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The inertial orientation dynamics of anisotropic particles in planar linear flows – CORRIGENDUM
- Navaneeth K. Marath, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 949 / 25 October 2022
- Published online by Cambridge University Press:
- 07 October 2022, E1
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The rotation of a sedimenting spheroidal particle in a linearly stratified fluid
- Arun Kumar Varanasi, Navaneeth K. Marath, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 933 / 25 February 2022
- Published online by Cambridge University Press:
- 24 December 2021, A17
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We derive analytically the angular velocity of a spheroid, of an arbitrary aspect ratio
$\kappa$, sedimenting in a linearly stratified fluid. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise settling in the limit
$Re, Ri_v \ll 1$, where
$Re = \rho _0UL/\mu$ and
$Ri_v =\gamma L^3\,g/\mu U$, the Reynolds and viscous Richardson numbers, respectively, are dimensionless measures of the importance of inertial and buoyancy forces relative to viscous ones. Here,
$L$ is the spheroid semi-major axis,
$U$ an appropriate settling velocity scale,
$\mu$ the fluid viscosity and
$\gamma \ (>0)$ the (constant) density gradient characterizing the stably stratified ambient, with the fluid density
$\rho_0$ taken to be a constant within the Boussinesq framework. A reciprocal theorem formulation identifies three contributions to the angular velocity: (1) an
$O(Re)$ inertial contribution that already exists in a homogeneous ambient, and orients the spheroid broadside-on; (2) an
$O(Ri_v)$ hydrostatic contribution due to the ambient stratification that also orients the spheroid broadside-on; and (3) a hydrodynamic contribution arising from the perturbation of the ambient stratification whose nature depends on
$Pe$;
$Pe = UL/D$ being the Péclet number with
$D$ the diffusivity of the stratifying agent. For
$Pe \ll 1$, this contribution is
$O(Ri_v)$ and orients prolate spheroids edgewise for all
$\kappa \ (>1)$. For oblate spheroids, it changes sign across a critical aspect ratio
$\kappa _c \approx 0.41$, orienting oblate spheroids with
$\kappa _c < \kappa < 1$ edgewise and those with
$\kappa < \kappa _c$ broadside-on. For
$Pe \ll 1$, the hydrodynamic component is always smaller in magnitude than the hydrostatic one, so a sedimenting spheroid in this limit always orients broadside-on. For
$Pe \gg 1$, the hydrodynamic contribution is dominant, being
$O(Ri_v^{{2}/{3}}$) in the Stokes stratification regime characterized by
$Re \ll Ri_v^{{1}/{3}}$, and orients the spheroid edgewise regardless of
$\kappa$. Consideration of the inertial and large-
$Pe$ stratification-induced angular velocities leads to two critical curves which separate the broadside-on and edgewise settling regimes in the
$Ri_v/Re^{{3}/{2}}$–
$\kappa$ plane, with the region between the curves corresponding to stable intermediate equilibrium orientations. The predictions for large
$Pe$ are broadly consistent with observations.
