[Paper Review] Note on the stability criteria for a new type of helical flows

Sergey V. Ershkov|arXiv (Cornell University)|Jul 9, 2018

Fluid Dynamics and Turbulent Flows55 references3 citations

TL;DR

This paper establishes stability criteria for non-stationary helical flows governed by the Navier-Stokes equations with spatially varying proportionality between velocity and vorticity. By leveraging an invariant Bernoulli function and deriving the pressure field from it, the study identifies sufficient conditions under which exact solutions remain stable, advancing analytical understanding of complex fluid dynamics in helical configurations.

ABSTRACT

In this paper, we proceed exploring the case of non-stationary helical flows of the Navier-Stokes equations for incompressible fluids with variable (spatially dependent) coefficient of proportionality between velocity and the curl field of flow. Meanwhile, the system of Navier-Stokes equations (including continuity equation) has been successfully explored previously with respect to the existence of analytical way for presentation of non-stationary helical flows of the aforementioned type. The main motivation of the current research is the exploring the stability of previously obtained helical flows. Conditions for the stability criteria of the exact solution for the aforementioned type of flows are obtained, for which non-stationary helical flow with invariant Bernoulli-function is considered. As it has been formulated before, the spatial part of the pressure field of the fluid flow should be determined via Bernoulli-function, if components of the velocity of the flow are already obtained.

Motivation & Objective

• To investigate the stability of previously derived exact solutions for non-stationary helical flows with spatially dependent vorticity-velocity proportionality.
• To address the challenge of ensuring long-term solution stability in non-stationary, incompressible fluid flows governed by the Navier-Stokes equations.
• To establish conditions under which the Bernoulli function remains invariant, enabling pressure field determination and stability analysis.
• To extend analytical methods for helical flows beyond existence to include dynamic stability assessment.

Proposed method

• The study employs the Navier-Stokes equations with an incompressibility constraint and a spatially dependent proportionality factor linking velocity and vorticity.
• It assumes the Bernoulli function remains invariant over time, which simplifies the pressure field determination.
• The pressure field is reconstructed from the Bernoulli function using the known velocity components.
• Stability criteria are derived by analyzing the behavior of the exact solution under small perturbations, relying on energy-based or Lyapunov-type arguments implied by the invariant Bernoulli function.
• The analysis focuses on the structural properties of the flow and the role of spatial variation in the vorticity-velocity coupling coefficient.

Experimental results

Research questions

• RQ1 Under what conditions does a non-stationary helical flow with spatially varying vorticity-velocity proportionality remain stable?
• RQ2 How does the invariance of the Bernoulli function influence the stability of exact solutions to the Navier-Stokes equations?
• RQ3 What is the role of the pressure field, derived from the Bernoulli function, in determining the stability of helical flows?
• RQ4 Can the stability of such flows be assessed without numerical simulation, using analytical criteria?

Key findings

• Stability of the exact solution is ensured when the Bernoulli function remains invariant throughout the flow domain.
• The pressure field is fully determined by the velocity components and the invariant Bernoulli function, enabling consistent stability analysis.
• The spatial dependence of the vorticity-velocity proportionality coefficient introduces constraints that must be satisfied for stability.
• The derived stability criteria are analytical and do not require numerical integration, offering a direct method to assess solution robustness.
• The results extend prior work on existence of solutions by providing conditions under which such solutions remain dynamically stable.