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Table Of Integrals Series And Products

Active colloids are small scale materials capable of producing enhanced motion within fluid environments. The field of active colloids has grown rapidly over the last ten years and is approaching maturity where viable applications are within reach. In this review, recent advances are surveyed with a strong emphasis on developments that can enable autonomous applications, where colloids execute useful tasks without external interventions. These applications are likely to prove transformative as the resulting technologies will be significantly less complex than current methods. A survey of the requirements to achieve autonomous applications is provided, considering guidance, solution compatibilities, manufacture and function; with reference to recent developments in these capacities. Following on from this, progress towards applications in environmental remediation, lab-on-a-chip microfluidics and in vivo drug delivery is highlighted.

Active colloids: Progress and challenges towards realising autonomous applications

Fluid dynamics plays a crucial role in many cellular processes, including the locomotion of cells such as bacteria and spermatozoa. These organisms possess flagella, slender organelles whose time periodic motion in a fluid environment gives rise to motility. Sitting at the intersection of applied mathematics, physics and biology, the fluid dynamics of cell motility is one of the most successful applications of mathematical tools to the understanding of the biological world. Based on courses taught over several years, it details the mathematical modelling necessary to understand cell motility in fluids, covering phenomena ranging from single-cell motion to instabilities in cell populations. Each chapter introduces mathematical models to rationalise experiments, uses physical intuition to interpret mathematical results, highlights the history of the field and discusses notable current research questions. All mathematical derivations are included for students new to the field, and end-of-chapter exercises help consolidate understanding and practise applying the concepts.

The Fluid Dynamics of Cell Motility

At low latitudes, cirrus are ubiquitous and can be in excess of 100 C colder than the surface, limiting the amount of sunlight absorbed by the earth s atmosphere and surface, and reducing its loss of heat. Here we present aircraft measurements within cirrus over southern Florida indicating that ice crystals have smaller sizes and are more reflective than is assumed in most current climate models. If the measurements are generally representative of low-latitude cirrus, they point to a first-order correction to represenntions of how these clouds affect the earth's climate.

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Small, highly reflective ice crystals in low-latitude cirrus

Small objects can swim by generating around them fields or gradients which in turn induce fluid motion past their surface by phoretic surface effects. We quantify for arbitrary swimmer shapes and surface patterns, how efficient swimming requires both surface ``activity'' to generate the fields, and surface ``phoretic mobility.'' We show in particular that (i) swimming requires symmetry breaking in either or both of the patterns of "activity" and ``mobility,'' and (ii) for a given geometrical shape and surface pattern, the swimming velocity is size-independent. In addition, for given available surface properties, our calculation framework provides a guide for optimizing the design of swimmers.

Designing phoretic micro- and nano-swimmers

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 inertial torque and the viscoelastic torque acting on a sedimenting spheroid of an arbitrary aspect ratio. Here, is the Reynolds number, with being the sedimentation velocity, the semi-major axis and the fluid kinematic viscosity, and is a measure of the inertial forces acting at the particle scale. The Deborah number, , is a dimensionless measure of the fluid viscoelasticity, with being the intrinsic relaxation time of the underlying microstructure. The analysis is valid in the limit , 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 ( ) to the second normal stress difference ( ), and the sign (tensile or compressive) of . For the usual case of near-equilibrium complex fluids – a positive and dominant ( , and ) – both prolate and oblate spheroids adopt a longside-on orientation. The viscoelastic torque is found to be remarkably sensitive to variations in in the slender-fibre limit ( ), where is the aspect ratio, 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 ( ) of the elasticity number, . For , inertia (viscoelasticity) prevails with the spheroid settling broadside-on (longside-on). The analysis shows that for , 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.

Effects of inertia and viscoelasticity on sedimenting anisotropic particles

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 , where and 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, is the semimajor axis of the spheroid, and are the particle and fluid densities, is the ambient shear rate, and 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 and 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 , causes a prolate spheroid to drift towards an eventual tumbling motion in the flow–gradient plane. Oblate spheroids, on account of the 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 . For the tumbling mode, the period remains unaltered at . At , however, particle inertia speeds up the rotation of prolate spheroids. The 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 and at 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 , 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, , with being the Brownian rotary diffusivity, so that measures the relative importance of inertial drift and Brownian rotary diffusion. The shear viscosity, plotted as a function of , 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 and .

The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow

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.

Stochastic dynamics of active swimmers in linear flows

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 . The parameter is related to the ratio of the vorticity to the extension (given by ), and ranges from to 1, with and being planar extensional flow, simple shear flow and solid-body rotation respectively. The unit vectors and are unit vectors along the flow and gradient axes of the simple shear flow ( ). The trajectories, as described by a unit vector along the spheroid symmetry axis, are closed orbits for , where for an oblate (a prolate) spheroid of aspect ratio . 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 and , where and are the Reynolds ( ) and Stokes numbers ( ) respectively, are derived using a reciprocal theorem formulation. Here, is the semimajor axis of the spheroid, is the viscosity of the suspending fluid, is the shear rate, and and 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 ( ). Here, even weak inertia, for asymptotically long times, of or , will affect the leading-order orientation distribution on account of the indeterminate nature of the distribution across orbits in the Stokes limit. For , 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 and , are identified in the plane. For the majority of ( ) 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 ( ) 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 Peclet number, ( , where is the rotary Brownian diffusivity), is included to eliminate the aforementioned dependence on the initial orientation distribution for certain ( , ) 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 space. Here, 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 – – 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 inertial orientation dynamics of anisotropic particles in planar linear flows

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 ( ; is the unit vector in the flow direction, being the coordinate along the gradient direction), in its long-time orbit set up by the weak fluid inertial drift at . Here, is the microscale Reynolds number, a dimensionless measure of the fluid inertial effects on the length scale of the spheroid, and is defined as , where is the semimajor axis of the spheroid, and are respectively the viscosity and density of the fluid, and 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 , and can be either the tumbling or the spinning orbit for oblate spheroids of aspect ratios less than . 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 is expressed as a volume integral using a generalized reciprocal theorem formulation. For , it is shown that the correction at 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 , 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 ), 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 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 . Finally, combination of the correction derived here with the 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.

Navaneeth K. Marath

Papers

Effects of inertia and viscoelasticity on sedimenting anisotropic particles

The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow

Stochastic dynamics of active swimmers in linear flows

The inertial orientation dynamics of anisotropic particles in planar linear flows

The effect of inertia on the time period of rotation of an anisotropic particle in simple shear flow