Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity

In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a quasi-longitudinal Primary (qP) wave and a quasi-transverse Secondary (qS) wave; and (2) a mainly thermal (qT) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 1: Linear Stability Analysis

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

Families of rational solutions of the <Emphasis Type="Italic">y</Emphasis>-nonlocal Davey–Stewartson II equation

The y-nonlocal Davey–Stewartson II equation is an extension of the usual DS II equation involving a partially parity-time-symmetric potential only with respect to the spatial variable y. By using the Hirota bilinear method, families of n-order rational solutions are obtained, which include lumps in the (x, y)-plane and the (y, t)-plane, growing-and-decaying line waves in the (x, t)-plane, and hybrid solutions of interacting line rogue waves and lumps in the (x, y)-plane.     

Surface waves under constrained deformation

The possibility of the existence of surface waves in a range of velocities greater than the velocity of transverse waves, but smaller than the velocity of longitudinal waves is shown. It turns out that, in the boundary value problem for an elastic half-space in this velocity range, there are the surface waves whose velocity is constant and equal to \(\sqrt 2 \) b, where b is the velocity of transverse waves. These waves as well as the Rayleigh surface waves have no dispersion. Their velocity is specified only by the elastic constants and density of the medium. It is also shown that the existence of such a velocity is possibly related to the velocity of surface waves that appear as unloading waves under constrained deformation.     

Wall slip and multi-tier yielding in capillary suspensions

Impact of wall slip on the yield stress measurement is examined for capillary suspensions consisting of cocoa powder as the dispersed phase, vegetable oil as the continuous primary fluid, and water as the secondary fluid using smooth and serrated parallel plates. Using dynamic oscillatory measurements, we investigated the yielding behavior of this ternary solid-fluid-fluid system with varying particle volume fraction, ?, from 0.45 to 0.65 and varying water volume fraction, ?_w, from 0.02 to 0.08. Yield stress is defined as the maximum in the elastic stress (G^′γ), which is obtained by plotting the product of elastic modulus (G^′) and strain amplitude (γ) as a function of applied strain amplitude. With serrated plates, which offer minimal slippage, capillary suspensions with ? ≥?0.45 and a fixed ?_w =?0.06 showed a two-step yielding behavior as indicated by two peaks in the plots of elastic stress as a function of strain amplitude. On the other hand with smooth plates, the capillary suspensions showed strong evidence of wall slip as evident by the presence of three distinct peaks and lowered first yield stresses for all ? and ?_w. These results can be interpreted based on the fact that a particle-depleted layer, which is known to be responsible for slip, is present in the vicinity of the smooth surfaces. The slip layer presents itself as an additional “pseudo-microstructure” (characteristic length scale) besides the two microstructures, aqueous bridges and solid particle agglomerates, that may occur in the system. With serrated plates, both the yield stresses (σ₁σ₂) and storage moduli plateau at lower strain (before the first yield point) and at higher strain (before the second yield point) (G\(^{\prime }_{p1}\), G\(^{\prime }_{p2}\)) were found to increase with ? (at a fixed ?_w =?0.06) following power-law dependences. Similarly with increasing ?_w (0.02 – 0.08) at a fixed ? =?0.62, the system behaved as a solid-like material in a jammed state with particles strongly held together as manifested by rapidly increasing σ₁ and σ₂. The usage of smooth surfaces primarily affected σ₁ which was reflected by an approximately 70–90% decrement in the measured σ₁ for all values of ?. By contrast, σ₂ and G\(^{\prime }_{p2}\) were found to be unaffected as shown by close agreement of values obtained using serrated geometry due to vanishing slip layers at higher strain amplitudes.     

Propagation of plane waves in thermoelastic cubic crystal material with two relaxation times

A problem concerned with the reflection and refraction of thermoelastic plane waves at an imperfect interface between two generalized thermally conducting cubic crystal solid half-spaces of different elastic and thermal properties with two relaxation times has been investigated. The generalized thermoelastic theory with two relaxation times developed by Green and Lindsay has been used to study the problem. The expressions for the reflection and refraction coefficients which are the ratios of the amplitudes of reflected and refracted waves to the amplitude of incident waves are obtained for an imperfect boundary and deduced for normal stiffness, transverse stiffness, thermal contact conductance, slip and welded boundaries. Amplitude ratios of different reflected and refracted waves for different boundaries with angle of emergence have been compared graphically for different incident waves. It is observed that the amplitude ratios of reflected and refracted waves are affected by the stiffness and thermal properties of the media.     

