Two-fluid pressure-driven channel flow with wall deposition and ageing effects

Two-fluid, stratified pressure-driven channel flow is studied in the limit of small viscosity ratios. Cases are considered in which the core fluid undergoes phase separation that results in the ‘precipitation’ of a distinct phase and the formation of a wall layer; these situations are common in the oil industry where ‘fouling’ deposits are formed during the flow. The thickness of this layer increases as a result of continual deposition through Stefan-like fluxes, which are related to the phase behaviour of the core fluid through a chemical equilibria model that treats the fluid as a bi-component mixture. The deposit also undergoes an ‘ageing’ process whereby its viscosity increases due to the build-up of internal structure; the latter is modelled here via a Coussot-type relation. Lubrication theory is used in the wall layer and an integral balance in the core fluid wherein inertial effects are important. By choosing appropriate semi-parabolic velocity and temperatures closures for the laminar flow in the channel core, and a closure relation for the wall layer rheology, evolution equations are derived that describe the flow dynamics. In the presence of ageing but absence of deposition, it is demonstrated how the time-varying deposit rheology alters the wave dynamics; for certain parameter ranges, these effects can give rise to the formation of steep waves and what appears to be finite-time ‘blow-up’. With both ageing and deposition, the spatio-temporal evolution of the deposit is shown together with the increase in the deposition rate with increasing temperature difference between the wall and the inlet.