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.     

A framework for synthetic validation of 3D echocardiographic particle image velocimetry

Particle image velocimetry (PIV) has been significantly advanced since its conception in early 1990s. With the advancement of imaging modalities, applications of 2D PIV have far expanded into biology and medicine. One example is echocardiographic particle image velocimetry that is used for in vivo mapping of the flow inside the heart chambers with opaque boundaries. Velocimetry methods can help better understanding the biomechanical problems. The current trend is to develop three-dimensional velocimetry techniques that take advantage of modern medical imaging tools. This study provides a novel framework for validation of velocimetry methods that are inherently three dimensional such as but not limited to those acquired by 3D echocardiography machines. This framework creates 3D synthetic fields based on a known 3D velocity field \({\mathbf{V}}\) and a given 3D brightness field \({\mathbf{B}}\). The method begins with computing the inverse flow \({\mathbf{V}}^{\varvec{*}} \) based on the velocity field \({\mathbf{V}}\). Then the transformation of \({\mathbf{B}}\), imposed by \({\mathbf{V}}\), is calculated using the computed inverse flow according to \({\mathbf{B}}^{\varvec{*}} \left( {\mathbf{x}} \right) = {\mathbf{B}}\left( {{\mathbf{x}} + {\mathbf{V}}^{\varvec{*}} \left( {\mathbf{x}} \right)} \right)\), where x is the coordinates of voxels in \({\mathbf{B}}^{\varvec{*}} \), with a 3D weighted average interpolation, which provides high accuracy, low memory requirement, and low computational time. To check the validity of the framework, we generate pairs of 3D brightness fields by employing Hill’s spherical vortex velocity field. \({\mathbf{B}}\) and the generated \({\mathbf{B}}^{\varvec{*}} \) are then processed by our in-house 3D particle image velocimetry software to obtain the interrelated velocity field. The results indicates that the computed and imposed velocity fields are in agreement.     

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.     

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.     

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.     

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.     

The role of angular momentum in the laminar motion of viscous fluids

In laminar flow, viscous fluids must exert appropriate elastic shear stresses normal to the flow direction. This is a direct consequence of the balance of angular momentum. There is a limit, however, to the maximum elastic shear stress that a fluid can exert. This is the ultimate shear stress, \(\tau _\mathrm{y}\), of the fluid. If this limit is exceeded, laminar flow becomes dynamically incompatible. The ultimate shear stress of a fluid can be determined from experiments on plane Couette flow. For water at \(20\,^{\circ }\hbox {C}\), the data available in the literature indicate a value of \(\tau _\mathrm{y}\) of about \(14.4\times 10^{-3}\, \hbox {Pa}\). This study applies this value to determine the Reynolds numbers at which flowing water reaches its ultimate shear stress in the case of Taylor–Couette flow and circular pipe flow. The Reynolds numbers thus obtained turn out to be reasonably close to those corresponding to the onset of turbulence in the considered flows. This suggests a connection between the limit to laminar flow, on the one hand, and the occurrence of turbulence, on the other.     

Transitional and Turbulent Flow in a Bed of Spheres as Measured with Stereoscopic Particle Image Velocimetry

Stereoscopic particle image velocimetry has been used to investigate inertia dominated, transitional and turbulent flow in a randomly packed bed of monosized PMMA spheres. By using an index-matched fluid, the bed is optically transparent and measurements can be performed in an arbitrary position within the porous bed. The velocity field observations are carried out for particle Reynolds numbers, \({Re}_{{p}}\), between 20 and 3220, and the sampling is done at a frequency of 75 Hz. Results show that, in porous media, the dynamics of the flow can vary significantly from pore to pore. At \({Re}_{{p}}\) around 400 the spatially averaged time fluctuations of total velocity reach a maximum and the spatial variation of the time-averaged total velocity, \(u_\mathrm{tot}\) increases up to about the same \({Re}_{{p}}\) and then it decreases. Also in the studied planes, a considerable amount of the fluid moves in the perpendicular directions to the main flow direction and the time-averaged magnitude of the velocity in the main direction, \(u_{x}\), has an averaged minimum of 40% of the magnitude of \(u_\mathrm{tot}\) at \({Re}_{{p}}\) about 400. For \({Re}_{{p}} > 1600\), this ratio is nearly constant and \(u_{x}\) is on average a little bit less than 50% of \(u_\mathrm{tot}\). The importance of the results for longitudinal and transverse dispersion is discussed.     

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.     