Dual yielding in capillary suspensions

Rheological measurements were performed to examine the yielding behavior of capillary suspensions prepared by mixing cocoa powder as dispersed phase, vegetable oil as the continuous primary fluid, and water as the secondary fluid. Here, we investigated the yielding behavior of solid-fluid-fluid systems with varying particle volume fraction, ?, spanning the regime from a low volume fraction (? = 0.25) to a highly filled regime (? = 0.65) using dynamic oscillatory measurements. While for ? ≤ 0.4 with a fixed water volume fraction (? _w ) of 0.06 as the secondary fluid, capillary suspensions exhibited a single yield point due to rupturing of aqueous capillary bridges between the particles, while capillary suspensions with ? ≥ 0.45 showed a two-step yielding behavior. On plotting elastic stress (G^′ γ) as a function of applied strain (γ), two distinct peaks, indicating two yield stresses, were observed. Both the yield stresses and storage modulus at low strains were found to increase with ? following a power law dependence. With increasing ? _w (0 – 0.08) at a fixed ? = 0.65, the system shifted to a frustrated, jammed state with particles strongly held together shown by rapidly increasing first and second yield stresses. In particular, the first yield stress was found to increase with ? _w following a power law dependence, while the second yield stress was found to increase exponentially with ? _w . Transient steady shear tests were also performed. The single stress overshoot for ? ≤ 0.4 with ? _w = 0.06 reflected one-step yielding behavior. In contrast, for high ? (≥ 0.45) values with ? _w = 0.06, two stress overshoots were observed in agreement with the two-step yielding behavior shown in the dynamic oscillatory measurements. Experiments on the effect of resting time on microstructure recovery demonstrated that aggregates could reform after resting under quiescent conditions.     

A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t _f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p₁ >0) and the duration t _f does not exceed 2, i.e., p₁t _f <2. The proposed criterion (p₁t _f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p₁t _f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.     

On $$\varvec{}$$-mixed-type soliton propagation in dispersive nonautonomous long waves with waveguides

In this paper N-soliton propagations for the Calogero–Bogoyavlenskii–Schiff (CBS) equation in an inhomogeneous media which describes the long nonautonomous waves are obtained. Here attention is focused to study the effect of the dispersion coefficient on the propagation solitons waves. It is found that N-bright-dark solitons are produced by periodic or coupled periodic and pulses waves. Solitons waves are propagated for two and three pulses with periodic oscillating. Further, the double-periodic and solitary waves are dispersive to broken-solitons waves for the graded-index with oscillating reflection components. These results are useful for the application for long-distance telecommunication and optical fiber.     

Application of scaling transformation to characterizing complex rheological behaviors and fractal derivative modeling

It has been long observed that cumbersome parameters are required for the traditional viscoelastic models to describe complex rheological behaviors. Inspired by the relationship between normal and anomalous diffusions, this paper tentatively employs t ^α to replace t, called as the scaling transformation, in the traditional creep compliance and relaxation modulus. With this methodology, the relaxation modulus is found to agree with the well-known Kohlrausch-Williams-Watts (KWW) stretched exponential function. The fitting results confirm that the proposed models accurately characterize rheological behaviors only with one more parameter α. Moreover, it is noted that the present formulations are directly related to the fractal derivative viscoelastic models and the index α is actually the order of the fractal derivative.     

Growths of mechanical elasticity and electrical conductance of graphene nanoplatelet/poly(lactic acid) composites under strong electric field: correlation with time evolution of higher order structure of graphene nanoplatelets

For a composite of poly(lactic acid) containing graphene nanoplatelets (GNPs) at a low concentration (0.34 vol%), this study examined growths of mechanical and electrical properties under an alternating current (AC) electric field, focusing on field-induced GNP structures governing those properties. The composite was subjected to the AC field (60 Hz) of various intensities E for various times t _E at 190 °C. A fraction of randomly oriented GNPs was aligned by the field and then connected into columns, as suggested from optical microscopy. This structural evolution led to qualitatively similar growths of low-frequency storage modulus and static electrical conductivity. The key quantity for understanding this growth was a time t _E* for occurrence of short circuit that detected formation of GNP columns conductively bridging the electrodes. The growths of both modulus and conductivity for various E were summarized as functions of a reduced variable, t _E/t _E*, confirming the growths commonly reflected the evolution of the GNP columns. However, the modulus grew fast and leveled off by t _E/t _E* ~ 1, whereas the conductivity kept growing gradually even at t _E/t _E* > 1. This difference was discussed in relation to the matrix chains and leftover GNPs out the column.     

