There are several models for the effective thermal conductivity of two-phase composite materials in terms of the conductivity of the solid and the disperse material. In this paper, we generalise three models of Maxwell type (namely, the classical Maxwell model and two generalisations of it obtained from effective medium theory and differential effective medium theory) so that the resulting effective thermal conductivity accounts for radiative heat transfer within gas voids. In the high-temperature regime, radiative transfer within voids strongly influences the thermal conductivity of the bulk material. Indeed, the utility of these models over classical Maxwell-type models is seen in the high-temperature regime, where they predict that the effective thermal conductivity of the composite material levels off to a constant value (as a function of temperature) at very high temperatures, provided that the material is not too porous, in agreement with experiments. This behaviour is in contrast to models which neglect radiative transfer within the pores, or lumped parameter models, as such models do not resolve the radiative transfer independently from other physical phenomena. Our results may be of particular use for industrial and scientific applications involving heat transfer within porous composite materials taking place in the high-temperature regime.Common models for calculating the effective thermal conductivity of a solid composite material include the Maxwell model (based on the pioneering work of Maxwell [3], where an expression for the effective thermal conductivity was derived via far-field perturbations to solutions of the steady heat equation), as well as variants thereof, including the effective medium theory (EMT) model and the differential effective medium theory (DEMT) model. EMT uses a similar approach to that used in deriving the Maxwell model, and can be applied to many other physical properties (see [4], for instance, for the electrical resistance problem). In [5,6], there is a comparison between the Maxwell model and EMT, and those works outline how certain bounds on the thermal conductivities can be obtained. The multipole expansion method [7] gives similar results. In applying the DEMT, one incrementally adds one of the materials to the composite, and considers the effect of an infinitesimal change in the composite material composition on the effective thermal conductivity, obtaining a differential equation for the effective thermal conductivity in terms of the volume fraction of inclusions; see [8,9]. Reviews of many current models and methodologies, including those outlined above, can be found in [10,11,12].Recent work has involved the application of Maxwell-like models to the study of composite materials [13], including polymer composites [14,15]. Such models have recently proven useful in understanding nanoflake thermal annealing [16], and in the understanding of effective thermal conductivity in a variety of materials, such as for a wood cell modelled as a constituent element of briquette c...

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form$t^{k}$, we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of$k$. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these.

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Abstract. We develop a network-based model of a catchment basin that incorporates the possibility of small-scale, in-channel, leaky barriers as flood attenuation features, on each of the edges of the network. The model can be used to understand effective risk reduction strategies considering the whole-system performance; here we focus on identifying network dam placements promoting effective dynamic utilisation of storage and placements that also reduce risk of breach or cascade failure of dams during high flows. We first demonstrate the model using idealised networks and explore risk of cascade failure using probabilistic barrier-fragility assumptions. The investigation highlights the need for robust design of nature-based measures, to avoid inadvertent exposure of communities to a flood risk, and we conclude that the principle of building the leaky barriers on the upstream tributaries is generally less risky than building on the main trunk, although this may depend on the network structure specific to the catchment under study. The efficient scheme permits rapid assessment of the whole-system performance of dams placed in different locations in real networks, demonstrated in application to a real system of leaky barriers built in Penny Gill, a stream in the West Cumbria region of Britain.

The plant hormone gibberellin (GA) regulates multiple developmental processes. It accumulates in the root elongating endodermis, but how it moves into this cell le and the signi cance of this accumulation are unclear. Here, we identi ed a monophyletic clade of NPF transporters required for GA and abscisic acid (ABA) translocation. We demonstrate that NPF2.14 is a subcellular GA/ABA transporter, the rst to be identi ed in plants, facilitating GA and ABA accumulation in the root endodermis to regulate suberization. Further, NPF2.12 and NPF2.13, closely related proteins, are plasma membrane-localized GA and ABA importers that facilitate shoot-to-root GA 12 translocation, regulating endodermal hormone accumulation. This work reveal that GA promotes root suberization and that GA and ABA can act nonantagonistically. We demonstrated how a clade of transporters mediates hormone ow while utilizing de ned cell-le-speci c vacuolar storage at the phloem unloading zone, allowing a hormone slow-release mechanism required for suberin formation in the maturation zone.