The inertial orientation dynamics of anisotropic particles in planar linear flows
- Navaneeth K. Marath, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 844 / 10 June 2018
- Published online by Cambridge University Press:
- 04 April 2018, pp. 357-402
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In the Stokes limit, the trajectories of neutrally buoyant torque-free non-Brownian spheroids in ambient planar linear flows are well known. These flows form a one-parameter family, with the velocity gradient tensor given by
$\unicode[STIX]{x1D735}\boldsymbol{u}^{\infty \dagger }=\dot{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{x}^{\prime }\mathbf{1}_{y}^{\prime }+\unicode[STIX]{x1D706}\mathbf{1}_{y}^{\prime }\mathbf{1}_{x}^{\prime })$. The parameter
$\unicode[STIX]{x1D706}$ is related to the ratio of the vorticity to the extension (given by
$(1-\unicode[STIX]{x1D706})/(1+\unicode[STIX]{x1D706})$), and ranges from
$-1$ to 1, with
$\unicode[STIX]{x1D706}=1\,,0$ and
$-1$ being planar extensional flow, simple shear flow and solid-body rotation respectively. The unit vectors
$\mathbf{1}_{x}^{\prime }$ and
$\mathbf{1}_{y}^{\prime }$ are unit vectors along the flow and gradient axes of the simple shear flow (
$\unicode[STIX]{x1D706}=0$). The trajectories, as described by a unit vector along the spheroid symmetry axis, are closed orbits for
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$, where
$\unicode[STIX]{x1D706}_{crit}=\unicode[STIX]{x1D705}^{2}(1/\unicode[STIX]{x1D705}^{2})$ for an oblate (a prolate) spheroid of aspect ratio
$\unicode[STIX]{x1D705}$. We investigate analytically the orientation dynamics of such a spheroid in the presence of weak inertial effects. The inertial corrections to the angular velocities at
$O(Re)$ and
$O(St)$, where
$Re$ and
$St$ are the Reynolds (
$Re=\unicode[STIX]{x1D70C}_{f}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}$) and Stokes numbers (
$St=\unicode[STIX]{x1D70C}_{p}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}$) respectively, are derived using a reciprocal theorem formulation. Here,
$L$ is the semimajor axis of the spheroid,
$\unicode[STIX]{x1D707}$ is the viscosity of the suspending fluid,
$\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate, and
$\unicode[STIX]{x1D70C}_{p}$ and
$\unicode[STIX]{x1D70C}_{f}$ are the particle and fluid densities respectively. A spheroidal harmonics formalism is then used to evaluate the reciprocal theorem integrals and obtain closed-form expressions for the inertial corrections. The detailed examination of these corrections is restricted to the aforementioned Stokesian closed-orbit regime (
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$). Here, even weak inertia, for asymptotically long times, of
$O(1/(\dot{\unicode[STIX]{x1D6FE}}Re))$ or
$O(1/(\dot{\unicode[STIX]{x1D6FE}}St))$, will affect the leading-order orientation distribution on account of the indeterminate nature of the distribution across orbits in the Stokes limit. For
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$, inertia results in a drift across the closed orbits in Stokes flow, and this orbital drift is characterized using a multiple time scale analysis. The orbits stabilized by the inertial drift, at
$O(Re)$ and
$O(St)$, are identified in the
$\unicode[STIX]{x1D706}{-}\unicode[STIX]{x1D705}$ plane. For the majority of (
$\unicode[STIX]{x1D706},\unicode[STIX]{x1D705}$) combinations, the stabilized orbit is either one confined to the plane of symmetry (the flow-gradient plane) of the ambient flow (the tumbling orbit) or one where the spheroid is aligned with the ambient vorticity vector (the spinning orbit). However, for some (
$\unicode[STIX]{x1D706},\unicode[STIX]{x1D705}$) combinations, depending on the initial orientation, the orbit stabilized can be either the spinning or the tumbling orbit, since both orbits have non-trivial basins of attraction, separated by a pair of unstable (repelling) limit cycles, on the unit sphere of orientations. A stochastic orientation decorrelation mechanism in the form of rotary Brownian motion, characterized by a Péclet number,
$Pe_{r}$ (
$Pe_{r}=\dot{\unicode[STIX]{x1D6FE}}/D_{r}$, where
$D_{r}$ is the rotary Brownian diffusivity), is included to eliminate the aforementioned dependence on the initial orientation distribution for certain (
$\unicode[STIX]{x1D706}$,
$\unicode[STIX]{x1D705}$) combinations. The unique steady-state orientation distribution determined by the combined effect of Brownian motion and inertia is obtained by solving a closed-orbit-averaged drift–diffusion equation. The steady-state orientation dynamics of an inertial spheroid in a planar linear flow, in the presence of weak thermal orientation fluctuations, has similarities to the thermodynamic description of a one-component system. Thus, we identify a tumbling–spinning transition in a
$C{-}\unicode[STIX]{x1D705}{-}Re\,Pe_{r}$ space. Here,
$C$ is the orbital coordinate that acts as a label for the closed orbits in the Stokes limit. This transition implies hysteretic orientation dynamics in certain regions in the
$C$–
$\unicode[STIX]{x1D705}$–
$Re\,Pe_{r}$ space, although the hysteretic volume shrinks rapidly on either side of simple shear flow. In the hysteretic region, one requires exceedingly large times to achieve the unique steady-state distribution (underlying the thermodynamic interpretation), and for durations relevant to experiments, the system may instead attain an initial-condition-dependent metastable distribution.
