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Static Equilibrium of Fluids and Interface
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
Equations (4.50) and (4.51) include the special case of a free surface when τAij=0, and pA= pgas. The mean curvature, 2H, of a surface is necessary in order to account for the role of surface tension that gives rise to normal stress discontinuities.
Morphological investigation of a tensegrity helicoid for display during the 2018 Biennale Architecture in Venice
Published in Paulo J.S. Cruz, Structures and Architecture: Bridging the Gap and Crossing Borders, 2019
K.A. Liapi, A. Papantoniou, A. Ioannidi
The term ‘tensegrity helicoid’ refers to a structure that belongs to minimal surface tensegrity networks, a new typology of tensegrity structures investigated and developed by two of the authors (a). Minimal surfaces which are surfaces with zero mean curvature can be also defined as surfaces of the smallest area spanned by a given boundary (Velimirovic et al. 2008, Osserman 2006). This definition suggests multiple applications in architecture, in particular when membrane surfaces are involved. The helicoid is a minimal surface having a helix as its boundary and it is the only ruled minimal surface other than the plane (Do a).
Problems with Explicit Solutions
Published in Vasilios Alexiades, Alan D. Solomon, Mathematical Modeling of Melting and Freezing Processes, 2018
Vasilios Alexiades, Alan D. Solomon
where Γ > 0 depends on the ratio of surface-tension coefficient to latent heat, see §2.4.F, also [PORTER-EASTERLING], [KURZ-FISHER], [TRIVEDI]. The mean curvature of a surface is the average of its two principal curvatures: 12κ=12(1R1+1R2), with R1, R2 the principal radii of curvature (in our notation κ denotes the sum of the principal curvatures). For a planar (flat) interface, we have κ = 0, so Tf = Tm, and the supercooled liquid must be warmed up to Tm in order to freeze. On the other hand, at a protrusion of curvature κ > 0, the freezing point is lowered by Γκ, so supercooled liquid of temperature Tm−Γκ will freeze on it making it grow. In fact, the more pointed the tip the more it is favored for growth! This is the mechanism by which columnar and dendritic growth takes place, creating a fascinating variety of morphologies, (Figure 1.1.2. Note that the Gibbs-Thomson effect is strictly multidimensional since in one-dimension the interface is just a point, having no curvature, so the effect cannot be felt there.
Phase behaviour of a simple fluid confined in a periodic porous material
Published in Molecular Physics, 2021
Daniel Stopper, Gerd E. Schröder-Turk, Klaus Mecke, Roland Roth
Since the walls of the pores considered in our study are minimal surfaces, the integrated mean curvature vanishes – in fact for a minimal surface the mean curvature vanishes at each point. also vanishes for the nodal approximation employed here, which are not strictly minimal surfaces, but are reasonable approximations thereof. This implies that the morphometric thermodynamics predicts that the grand potential for a single phase of a fluid confined by a minimal surface (ms) reduces to i.e. has a volume, a surface and a topological term. If one compares this to the grand potential of a fluid in a slit pore (slit) of two infinite parallel planar walls [3] one notices that the main difference is due to the topological term in Equation (9). In fact it is straightforward to construct an equivalent slit pore with a unit cell that possesses the same volume and surface area as a unit cell of the minimal surface and one finds and i = gyr, p.
Design, analysis and manufacturing of lattice structures: an overview
Published in International Journal of Computer Integrated Manufacturing, 2018
A minimal surface is defined as a surface in hyperbolic space which possesses a mean curvature or zero. TPMS are defined as minimal surfaces that are infinitely periodic in all three axes in 3D space, with a crystallographic space group as its symmetry group (Wang 2007; Gandy et al. 2001). They also have their topology partitioned into two regions that penetrate each other to form a single surface (Lord and Mackay 2003). TPMS are described as surfaces that are free from self-intersections, which proves to be useful to additive manufacturing. A surface that is free of self-intersection is said to have the whole surface contained within bounding constraints denoted as the convex hull of the surface’s boundary curve (Schoen 1970). Although TPMS are always defined to be periodic, they are not necessarily always non-self-intersecting (Schwarz 1890). Examples of such structures can be seen in Figure 8.
A priori estimates of the electrohydrodynamic waves with vorticity: horizontal electric field
Published in Applicable Analysis, 2023
The hydrodynamic boundary conditions at the interface are, a kinematic condition and at the interface , we have where with , represents the surface tension coefficient of the liquid, denotes the mean curvature of the interface, and and we define as follows where can be real numbers, functions, etc. It is easy to get that We now introduce a modified voltage potential by defining , then and becomes Set are the Dirichlet–Neumann operators corresponding to two fluid layers (see Section 4 for the precise definition). Following the calculation of [13], the details see [31], we can rewrite as follows: where Hence, we only need study the following problem where .