A Simple Method for the Determination of Adsorption Kinetic Parameters Using Recycle Fixed-Bed Adsorber

This study focuses on the development of a novel analysis technique for determining the intraparticle diffusivity \((D_{\mathrm{S}})\) and fluid film mass transfer coefficient \((k_\mathrm{F})\) from a concentration history curve of a recycle fixed-bed reactor without using the linear driving force approximation and empirical equations for the estimation of \(k_\mathrm{F}\). The recycle fixed-bed method requires lesser amounts of working fluid for experiment purposes, which is an advantage over the usual fixed-bed method. Based on the characterization results of the concentration history curves, simulated by rigorous numerical calculations, a novel analysis technique was established. The \(D_{\mathrm{S}}\) value can be determined from the experimentally obtained time at which the concentration of the curve is minimal \((t_{C\mathrm{min}})\). The \(k_\mathrm{F}\) value can be determined from the \(D_{\mathrm{S}}\) value and Biot number (Bi), which can be estimated from the experimental ratio of the maximum concentration to the minimum concentration \((c_{\max }/c_{\min })\). The \(D_{\mathrm{S}}\) and \(k_\mathrm{F}\) values of phenol adsorption on an activated carbon material were determined experimentally using the proposed analysis method. This method enables the determination of reliable adsorption kinetic parameters through a simple and economical experiment. However, appropriate experimental data must be acquired under regulated experimental conditions, especially in the case of fluctuation of the concentration history curve.     

In Situ Chemically-Selective Monitoring of Multiphase Displacement Processes in a Carbonate Rock Using 3D Magnetic Resonance Imaging

Accurate monitoring of multiphase displacement processes is essential for the development, validation and benchmarking of numerical models used for reservoir simulation and for asset characterization. Here we demonstrate the first application of a chemically-selective 3D magnetic resonance imaging (MRI) technique which provides high-temporal resolution, quantitative, spatially resolved information of oil and water saturations during a dynamic imbibition core flood experiment in an Estaillades carbonate rock. Firstly, the relative saturations of dodecane (\(S_{\mathrm{o}})\) and water (\(S_{\mathrm{w}})\), as determined from the MRI measurements, have been benchmarked against those obtained from nuclear magnetic resonance (NMR) spectroscopy and volumetric analysis of the core flood effluent. Excellent agreement between both the NMR and MRI determinations of \(S_{\mathrm{o}}\) and \(S_{\mathrm{w}}\) was obtained. These values were in agreement to 4 and 9% of the values determined by volumetric analysis, with absolute errors in the measurement of saturation determined by NMR and MRI being 0.04 or less over the range of relative saturations investigated. The chemically-selective 3D MRI method was subsequently applied to monitor the displacement of dodecane in the core plug sample by water under continuous flow conditions at an interstitial velocity of \(1.27\times 10^{-6}\,\hbox {m}\,\hbox {s}^{-1}\) (\(0.4\,\hbox {ft}\,\hbox {day}^{-1})\). During the core flood, independent images of water and oil distributions within the rock core plug at a spatial resolution of \(0.31\,\hbox {mm}\times 0.39\,\hbox {mm} \times 0.39\,\hbox {mm}\) were acquired on a timescale of 16 min per image. Using this technique the spatial and temporal dynamics of the displacement process have been monitored. This MRI technique will provide insights to structure–transport relationships associated with multiphase displacement processes in complex porous materials, such as those encountered in petrophysics research.     

