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Viscosity measures the resistance a fluid offers to flow and deformation. It results from internal friction between layers of fluid moving relative to one another. Dynamic viscosity, denoted by the Greek letter mu (μ), quantifies the force needed to move one fluid layer over another. For Newtonian fluids like water and air, the relationship between the shearing stress and the rate of shearing strain is linear, meaning their viscosity remains constant regardless of the applied stress.
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Divergence-Conforming Velocity and Vorticity Approximations for Incompressible Fluids Obtained with Minimal Facet

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Two novel lowest order methods approximate incompressible flows using divergence-conforming and Raviart-Thomas spaces. These methods ensure divergence-free velocity solutions and optimal, pressure-robust error estimates for fluid dynamics simulations.

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Area of Science:

  • Computational fluid dynamics
  • Numerical analysis
  • Finite element methods

Background:

  • Approximating incompressible flows is crucial in computational fluid dynamics.
  • Existing methods often face challenges with divergence-free constraints and pressure robustness.
  • Accurate simulation of fluid behavior requires robust numerical techniques.

Purpose of the Study:

  • Introduce two new lowest order numerical methods for incompressible flow approximation.
  • Develop methods that provide exactly divergence-free discrete velocity solutions.
  • Achieve optimal error estimates that are robust with respect to pressure variations.

Main Methods:

  • Utilize a mixed method and a hybrid discontinuous Galerkin method.
  • Employ divergence-conforming linear Brezzi-Douglas-Marini space for velocity approximation.
  • Use the lowest order Raviart-Thomas space for vorticity approximation.
  • Incorporate the physically correct viscous stress tensor involving the symmetric gradient of velocity.

Main Results:

  • The proposed methods yield exactly divergence-free discrete velocity solutions.
  • Optimal error estimates that are pressure robust have been achieved.
  • Stability analysis is grounded in a Korn-like inequality for vector finite elements.
  • Numerical examples validate theoretical findings and compare method condition numbers.

Conclusions:

  • The developed methods offer accurate and robust approximations for incompressible flows.
  • These methods provide significant advantages in handling divergence-free constraints and pressure robustness.
  • The findings contribute to advancing numerical techniques in computational fluid dynamics.