The effect of inertia on the time period of rotation of an anisotropic particle in simple shear flow
- Navaneeth K. Marath, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 830 / 10 November 2017
- Published online by Cambridge University Press:
- 29 September 2017, pp. 165-210
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We calculate the leading-order correction to the time period of rotation of a neutrally buoyant spheroid of arbitrary aspect ratio, in a simple shear flow (
$\boldsymbol{u}^{\infty }=\dot{\unicode[STIX]{x1D6FE}}y\mathbf{1}_{1}$;
$\mathbf{1}_{1}$ is the unit vector in the flow direction,
$y$ being the coordinate along the gradient direction), in its long-time orbit set up by the weak fluid inertial drift at
$O(Re)$. Here,
$Re$ is the microscale Reynolds number, a dimensionless measure of the fluid inertial effects on the length scale of the spheroid, and is defined as
$Re=\dot{\unicode[STIX]{x1D6FE}}L^{2}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D707}$, where
$L$ is the semimajor axis of the spheroid,
$\unicode[STIX]{x1D707}$ and
$\unicode[STIX]{x1D70C}$ are respectively the viscosity and density of the fluid, and
$\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate. This long-time orbit is the tumbling orbit for prolate spheroids; for oblate spheroids, it is the spinning orbit for aspect ratios greater than
$0.137$, and can be either the tumbling or the spinning orbit for oblate spheroids of aspect ratios less than
$0.137$. We also calculate the leading-order correction to the time period of rotation of a neutrally buoyant triaxial ellipsoid in a simple shear flow, rotating with its intermediate principal axis aligned along the vorticity of the flow; the latter calculation is in light of recent evidence, by way of numerical simulations (Rosen, PhD dissertation, 2016, Stockholm), of the aforementioned rotation being stabilized by weak inertia. The correction to the time period for arbitrary
$Re$ is expressed as a volume integral using a generalized reciprocal theorem formulation. For
$Re\ll 1$, it is shown that the correction at
$O(Re)$ is zero for spheroids (with aspect ratios of order unity) as well as triaxial ellipsoids in their long-time orbits. The first correction to the time period therefore occurs at
$O(Re^{3/2})$, and has a singular origin, arising from fluid inertial effects in the outer region (distances from the spheroid or triaxial ellipsoid of the order of the inertial screening length of
$O(LRe^{-1/2})$), where the leading-order Stokes approximation ceases to be valid. Since the correction comes from the effects of inertia in the far field, the rotating spheroid (triaxial ellipsoid) is approximated as a time-dependent point-force-dipole singularity, allowing for the reciprocal theorem integral to be evaluated in Fourier space. It is shown for all relevant cases that fluid inertia at
$O(Re^{3/2})$ leads to an increase in the time period of rotation compared with that in the Stokes limit, consistent with the results of recent numerical simulations at finite
$Re$. Finally, combination of the
$O(Re^{3/2})$ correction derived here with the
$O(Re)$ correction derived earlier by Dabade et al. (J. Fluid Mech., vol. 791, 2016, 631703) yields a uniformly valid description of the first effects of inertia for spheroids of all aspect ratios, including prediction of the arrest of rotation for extreme-aspect-ratio spheroids.
