[Paper Review] Critical curves of plane Poiseuille flow with slip boundary conditions

Andreas Spille, Alexander Rauh+1 more|ArXiv.org|Jan 3, 2001

Fluid Dynamics and Turbulent Flows19 citations

TL;DR

This study investigates the linear stability of 2D plane Poiseuille flow under slip boundary conditions, where slip is defined as the ratio of wall tangential velocity to maximum flow velocity. Using the Orr-Sommerfeld eigenvalue formulation with Chebyshev spectral collocation, the authors show that the critical Reynolds number increases sharply with slip, rising from 5772 at no-slip to over 23,000 at 6% slip, indicating strong stabilizing effects of slip on flow instability.

ABSTRACT

We investigate the linear stability of plane Poiseuille flow in 2D under slip boundary conditions. The slip s is defined by the tangential velocity at the wall in units of the maximal flow velocity. As it turns out, the critical Reynolds number depends smoothly on s but increases quite rapidly.

Motivation & Objective

• To examine how slip at the wall affects the linear stability of plane Poiseuille flow.
• To quantify the dependence of the critical Reynolds number on the slip parameter s.
• To determine how the slip of the critical unstable mode (sc) evolves with increasing wall slip.
• To provide a numerical framework for solving the Orr-Sommerfeld equation under slip boundary conditions.
• To assess the stabilizing effect of slip on the onset of hydrodynamic instability in channel flows.

Proposed method

• Formulates the Navier-Stokes equations in dimensionless form using a stream function Ψ for incompressible 2D flow.
• Decomposes the flow into a base state Ψb and a Fourier-mode disturbance Ψq with wave number α.
• Derives the linearized Orr-Sommerfeld equation with slip boundary conditions: ∂Ψ/∂z ± bΨ = 0 at z = ±1.
• Expresses the slip s in terms of the boundary parameter b: s = 2/(2 + b), with s → 0 as b → ∞.
• Solves the resulting generalized eigenvalue problem numerically using up to 70 Chebyshev polynomials as basis functions.
• Identifies the critical curve as the set of (R, α) where the most unstable mode has zero real part and all others decay.

Experimental results

Research questions

• RQ1 How does the critical Reynolds number R_c for the onset of instability vary with increasing slip s?
• RQ2 What is the slip s_c of the critical mode at the wall, and how does it scale with the imposed slip s?
• RQ3 Does the presence of slip significantly alter the neutral stability curve of plane Poiseuille flow?
• RQ4 How does the stabilizing effect of slip compare to the classical no-slip case (R_c ≈ 5772)?
• RQ5 What is the quantitative relationship between the slip parameter b and the resulting flow stability?

Key findings

• The critical Reynolds number R_c increases smoothly but sharply with increasing slip s, rising from 5772 at s = 0% to 23,230 at s = 6%.
• The slip s_c of the critical mode at the wall increases with s, reaching 55% at s = 6%, indicating that the most unstable mode itself exhibits significant slip.
• For s = 0.1%, R_c increases to 5773, showing even small slips have a measurable stabilizing effect.
• At s = 1%, R_c reaches 6070, and at s = 5%, it exceeds 15,000, demonstrating a strong nonlinear stabilizing influence.
• The stabilizing effect is most pronounced at higher slip values, with R_c growing from 6,960 at s = 2% to 23,230 at s = 6%.
• The results confirm that slip boundary conditions significantly delay the onset of instability in plane Poiseuille flow.