Role of slip in the stability of viscoelastic liquid flow through a channel

Abstract

For a horizontal channel flow of a weakly viscoelastic liquid that follows the constitutive model of Walters’ liquid B′′, a linear stability analysis is performed when the channel walls are slippery. In this case, the viscoelastic channel flow is driven by the streamwise pressure gradient. We explore the primary instability for a symmetric slip flow with the same slip length of the channel walls, an asymmetric slip flow with different slip lengths of the channel walls, and an analogy to the Poiseuille–Couette flow of a viscoelastic liquid with a slip effect on the lower wall but no slip effect on the upper wall. We observe that the viscoelastic parameter shows a destabilizing impact, but the slip length shows a stabilizing impact on the most unstable shear mode in all three flow configurations. Moreover, the Poiseuille and Poiseuille–Couette flows for the viscoelastic liquid are linearly more unstable to infinitesimal disturbances than those of the Newtonian liquid. As a result, wall velocity needs to be higher than in the Newtonian liquid to stabilize the viscoelastic Poiseuille–Couette flow. Using the asymptotic analysis, we have determined the critical value of the slip length, or equivalently, the critical value of the lower wall velocity, above which the asymmetric slip flow and the Poiseuille–Couette flow of the viscoelastic liquid are permanently stable to infinitesimal disturbances.