An orientational order transition in a sheared suspension of anisotropic particles
- Navaneeth K. Marath, Ruchir Dwivedi, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 811 / 25 January 2017
- Published online by Cambridge University Press:
- 12 December 2016, R3
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Under Stokesian conditions, a neutrally buoyant non-Brownian spheroid in simple shear flow rotates indefinitely in any of a one-parameter family of closed (Jeffery) orbits characterized by an orbit constant
$C$. The limiting values,
$C=0$ and
$C=\infty$, correspond to spinning and tumbling modes respectively. Hydrodynamics alone does not determine the distribution of spheroid orientations across Jeffery orbits in the absence of interactions, and the rheology of a dilute suspension of spheroids remains indeterminate. A combination of inertia and stochastic orientation fluctuations eliminates the indeterminacy. The steady-state Jeffery-orbit distribution arising from a balance of inertia and thermal fluctuations is shown to be of the Boltzmann equilibrium form, with a potential that depends on
$C$, the particle aspect ratio (
$\unicode[STIX]{x1D705}$), and a dimensionless shear rate (
$Re\,Pe_{r}$,
$Re$ and
$Pe_{r}$ being the Reynolds and rotary Péclet numbers), and therefore lends itself to a novel thermodynamic interpretation in
$C{-}\unicode[STIX]{x1D705}{-}Re\,Pe_{r}$ space. In particular, the transition of the potential from a single to a double-well structure, below a critical
$\unicode[STIX]{x1D705}$, has similarities to a thermodynamic phase transition, and the small-
$C$ and large-
$C$ minima are therefore identified with spinning and tumbling phases. The hysteretic dynamics within the two-phase tumbling–spinning envelope renders the rheology sensitively dependent on the precise shear rate history, the signature in simple shear flow being a multivalued viscosity at a given shear rate. The tumbling–spinning transition identified here is analogous to the coil–stretch transition in the polymer physics literature. It should persist under more general circumstances, and has implications for the suspension stress response in inhomogeneous shearing flows.
The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow
- Vivekanand Dabade, Navaneeth K. Marath, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 791 / 25 March 2016
- Published online by Cambridge University Press:
- 24 February 2016, pp. 631-703
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It is well known that, under inertialess conditions, the orientation vector of a torque-free neutrally buoyant spheroid in an ambient simple shear flow rotates along so-called Jeffery orbits, a one-parameter family of closed orbits on the unit sphere centred around the direction of the ambient vorticity (Jeffery, Proc. R. Soc. Lond. A, vol. 102, 1922, pp. 161–179). We characterize analytically the irreversible drift in the orientation of such torque-free spheroidal particles of an arbitrary aspect ratio, across Jeffery orbits, that arises due to weak inertial effects. The analysis is valid in the limit
$Re,St\ll 1$, where
$Re=(\dot{{\it\gamma}}L^{2}{\it\rho}_{f})/{\it\mu}$ and
$St=(\dot{{\it\gamma}}L^{2}{\it\rho}_{p})/{\it\mu}$ are the Reynolds and Stokes numbers, which, respectively, measure the importance of fluid inertial forces and particle inertia in relation to viscous forces at the particle scale. Here,
$L$ is the semimajor axis of the spheroid,
${\it\rho}_{p}$ and
${\it\rho}_{f}$ are the particle and fluid densities,
$\dot{{\it\gamma}}$ is the ambient shear rate, and
${\it\mu}$ is the suspending fluid viscosity. A reciprocal theorem formulation is used to obtain the contributions to the drift due to particle and fluid inertia, the latter in terms of a volume integral over the entire fluid domain. The resulting drifts in orientation at
$O(Re)$ and
$O(St)$ are evaluated, as a function of the particle aspect ratio, for both prolate and oblate spheroids using a vector spheroidal harmonics formalism. It is found that particle inertia, at
$O(St)$, causes a prolate spheroid to drift towards an eventual tumbling motion in the flow–gradient plane. Oblate spheroids, on account of the
$O(St)$ drift, move in the opposite direction, approaching a steady spinning motion about the ambient vorticity axis. The period of rotation in the spinning mode must remain unaltered to all orders in
$St$. For the tumbling mode, the period remains unaltered at
$O(St)$. At