Effect of tube diameter and capillary number on platelet margination and near-wall dynamics

I investigate the effect of tube diameter D and red blood cell capillary number Ca (i.e. the ratio of viscous to elastic forces) on platelet margination in blood flow at ≈37 % tube haematocrit. The system is modelled as three-dimensional suspension of deformable red blood cells and nearly rigid platelets using a combination of the lattice-Boltzmann, immersed boundary and finite element methods. Results of simulations during the dynamics before the steady state has been reached show that a non-diffusive radial platelet transport facilitates margination. This non-diffusive effect is important near the edge of the cell-free layer, but only for Ca > 0.2, when red blood cells are tank-treading. I also show that platelet trapping in the cell-free layer is reversible for Ca ≤ 0.2. Margination is essentially independent of Ca only for the smallest investigated tube diameter (D = 10 μm). Once platelets have reached the cell-free layer, they tend to slide rather than tumble. The tumbling rate is essentially independent of Ca but increases with D. Strong confinement suppresses tumbling due to the relatively small cell-free layer thickness at ≈ 37 % tube haematocrit.     

Transverse wave at a plane interface in thermo-elastic materials with voids

The problem of reflection and refraction of elastic waves for an incident transverse wave at a plane interface between two dissimilar half-spaces of thermo-elastic materials with voids has been investigated. Using the theory developed by Iesan (Acta Mech 60:67–89, 1986), the formulae corresponding to the amplitude and energy ratios of reflected and refracted elastic waves have been obtained. The results similar to Singh and Tomar (Mech Materials 39:932–940, 2007) are recovered from the present analysis. The amplitude and energy ratios are computed numerically for a particular model.     

Simultaneous impacts of MHD and variable wall temperature on transient mixed Casson nanofluid flow in the stagnation point of rotating sphere

A numerical analysis is provided to scrutinize time-dependent magnetohydrodynamics(MHD) free and forced convection of an electrically conducting non-Newtonian Casson nanofluid flow in the forward stagnation point region of an impulsively rotating sphere with variable wall temperature. A single-phase flow of nanofluid model is reflected with a number of experimental formulae for both effective viscosity and thermal conductivity of nanofluid. Exceedingly nonlinear governing partial differential equations(PDEs)subject to their compatible boundary conditions are mutated into a system of nonlinear ordinary differential equations(ODEs). The derived nonlinear system is solved numerically with implementation of an implicit finite difference procedure merging with a technique of quasi-linearization. The controlled parameter impacts are clarified by a parametric study of the entire flow regime. It is depicted that from all the exhibited nanoparticles,Cu possesses the best convection. The surface heat transfer and surface shear stresses in the x-and z-directions are boosted with maximizing the values of nanoparticle solid volume fraction ? and rotation λ. Besides, as both the surface temperature exponent n and the Casson parameter γ upgrade, an enhancement of the Nusselt number is given.     

Slow gas motions and the negative drag of a strongly heated spherical particle

In the slow flows of a strongly and nonuniformly heated gas, in the continuum regime (Kn → 0) thermal stresses may be present. The theory of slow nonisothermal continuum gas flows with account for thermal stresses was developed in 1969–1974. The action of the thermal stresses on the gas results in certain paradoxical effects, including the reversal of the direction of the force exerted on a spherical particle in Stokes flow. The propulsion force effect is manifested at large but finite temperature differences between the particle and the gas. This study is devoted to the thermal-stress effect on the drag of a strongly heated spherical particle traveling slowly in a gas for small Knudsen numbers (M ～ Kn → 0), small but finite Reynolds numbers (Re ≤ 1), a linear temperature dependence of the transport coefficients µ ∝ T, and large but finite temperature differences ((T _w ? T _∞ )/T _M8 ～ 1). Two different systems of equations are solved numerically: the simplified Navier-Stokes equations and the modified Navier-Stokes equations with account for the thermal stresses.     

Operator method for solving the plane elasticity problem for a strip with periodic cuts

In the present paper, we use the conformal mapping z/c = ζ?2a sin ζ (a, c?const, ζ = u + iv) of the strip {|v| ≤ v ₀, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = C ψ ₀ + S ψ*₀ + x(C ψ ₁ ? S ψ ₂) + y(C ψ ₂ + S ψ ₁), where C(v) and S(v) are the operator functions described in [1] and ψ ₀(u), …, ψ ₂(u) are the desired functions. The boundary conditions for the function F posed for v = ±v ₀ are equivalent to two operator equations for ψ ₁(u) and ψ ₂(u) and to two ordinary differential equations of first order for ψ ₀(u) and ψ*₀(u) [2]. By finding the functions ψ _j (u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v ₀ = 1) loaded on the boundary v = v ₀ by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X ^∞ and to the transverse and pure bending of the strip due to the transverse force Y ^∞ and the constant moment M ^∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.     

