The interaction of a planar shock wave with a polygonal N 2 volume surrounded by SF 6 is investigated experimentally and numerically. Three polygonal interfaces (square, triangle and diamond) are formed by the soap film technique developed in our previous work, in which thin pins are introduced as angular vertexes to connect adjacent sides of polygonal soap films. The evolutions of the shock-accelerated polygonal interfaces are then visualized by a high-speed schlieren system. Wave systems and interface structures can be clearly identified in experimental schlieren images, and agree well with the numerical ones. Quantitatively, the movement of the distorted interface, and the length and height of the interface structures are further compared and good agreements are achieved between experimental and numerical results. It is found that the evolution of these polygonal interfaces is closely related to their initial shapes. In the square interface, two vortices are generated shortly after the shock impact around the left corner and dominate the flow field at late stages. In the triangular and diamond cases, the most remarkable feature is the small 'SF 6 jet' which grows constantly with time and penetrates the downstream boundary of the interface, forming two independent vortices. These distinct morphologies of the three polygonal interfaces also lead to the different behaviours of the interface features including the length and height. It is also found that the velocities of the vortex pair predicted from the theory of Rudinger and Somers (J. Fluid Mech., vol. 7, 1960, pp. 161-176) agree with the experimental ones, especially for the square case. Typical free precursor irregular refraction phenomena and the transitions among them are observed and analysed, which gives direct experimental evidence for wave patterns and their transitions at a slow/fast interface. The velocities of triple points and shocks are experimentally measured. It is found that the transmitted shock near the interface boundary has weakened into an evanescent wave.

Diabetes mellitus has become a prevalent disease in the world. Diagnostic protocol for the onset of diabetes mellitus is the initial step in the treatments. The intravenous glucose tolerance test (IVGTT) has been considered as the most accurate method to determine the insulin sensitivity and glucose effectiveness. It is well known that there exists a time delay in insulin secretion stimulated by the elevated glucose concentration level. However, the range of the length of the delay in the existing IVGTT models are not fully discussed and thus in many cases the time delay may be assigned to a value out of its reasonable range. In addition, several attempts had been made to determine when the unique equilibrium point is globally asymptotically stable. However, all these conditions are delay-independent. In this paper, we discuss the range of the time delay and provide easy-to-check delay-dependent conditions for the global asymptotic stability of the equilibrium point for a recent IVGTT model through Liapunov function approach. Estimates of the upper bound of the delay for global stability are given in corollaries. In addition, the numerical simulation in this paper is fully incorporated with functional initial conditions, which is natural and more appropriate in delay differential equation system.

The interaction of a planar shock wave (M ≈ 1.2) with an SF 6 polygonal inhomogeneity surrounded by air is experimentally investigated. Six polygons including a square, two types of rectangle, two types of triangle, and a diamond are generated by the soap film technique developed in our previous work, in which thin pins are used as angular vertexes to avoid the pressure singularities caused by the surface tension. The evolutions of the shock-accelerated SF 6 polygons are captured by a high-speed schlieren system from which wave systems and the interface characteristics can be clearly identified. Both regular and irregular refraction phenomena are observed outside the volume, and more complex wave patterns, including transmitted shock, refracted shock, Mach stem and the interactions between them, are found inside the volume. Two typical irregular refraction phenomena (free precursor refraction, FPR, and free precursor von Neumann refraction, FNR) are observed and analysed, and the transition from FPR to FNR is found, providing the experimental evidence for the transition between different wave patterns numerically found in the literature. Combined with our previous work (Zhai et al., J. Fluid Mech., vol. 757, 2014, pp. 800-816), the reciprocal transitions between FPR and FNR are experimentally confirmed. The velocities and trajectories of the triple points are further measured and it is found that the motions of the triple points are self-similar or pseudo-stationary. Besides the shock dynamics phenomena, the evolutions of these shocked heavy polygonal volumes, which are quite different from the light ones, are captured and found to be closely related to their initial shapes. Specifically, for square and rectangular geometries, the different width-height ratios result in different behaviours of shock-shock interaction inside the volume, and subsequently different features for the outward jet and the interface. Quantitatively, the time-variations of the interface scales, such as the width and the normalized displacements of the edges, are obtained and compared with those from previous work. The comparison illustrates the superiority of the interface formation method and the significant effect of the initial interface shape on the interface features. Furthermore, the characteristics of the vortex core, including the velocity and vortex spacing, are experimentally measured, and the vortex velocity is compared with those from some circulation models to check the † Email address for correspondence: sanjing@ustc.edu.cn Interaction of planar shock waves with SF 6 polygons 367 validity of the models. The results in the present work enrich understanding of the shock refraction phenomenon and the database of research into Richtmyer-Meshkov instability (RMI).

