Moment-based formulation of Navier–Maxwell slip boundary conditions for lattice Boltzmann simulations of rarefied flows in microchannels

We present an implementation of first-order Navier–Maxwell slip boundary conditions for simulating near-continuum rarefied flows in microchannels with the lattice Boltzmann method. Rather than imposing boundary conditions directly on the particle velocity distribution functions, following the existing discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory, we use a moment-based method to impose the Navier–Maxwell slip boundary conditions that relate the velocity and the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the\ud domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. The results are in excellent agreement with asymptotic solutions of the compressible Navier-Stokes equations for microchannel flows in the slip regime. Our moment formalism is also valuable for analysing the existing boundary conditions, and explains the origin of numerical slip in the bounce-back and other common boundary conditions that impose explicit conditions on the higher moments instead of on the local tangential velocity

A random projection method for sharp phase boundaries in lattice Boltzmann simulations

Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting

Roughness exponents and grain shapes

In surfaces with grainy features, the local roughness $w$ shows a crossover at a characteristic length $r_c$, with roughness exponent changing from $\alpha_1\approx 1$ to a smaller $\alpha_2$. The grain shape, the choice of $w$ or height-height correlation function (HHCF) $C$, and the procedure to calculate root mean-square averages are shown to have remarkable effects on $\alpha_1$. With grains of pyramidal shape, $\alpha_1$ can be as low as 0.71, which is much lower than the previous prediction 0.85 for rounded grains. The same crossover is observed in the HHCF, but with initial exponent $\chi_1\approx 0.5$ for flat grains, while for some conical grains it may increase to $\chi_1\approx 0.7$. The universality class of the growth process determines the exponents $\alpha_2=\chi_2$ after the crossover, but has no effect on the initial exponents $\alpha_1$ and $\chi_1$, supporting the geometric interpretation of their values. For all grain shapes and different definitions of surface roughness or HHCF, we still observe that the crossover length $r_c$ is an accurate estimate of the grain size. The exponents obtained in several recent experimental works on different materials are explained by those models, with some surface images qualitatively similar to our model films.Comment: 7 pages, 6 figures and 2 table

Phases of granular segregation in a binary mixture

We present results from an extensive experimental investigation into granular segregation of a shallow binary mixture in which particles are driven by frictional interactions with the surface of a vibrating horizontal tray. Three distinct phases of the mixture are established viz; binary gas (unsegregated), segregation liquid and segregation crystal. Their ranges of existence are mapped out as a function of the system's primary control parameters using a number of measures based on Voronoi tessellation. We study the associated transitions and show that segregation can be suppressed is the total filling fraction of the granular layer, $C$, is decreased below a critical value, $C_{c}$, or if the dimensionless acceleration of the driving, $\gamma$, is increased above a value $\gamma_{c}$.Comment: 12 pages, 12 figures, submitted to Phys. Rev.

A Rapidly Spinning Black Hole Powers the Einstein Cross

Observations over the past 20 years have revealed a strong relationship between the properties of the supermassive black hole (SMBH) lying at the center of a galaxy and the host galaxy itself. The magnitude of the spin of the black hole will play a key role in determining the nature of this relationship. To date, direct estimates of black hole spin have been restricted to the local Universe. Herein, we present the results of an analysis of $\sim$ 0.5 Ms of archival Chandra observations of the gravitationally lensed quasar Q 2237+305 (aka the "Einstein-cross"), lying at a redshift of z = 1.695. The boost in flux provided by the gravitational lens allows constraints to be placed on the spin of a black hole at such high redshift for the first time. Utilizing state of the art relativistic disk reflection models, the black hole is found to have a spin of $a_* = 0.74^{+0.06}_{-0.03}$ at the 90% confidence level. Placing a lower limit on the spin, we find $a_* \geq 0.65$ (4$\sigma$). The high value of the spin for the $\rm \sim 10^9~M_{\odot}$ black hole in Q 2237+305 lends further support to the coherent accretion scenario for black hole growth. This is the most distant black hole for which the spin has been directly constrained to date.Comment: 5 pages, 3 figures, 1 table, formatted using emulateapj.cls. Accepted for publication in ApJ

Finite-size effects in roughness distribution scaling

We study numerically finite-size corrections in scaling relations for roughness distributions of various interface growth models. The most common relation, which considers the average roughness $$as scaling factor, is not obeyed in the steady states of a group of ballistic-like models in 2+1 dimensions, even when very large system sizes are considered. On the other hand, good collapse of the same data is obtained with a scaling relation that involves the root mean square fluctuation of the roughness, which can be explained by finite-size effects on second moments of the scaling functions. We also obtain data collapse with an alternative scaling relation that accounts for the effect of the intrinsic width, which is a constant correction term previously proposed for the scaling of$$. This illustrates how finite-size corrections can be obtained from roughness distributions scaling. However, we discard the usual interpretation that the intrinsic width is a consequence of high surface steps by analyzing data of restricted solid-on-solid models with various maximal height differences between neighboring columns. We also observe that large finite-size corrections in the roughness distributions are usually accompanied by huge corrections in height distributions and average local slopes, as well as in estimates of scaling exponents. The molecular-beam epitaxy model of Das Sarma and Tamborenea in 1+1 dimensions is a case example in which none of the proposed scaling relations works properly, while the other measured quantities do not converge to the expected asymptotic values. Thus, although roughness distributions are clearly better than other quantities to determine the universality class of a growing system, it is not the final solution for this task.Comment: 25 pages, including 9 figures and 1 tabl

Activity of water in aqueous systems; A frequently neglected property

In this critical review, the significance of the term ‘activity’ is examined in the context of the properties of aqueous solutions. The dependence of the activity of water(ℓ) at ambient pressure and 298.15 K on solute molality is examined for aqueous solutions containing neutral solutes, mixtures of neutral solutes and salts. Addition of a solute to water(ℓ) always lowers its thermodynamic activity. For some solutes the stabilisation of water(ℓ) is less than and for others more than in the case where the thermodynamic properties of the aqueous solution are ideal. In one approach this pattern is accounted for in terms of hydrate formation. Alternatively the pattern is analysed in terms of the dependence of practical osmotic coefficients on the composition of the aqueous solution and then in terms of solute–solute interactions. For salt solutions the dependence of the activity of water on salt molalities is compared with that predicted by the Debye–Hückel limiting law. The analysis is extended to consideration of the activities of water in binary aqueous mixtures. The dependence on mole fraction composition of the activity of water in binary aqueous mixtures is examined. Different experimental methods for determining the activity of water in aqueous solutions are critically reviewed. The role of water activity is noted in a biochemical context, with reference to the quality, stability and safety of food and finally with regard to health science.