Bulk and interfacial modes of instability in channel flow of thixotropic-viscoelasto-plastic fluids with shear-banding

We carry out a linear stability analysis of the generalised BMP model, which incorporates the shear-banding phenomenon into flows of thixotropic-viscoelasto-plastic fluids: common behaviours observed in wormlike micellar solutions. We introduce a new dimensionless parameter governing shear-banding, and find that when a flow is unstable in the absence of shear-banding, this parameter has a stabilising effect. Above a critical value of the shear-banding parameter, shear bands appear in the flow gradient direction. Here two different modes of instability are observed: a bulk mode (which is the same as that seen in the absence of shear banding) and an interfacial mode. We found that the former dominates over the latter at low values of the shear-banding parameter, but the interfacial instability becomes dominant at very high values of the parameter. In particular, the interfacial modes are more unstable in flows where the low shear band has almost constant viscosity. The two modes of instability depend on different material timescales: interfacial modes become more dangerous if the interface location is near the channel wall, which is seen when the structural relaxation time is much larger than that of plasticity, while bulk modes (as seen in previous work) are highly unstable if the timescale of viscoelasticity is much longer than both those of plasticity and thixotropy.