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Stream Function01:20

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Green's function for a generalized two-dimensional fluid.

Takahiro Iwayama1, Takeshi Watanabe

  • 1Department of Earth and Planetary Sciences, Graduate School of Science, Kobe University, Kobe 657-8501, Japan. iwayama@kobe-u.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary
This summary is machine-generated.

This study explores the Green's function for generalized 2D fluids, revealing its Riesz potential form and logarithmic corrections for even α values. Physically realizable systems exist only for α ≤ 3.

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Area of Science:

  • Fluid dynamics
  • Mathematical physics

Background:

  • The study addresses the Green's function for generalized two-dimensional (2D) fluids, known as the α turbulence system.
  • This system is defined by the relationship q = -(-Δ){α/2}ψ, where q is vorticity and ψ is the stream function.

Purpose of the Study:

  • To analyze the Green's function for a generalized 2D fluid in an unbounded domain.
  • To explain the transition in small-scale behavior at α=2 and derive azimuthal velocity.

Main Methods:

  • The Green's function is defined as the stream function generated by a unit point vortex (delta-functional vorticity distribution).
  • The study derives the functional form of the Green's function, G{(α)}(r) ∝ r{α-2}, identifying it as the Riesz potential.
  • Logarithmic corrections and behavior for even and negative α values are analyzed.

Main Results:

  • The Green's function follows a Riesz potential form, G{(α)}(r) ∝ r{α-2}, with logarithmic corrections for positive even α.
  • For negative even α, the Green's function is related to higher-order Laplacians of the delta function.
  • The azimuthal velocity derived from the Green's function indicates physical realizability only for α ≤ 3.

Conclusions:

  • The Green's function provides insight into the small-scale behavior transitions in α turbulence.
  • The findings establish constraints for physically realizable generalized 2D fluid systems.
  • The study extends to anisotropic generalized 2D fluids, presenting their Green's function and realizable systems.