Viscoelastic Pipe Flow is Linearly Unstable

Newtonian pipe flow is known to be linearly stable at all Reynolds numbers. We report, for the first time, a linear instability of pressure-driven pipe flow of a viscoelastic fluid, obeying the Oldroyd-B constitutive equation commonly used to model dilute polymer solutions. The instability is shown to exist at Reynolds numbers significantly lower than those at which transition to turbulence is typically observed for Newtonian pipe flow. Our results qualitatively explain experimental observations of transition to turbulence in pipe flow of dilute polymer solutions at flow rates where Newtonian turbulence is absent. The instability discussed here should form the first stage in a hitherto unexplored dynamical pathway to turbulence in polymer solutions. An analogous instability exists for plane Poiseuille flow.

Figure 1

Eigenspectrum for pipe flow of an Oldroyd-B fluid for $\mathrm{Re}=800$, $\mathrm{Wi}=65$, $\beta =0.65$, and $k=1$ (for $N=200$ and 400); the inset zooms into the region around the unstable mode.

Figure 2

Perturbation velocity (left) and polymer force (right) fields for the unstable mode for $\mathrm{Re}=800$, $\mathrm{Wi}=65$, $\beta =0.6$, and $k=1$.

Figure 3

The unstable center-mode eigenfunctions for the axial velocity (left) and radial velocity (right) in scaled boundary-layer coordinates in the limit $\mathrm{Re}\to \infty$ and $\mathrm{Wi}\to \infty$ for a fixed $\mathrm{Wi}/{\mathrm{Re}}^{1/2}$ ($k=1$ and $\beta =0.5$).

Figure 4

The critical Reynolds number, ${\mathrm{Re}}_c$, as a function of $E$ for different viscosity ratios, $\beta$. The inset shows neutral curves in the Re-$k$ plane for $\beta =0.8$ for different $E$.

Figure 5

The critical Reynolds number, ${\mathrm{Re}}_c$, as a function of the viscosity ratio $\beta$ for $E=0.01$. The inset shows the scaling behavior in the dual limit $E\to 0$, $\beta \to 1$.