Plane Couette flow of viscoplastic materials along a slippery vibrating wall

Abstract This study looks at the influence of slip at the wall on plane Couette flows of viscous and yield stress fluids with ultrasonic wall motion. These fluids are used in coating processes. A constant speed V at one wall creates the flow, and vibrations and slip take place at the other wall. Isothermal conditions and arbitrary (longitudinal or transverse) vibrations are considered, with negligible vibrational inertia. For the Bingham model, due to its nonlinearity, whatever the vibration direction and the wall slipperiness, significant decreases occur in the average stress as soon as moderate values of the dimensionless vibration velocity amplitude are involved. Such effects are associated with adherent or slippery walls, even with linear friction laws. They do not occur with linear viscous (Newtonian) models. Average stress reductions can reach nearly 100% for very high Oldroyd numbers, i.e. for stress values without vibration close to the yield limit. Slip velocity also decreases. The cost in terms of the power dissipated remains relatively less than in the Newtonian case, and may contribute to a change in the temperature field. Even when the flow without vibration is a pure slip one, large enough amplitude vibrations, either longitudinal or transverse, applied at the wall can reduce the average shear stress and slip velocity, giving rise to an average axial shear flow. Hence vibrations of moderate or high-velocity amplitude applied to adherent or slippery walls enhance plane Couette flow rates for viscoplastic materials. With moderate values of this amplitude, longitudinal vibrations may be 1.5–2 times more efficient than transverse vibrations with an equivalent cost. However, if for technological reasons transverse vibrations have to be preferred, they can also produce significant results. In any case, coating flows should benefit from an adequate application of ultrasound at the wall.