A framework for diagnosing inertial lift generation in wall-bounded flows: application to eccentric rotating cylinders in Newtonian and shear-thinning fluids

Abstract

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.

Figures (12)

• Figure 1: Schematic of the flow between eccentric rotating cylinders. The gap between the inner and outer cylinders is filled with a shear-thinning fluid. Here, $\varepsilon$ denotes the centre-to-centre distance between the cylinders, and $R_{\mathrm{i}}$ and $R_{\mathrm{o}}$ are the radii of the inner and outer cylinders, respectively. The angular velocities $\Omega$ and $\omega$ denote the orbital motion about the outer-cylinder centre and the rotation of the inner cylinder about its own centre, respectively. The origin of the Cartesian coordinate system $(x,y)$ is located at the centre of the outer cylinder.
• Figure 2: Dependence of the lift force $F_x$ on the eccentricity $e$ in an eccentric rotating-cylinder system. Red symbols indicate the results of the present numerical simulations. (a) Newtonian fluid case ($n = 1.0$) at $\operatorname{Re} = 1$ and $\Omega/\omega = 1$. The solid and dashed lines represent the analytical solutions of brindley1979flow and kazakia1978flow, respectively. (b) Shear-thinning power-law fluid cases at $\operatorname{Re} = 100$ with $n = 0.5$ and $0.2$, where the inner cylinder rotates without orbital motion ($\Omega = 0$). Grey symbols indicate the numerical results reported by podryabinkin2011moment.
• Figure 3: Flow fields at eccentricity $e = 0.4$ for several power-law indices $n$: $n=1$, $0.6$, and $0.4$. Panels (a) and (b) correspond to the rotation-rate ratios $\Omega/\omega = 0.02$ and $2$, respectively. In each panel, the top row shows colour contours of the pressure $p$ with black arrows indicating the velocity vectors, while the bottom row shows the spatial distribution of the viscosity ratio $\mu/\mu_0$, where $\mu_0$ is the viscosity of the Newtonian fluid.
• Figure 4: Dependence of the drag force $F_y$ on the eccentricity $e$ in a power-law fluid for two different rotation-rate ratios: (a) rotation-dominated regime $(\Omega / \omega = 0.02)$ and (b) orbital-dominated regime $(\Omega / \omega = 2)$. Each curve corresponds to a different power-law index $n$. The inset shows the normalised variation $\Delta F_y / (1-n)$, where $\Delta F_y = F_y - F_{y,0}$ and $F_{y,0}$ denotes the drag force in a Newtonian fluid. The red dashed curve indicates the leading-order asymptotic expression in integral form, derived in the limit $1-n \ll 1$ (see \ref{['sec:app_asymp']}).
• Figure 5: Dependence of the lift force $F_x$ on the eccentricity $e$ in a power-law fluid for three different rotation-rate ratios: (a) $\Omega / \omega = 0.02$, (b) $\Omega / \omega = 0.1$, (c) $\Omega / \omega = 0.4$ and (d) $\Omega / \omega = 2$. Each curve corresponds to a different power-law index $n$.