Time-dependent lift and drag on a rigid body in a viscous steady linear flow

TL;DR

This paper develops a method to compute time-dependent inertial forces on rigid bodies in linear flows, useful for understanding particle dynamics in turbulent fluids, by solving flow disturbance equations with matched asymptotic expansions.

Contribution

It introduces a formulation for calculating inertial corrections to forces on arbitrarily-shaped bodies in general linear flows using co-moving coordinates and asymptotic methods.

Findings

Derived explicit force expressions for spheres in canonical flows.

Analyzed short-time and quasi-steady inertial effects.

Illustrated applications with spheroids in shear flow.

Abstract

We compute the leading-order inertial corrections to the instantaneous force acting on a rigid body moving with a time-dependent slip velocity in a linear flow field, assuming that the variation of the undisturbed flow at the body scale is much larger than the slip velocity between the body and the fluid. Motivated by applications to turbulent particle-laden flows, we seek a formulation allowing this force to be determined for an arbitrarily-shaped body moving in a general linear flow. We express the equations governing the flow disturbance in a non-orthogonal coordinate system moving with the undisturbed flow and solve the problem using matched asymptotic expansions. The use of the co-moving coordinates enables the leading-order inertial corrections to the force to be obtained at any time in an arbitrary linear flow field. We then specialize this approach to compute the time-dependent force components for a sphere moving in three canonical flows: solid body rotation, planar elongation, and uniform shear. We discuss the behaviour and physical origin of the different force components in the short-time and quasi-steady limits. Last, we illustrate the influence of time-dependent and quasi-steady inertial effects by examining the sedimentation of prolate and oblate spheroids in a pure shear flow.