On Kelvin-Stuart vortices in a viscous fluid

Area of Science:

• Fluid dynamics
• Computational fluid dynamics
• Theoretical fluid mechanics

Background:

• J. T. Stuart's 1967 discovery of a one-parameter family of steady inviscid shear flows.
• The conceptual similarity between these inviscid flows and viscous fluid diffusion with reversed time.
• The need to quantify the resemblance between inviscid shear flows and viscous diffusion.

Purpose of the Study:

• To explore the Navier-Stokes solutions for a row of point vortices as an initial condition.
• To determine the quantitative similarity between inviscid shear flows and viscous fluid diffusion.
• To analyze the flow behavior in specific limiting cases for explicit solutions.

Main Methods:

• Numerical exploration of Navier-Stokes equations.
• Analysis of initial conditions based on classical irrotational flow of a point vortex row.
• Focus on two tractable cases: small time with arbitrary Reynolds number, and small Reynolds number with arbitrary time.

Main Results:

• The study initiates an exploration into the Navier-Stokes solutions for a specific initial condition.
• Explicit analytical or semi-analytical results are obtained for the identified limiting cases.
• The degree of resemblance between the inviscid shear flow model and viscous diffusion is assessed.

Conclusions:

• The paper provides insights into the relationship between inviscid shear flows and viscous diffusion phenomena.
• The findings are based on the analysis of specific, simplified scenarios of the Navier-Stokes equations.
• This work contributes to understanding the applicability of inviscid flow models in viscous fluid dynamics contexts.