Measurements of Transitional and Turbulent Flow in a Randomly Packed Bed of Spheres with Particle Image Velocimetry

Particle image velocimetry (PIV) has been used to investigate transitional and turbulent flow in a randomly packed bed of mono-sized transparent spheres at particle Reynolds number, \(20<{{ Re}}_{\mathrm{p}}< 3220\). The refractive index of the liquid is matched with the spheres to provide optical access to the flow within the bed without distortions. Integrated pressure drop data yield that Darcy law is valid at \({{ Re}}_{\mathrm{p}} \approx 80\). The PIV measurements show that the velocity fluctuations increase and that the time-averaged velocity distribution start to change at lower \({{ Re}}_{\mathrm{p}}\). The probability for relatively low and high velocities decreases with \({{ Re}}_{\mathrm{p}}\) and recirculation zones that appear in inertia dominated flows are suppressed by the turbulent flow at higher \({{ Re}}_{\mathrm{p}}\). Hence there is a maximum of recirculation at about \({{ Re}}_{\mathrm{p}} \approx 400\). Finally, statistical analysis of the spatial distribution of time-averaged velocities shows that the velocity distribution is clearly and weakly self-similar with respect to \({{ Re}}_{\mathrm{p}}\) for turbulent and laminar flow, respectively.     

Uniaxial experimental study of the acoustic emission and deformation behavior of composite rock based on 3D digital image correlation (DIC)

In this paper, uniaxial compression tests were carried out on a series of composite rock specimens with different dip angles, which were made from two types of rock-like material with different strength. The acoustic emission technique was used to monitor the acoustic signal characteristics of composite rock specimens during the entire loading process. At the same time, an optical non-contact 3 D digital image correlation technique was used to study the evolution of axial strain field and the maximal strain field before and after the peak strength at different stress levels during the loading process. The effect of bedding plane inclination on the deformation and strength during uniaxial loading was analyzed. The methods of solving the elastic constants of hard and weak rock were described. The damage evolution process, deformation and failure mechanism, and failure mode during uniaxial loading were fully determined. The experimental results show that the θ = 0?–45?specimens had obvious plastic deformation during loading, and the brittleness of the θ = 60?–90?specimens gradually increased during the loading process. When the anisotropic angle θincreased from 0?to 90?, the peak strength, peak strain,and apparent elastic modulus all decreased initially and then increased. The failure mode of the composite rock specimen during uniaxial loading can be divided into three categories:tensile fracture across the discontinuities(θ = 0?–30?), slid-ing failure along the discontinuities(θ = 45?–75?), and tensile-split along the discontinuities(θ = 90?). The axial strain of the weak and hard rock layers in the composite rock specimen during the loading process was significantly different from that of the θ = 0?–45?specimens and was almost the same as that of the θ = 60?–90?specimens. As for the strain localization highlighted in the maximum principal strain field, the θ = 0?–30?specimens appeared in the rock matrix approximately parallel to the loading direction,while in the θ = 45?–90?specimens it appeared at the hard and weak rock layer interface.     

Synchronization transitions induced by partial time delay in a excitatory–inhibitory coupled neuronal network

Information transmission delays are an inherent factor of neuronal systems as a consequence of the finite propagation speeds and time lapses occurring by both dendritic and synaptic processes. In real neuronal systems, some delay between two neurons is too small and can be ignored, which results in partial time delay. In this paper, we focus on investigating influences of partial time delay on synchronization transitions in a excitatory–inhibitory (E–I) coupled neuronal networks. Here, we suppose time delay between two neurons equals to \(\tau \) with probability \(p_{\mathrm{delay}}\) and investigate effect of partial time delay on synchronization transitions of the neuronal networks by controlling \(\tau \) and \(p_{\mathrm{delay}}\) under three cases. In these three cases, excitatory synapses are always considered to delayed with probability \(p_{\mathrm{delay}}\), while inhibitory synapses are considered to be without delays (case I), delayed with probability \(p_{\mathrm{delay}}\) (case II), and always delayed (case III), respectively. It is revealed that, in the first two cases, partial time delay has little influences on synchronization of the neuronal network for small \(p_{\mathrm{delay}}\), while it could induce synchronization transitions at \(\tau \) around integer multiples of the period of individual neuron T when \(p_{\mathrm{delay}}\) is large enough, while in the case III, partial time delay could induce synchronization transitions at \(\tau \) being around odd integer multiples of T / 2 for small \(p_{\mathrm{delay}}\) and at \(\tau \) being around integer multiples of T for large \(p_{\mathrm{delay}}\). Most interesting observation is that partial time delay could induce frequent synchronization transitions at \(\tau \) being around integer multiples of T / 2 for intermediate \(p_{\mathrm{delay}}\). Moreover, effect of rewiring probability on synchronization transitions induced by partial time delay has been discussed. It is found that synchronization transitions induced by partial time delay are robust to rewiring probability for large \(p_{\mathrm{delay}}\) under the three cases.     

