The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials

In this paper, the basic equations of motion, of Gauss and of heat conduction, together with constitutive relations for pyro- and piezoelectric media, are presented. Three thermoelastic theories are considered: classical dynamical coupled theory, the Lord–Shulman theory with one relaxation time and Green and Lindsay theory with two relaxation times. For incident elastic longitudinal, potential electric and thermal waves, referred to as qP, φ-mode and T-mode waves, which impinge upon the interface between two different transversal isotropic media, reflection and refraction coefficients are obtained by solving a set of linear algebraic equations. A case study is investigated: a system formed by two semi-infinite, hexagonal symmetric, pyroelectric–piezoelectric media, namely Cadmium Selenide (CdSe) and Barium Titanate (BaTiO₃). Numerical results for the reflection and refraction coefficients are obtained, and their behavior versus the incidence angle is analyzed. The interaction with the interface give rises to different kinds of reflected and refracted waves: (i) two reflected elastic waves in the first medium, one longitudinal (qP-wave) and the other transversal (qSV-wave), and a similar situation for the refracted waves in the second medium; (ii) two reflected potential electric waves and a similar situation for the refracted waves; (iii) two reflected thermal waves and a similar situation for the refracted waves. The amplitudes of the reflected and refracted waves are functions of the incident angle, of the thermal relaxation times and of the media elastic, electric, thermal constants. This study is relevant to signal processing, sound systems, wireless communications, surface acoustic wave devices and military defense equipment.     

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.     

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.     

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.     

Effect of the mode of gas-surface interaction on intensive monatomic gas evaporation

The direct Monte Carlo simulation method is used for investigating the effect of the thermal accommodation coefficient α _E on the relations on the Knudsen layer boundary in the presence of intensive evaporation. The model of mirror reflection of molecules from the surface is considered. It is shown that diffuse reflection with α _E = 0 leads to almost the same relations on the Knudsen layer boundary as mirror reflection. The accuracy of the moment method is estimated in application to the problems of intensive evaporation with diffuse and mirror reflection from the surface.     

Kinetic Fokker-Planck equation for free-molecular,thermally nonequilibrium Brownian particles in an inhomogeneous gas

A kinetic equation for the translational and angular velocity distribution function of spherical rigid Brownian particles in an inhomogeneous monatomic gas is derived. The particle diameters are much smaller than the average free path of the gas molecules and the interaction between the particles and their effect on the carrier (gas) phase are neglected. The particle temperatures T _p are the same and differ from the local gas temperature T. The molecular velocity distribution function is specified by the first approximation of the Chapman-Enskog method. The difference between the characteristic phase velocities is small as compared with the mean thermal molecular velocity. The dependences of the diffusion coefficients in velocity space on the ratio T _p/T, which characterize the effect of thermal nonequilibrium, i.e., violation of the thermodynamic equilibrium between the phases of the disperse system, are found using a specular-diffuse law of reflection of the molecules from the particle surface.     

Suspensions of titania nanoparticle networks in nematic liquid crystals: rheology and microstructure

We study the influence of confinement on the rheology and structure of nematic liquid crystals (NLCs). NLCs get confined in networks of titania (TiO₂, primary particle size = 21 nm) nanoparticles in suspensions of TiO₂ and NLC, N-(4-methoxybenzylidene)-4-butylaniline (MBBA). Suspensions with TiO₂ nanoparticle volume fraction (?) of 0.006–0.017, form viscoelastic solids with low elastic modulus (G′) of 10¹ Pa–10² Pa and short relaxation times. Increase in TiO₂ nanoparticle ? leads to a rise in G′ with TiO₂ nanoparticles forming a percolating network at a critical volume fraction (? _c) = 0.023, and G′ of ~10³ Pa. TiO₂/MBBA NLC suspensions at and above ? _c = 0.023 show G′ ~ ω ^x?1 scaling, where ω is the angular frequency and the minimum in loss modulus (G′′) with ω. The effective noise temperature, x decreases and approaches 1 with the increase in the TiO₂ nanoparticle ? from 0.023–0.035, is indicative of an increase in the glassy dynamics. Through the polarized light microscopy and differential scanning calorimetry experiments, we propose that the progressive addition of TiO₂ nanoparticles introduces a quenched random disorder (QRD) in the NLC medium which disturbs the nematic order. This results in metastable TiO₂/MBBA NLC suspensions in which NLC domains get confined in the network of flocs of TiO₂ nanoparticles. We also show that the salient rheological signatures of soft glassy rheology develop only in the presence of NLC MBBA and are absent in the isotropic phase of MBBA.     

