A pressure-driven two-layer channel flow of a Newtonian fluid with constant viscosity (top layer) and a fluid with a time-dependent viscosity (bottom layer) is numerically investigated. The bottom layer goes through an ageing process in which its viscosity increases due to the formation of internal structure, which is represented by a Coussot-type relationship. The resultant flow dynamics is the consequence of the competition between structuration and de-structuration, as characterised by the dimensionless timescale for structuration (τ) and the dimensionless material property (β) of the bottom fluid. The development of Kelvin-Helmholtz type instabilities (roll-up structures) observed in the Newtonian constant viscosity case was found to be suppressed as the viscosity of the bottom layer increased over time. It is found that, for the set of parameters considered in the present study, the bottom layer almost behaves like a Newtonian fluid with constant viscosity for τ> 10 and β> 1. It is also shown that decreasing the value of the Froude number stabilises the interfacial instabilities. The wavelength of the interfacial wave increases as the capillary number increases. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.