The Nusselt number of a hot sphere levitated by a volatile pool

When placed at the surface of a volatile liquid, a sphere of hot dense non-volatile material remains suspended until it cools sufficiently. The duration of this ‘inverse Leidenfrost’ phenomenon depends on the Nusselt number $Nu$ of the sphere, itself determined by flow in the film of vapour separating particle and liquid. It is shown that provided the Nusselt number is large, it can be calculated numerically using only the Laplace relation and the equations governing the thin film; patching to a solution for the outer thick film is not necessary. This method is demonstrated by using it to determine $Nu$ for a sphere sufficiently small that in the governing equations, the acceleration due to gravity is negligible except where multiplied by the density of the sphere. Numerical results giving $Nu$ as a function of a dimensionless measure of sphere weight are supplemented with analysis showing that, when the weight is of the order of the maximum supportable by surface tension alone, the film consists of a spherical bubble cap bounded by its contact rim. The solutions for these regions are coupled: although the apparent contact angle $\chi$ for the cap is determined within the rim, its value depends on the flow rate arriving from the cap as well as on the additional evaporation from the rim. The latter acts to reduce $\chi$ from the value it would otherwise have, thereby reducing the thickness of the entire cap. For the example treated here, the value of $Nu$ is doubled by this mechanism

The relation of steady evaporating drops fed by an influx and freely evaporating drops

We discuss a thin film evolution equation for a wetting evaporating liquid on a smooth solid substrate. The model is valid for slowly evaporating small sessile droplets when thermal effects are insignificant, while wettability and capillarity play a major role. The model is first employed to study steady evaporating drops that are fed locally through the substrate. An asymptotic analysis focuses on the precursor film and the transition region towards the bulk drop and a numerical continuation of steady drops determines their fully non-linear profiles. Following this, we study the time evolution of freely evaporating drops without influx for several initial drop shapes. As a result we find that drops initially spread if their initial contact angle is larger than the apparent contact angle of large steady evaporating drops with influx. Otherwise they recede right from the beginning

Exploration of Shared Genetic Architecture Between Subcortical Brain Volumes and Anorexia Nervosa

In MRI scans of patients with anorexia nervosa (AN), reductions in brain volume are often apparent. However, it is unknown whether such brain abnormalities are influenced by genetic determinants that partially overlap with those underlying AN. Here, we used a battery of methods (LD score regression, genetic risk scores, sign test, SNP effect concordance analysis, and Mendelian randomization) to investigate the genetic covariation between subcortical brain volumes and risk for AN based on summary measures retrieved from genome-wide association studies of regional brain volumes (ENIGMA consortium, n = 13,170) and genetic risk for AN (PGC-ED consortium, n = 14,477). Genetic correlations ranged from − 0.10 to 0.23 (all p > 0.05). There were some signs of an inverse concordance between greater thalamus volume and risk for AN (permuted p = 0.009, 95% CI: [0.005, 0.017]). A genetic variant in the vicinity of ZW10, a gene involved in cell division, and neurotransmitter and immune system relevant genes, in particular DRD2, was significantly associated with AN only after conditioning on its association with caudate volume (pFDR = 0.025). Another genetic variant linked to LRRC4C, important in axonal and synaptic development, reached significance after conditioning on hippocampal volume (pFDR = 0.021). In this comprehensive set of analyses and based on the largest available sample sizes to date, there was weak evidence for associations between risk for AN and risk for abnormal subcortical brain volumes at a global level (that is, common variant genetic architecture), but suggestive evidence for effects of single genetic markers. Highly powered multimodal brain- and disorder-related genome-wide studies are needed to further dissect the shared genetic influences on brain structure and risk for AN

A new continuum model of the incoherent interface compared with growth of a spinel rim on an olivine grain

