Nonlinear viscosity derived by means of Grad's moment method

Area of Science:

• Fluid dynamics
• Statistical mechanics
• Non-equilibrium thermodynamics

Background:

• Existing stress tensor evolution equations by Uribe and Garcia-Colin (1999) were based on vanishing transversal velocity gradients.
• The Boltzmann equation and Grad moment expansion are foundational for kinetic theory and fluid dynamics.

Purpose of the Study:

• To derive a more general stress tensor evolution equation by removing the vanishing transversal velocity gradient assumption.
• To compare the derived equations with those of Uribe and Garcia-Colin.
• To investigate the relationship between hydrodynamic and kinetic equation dimensionalities.

Main Methods:

• Derivation of stress tensor evolution equations from the Boltzmann equation using the Grad moment expansion.
• Application of the Grad method under uniform temperature conditions.
• Specialization of the derived equations to steady unidirectional flow in a square channel.

Main Results:

• A more general stress tensor evolution equation is derived, applicable even when transversal velocity gradients are non-zero.
• The derived equation for the xy component of the stress tensor differs from Uribe and Garcia-Colin's due to a missing term in their formulation.
• Nonlinear viscosity formulas obtained are also distinct from previous results.

Conclusions:

• The derived stress tensor evolution equations provide a more comprehensive description for unidirectional flows compared to previous models.
• Discrepancies with Uribe and Garcia-Colin's work highlight the importance of the transversal velocity gradient assumption.
• The dimensionality of hydrodynamic equations does not necessarily match that of the underlying kinetic equations.