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CHAPTER 2

Shape and structure morphing of

systems with internal flows

2.1 Natural form and organization—quandary, observation, and
rationale

The quest for a principle that could rule the shape or “geometric” organization of the
macroscopic entities that regenerate everywhere in nature is prevalent in the rational,
deterministic understanding of the natural self-organization.
More complex systems may host fluxes of different nature. Convection-diffusion
processes are such examples, but the list may continue: coupled gradient-flux systems
(piezoelectric, piezomagnetic, thermoelectric), electric and magnetic fluxes, etc. For
instance, the spatiotemporal structure of “convection diffusion wave” made by a nat-
ural heat and mass convection system may show off (in experiment and numerical sim-
ulation, Nishimura et al., 2000a,b) similarity with the reaction-diffusion wave,
presented by the mathematical Turing’s model (Turing, 1952). This resemblance gives
perception of the existence of a spatial pattern, which appears in convection diffusion
system similar to that of the Turing model, and has an eigen dimension, independent
of the scale of experimental apparatus resembling the Turing reaction-diffusion wave.
An optimal weaving of interlaced internal detailsfor compactheatexchangers—fins,
hydraulic diameters, channels, etc.—is observed in the biological organization too. The
volume of the system (device) is limited to perform its objective, which may be to mini-
mize the global, internal thermal resistance between the solid parts “packed” in the system
and the fluid flow that bathes the system. Moreover, in many natural volume-to-point
flows the high permeability paths are empty spaces: cracks (fissures, channels) in bidimen-
sional systems and vessels (tubes, ducts) in tridimensional systems.
Certain shapes that are observed in natural, “alive” systems with flows and leading
gradients (pressure, temperature, etc.) are manifestations of disequilibrium, but the
number of identifiable flow shapes that are seen inside and outside the animated and
unanimated natural entities is limited to just three: tree-like arborescences, round, and
slice (lemon-slice like) cross-sections.
Natural systems may be similar in shape but never identical (as the fingerprints).
For instance, vascular trees are always different, as their vessels cross-sections are round
shapes, but never a perfect circle.

Computational Modeling in Biomedical Engineering and Medical Physics r 2021 Elsevier Inc.
DOI: https://doi.org/10.1016/B978-0-12-817897-3.00002-6 All rights reserved. 43