In a polymorphic change in which the phases differ only by a reversible difference in specific volume, kinematics requires a unit mass to suffer deviatoric strain in the instant it is transformed. Unlike the Eshelby stress–free strain, this strain is a property of the motion. Its existence must be considered when formulating the constitutive relation for the product of an incoherent transformation. To show this, two models are compared: in both, the (Nabarro) condition of vanishing shear stress is imposed at the incoherent interface; they differ only in the treatment of the deviatoric strain at issue. In the existing model, deviatoric stress within a unit mass of product is determined by total deviatoric strain from its initial state as parent phase. In the new model, lattice reconstruction is assumed to erase all memory within the unit mass of deviatoric strain suffered before, or during, its transformation. The existing model is not consistent with experiments on the olivine spinel–phase change in single crystals. It predicts that when the pressure applied exceeds a critical value, samples should transform completely at almost constant rate; instead, growth is seen to slow, and may even cease. The new model predicts this. Without adjustable constants, fair agreement is obtained with experiments on samples having 75–200 ppmw of water. Because elastic deformation by itself can explain those observations, the very thin rims seen on even drier samples suggest that water may be essential to lattice reconstruction in this phase change

The evaporating meniscus in a channel

We consider the evaporating meniscus of a perfectly wetting liquid in a channel whose superheated walls are at common temperature. Heat flows by pure conduction from the walls to the phase interface; there, evaporation induces a small-scale liquid flow concentrated near the contact lines. Liquid is continually fed to the channel, so that the interface is stationary, but distorted by the pressure differences caused by the small-scale flow. To determine the heat flow, we make a systematic analysis of this free-boundary problem in the limit of vanishing capillary number based on the velocity of the induced flow. Because surface tension is then large, the induced flow can distort the phase interface only in a small inner region near the contact lines; the effect is to create an apparent contact angle Θ depending on capillary number. Though, in general, there can be significant heat flow within that small inner region, the presence of an additional small parameter in the problem implies that, in practice, heat flow is significant only within the large outer region where the interface shape is determined by hydrostatics and Θ. We derive a formula for the heat flow, and show that the channel geometry affects the heat flow only through the value of the interface curvature at the contact line. Consequently, the heat flow relation for a channel can be applied to other geometries

The evaporating meniscus in a channel

We consider the evaporating meniscus of a perfectly wetting liquid in a channel whose superheated walls are at common temperature. Heat flows by pure conduction from the walls to the phase interface; there, evaporation induces a small-scale liquid flow concentrated near the contact lines. Liquid is continually fed to the channel, so that the interface is stationary, but distorted by the pressure differences caused by the small-scale flow. To determine the heat flow, we make a systematic analysis of this free-boundary problem in the limit of vanishing capillary number based on the velocity of the induced flow. Because surface tension is then large, the induced flow can distort the phase interface only in a small inner region near the contact lines; the effect is to create an apparent contact angle Θ depending on capillary number. Though, in general, there can be significant heat flow within that small inner region, the presence of an additional small parameter in the problem implies that, in practice, heat flow is significant only within the large outer region where the interface shape is determined by hydrostatics and Θ. We derive a formula for the heat flow, and show that the channel geometry affects the heat flow only through the value of the interface curvature at the contact line. Consequently, the heat flow relation for a channel can be applied to other geometries

On the contact region of a diffusion-limited evaporating drop: A local analysis

Motivated by experiments showing that a sessile drop of volatile perfectly wetting liquid initially advances over the substrate, but then reverses, we formulate the problem describing the contact region at reversal. Assuming a separation of scales, so that the radial extent of this region is small compared with the instantaneous radius a of the apparent contact line, we show that the time scale characterizing the contact region is small compared with that on which the bulk drop is evolving. As a result, the contact region is governed by a boundary-value problem, rather than an initial-value problem: the contact region has no memory, and all its properties are determined by conditions at the instant of reversal. We conclude that the apparent contact angle θ is a function of the instantaneous drop radius a, as found in the experiments. We then non-dimensionalize the boundary-value problem, and find that its solution depends on one parameter 葦, a dimensionless surface tension. According to this formulation, the apparent contact angle is well-defined: at the outer edge of the contact region, the film slope approaches a limit that is independent of the curvature of bulk drop. In this, it differs from the dynamic contact angle observed during spreading of non-volatile drops. Next, we analyse the boundary-value problem assuming 葦 to be small. Though, for arbitrary 葦, determining θ requires solving the steady diffusion equation for the vapour, there is, for small 葦, a further separation of scales within the contact region. As a result, θ is now determined by solving an ordinary differential equation. We predict that θ varies as a-1/6, as found experimentally for small drops (a < 1 mm). For these drops, predicted and measured angles agree to within 10-30 %. Because the discrepancy increases with a, but 葦 is a decreasing function of a, we infer that some process occurring outside the contact region is required to explain the observed behaviour of larger drops having a > 1 mm