New insights on fumed colloidal rheology—shear thickening and vorticity-aligned structures in flocculating dispersions

We investigate the rheology of dilute dispersions of fumed colloidal particles with attractive interactions in hydrocarbon liquids. Surprisingly, these systems display shear thickening due to the breakdown of densified flocs and a concomitant increase in the effective volume fraction of the fractal particles in the fluid. We show that this shear thickening is controlled by a critical stress and accompanied by a positive increase in the first normal stress difference, N ₁, at the shear thickening transition. This is in contrast to the well-known hydrocluster mechanism of shear thickening in concentrated hard-sphere and repulsive systems. Gel elasticity depends strongly on the stress applied to suspensions in preshear, scaling roughly as \(G'\sim\sigma_{\text{preshear}}^{2}\). We propose a simple model to account for these results in terms of the cluster number density determined by the preshear stress. At low shear rates, vorticity-aligned aggregates are present at \(\dot\gamma\approx 10^0 {\rm{s}}^{-1}\) . In this regime, the system displays a small but noticeable increase in viscosity on increasing shear rate. We investigate the effect of tool roughness and find that wall slip is not responsible for the observed phenomena. Instead, the increase in the apparent viscosity results from increased flow resistance due to the presence of gap-spanning log-like flocs in rolling flow.     

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.     

The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

In the paper [Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, JDDE 23 (2011), no. 4, 727–790], we have constructed large-amplitude periodic orbits for an equation with delayed monotone positive feedback. We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides spindle-like structures. In this paper we focus on a large-amplitude periodic orbit \({\mathcal {O}}_{p}\) with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \). We prove that \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) is a three-dimensional \(C^{1}\)-submanifold of the phase space and admits a smooth global graph representation. Within \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \), there exist heteroclinic connections from \({\mathcal {O}}_{p}\) to three different periodic orbits. These connecting sets are two-dimensional \(C^{1}\)-submanifolds of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) and homeomorphic to the two-dimensional open annulus. They form \(C^{1}\)-smooth separatrices in the sense that they divide the points of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) into three subsets according to their \(\omega \)-limit sets.     

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.     

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.     

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.     

Determination of Nanoparticle Macrotransport Coefficients from Pore Scale Processes

Nanoparticle transport in porous media is modeled using a hierarchical set of differential equations corresponding to pore scale and macroscale. At the pore scale, movement and interaction of a single particle with a solid matrix is modeled using the advection–dispersion–sorption equation. A single nanoparticle entering the space encounters viscous, diffusion and surface forces. Surface forces (electrostatic and van der Waals forces) between nanoparticles and mineral grains appear as sorption propensity on solid matrix boundary condition. These local events are then transformed into a macroscale continuum by imposing periodic boundary conditions for contiguous unit cells representing porous media and using a scheme of moment analysis. At the macroscale, propagation and retention of particles are characterized by three position-independent coefficients: mean nanoparticle velocity vector \({\bar{\mathbf{U}}}^*\), macroscopic dispersion coefficient \({\bar{\mathbf{D}}}^*\), and mean nanoparticle retention rate constant \({\bar{K}}^*\). The modeling results are validated with a set of nanoparticle transport tests in porous microchips. We also present simulations of realistic porous media, where an actual image of sandstone samples is processed into binary tones. The representative unit cells are constructed from the resulting binary image by searching for areas within the sample with maximum similarities to the whole sample in terms of porosity and specific surface area, which are found to show strong correlations with the resulting \({\bar{\mathbf{U}}}^*\) and \({\bar{K}}^*\), respectively.     

Density enhancement mechanism of upwind schemes for low Mach number flows

Many all-speed Roe schemes have been proposed to improve performance in terms of low speeds. Among them,the F-Roe and T-D-Roe schemes have been found to get incorrect density fluctuation in low Mach flows, which is expected to be with the square of Mach number. Asymptotic analysis presents the mechanism of how the density fluctuation problem relates to the incorrect order of terms in the energy equation UΔU. It is known that changing the upwind scheme coefficients of the pressure-difference dissipation term DPand the velocity-difference dissipation term in the momentum equation D~(ρU)to the order of O(c~(-1))and O(c~0) can improve the level of pressure and velocity accuracy at low speeds. This paper shows that corresponding changes in energy equation can also improve the density accuracy in low speeds. We apply this modification to a recently proposed scheme, TV-MAS, to get a new scheme,TV-MAS2. Unsteady Gresho vortex flow, double shear-layer flow, low Mach number flows over the inviscid cylinder, and NACA0012 airfoil show that energy equation modification in these schemes can obtain the expected square Ma scaling of density fluctuations, which is in good agreement with corresponding asymptotic analysis. Therefore, this density correction is expected to be widely implemented into allspeed compressible flow solvers.     

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.     

Uniqueness of Positive Ground State Solutions of the Logarithmic Schrödinger Equation

We prove the uniqueness of positive ground state solutions of the problem \({ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln(|u|) = 0}\), \({u(r) > 0~\forall r \ge 0}\), and \({(u(r),u'(r)) \to (0, 0)}\) as \({r \to \infty}\). This equation is derived from the logarithmic Schrödinger equation \({{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left(|u|^{2}\right)}\), and also from the classical equation \({{\frac {\partial u}{\partial t}} = {\Delta} u +u \left(|u|^{p-1}\right) -u}\). For each \({n \ge 1}\), a positive ground state solution is \({ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}\). We combine \({u_{0}(r)}\) with energy estimates and associated Ricatti equation estimates to prove that, for each \({n \in \left[1, 9 \right]}\), \({u_{0}(r)}\) is the only positive ground state. We also investigate the stability of \({u_{0}(r)}\). Several open problems are stated.