Direct Numerical Simulation of Head-On Quenching of Statistically Planar Turbulent Premixed Methane-Air Flames Using a Detailed Chemical Mechanism

A three-dimensional compressible Direct Numerical Simulation (DNS) analysis has been carried out for head-on quenching of a statistically planar stoichiometric methane-air flame by an isothermal inert wall. A multi-step chemical mechanism for methane-air combustion is used for the purpose of detailed chemistry DNS. For head-on quenching of stoichiometric methane-air flames, the mass fractions of major reactant species such as methane and oxygen tend to vanish at the wall during flame quenching. The absence of \(\text {OH}\) at the wall gives rise to accumulation of carbon monoxide during flame quenching because \(\text {CO}\) cannot be oxidised anymore. Furthermore, it has been found that low-temperature reactions give rise to accumulation of \(\text {HO}_{2}\) and \(\mathrm {H}_{2}\mathrm {O}_{2}\) at the wall during flame quenching. Moreover, these low temperature reactions are responsible for non-zero heat release rate at the wall during flame-wall interaction. In order to perform an in-depth comparison between simple and detailed chemistry DNS results, a corresponding simulation has been carried out for the same turbulence parameters for a representative single-step Arrhenius type irreversible chemical mechanism. In the corresponding simple chemistry simulation, heat release rate vanishes once the flame reaches a threshold distance from the wall. The distributions of reaction progress variable c and non-dimensional temperature T are found to be identical to each other away from the wall for the simple chemistry simulation but this equality does not hold during head-on quenching. The inequality between c (defined based on \(\text {CH}_{4}\) mass fraction) and T holds both away from and close to the wall for the detailed chemistry simulation but it becomes particularly prominent in the near-wall region. The temporal evolutions of wall heat flux and wall Peclet number (i.e. normalised wall-normal distance of \(T = 0.9\) isosurface) for both simple and detailed chemistry laminar and turbulent cases have been found to be qualitatively similar. However, small differences have been observed in the numerical values of the maximum normalised wall heat flux magnitude \(\left ({\Phi }_{\max } \right )_{\mathrm {L}}\) and the minimum Peclet number \((Pe_{\min })_{\mathrm {L}}\) obtained from simple and detailed chemistry based laminar head-on quenching calculations. Detailed explanations have been provided for the observed differences in behaviours of \(\left ({\Phi }_{\max }\right )_{\mathrm {L}}\) and \((Pe_{\min })_{\mathrm {L}}\). The usual Flame Surface Density (FSD) and scalar dissipation rate (SDR) based reaction rate closures do not adequately predict the mean reaction rate of reaction progress variable in the near-wall region for both simple and detailed chemistry simulations. It has been found that recently proposed FSD and SDR based reaction rate closures based on a-priori DNS analysis of simple chemistry data perform satisfactorily also for the detailed chemistry case both away from and close to the wall without any adjustment to the model parameters.     

