A framework for diagnosing inertial lift generation in wall-bounded flows: application to eccentric rotating cylinders in Newtonian and shear-thinning fluids
A body moving in a wall-bounded flow often experiences a hydrodynamic lift force normal to the wall, which plays an important role in many fluid systems. In this study, we develop a framework for diagnosing steady inertial lift from the internal structure of the flow field. Based on the generalised reciprocal theorem for finite-Reynolds-number flows, the lift is expressed as a volume integral that identifies both the dominant contributions and the regions from which they arise. We apply this framework to numerically obtained steady flows of Newtonian and shear-thinning fluids between eccentric rotating cylinders, and analyse the lift acting on the inner cylinder undergoing rotation and orbital motion. In particular, we focus on lift reversal induced by increasing eccentricity in a Newtonian fluid and on lift reversal induced by stronger shear-thinning behaviour at high eccentricity. The volume-integral expression decomposes the lift into a vortex-force contribution associated with inertia and a viscous stress contribution associated with the non-uniform viscosity field, and shows that the former dominates over the parameter range considered here. As the eccentricity increases, negative relative vorticity, and in some cases tangential velocity, become stronger in the narrow-gap region, thereby enhancing the negative local vortex-force contribution and inducing lift reversal. Stronger shear-thinning behaviour, on the other hand, amplifies negative relative vorticity near the inner cylinder, thereby increasing the positive local vortex-force contribution and inducing lift reversal. These results demonstrate that the proposed framework is useful for diagnosing and interpreting steady inertial lift in wall-bounded flows.