Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers

Abstract

The recently-discovered centre-mode instability of rectilinear viscoelastic shear flow (Garg et al. Phy. Rev. Lett. 121, 024502, 2018) has offered an explanation for the origin of elasto-inertial turbulence (EIT) which occurs at lower Weissenberg ($Wi$) numbers. In support of this, we show using weakly nonlinear analysis that the subcriticality found in Page et al. (Phys. Rev. Lett. 125, 154501, 2020) is generic across the neutral curve with the instability only becoming supercritical at low Reynolds ($Re$) numbers and high $Wi$. We demonstrate that the instability can be viewed as purely elastic in origin even for $Re=O(10^3)$, rather than `elasto-inertial', as the underlying shear does not energise the instability. It is also found that the introduction of a realistic maximum polymer extension length, $L_{max}$, in the FENE-P model moves the neutral curve closer to the inertialess $Re=0$ limit at a fixed ratio of solvent-to-solution viscosities, $\beta$. In the dilute limit ($\beta \to 1$) with $L_{max}=O(100)$, the linear instability can brought down to more physically-relevant $Wi\gtrsim 110$ at $\beta =0.98$, compared with the threshold $Wi=O(10^3)$ at $\beta =0.994$ reported recently by Khalid et al. (arXiv: 2103.06794) for an Oldroyd-B fluid. Again the instability is subcritical implying that inertialess rectilinear viscoelastic shear flow is nonlinearly unstable - i.e. unstable to finite amplitude disturbances - for even lower $Wi$.

References in corpus (10)

• Colloquium: Theory of Drag Reduction by Polymers in Wall Bounded Turbulence
• Understanding viscoelastic flow instabilities: Oldroyd-B and beyond
• The center-mode instability of viscoelastic plane Poiseuille flow
• Exact travelling wave solutions in viscoelastic channel flow
• A continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow
• Linear instability of viscoelastic pipe flow
• A first coherent structure in elasto-inertial turbulence
• Finite-amplitude elastic waves in viscoelastic channel flow from large to zero Reynolds number
• Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows
• Experimental observation of the origin and structure of elasto-inertial turbulence

Cited by in corpus (10)

• Understanding viscoelastic flow instabilities: Oldroyd-B and beyond
• Coherent structures in plane channel flow of dilute polymer solutions with vanishing inertia
• Inertial enhancement of the polymer diffusive instability
• Finite-amplitude elastic waves in viscoelastic channel flow from large to zero Reynolds number
• Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows
• A unified view of elastic and elasto-inertial turbulence in channel flows at low and moderate Reynolds numbers
• Linear stability analysis of purely elastic travelling wave solutions in pressure driven channel flows
• On the large-Weissenberg-number scaling laws in viscoelastic pipe flows
• Nature of continuous spectra in wall-bounded shearing flows of FENE-P fluids
• Vorticity amplification in viscoelastic channel flows with long-wave surface distortions