Dynamic Evolution of an Evaporating Liquid Meniscus from Structured Screen Meshes

Evaporation from screen mesh is a fundamental phenomenon that plays a vital role in thermal transport devices like heat pipes. Here, a laser scanning confocal microscopy setup has been utilized to study the effect of screen mesh wettability and its surface morphology on the evolution of evaporating liquid menisci from its liquid saturated pores. Stainless steel screen meshes (mesh #100 and #200) that are inherently hydrophobic in nature are turned super-hydrophilic by controlled heat treatment. The heat treatment leads to growth of oxide layer on the wire mesh and a change in surface morphology via formation of microscale pores, which improves the wettability of the screen mesh. The evaporation of liquid meniscus from pores of these untreated and heat-treated meshes is captured through confocal microscopy, and the dynamic evolution of the radius of curvature of the liquid meniscus is evaluated. A simple geometrical model is developed to predict the minimum radius of curvature \((R_{\mathrm{min}})\) of liquid meniscus in mesh pores just before its rupture (pore dryout). Meshes with high wettability, and smaller pore spacing-to-wire diameter ratio, are found to encounter a smaller \(R_{\mathrm{min}}\) before meniscus rupture. In addition, the water pore saturation inventory of the screen meshes are also measured to evaluate the effect of wettability on their water-holding capacity of screen meshes. Increase in screen mesh wettability is found to increase its pore saturation inventory. This increase in mesh pore saturation inventory coupled with the lower \(R_{\mathrm{min}}\) for super-hydrophilic mesh delays the liquid meniscus rupture (mesh dryout), leading to a much longer evaporation timescale for high wettability screen meshes.     

Three-Dimensional Lattice Boltzmann Simulations of Single-Phase Permeability in Random Fractal Porous Media with Rough Pore–Solid Interface

Single-phase permeability k has intensively been investigated over the past several decades by means of experiments, theories and simulations. Although the effect of surface roughness on fluid flow and permeability in single pores and fractures as well as networks of fractures was studied previously, its influence on permeability in a random mass fractal porous medium constructed of pores of different sizes remained as an open question. In this study, we, therefore, address the effect of pore–solid interface roughness on single-phase flow in random fractal porous media. For this purpose, we apply a mass fractal model to construct porous media with a priori known mass fractal dimensions \(2.579 \le D_{\mathrm{m}} \le 2.893\) characterizing both solid matrix and pore space. The pore–solid interface of the media is accordingly roughened using the Weierstrass–Mandelbrot approach and two parameters, i.e., surface fractal dimension \(D_{\mathrm{s}}\) and root-mean-square (rms) roughness height. A single-relaxation-time lattice Boltzmann method is applied to simulate single-phase permeability in the corresponding porous media. Results indicate that permeability decreases sharply with increasing \(D_{\mathrm{s}}\) from 1 to 1.1 regardless of \(D_{\mathrm{m}}\) value, while k may slightly increase or decrease, depending on \(D_{\mathrm{m}}\), as \(D_{\mathrm{s}}\) increases from 1.1 to 1.6.     

Bifurcations of a creeping air–water flow in a conical container

This numerical study describes the eddy emergence and transformations in a slow steady axisymmetric air–water flow, driven by a rotating top disk in a vertical conical container. As water height \(H_{\mathrm{w}}\) and cone half-angle \(\beta \) vary, numerous flow metamorphoses occur. They are investigated for \(\beta =30^{\circ }, 45^{\circ }\), and \(60^{\circ }\). For small \(H_{\mathrm{w}}\), the air flow is multi-cellular with clockwise meridional circulation near the disk. The air flow becomes one cellular as \(H_{\mathrm{w}}\) exceeds a threshold depending on \(\beta \). For all \(\beta \), the water flow has an unbounded number of eddies whose size and strength diminish as the cone apex is approached. As the water level becomes close to the disk, the outmost water eddy with clockwise meridional circulation expands, reaches the interface, and induces a thin layer with anticlockwise circulation in the air. Then this layer expands and occupies the entire air domain. The physical reasons for the flow transformations are provided. The results are of fundamental interest and can be relevant for aerial bioreactors.     

