Start-up and inertialess instability of elasto-viscoplastic channel flow

An exploration is presented of the start-up and linear stability of pressure-driven channel flow of an elasto-viscoplastic fluid described by Saramito's constitutive law. Streamwise uniform base states are non-unique, depending on the initial stress configuration, and develop discontinuities in the normal stresses and shear rate at the yield surfaces over infinite times. Such stress discontinuities can be eliminated by introducing a sufficient extensional pre-stress; true plugs bordered by stress jumps then become replaced by marginally yielded, plug-like flow, or pseudo-plugs. To examine the stability of all of these state, the linear initial-value problem is solved along with the evolving base states. Because this analysis is performed for finite times, the base states remain continuous and there is no need to perturb any stress discontinuities. Armed with the insights provided, stability is then analyzed as a normal-mode problem for the final states, building in perturbations to the stress discontinuities via certain jump conditions across any yield surfaces. Regardless of whether the base flows contain true plugs or pseudo-plugs, the base states are found to be linearly unstable at zero Reynolds number. The most unstable perturbations possess the highest streamwise wavenumbers and become spatially localized to the regions where stresses lie close to the yield stress.