Water-sediment regulation (WSR) of the Xiaolangdi Reservoir in the Yellow River is different from other water conservancy projects, with sediment resuspending along the river downstream of the reservoir during water regulation while some suspended sediment depositing during sediment regulation. In this study, samples were collected before, during, and after WSR to investigate the effect of WSR on the suspended sediment and organic carbon downstream of the reservoir. The suspended sediment concentration ([SPS]) increased with the river flow velocity (V) as a power function ([SPS]=1.348V(2.519)) during the three periods. The suspended sediment grain size decreased along the river during water and sediment regulations and after WSR; they were generally below 200μm with the fine particles (<50μm) of 68.0%-93.7% and positively correlated with the flow velocity. The black carbon content in suspended sediment elevated along the river during both water and sediment regulations, and it increased with 2-50μm fraction during water regulation and with <2μm fraction during sediment regulation, suggesting that black carbon mainly exists in fine particles and is influenced by both suspended sediment source and characteristics. There was no significant difference in dissolved organic carbon (DOC) concentration during water regulation, sediment regulation, and after WSR, inferring that the effect of sediment resuspension/deposition on DOC concentration was insignificant. The contribution of DOC flux (27.3%) during WSR period to the annual flux was comparable to that (22.6%) of water, but lower than the sediment (32.5%) and particulate organic carbon (POC) (49.5%). This study suggests that WSR will exert significant influence on the concentrations, characteristics and fluxes of POC (p<0.05) and sediment (p<0.05) but have no significant influence on DOC (p>0.1) of the Yellow River.

Insulin signaling is essential for -cell survival and proliferation in vivo. Insulin also has potent mitogenic and anti-apoptotic actions on cultured -cells, with maximum effect in the high picomolar range and diminishing effect at high nanomolar doses. In order to understand whether these effects of insulin are constitutive or can be subjected to physiological modulation, it is essential to estimate the extracellular concentration of monomeric insulin within an intact islet. Unfortunately, the in vivo concentration of insulin monomers within the islet cannot be measured directly with current technology. Here, we present the first mathematical model designed to estimate the levels of monomeric insulin within the islet extracellular space. Insulin is released as insoluble crystals that exhibit a delayed dissociation into hexamers, dimers, and eventually monomers, which only then can act as signaling ligands. The rates at which different forms of insulin dissolve in vivo have been estimated from studies of peripheral insulin injection sites. We used this and other information to formulate a mathematical model to estimate the local insulin concentration within a single islet as a function of glucose. Model parameters were estimated from existing literature. Components of the model were validated using experimental data, if available. Model analysis predicted that the majority of monomeric insulin in the islet is that which has been returned from the periphery, and the concentration of intra-islet monomeric insulin varies from 50–300 pM when glucose is in the physiological range. Thus, our results suggest that the local concentration of monomeric insulin within the islet is in the picomolar ‘sweet spot’ range of insulin doses that activate the insulin receptor and have the most potent effects on -cells in vitro. Together with experimental data, these estimations support the concept that autocrine/paracrine insulin signalling within the islet is dynamic, rather than constitutive and saturated.