Physical Clogging of Uniformly Graded Porous Media Under Constant Flow rates

This study investigated the physical clogging of uniformly graded porous media under constant flow rates using natural porous media and suspensions. The porous media selected for this experimental study was a fine-to-medium sandy soil fractioned into thirteen uniformly graded beds: seven unisize beds and six uniform beds. The physical clogging of the beds was studied using two types of silt suspensions as along with two suspension concentrations and three water discharges. It was found that the permeability reduction due to physical clogging \([(K_\mathrm{i} - K_\mathrm{t})/K_\mathrm{i}]\) increased with decreasing \({D}_{15}/{d}_{85}\) ratios until a critical value of \({D}_{15}/{d}_{85}\), after which a surface mat of suspension was formed on the porous media. It was also found that the value of reduced permeability at any time (at any number of pore volumes of injected suspension-laden water), \(K_\mathrm{t}\), is directly proportional to square of \({D}_{15}\) and inversely proportional to \({C}_{\mathrm{u}}\) of the porous media and \({d}_{85}\) of suspensions. The effects of suspension type and flow rates on physical clogging seemed to depend on the size of the pores in the porous media.     

A Spectral Criterion for Stability of a Steady Viscous Incompressible Flow Past an Obstacle

We show that the question of stability of a steady incompressible Navier-Stokes flow \({\mathrm{V}}\) in a 3D exterior domain \({\Omega}\) is essentially a finite-dimensional problem (Theorem 3.2). Although the associated linearized operator has an essential spectrum touching the imaginary axis, we show that certain assumptions on the eigenvalues of this operator guarantee the stability of flow \({\mathrm{V}}\) (Theorem 4.1). No assumption on the smallness of the steady flow \({\mathrm{V}}\) is required.     

Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

Characterization for elastic constants of fused deposition modelling-fabricated materials based on the virtual fields method and digital image correlation

Fused deposition modelling (FDM), a widely used rapid prototyping process, is a promising technique in manufacturing engineering. In this work, a method for characterizing elastic constants of FDM-fabricated materials is proposed. First of all, according to the manufacturing process of FDM, orthotropic constitutive model is used to describe the mechanical behavior. Then the virtual fields method (VFM) is applied to characterize all the mechanical parameters \((Q_{11}\), \(Q_{22}\), \(Q_{12}\), \(Q_{66})\) using the full-field strain, which is measured by digital image correlation (DIC). Since the principal axis of the FDM-fabricated structure is sometimes unknown due to the complexity of the manufacturing process, a disk in diametrical compression is used as the load configuration so that the loading angle can be changed conveniently. To verify the feasibility of the proposed method, finite element method (FEM) simulation is conducted to obtain the strain field of the disk. The simulation results show that higher accuracy can be achieved when the loading angle is close to \(30^{\circ }\). Finally, a disk fabricated by FDM was used for the experiment. By rotating the disk, several tests with different loading angles were conducted. To determine the position of the principal axis in each test, two groups of parameters \((Q_{11}\), \(Q_{22}\), \(Q_{12}\), \(Q_{66})\) are calculated by two different groups of virtual fields. Then the corresponding loading angle can be determined by minimizing the deviation between two groups of the parameters. After that, the four constants \((Q_{11}\), \(Q_{22}\), \(Q_{12}\), \(Q_{66})\) were determined from the test with an angle of \(27^{\circ }\).     

Initial and Boundary Blow-U Problem for $$$$-Lalacian Parabolic Equation with General Absortion

In this article, we investigate the initial and boundary blow-up problem for the \(p\)-Laplacian parabolic equation \(u_t-\Delta _p u=-b(x,t)f(u)\) over a smooth bounded domain \(\Omega \) of \(\mathbb {R}^N\) with \(N\ge 2\), where \(\Delta _pu=\mathrm{div}(|\nabla u|^{p-2}\nabla u)\) with \(p>1\), and \(f(u)\) is a function of regular variation at infinity. We study the existence and uniqueness of positive solutions, and their asymptotic behaviors near the parabolic boundary.     