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.     

Plane acoustic wave propagation through a composite of elastic and Kelvin–Voigt viscoelastic material layers

The problem of plane wave propagation through a plane composite layer of thickness h is considered. The composite consists of periodically repeated elastic and Kelvin–Voigt viscoelastic material layers, and all layers are either parallel or perpendicular to the incident wave front. Moreover, it is assumed that the thickness of each separate layer of the composite is much less than the acoustic wave length and the thickness h of the entire composite. We study the problem by using a homogenized model of the composite, which allows us to find the reflection and transmission factors and the variation in the sound intensity level as it propagates though the composite layer of thickness h.     

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.     

Effect of incomplete thermal accommodation on intensive subsonic gas condensation

The direct Monte Carlo simulation method is used for investigating the effect of the thermal accommodation coefficient α _E on the relation for the Knudsen layer in the presence of intensive subsonic condensation. It is shown that the deviation of α _E from unity may significantly affect the flow parameters, in particular, M_lim, the Mach number value limiting for subsonic condensation. It is shown that a decrease in α _E leads to an increase in M_lim (M_lim < 1) if the relative flow temperature (ratio of the outer Knudsen layer boundary to the surface temperature) T < 1 and to a decrease in M_lim if T > 1. It is shown that for mirror reflection of molecules from the surface this effect may intensify.     

The Oberbeck–Boussinesq approximation as a constitutive limit

We derive the usual Oberbeck–Boussinesq approximation as a constitutive limit of the full system describing the motion of an compressible linearly viscous fluid. To this end, the starting system is written, using the Gibbs free energy, in the variables v, θ and p. The Oberbeck–Boussinesq system is then obtained as the thermal expansion coefficient α and the isothermal compressibility coefficient β tend to zero.     

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.     

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.     

Experimental Study on Three-Dimensional Deformation Field of Portevin–Le Chatelier Effect Using Digital Image Correlation

In view of its high precision and high efficiency, three-dimensional digital image correlation (3D-DIC) is widely used to accurately measure full-field deformation. A spatiotemporal experimental study using 3D-DIC to explore the Portevin–Le Chatelier (PLC) deformational behavior, provides a new insight into the whole 3D deformation field, including the out-of-plane displacement, and in particular the relationship between the serrations and the strain field in the deformation bands corresponding to individual serrations. Specimens 1, 2 and 3 mm thick of 5456 Al-based alloy were tested in uniaxial tension at room temperature at strain rates from 1.8 × 10^?4 to 9.1 × 10^?3s^?1. The spatial and temporal characteristics of the strain localization were quantitatively analyzed. The out-of-plane displacement increment field (w) of the localized bands was observed by 3D-DIC, and found to be related to the specimen thickness and the in-plane strain increment. The largest displacement increments were respectively 15, 10 and 5 μm for 3, 2 and 1 mm specimens at maximum strain increment of about 12000 με. The elastic shrinkage outside the deformation bands was found to be an essential characteristic of the PLC effect. The width of the PLC band (w_band) increased with increasing thickness; the angle of the PLC band (??_band) was not affected by either specimen thickness or serration amplitude. Temporally, the serrations in the plots both of in-plane strain and out-of-plane displacement vs. time coincided throughout the entire loading procedure. When PLC banding occurred, the serration amplitude within the bands was found to be proportional to the maximum strain increment in the direction of the applied tensile force (??_max).     

