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Related Concept Videos

Surface Tension of Fluid01:22

Surface Tension of Fluid

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.
Surface tension varies with...
Surface Tension01:24

Surface Tension

Surface tension is defined as the force per unit length (γ) acting along the surface of a liquid. It arises due to strong intermolecular forces of attraction. A molecule located inside the bulk of the liquid is surrounded by other molecules and experiences equal forces in all directions. However, a molecule at the surface experiences unbalanced forces because there are more neighboring molecules below than above. This creates a net inward force that pulls surface molecules toward the interior,...
Uniform Depth Channel Flow01:27

Uniform Depth Channel Flow

Uniform depth channel flow keeps fluid depth consistent along channels such as irrigation canals. In natural channels, such as rivers, approximate uniform flow is often assumed. This condition occurs when the channel’s bottom slope matches the energy slope, balancing potential energy lost from gravity with head loss due to shear stress. This balance prevents depth changes along the channel length, resulting in a steady, uniform flow.Uniform flow in open channels with a constant cross-section...
Uniform Depth Channel Flow: Problem Solving01:18

Uniform Depth Channel Flow: Problem Solving

To calculate the flow rate for a trapezoidal channel, first, identify the bottom width, side slope, and flow depth of the channel. The cross-sectional area (A) corresponding to the depth of flow (y), channel bottom width (B), and side slope (θ) is determined by:Next, calculate the wetted perimeter, which includes the bottom width and the sloped side lengths in contact with the water. Using the values of the cross-sectional area and the wetted perimeter, determine the hydraulic radius by...
Gradually Varying Flow01:29

Gradually Varying Flow

Gradually varying flow (GVF) in open channels describes situations where water depth changes slowly along the channel due to factors like non-uniform bed slope, channel shape variations, or obstructions. This flow type occurs when the depth adjusts gradually to balance gravitational forces, shear forces, and energy requirements, resulting in a low rate of depth change.Characteristics of Gradually Varying FlowGVF is commonly observed in natural streams, rivers, and canals, where flow depth...
Irrotational Flow01:28

Irrotational Flow

Irrotational flow is characterized by fluid motion where particles do not rotate around their axes, resulting in zero vorticity. For a flow to be irrotational, the curl of the velocity field must be zero. This imposes specific conditions on velocity gradients. For instance, to maintain zero rotation about the z-axis, the gradient condition:

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Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp
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Published on: February 3, 2014

DIFFEOMORPHIC SURFACE FLOWS: A NOVEL METHOD OF SURFACE EVOLUTION.

Sirong Zhang1, Laurent Younes, John Zweck

  • 1Center for Imaging Science, Johns Hopkins University, Baltimore, MD, 21218-2686 ( sirongzhang@jhu.edu ).

SIAM Journal on Applied Mathematics
|September 28, 2011
PubMed
Summary
This summary is machine-generated.

We introduce diffeomorphic surface flows, a novel topology-invariant and singularity-free method for surface evolution. This approach offers advantages over classical flows in computational anatomy and computer graphics.

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Area of Science:

  • Differential Geometry
  • Computational Anatomy
  • Computer Graphics

Background:

  • Classical surface partial differential equation (PDE) flows, like mean curvature flow, can suffer from topological changes and singularities.
  • These limitations hinder their application in fields requiring robust shape analysis and manipulation.

Purpose of the Study:

  • Introduce a new class of surface flows: diffeomorphic surface flows.
  • Demonstrate their topology-invariant and singularity-free properties.
  • Explore their potential in computational anatomy and computer graphics.

Main Methods:

  • Derive Euler-Lagrange equations for elastic energy in diffeomorphic surface flows.
  • Analyze diffeomorphic mean curvature flow, proving short-time existence and uniqueness.
  • Investigate long-time existence for surfaces of revolution.

Main Results:

  • Diffeomorphic surface flows are solutions to integro-differential equations within the group of diffeomorphisms.
  • The derived flows act as smoothed versions of classical surface flows.
  • Numerical experiments on synthetic and neuroimaging data show promising results.

Conclusions:

  • Diffeomorphic surface flows offer a robust alternative to classical methods, preserving topology and avoiding singularities.
  • Potential applications span computational anatomy, computer graphics, and medical imaging analysis.
  • Further research is needed to address unresolved issues and expand applications.