The stress state of a plate subjected to a piecewise homogeneous load

We consider the stress-strain state of a plate having a doubly connected domain S bounded from the outside by a circle of radius R and from the inside by an ellipse with two rectilinear cuts. The cuts lie symmetrically on the x-axis. The plate is subjected to various forces: the hole contour (the ellipse) is under the action of uniformly distributed forces of intensity q, and the cut shores are free of loads; at the points ±ib of the imaginary axis, the plate is under the action of a lumped force P.The solution of the problem is reduced to determining two analytic functions φ(z) and ψ(z) satisfying certain boundary conditions (depending on the type of the acting loads).We use the Kolosov-Muskhelishvili method to reduce the problem to a system of linear algebraic equations for the coefficients in the expansions of the functions φ(z) and ψ(z). The solution thus obtained is illustrated by numerical examples.     

On the orders of the best approximations of integrals of functions by integrals of rank <Emphasis Type="Italic">σ</Emphasis>

We study the values e _σ(f) of the best approximation of integrals of functions from the spaces L _p (A, dμ) by integrals of rank σ. We determine the orders of the least upper bounds of these values as σ → ∞ in the case where the function ? is the product of two nonnegative functions one of which is fixed and the other varies on the unit ball U _p (A) of the space L _p (A, dμ). We consider applications of the obtained results to approximation problems in the spaces S _p ^? .     

Stability of Traveling Waves of Nonlinear Schrödinger Equation with Nonzero Condition at Infinity

We study effective elastic behavior of the incompatibly prestrained thin plates, where the prestrain is independent of thickness and uniform through the plate’s thickness h. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric G, and seek the limiting behavior as \({h \to 0}\). We first establish that when the energy per volume scales as the second power of h, the resulting \({\Gamma}\) -limit is a Kirchhoff-type bending theory. We then show the somewhat surprising result that there exist non-immersible metrics G for whom the infimum energy (per volume) scales smaller than h². This implies that the minimizing sequence of deformations carries nontrivial residual three-dimensional energy but it has zero bending energy as seen from the limit Kirchhoff theory perspective. Another implication is that other asymptotic scenarios are valid in appropriate smaller scaling regimes of energy. We characterize the metrics G with the above property, showing that the zero bending energy in the Kirchhoff limit occurs if and only if the Riemann curvatures R₁₂₁₃, R₁₂₂₃ and R₁₂₁₂ of G vanish identically. We illustrate our findings with examples; of particular interest is an example where \({G_{2 \times 2}}\), the two-dimensional restriction of G, is flat but the plate still exhibits the energy scaling of the Föppl–von Kármán type. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for \({G = Id_{3} + \gamma n \otimes n}\) given in terms of the inhomogeneous unit director field distribution \({ n \in \mathbb{R}^3}\).     

Re-examination of terminal relaxation behavior of high-molecular-weight ring polystyrene melts

For high-molecular-weight (M) ring polymers with low contamination of linear chains, recent viscoelastic tests revealed broad terminal relaxation associated with no clear entanglement plateau. This relaxation behavior is qualitatively similar to that deduced from molecular models (double-folded lattice-animal model and the fractal loopy globule model) for entangled ring polymers, but quantitatively important differences are also noted: For example, the full terminal relaxation of those polymers is slower than the model prediction. This study re-examined the viscoelastic data of entangled high-M ring polystyrene (PS) samples (coded as R-240; M = 244×10³) specifically for two points: the purity of the ring samples after the viscoelastic tests and the molecular origin of the stress. For the first point, the R-240 samples contaminated with linear chains at low but different levels were prepared by tuning either the purification efficiency or the retention time of the sample at high temperature (T) before/during the viscoelastic test. The fraction w _L of the linear contaminant, determined after the viscoelastic measurement, was ranging from 0.7 to 4.9%, and the extrapolation of the modulus data to w _L = 0 gave the data for the ideally pure ring melt. This pure ring melt exhibited broad terminal relaxation that started faster but completed slower compared to the model prediction, indicating that the ring relaxation is not well described by the current model(s) even in the absence of linear contaminant. For the second point, dynamic birefringence measurements were conducted for the R-240 samples with w _L = 4.6 and 1.0%. These samples obeyed the stress-optical rule, and their stress-optical coefficient was indistinguishable from that for linear PS samples, revealing that the stress of the ring PS chains reflects the orientational anisotropy of the chains (as is the case also for linear chains). The relaxation behavior of pure ring PS melt is discussed on the basis of these findings, with the focus being placed on the ring-ring threading not considered in the models.