  Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently
found to be linearly unstable by Garg et al. Phys. Rev. Lett., 121.024502
(2018). 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 evidences 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
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
$\beta\lessapprox0.785$), the nonlinearity stabilises 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
$\beta\approx0.855$. The pipe flow of UCM 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 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.
