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Navier–Stokes Equations01:28

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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
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The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
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Stochastic Navier-Stokes Equations on a Thin Spherical Domain.

Zdzisław Brzeźniak1, Gaurav Dhariwal2, Quoc Thong Le Gia3

  • 1Department of Mathematics, University of York, Heslington, York, YO10 5DD UK.

Applied Mathematics and Optimization
|November 1, 2021
PubMed
Summary

We studied incompressible Navier-Stokes equations on thin spherical domains. The study establishes the convergence of solutions on thin domains to those on a full sphere as thickness approaches zero.

Keywords:
Navier–Stokes equations on a sphereSingular limitStochastic Navier–Stokes equations

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Area of Science:

  • Fluid dynamics
  • Stochastic partial differential equations
  • Mathematical physics

Background:

  • The study addresses the incompressible Navier-Stokes equations, fundamental to fluid dynamics.
  • It focuses on a thin spherical domain, a specific geometric configuration.
  • The research incorporates random forcing and free boundary conditions.

Purpose of the Study:

  • To analyze the behavior of incompressible Navier-Stokes equations on a thin spherical domain.
  • To establish the convergence of martingale solutions as the domain thickness diminishes.
  • To relate solutions on a thin domain to those on a standard spherical domain.

Main Methods:

  • Consideration of incompressible Navier-Stokes equations with free boundary conditions.
  • Application of random forcing to the equations.
  • Mathematical analysis of martingale solutions and their convergence properties.

Main Results:

  • The convergence of the martingale solution for the thin spherical domain is established.
  • This convergence is shown with respect to the martingale solution of stochastic Navier-Stokes equations on a sphere.
  • The limit is achieved as the domain thickness approaches zero.

Conclusions:

  • The study successfully demonstrates the mathematical link between fluid behavior in thin spherical shells and a full sphere.
  • This provides a theoretical foundation for modeling fluid dynamics in such geometries.
  • The findings are relevant for understanding fluid phenomena in systems with reduced dimensionality.