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

TL;DR

This work develops a general matched asymptotic framework to compute leading-order inertial corrections to the instantaneous force and torque on arbitrarily shaped rigid bodies moving with time-dependent slip in steady linear flows. By recasting the disturbance problem in time-dependent co-moving coordinates, the authors reduce the governing equations to tractable ordinary differential equations and obtain a universal time-dependent kernel $\bf K(t)$ that encodes history effects, with body shape entering solely through resistance tensors. They derive explicit kernels for solid-body rotation, planar elongation, and linear shear, and apply the theory to non-spherical spheroids to reveal how time-dependent inertia alters translation, while rotation follows Jeffery-like dynamics at this order. The results quantify when quasi-steady approximations fail in turbulent-like flows and provide a pathway to extend low-Reynolds-number models toward general linear carrying flows and non-spherical particles, enhancing predictions in particle-laden turbulence.

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.

Figures (8)

• Figure 1: A particle is moving in a steady linear flow. The body shown in this sketch is a spheroid (with symmetry vector $\hbox{\boldmath$n$}$), but the method applies to arbitrary body shapes.
• Figure 2: Time variation of the kernel $\mathbb{K}$ in a solid-body rotation flow. Solid line: $6\pi {[\mathbb{K}]^1}_1=6\pi {[\mathbb{K}]^2}_2$; dashed line: $6 \pi {[\mathbb{K}]^3}_3$; grey dashed line: $t^{-1/2}$ short-time behaviour resulting from the contribution ${\mathbb{K}}_h(t)$ in (\ref{['Krot']}); black dash-dotted line: $6 \pi { |[\mathbb{K}]^1}_2$; grey dash-dotted line: short-time expansion $6 \pi {[\mathbb{K}]^1}_2 \sim \frac{1}{75\sqrt\pi} t^{5/2}$.
• Figure 3: Time variation of the kernel $\mathbb{K}$ in a planar elongational flow. Solid line: $6\pi {[\mathbb{K}]^1}_1$; dashed line: $6 \pi {[\mathbb{K}]^3}_3$; dash-dotted line: $6 \pi { |[\mathbb{K}]^2}_2 |$; grey dashed line: $t^{-1/2}$ short-time behaviour. The component ${[\mathbb{K}]^2}_2$ switches from positive to negative at $t\approx1.4$.
• Figure 4: Time variation of the kernel $\mathbb{K}$ in a linear shear flow. Black lines correspond to the diagonal components (i.e. the inertial corrections to the drag force), with $6\pi{[\mathbb{K}]^1}_1$ (solid line), $6\pi{[\mathbb{K}]^2}_2$ (dash-dotted line), and $6\pi{[\mathbb{K}]^3}_3$ (dashed line). Dark grey lines correspond to the off-diagonal components, with $6 \pi{[\mathbb{K}]^1}_3$ (dashed line), and $6\pi{[\mathbb{K}]^3}_1$ (solid line); the latter is the time-dependent counterpart of the Saffman lift force. Circles correspond to the inverse Fourier transform of the results obtained in the frequency domain by Asmolov99. Pale grey lines correspond to the $t^{-1/2}$-Basset-Boussinesq kernel (dashed line), and to the off-diagonal components of the kernel derived by Miyazaki95a in the short time limit (dash-dotted line).
• Figure 5: Evolution of $(a)$ the $\hbox{\boldmath$e$}_1$-component, and $(b)$ the $\hbox{\boldmath$e$}_3$-component of the slip velocity, $\hbox{\boldmath$u$}_s$, of a prolate spheroid with aspect ratio $\lambda=2$, as predicted using different approximations. Black line: present unsteady theory; dash-dotted line: present quasi-steady theory; dashed line: prediction based on the Stokes quasi-steady drag; grey line: prediction based on the sum of the Stokes quasi-steady drag and the Basset-Boussinesq force.