limited period. Four brief and complex periods through which the history of the particle evolves from a heat-up period to the diffusion-limited period are described. A comparison between results of activation energy asymptotics and exact numerical solutions is given. The agreement is considered satisfactory.     

Effects of cerium incorporation on the catalytic oxidation of benzene over flame-made perovskite La1−xCexMnO3 catalysts

Perovskite-type La_1−xCe_xMnO₃ (x = 0–10%) catalysts were prepared by flame spray pyrolysis and their activities during the catalytic oxidation of benzene were examined over the temperature range of 100–450 °C. The structural properties and reducibility of these materials were also characterized by X-ray diffraction (XRD), N₂ adsorption/desorption, H₂ temperature-programmed reduction (H₂-TPR) and X-ray photoelectron spectroscopy (XPS). The incorporation of Ce was found to improve the benzene oxidation activity, and the perovskite in which x was 0.1 exhibited the highest activity. Phase composition and surface elemental analyses indicated that non-stoichiometric compounds were present. The incorporation of Ce had a negligible effect on the specific surface area of the perovskites and hence this factor has little impact on the catalytic activity. Introduction of Ce⁴⁺ resulted in modification of the chemical states of both B-site ions and oxygen species and facilitated the reducibility of the perovskite. The surface Mn⁴⁺/Mn³⁺ ratio was increased as a result of Ce⁴⁺ substitution, while a decrease in the surface-adsorbed O/lattice O (O_ads/O_latt) ratio was observed. The relationship between the surface elemental ratios and catalytic activity was established to allow a better understanding of the process by which benzene is oxidized over perovskites.     

Synthesis and characterization of BaMgAl10O17：Eu2＋ phosphor prepared by homogeneous precipitation

A homogeneous precipitation process based on urea hydrolysis reaction was exploited to synthesize BaMgAl10O17：Eu2＋ phosphor. The process parameters, such as the dosage of urea, the calcination tem- peratures and the concentration of Eu2＋, were refined in light of the characterization of the products. The experimental results revealed that pure and well-crystallized BaMgAl10O17：Eu2＋ phosphor could be obtained at 1250℃, a much lower temperature than that for traditional solid-state reaction. The as-prepared phosphor particles were small in grain size, regular in morphology, and uniform in size distribution. Because of the high homogeneity of the process, the as-prepared phosphor exhibited stronger emission intensity and higher thermal stability than the sample prepared by solid-state reaction at 1600℃.     

Nonlinear Taylor stability of viscoelastic fluids

Using Stuart's energy method, the torque on the inner cylinder, for a second order fluid, in the supercritical regime is calculated. It is found that when the second normal stress difference is negative, the flow is more stable than for a Newtonian fluid and the torque is reduced. If the second normal stress difference is positive, then the flow is more stable and there is no torque reduction. Experimental data related to the present work are discussed.Nomenclature a amplitude of the fundamentals - A _ij ⁽¹⁾ , A _ij ⁽²⁾ first and second Rivlin-Ericksen tensors - d r ₂–r ₁ - D d/dx - E - F - g _ij metric tensor - G torque on the inner cylinder in the supercritical regime - h height of the cylinders - k ₀ /d ² - k ₁ /d ² - I ₁ - I ₂ - I ₃ - I ₄ - r ₁, r ₂ radii of inner and outer cylinders respectively - r ₀ 1/2(r ₁+r ₂) - R Reynolds number ₁ r ₁ d/ ₀ - R _c critical Reynolds number - T Taylor number r _{1 1} ² d ^{3 2}/ ₀ ² *) - T _c critical Taylor number - u ₁, v ₁, w ₁ Fundamentals of the disturbance - u _i , v _i , w _i , (i>1) harmonics - mean velocity (not laminar velocity) - u –u ₁/ar _{1 1} - v v ₁/Rar _{1 1} - x (r–r ₀)/d - , material constants - 0 viscosity - wave number d - density - ₁ angular velocity of inner cylinder - tilde denotes complex conjugate     

Presentation of the creep curves of solid biological materials by a simplified mathematical version of the generalized Kelvin-Voigt model

Published creep equations, derived from the Kelvin-Voigt model with a discrete retardation time spectrum were used to generate data for linear presentation of the creep function. It is shown that 2–3 term creep functions, containing 4–6 constants, could be reduced to an algebraic expression having two constants only. Consequently, entire creep curves of a variety of biological and food materials could be described by a single type of equation in the form of:P (t) = k ₀ + k₁ t + t/(k₂ + k₃t) whereP (t) is the creep parameter (compliance, strain or deformation),k ₀ a constant representing the instantaneous compliance,k ₁ a constant representing the steady-state flow, andk ₂ andk ₃ the characteristic constants of the creep function (t) obtained by regression from the linear relationship oft/ (t) vst.     

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

About the singularities and bifurcations of double indices recursion sequences

This paper reports analytical and numerical investigations about the singularities and bifurcations of a new class of discrete dynamical systems defined generally by the following 2-D discrete systems: y _m+1,n+1=f(y _m,n ,y _m+1,n ,y _m,n+1); we consider particularly the case y_m+1,n+1 = f(y_m,n, y_m+1,n)y_{m+1,n+1} = f(y_{m,n}, y_{m+1,n}).     

The isotope effect of gaseous hydrogen under shock compression

The shock compression method has been used to measure the Hugoniot data and shock temperature for gaseous hydrogen samples, covering the pressure range of 55-140 MPa and the temperature range of 3400-4500 K and with the initial conditions of P ₀ = 0.6 MPa, 1.2 MPa and T ₀ at room temperature. Spectral radiance histories emitted from shocked D ₂ and H ₂ + D ₂ (equimolar mixture) are monitored by a pyrometer system with seven wavelength channels. Theoretical calculations based on the Saha model with Debye-Hückel correction for the shock compression behavior of shocked gaseous samples are in good agreement with the measured Hugoniot data, but show slightly higher values for the shock temperature when comparing with experiments. An isotope effect relevant to these shocked hydrogen species has been found in the linear shock velocity vs particle velocity relation, in which the correlation factor between these hydrogen isotopes or hydrogen mixtures is simply of initial density dependence.Received: 8 December 2002, Accepted: 8 May 2003, Published online: 2 September 2003PACS: 62.50 + p, 31.30.GS, 51.90. + r     

Exact solution and dynamic buckling analysis of a beam-column system having the elliptic type loading

This paper presents a closed form solution to the dynamic stability problem of a beam-column system with hinged ends loaded by an axial periodically time-varying compressive force of an elliptic type,i.e.,a1cn 2（τ,k 2）＋a2sn 2（τ,k 2）＋a3dn 2（τ,k 2）.The solution to the governing equation is obtained in the form of Fourier sine series.The resulting ordinary differential equation is solved analytically.Finding the exact analytical solutions to the dynamic buckling problems is difficult.However,the availability of exact solutions can provide adequate understanding for the physical characteristics of the system.In this study,the frequency-response characteristics of the system,the effects of the static load,the driving forces,and the frequency ratio on the critical buckling load are also investigated.