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Deformations in a Symmetric Member in Bending01:18

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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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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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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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Dean instability in double-curved channels.

J-D Debus1, M Mendoza1, H J Herrmann1

  • 1ETH Zürich, Computational Physics for Engineering Materials, Institute for Building Materials, Wolfgang-Pauli-Str. 27, HIT, CH-8093 Zürich, Switzerland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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Summary

We investigated Dean instability in curved channels using a lattice Boltzmann model. The study found that channel curvature significantly influences flow transitions, with equal radii minimizing the critical Dean number for two-cell vortex flow.

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Area of Science:

  • Fluid Dynamics
  • Computational Physics

Background:

  • Dean instability is a key phenomenon in fluid mechanics, occurring in curved channel flows.
  • Understanding flow transitions is crucial for designing efficient microfluidic devices and understanding astrophysical phenomena.

Purpose of the Study:

  • To investigate Dean instability in both singly and doubly curved channels.
  • To analyze the impact of channel geometry, specifically curvature radii, on flow stability and vortex formation.
  • To validate the lattice Boltzmann method for simulating complex fluid flows in generalized metrics.

Main Methods:

  • Utilized the lattice Boltzmann model adapted for generalized metrics.
  • Validated the model by simulating flow in a streamwise curved rectangular channel and comparing critical Dean numbers with literature.
  • Employed ellipsoidal coordinates to model fluid flow in a double-curved channel.

Main Results:

  • The lattice Boltzmann model accurately predicted critical Dean numbers for laminar to vortex flow transitions in singly curved channels.
  • In double-curved channels, transitions to two-cell, four-cell, and six-cell vortex flows were observed.
  • A minimum critical Dean number for the transition to two-cell vortex flow was identified when the two perpendicular curvature radii were approximately equal.

Conclusions:

  • The study successfully validates the lattice Boltzmann method for complex curved channel flows.
  • Channel curvature plays a critical role in determining the onset and type of vortex flow.
  • Geometric optimization, specifically balancing curvature radii, can influence flow stability and potentially reduce energy dissipation.