(354g) Radial Lift Force On a Suspended Sphere Due to Fluid Inertia for Low but Finite Reynolds Number Flow through a Cylinder

It is well known that when a spherical non-Brownian particle moves through a cylinder in a viscous liquid, its equilibrium position is at an intermediate radial distance in between the cylinder-axis and the bounding wall. It is a surprising phenomenon because the symmetry of Stokes equation dictates zero radial force on the sphere for low Reynolds number flow. However, the Stokesian symmetry breaks down when the Reynolds number is finite leading to a radial lift force on the suspended body due to fluid inertia. In this talk, we present our study describing this inertial lift force on the transporting sphere.

In our analysis, we apply perturbation theory to determine the lift force. For free-space and planar wall geometry, such approach requires singular perturbation to avoid Whitehead paradox. We recognize that as the scattered flow from the sphere in a cylinder is exponentially screened in the axial direction, simpler regular perturbation can yield the required result without confronting Whitehead paradox. Accordingly, first, we accurately solve for the Stokesian fields around a freely suspended sphere inside a cylinder. Then, we evaluate the fluid acceleration in the domain based on the Stokesian solution. Finally, we use the leading order flow inertia as a source term for the next order equation in Reynolds number to determine the inertia-corrected flow-field and the consequent lift force. We plot this force as a function of radial distance for different cylinder to sphere size ratios.

Our results confirm that at the cylinder-axis the lift force is zero, but the force is such that the axisymmetric position corresponds to an unstable equilibrium. In contrast, there is an intermediate position with zero radial force where the radial variation in force ensures stable equilibrium. First, we present this equilibrium radius as a function of cylinder to sphere size-ratio considering only leading order perturbation terms. Then, we use higher order perturbations to describe the equilibrium radius as a function of both size-ratio and Reynolds number. Our results agree well with known experimental findings.