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Related Concept Videos

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Hydrostatic pressure on curved surfaces is a fundamental concept in fluid mechanics with broad applications in the civil engineering field. When fluid is in contact with a curved surface, as in a reservoir, dam, or storage tank, it exerts pressure that varies in magnitude and direction along the curved surface. To assess the total hydrostatic force exerted by the fluid on a curved structure, engineers typically isolate the fluid volume adjacent to the surface and analyze the forces acting on...
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In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.
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Surface tension is a fundamental property of fluids, occurring at the boundary between a liquid and a gas or between two immiscible liquids. This phenomenon arises from the cohesive forces between molecules at the fluid's surface, creating an effect similar to a stretched elastic membrane. Inside each fluid, molecules are equally attracted in all directions by neighboring molecules, but surface molecules experience a net inward force, resulting in surface tension.
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The mechanics of deformation in curved members, such as beams or arches, under bending moments, involve complex responses. When such a member, symmetric about the y-axis and shaped like a segment of a circle centered at point C, is subjected to equal and opposite forces, its curvature and surface lengths change significantly. This alteration results in the shift of the curvature's center from C to C', indicating a tighter curve.
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When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
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Pattern Propagation Driven by Surface Curvature.

Ryosuke Nishide1, Shuji Ishihara1,2

  • 1Graduate School of Arts and Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan.

Physical Review Letters
|June 17, 2022
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Summary
This summary is machine-generated.

A static Turing pattern can propagate on curved surfaces, a novel finding driven by surface geometry. This research reveals how curvature and pattern symmetries initiate pattern propagation, offering new insights into natural pattern formation.

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Area of Science:

  • * Mathematical Biology
  • * Pattern Formation
  • * Surface Geometry

Background:

  • * Pattern dynamics in nature are influenced by surface geometry, but a complete understanding remains elusive.
  • * Previous studies assumed patterns remain static regardless of surface curvature.

Purpose of the Study:

  • * To investigate how static Turing patterns behave on curved surfaces.
  • * To elucidate the role of surface geometry in pattern propagation.
  • * To explore reaction-diffusion systems on axisymmetric curved surfaces.

Main Methods:

  • * Numerical analysis of reaction-diffusion systems.
  • * Theoretical analysis of pattern dynamics on curved surfaces.
  • * Investigation on axisymmetric curved surfaces.

Main Results:

  • * Demonstrated for the first time that static Turing patterns can propagate on curved surfaces.
  • * Identified surface and pattern symmetries as key factors in initiating pattern propagation.
  • * Revealed a novel mechanism for pattern propagation driven by surface curvature.

Conclusions:

  • * Surface geometry significantly impacts pattern dynamics, enabling propagation where previously thought static.
  • * The interplay of surface and pattern symmetries governs pattern propagation.
  • * Provides fundamental insights into the role of geometry in biological and chemical pattern formation.