Flow visualization of supersonic laminar flow over a backward-facing step via NPLS

An experimental study on a supersonic laminar flow over a backward-facing step of 5 mm height was undertaken in a low-noise indraft wind tunnel. To investigate the fine structures of Ma = 3.0 and 3.8 laminar flow over a backward-facing step, nanotracer planar laser scattering was adopted for flow visualization. Flow structures, including supersonic laminar boundary layer, separation, reattachment, redeveloping turbulent boundary layer, expansion wave fan and reattachment shock, were revealed in the transient flow fields. In the Ma = 3.0 BFS (backward-facing step) flow, by measuring four typical regions, it could be found that the emergence of weak shock waves was related to the K–H (Kelvin–Helmholtz) vortex which appeared in the free shear layer and that the convergence of these waves into a reattachment shock was distinct. Based on large numbers of measurements, the structure of time-averaging flow field could be gained. Reattachment occurred at the location downstream from the step, about 7–7.5 h distance. After reattachment, the recovery boundary layer developed into turbulence quickly and its thickness increased at an angle of 4.6°. At the location of X = 14h, the redeveloping boundary layer was about ten times thicker than its original thickness, but it still had not changed into fully developed turbulence. However, in the Ma = 3.8 flow, the emergence of weak shock waves could be seen seldom, due to the decrease of expansion. The reattachment point was thought to be near X = 15h according to the averaging result. The reattachment shock was not legible, which meant the expansion and compression effects were not intensive.     

On Properties of the Generalized Wasserstein Distance

The Wasserstein distances W_p (p \({\geqq}\) 1), defined in terms of a solution to the Monge–Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou–Brenier formula characterizes the square of the Wasserstein distance W₂ as the infimum of the kinetic energy, or action functional, of all vector fields transporting one measure to the other. Another important property of the Wasserstein distances is the Kantorovich–Rubinstein duality, stating the equality between the distance W₁(μ, ν) of two probability measures μ, ν and the supremum of the integrals in d(μ ?ν) of Lipschitz continuous functions with Lipschitz constant bounded by one. An intrinsic limitation of Wasserstein distances is the fact that they are defined only between measures having the same mass. To overcome such a limitation, we recently introduced the generalized Wasserstein distances \({W_p^{a,b}}\), defined in terms of both the classical Wasserstein distance W_p and the total variation (or L¹) distance, see (Piccoli and Rossi in Archive for Rational Mechanics and Analysis 211(1):335–358, 2014). Here p plays the same role as for the classic Wasserstein distance, while a and b are weights for the transport and the total variation term. In this paper we prove two important properties of the generalized Wasserstein distances: (1) a generalized Benamou–Brenier formula providing the equality between \({W_2^{a,b}}\) and the supremum of an action functional, which includes a transport term (kinetic energy) and a source term; (2) a duality à la Kantorovich–Rubinstein establishing the equality between \({W_1^{1,1}}\) and the flat metric.     

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.     

On some differences between dynamic- and static-stress distributions

Dynamic-stress concentrations due to elastic waves are analyzed by high-speed photoelasticity, and some differences between dynamic- and static-stress distributions are clarified. Specimens such as long struts with shoulders or notches are loaded dynamically by an elastic wave behind the front of a rectangular pulse with a plateau of constant stress ρ _cv (ρ=density,c=elasticwave velocity,v=tensile velocity) and of long duration. Dynamic isochromatic patterns caused in polyurethanerubber specimens are recorded with a “Himac 16H” high-speed camera (framing speed: 10,000 pictures/ sec). It is found that dynamic-stress-concentration factors, that is, maximum fringe order in passage of the wave front and the initial part of the plateau, divided by the fringe order corresponding to the height of the plateau, are about 0.5–0.6 times the static-stress-concentration factors, for the specimen dimensions considered. An approximate theory of dynamic-stress concentration factor is derived by considering notches as the discontinuities in the cross-sectional area of strut. This theoretical consideration correlates somewhat with the values of dynamic-stress-concentration factors obtained experimentally.