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Microemulsions—A Historical Overview
Published in Promod Kumar, K. L. Mittal, Handbook of Microemulsion Science and Technology, 2018
Björn Lindman, Stig E. Friberg
The modeling of microemulsions is still somewhat controversial, but a very powerful tool in establishing the relation between the geometrical description and the thermodynamics has been found in the flexible surface model of Helfrich [39]. The local energy is given by the local curvature, and the total curvature free energy is obtained by integrating over the surface. Talmon and Prager [40] pioneered the modeling of microemulsions and provided a thermodynamic model for the bicontinuous phase based on a subdivision of space into Voronoi polyhedra; they predicted a rich phase behavior, including the three-phase coexistence. Further important developments were due to de Gennes and coworkers [42,43]—who used a cubic lattice randomly filled with water and oil, with the lattice size chosen as the persistence length related to the bending energy of the surfactant film—and by Widom [44,45]. More recently, Wennerstrom and coworkers [46–48] demonstrated that with the flexible surface model a general form of the free energy of both balanced microemulsions and the sponge phase can be obtained. For the first time, the significant features of the stability of the balanced microemulsions with respect to other phases could be captured. In Fig. 8, we compare the theoretically predicted phase diagram with that of an experimental investigation [49]. Important characteristics such as the three-phase coexistence and the limited swelling of the bicontinuous microemulsion but not of the lamellar phase are nicely reproduced.
Plates under Combined Loads
Published in Ansel C. Ugural, Plates and Shells, 2017
However, as might be anticipated on physical grounds, the load-carrying capacity and deformation of a plate under in-plane and lateral loading are significantly affected by any initial curvature. To take into account the extent of this influence, it can be shown that Equation 9.3 can be modified as follows [4]: ∇4w1=1D(p+Nx∂2w∂x2+Ny∂2w∂y2+2Nxy∂2w∂x∂y) where w = w0 + w1. This is the governing differential equation for deflection of thin plates with small initial curvature. Note that the left- and the right-hand sides of the above depend on the change in the curvature and the total curvature of the plate, respectively.
Characteristic classes
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
so we can recover the total covariant derivative. Furthermore, the total curvature takes the form: () Ω=12ei∧ej⊗Ω(ei,ej).
Curvature: An Indicator of the Mechanical Condition of Old Prestressed Concrete Bridges
Published in Structural Engineering International, 2018
Bernard Tonnoir, Christophe Carde, David Banant
Regarding the second observation, the following explanation is proposed. The temperature of the pavement quickly rises as it is heated by direct sunlight; it is only later that the concrete of the beams starts to heat up, by way of vertical conduction. The fact that the timing of the curvature change is ahead of the gradient evaluated in the concrete attests to the proposition that it is actually influenced by the temperature of the pavement. The logical conclusion is that the pavement participates in the inertia of the beams’ bending, since it influences their deformations. Another consequence is that the thermal gradient determined from the temperatures measured in just the concrete cannot be related to the measured curvatures, as the curvatures are also influenced by the pavement. Thus, applying thermal correction to the total curvature in order to obtain the residual curvature, as envisaged above, is rendered—for the moment—invalid. This topic is returned to in the conclusion.
Effects of Karlovitz Number on Flame Surface Wrinkling in Turbulent Lean Premixed Methane-Air Flames
Published in Combustion Science and Technology, 2018
Figure 11a shows the conditional mean of Sd on the cylindrical surfaces alone, i.e., with a shape factor equal to 0, as a function of surface curvatures H. In the analysis, the cylindrical surfaces are selected by isolating surfaces with a shape factor between –0.1 and 0.1, that is, the larger principal curvature is at least 10 times the smaller principal curvature. The flame displacement speeds on the cylindrical surfaces are negatively correlated with surface curvature. Furthermore, the mean Sd on the cylindrical surfaces in Figure 11a agrees within 5% with mean Sd of all surface topologies shown in Figure 10c when it is conditioned on the total curvature H. Figure 11b shows the conditional mean of Sd as a function of surface curvatures H when considering only the spherical surfaces. The analysis is carried out on surfaces with a shape factor greater than 0.8. Again, it is evident that the conditional mean Sd on the spherical surfaces agrees well with the mean Sd when all surface topologies are considered. This suggests that flame displacement speed on average is independent of the shape factor. In other words, the total curvature H (or the mean curvature κm) alone is sufficient to account for Sd variations on the flame surface.
Tangential developable and hydrodynamic surfaces for early stage of ship shape design
Published in Ships and Offshore Structures, 2023
A stern having the form of zero total curvature surfaces was worked out in Japan (Chida and Davies 2013). This stern is an example of the lines well adapted for production potentialities of ship-building yards. River vessels with hulls from the strips of tangential developable surfaces were designed and made during the years 1950 and 1960. These engineering developments are being widely used in the present times (Figure 2). Developable surfaces are especially important for boat builders because they often work with sheet materials like plywood, steel or aluminium.