$O(St^{2})$, however, particle inertia speeds up the rotation of prolate spheroids. The
$O(Re)$ drift due to fluid inertia drives a prolate spheroid towards a tumbling motion in the flow–gradient plane for all initial orientations and for all aspect ratios. Interestingly, for oblate spheroids, there is a bifurcation in the orientation dynamics at a critical aspect ratio of approximately 0.14. Oblate spheroids with aspect ratios greater than this critical value drift in a direction opposite to that for prolate spheroids, and eventually approach a spinning motion about the ambient vorticity axis starting from any initial orientation. For smaller aspect ratios, a pair of non-trivial repelling orbits emerge from the flow–gradient plane, and divide the unit sphere into distinct basins of orientations that asymptote to the tumbling and spinning modes. With further decrease in the aspect ratio, these repellers move away from the flow–gradient plane, eventually coalescing onto an arc of the great circle in which the gradient–vorticity plane intersects the unit sphere, in the limit of a vanishing aspect ratio. Thus, sufficiently thin oblate spheroids, similar to prolate spheroids, drift towards an eventual tumbling motion irrespective of their initial orientation. The drifts at
$O(St)$ and at
$O(Re)$ are combined to obtain the drift for a neutrally buoyant spheroid. The particle inertia contribution remains much smaller than the fluid inertia contribution for most aspect ratios and density ratios of order unity. As a result, the critical aspect ratio for the bifurcation in the orientation dynamics of neutrally buoyant oblate spheroids changes only slightly from its value based only on fluid inertia. The existence of Jeffery orbits implies a rheological indeterminacy, and the dependence of the suspension shear viscosity on initial conditions. For prolate spheroids and oblate spheroids of aspect ratio greater than 0.14, inclusion of inertia resolves the indeterminacy. Remarkably, the existence of the above bifurcation implies that, for a dilute suspension of oblate spheroids with aspect ratios smaller than 0.14, weak stochastic fluctuations (residual Brownian motion being analysed here as an example) play a crucial role in obtaining a shear viscosity independent of the initial orientation distribution. The inclusion of Brownian motion leads to a new smaller critical aspect ratio of approximately 0.013. For sufficiently large
$Re\,Pe_{r}$, the peak in the steady-state orientation distribution shifts rapidly from the spinning- to the tumbling-mode location as the spheroid aspect ratio decreases below this critical value; here,
$Pe_{r}=\dot{{\it\gamma}}/D_{r}$, with
$D_{r}$ being the Brownian rotary diffusivity, so that
$Re\,Pe_{r}$ measures the relative importance of inertial drift and Brownian rotary diffusion. The shear viscosity, plotted as a function of
$Re\,Pe_{r}$, exhibits a sharp transition from a shear-thickening to a shear-thinning behaviour, as the oblate spheroid aspect ratio decreases below 0.013. Our results are compared in detail to earlier analytical work for limiting cases involving either nearly spherical particles or slender fibres with weak inertia, and to the results of recent numerical simulations at larger values of
$Re$ and
$St$.
Effects of inertia and viscoelasticity on sedimenting anisotropic particles
- Vivekanand Dabade, Navaneeth K. Marath, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 778 / 10 September 2015
- Published online by Cambridge University Press:
- 30 July 2015, pp. 133-188
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An axisymmetric particle sedimenting in an otherwise quiescent Newtonian fluid, in the Stokes regime, retains its initial orientation. For the special case of a spheroidal geometry, we examine analytically the effects of weak inertia and viscoelasticity in driving the particle towards an eventual steady orientation independent of initial conditions. The generalized reciprocal theorem, together with a novel vector spheroidal harmonics formalism, is used to find closed-form analytical expressions for the
$O(\mathit{Re})$ inertial torque and the
$O(\mathit{De})$ viscoelastic torque acting on a sedimenting spheroid of an arbitrary aspect ratio. Here,
$\mathit{Re}=UL/{\it\nu}$ is the Reynolds number, with
$U$ being the sedimentation velocity,
$L$ the semi-major axis and
${\it\nu}$ the fluid kinematic viscosity, and is a measure of the inertial forces acting at the particle scale. The Deborah number,
