Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows

Abstract

Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garg et al. (Phys. Rev. Lett., vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio $\beta \lessapprox 0.785$ ), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ( $\beta \gtrapprox 0.785$ ), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical $\beta \approx 0.855$ . The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of $\beta$ , the Landau coefficient scales with the inverse of the Weissenberg number ( $Wi$ ) when $Wi$ is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.

Understanding viscoelastic flow instabilities: Oldroyd-B and beyond

“…The relation between this earlier edge-state-based hypothesis, and the more recent TS-analog-based hypothesis, needs further investigation. [67] for channel flow and Wan et al [89] for pipe flow further confirm that the centermode instability is likely subcritical in large parts of the parameter space. Despite these developments, it is relevant to point out that there still remain vast tracts of the viscoelastic parameter space where the mechanism of transition is not understood.…”

“…Recent efforts have taken a step towards answering these questions by showing that the elastoinertial center-mode instability is subcritical, especially for dilute solutions (β > 0.8), both in channel [56,67] and pipe [89] flow, in turn showing that the instability could be relevant even in parameter regimes where the flow is linearly stable; see dashed curve in Fig. 11.…”

“…The instability ceases to exist for β < 0.5 in channel flow [64], while continuing down to β ∼ 10 −3 for pipe flow [32]. Recent work by Wan et al [89] has found that the center-mode instability is present even in the UCM limit (β = 0) for some isolated regions in the Re-W i space. The more interesting limit is that corresponding to dilute solutions with β → 1.…”

“…Below, we review theoretical and experimental efforts to discover such instabilities, and discuss their implications for our understanding of elastic turbulence. While there have been very recent efforts [89,67] that have carried out weakly nonlinear analyses for the elastoinertial center-mode instability at finite Re (see Sec. 2.4), the discussion in the present section will confine itself to the earlier body of work that has focused on inertialess flows.…”

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

“…While this work was going through review, we became aware of the complementary work of Wan, Sun & Zhang (2021) on the weakly nonlinear analysis of axisymmetric pipe flow. Their findings are consistent with those reported here for channel flow.…”