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

     

Inertial Sensitivity of Porous Microstructures

Fluid flows through porous media are subject to different regimes, ranging from linear creeping flows to unsteady, chaotic turbulence. These different flow regimes at the pore scale have repercussions at larger scales, with the macroscale drag force experienced by a fluid moving through the medium becoming a nonlinear function of the average velocity beyond the creeping flow regime. Accurate prediction of the transition between different flow regimes is an important challenge with repercussions onto many engineering applications. Here, we are interested in the first deviation from Darcy’s law, when inertia effects become sizeable. Our goal is to define a Reynolds number, \(Re_{\mathrm{C}}\), so that the inertial deviation occurs when \(Re_{\mathrm{C}}\sim 1\) for any microstructure. The difficulty in doing so is to reduce the multiple length scales characterizing the geometry of the porous structure to a single length scale, \(\ell \). We analyze the problem using the method of volume averaging and identify a length scale in the form \(\ell =C_\lambda \sqrt{\nicefrac {K_\lambda }{\epsilon _\beta }}\), with \(C_\lambda \) a parameter that indicates the sensitivity of the microstructure to inertia. The main advantage of this definition is that an explicit formula for \(C_\lambda \) is given; \(C_\lambda \) is computed from a creeping flow simulation in the porous medium; and \(Re_{\mathrm{C}}\) can be used to predict the transition to a non-Darcian regime more accurately than by using Reynolds numbers based on alternative length scales. The theory is validated numerically with data from flow simulations for a variety of microstructures.     

A new facility for studying shock-wave passage over dust layers

Dust explosion hazards in areas where coal and other flammable materials are found have caused unnecessary loss of life and halted business operations in some instances. The elimination of secondary dust explosion hazards, i.e., reducing dust dispersion, can be characterized in shock tubes to understand shock–dust interactions. For this reason, a new shock-tube test section was developed and integrated into an existing shock-tube facility. The test section has large windows to allow for the use of the shadowgraph technique to track dust-layer growth behind a passing normal shock wave, and it is designed to handle an initial pressure of 1 atm with an incident shock wave Mach number as high as 2 to mimic real-world conditions. The test section features an easily removable dust pan with inserts to allow for adjustment of the dust-layer thickness. The design also allows for changing the experimental variables such as initial pressure, shock Mach number \((M_{\mathrm{s}})\), dust-layer thickness, and the characteristics of the dust itself. The characterization experiments presented herein demonstrate the advantages of the authors’ test techniques toward providing new physical insights over a wider range of data than what have been available heretofore in the literature. Limestone dust with a layer thickness of 3.2 mm was subjected to \(M_{\mathrm{s}} = 1.23,\, 1.32\), and 1.6 shock waves, and dust-layer rise height was mapped with respect to time after shock passage. Dust particles subjected to a \(M_{\mathrm{s}} = 1.6\) shock wave rose more rapidly and to a greater height with respect to shock wave propagation than particles subjected to \(M_{\mathrm{s}} = 1.23\) and 1.32 shock waves. Although these results are in general agreement with the literature, the new data also highlight physical trends for dust-layer growth that have not been recorded previously, to the best of the authors’ knowledge. For example, the dust-layer height rises linearly until a certain time where the growth rate is dramatically reduced, and in this second regime there is clear evidence of surface vertical structures at the dust–air interface.     

Generic Hamiltonian Dynamics

In this paper we contribute to the generic theory of Hamiltonians by proving that there is a \(C^2\)-residual \({\mathcal {R}}\) in the set of \(C^2\) Hamiltonians on a closed symplectic manifold \(M\), such that, for any \(H\in {\mathcal {R}}\), there is a full measure subset of energies \(e\) in \(H(M)\) such that the Hamiltonian level \((H,e)\) is topologically mixing; moreover these level sets are homoclinic classes.     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).