Layered Growth of 3D Snowflake Subject to Membrane Effect and More than One Nucleation Center by Means of Cellular Automata

• Published in: Mathematics (Basel) Vol. 13; no. 3; p. 433
• Main Authors: Acosta, César Renán, Martín, Irma, Rivadeneyra, Gabriela
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.02.2025
• Subjects: Approximation; Cellular automata; cellular automaton; Cellular structure; discrete Laplacian; Flying-machines; hexagonal lattice; Hexagons; Membranes; Nucleation; Oblate spheroids; Parameters; Pressure effects; Robots; snowflake; Snowflakes; Thermodynamics; Water drops; Water vapor
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math13030433

In this work, it is taken into account that in nature, due to pressure and temperature, water drops in general are either spherical or ellipsoidal. Thus, starting from a more general structure, a 3D elliptical surface (oblate spheroid) is constructed, which, by means of parameters, can be turned into a spherical shape. Hexagons are built on a rectangular horizontal plane, then this plane is passed through an elliptical surface at height h, which is determined by a parameter θ. As a result of the cutting of these surfaces, a curve and a plane are obtained, both horizontal ellipsoidal; if these hexagons are within the perimeter of the horizontal ellipse obtained as a function of θ, they are marked with an N, and if they are outside the perimeter, they are marked with an E. Several frozen nucleation centers are established, either in the same layer or in different planes, marking them with an F and their first eight neighbors with a B. The calculations based on a modified snowflake model are carried out tile by tile and layer by layer, governed by the thermodynamic factors α, β, and γ, leading to results that depend on the position of the nucleator, which can be symmetrical or asymmetrical for a snowflake with more than one nucleation center and an external surface formed by water vapor that functions as a membrane.