$\mathit{De}=({\it\lambda}U)/L$, is a dimensionless measure of the fluid viscoelasticity, with
${\it\lambda}$ being the intrinsic relaxation time of the underlying microstructure. The analysis is valid in the limit
$\mathit{Re},\mathit{De}\ll 1$, and the effects of viscoelasticity are therefore modelled using the constitutive equation of a second-order fluid. The inertial torque always acts to turn the spheroid broadside-on, while the final orientation due to the viscoelastic torque depends on the ratio of the magnitude of the first (
$N_{1}$) to the second normal stress difference (
$N_{2}$), and the sign (tensile or compressive) of
$N_{1}$. For the usual case of near-equilibrium complex fluids – a positive and dominant
$N_{1}$ (
$N_{1}>0$,
$N_{2}<0$ and
$|N_{1}/N_{2}|>1$) – both prolate and oblate spheroids adopt a longside-on orientation. The viscoelastic torque is found to be remarkably sensitive to variations in
${\it\kappa}$ in the slender-fibre limit (
${\it\kappa}\gg 1$), where
${\it\kappa}=L/b$ is the aspect ratio,
$b$ being the radius of the spheroid (semi-minor axis). The angular dependence of the inertial and viscoelastic torques turn out to be identical, and one may then characterize the long-time orientation of the sedimenting spheroid based solely on a critical value (
$\mathit{El}_{c}$) of the elasticity number,
$\mathit{El}=\mathit{De}/\mathit{Re}$. For
$\mathit{El}<\mathit{El}_{c}~({>}\mathit{El}_{c})$, inertia (viscoelasticity) prevails with the spheroid settling broadside-on (longside-on). The analysis shows that
$\mathit{El}_{c}\sim O[(1/\text{ln}\,{\it\kappa})]$ for
${\it\kappa}\gg 1$, and the viscoelastic torque thus dominates for a slender rigid fibre. For a slender fibre alone, we also briefly analyse the effects of elasticity on fibre orientation outside the second-order fluid regime.
Stochastic dynamics of active swimmers in linear flows
- Mario Sandoval, Navaneeth K. Marath, Ganesh Subramanian, Eric Lauga
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- Journal:
- Journal of Fluid Mechanics / Volume 742 / 10 March 2014
- Published online by Cambridge University Press:
- 21 February 2014, pp. 50-70
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Most classical work on the hydrodynamics of low-Reynolds-number swimming addresses deterministic locomotion in quiescent environments. Thermal fluctuations in fluids are known to lead to a Brownian loss of the swimming direction, resulting in a transition from short-time ballistic dynamics to effective long-time diffusion. As most cells or synthetic swimmers are immersed in external flows, we consider theoretically in this paper the stochastic dynamics of a model active particle (a self-propelled sphere) in a steady general linear flow. The stochasticity arises both from translational diffusion in physical space, and from a combination of rotary diffusion and so-called run-and-tumble dynamics in orientation space. The latter process characterizes the manner in which the orientation of many bacteria decorrelates during their swimming motion. In contrast to rotary diffusion, the decorrelation occurs by means of large and impulsive jumps in orientation (tumbles) governed by a Poisson process. We begin by deriving a general formulation for all components of the long-time mean square displacement tensor for a swimmer with a time-dependent swimming velocity and whose orientation decorrelates due to rotary diffusion alone. This general framework is applied to obtain the convectively enhanced mean-squared displacements of a steadily swimming particle in three canonical linear flows (extension, simple shear and solid-body rotation). We then show how to extend our results to the case where the swimmer orientation also decorrelates on account of run-and-tumble dynamics. Self-propulsion in general leads to the same long-time temporal scalings as for passive particles in linear flows but with increased coefficients. In the particular case of solid-body rotation, the effective long-time diffusion is the same as that in a quiescent fluid, and we clarify the lack of flow dependence by briefly examining the dynamics in elliptic linear flows. By comparing the new active terms with those obtained for passive particles we see that swimming can lead to an enhancement of the mean-square displacements by orders of magnitude, and could be relevant for biological organisms or synthetic swimming devices in fluctuating environmental or biological flows.
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