38 results
Inertial migration of a sphere in plane Couette flow
- Prateek Anand, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 977 / 25 December 2023
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- 19 December 2023, A33
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We study the inertial migration of a torque-free neutrally buoyant sphere in wall-bounded plane Couette flow over a wide range of channel Reynolds numbers $Re_c$ in the limit of small particle Reynolds number ($Re_p\ll 1$) and confinement ratio ($\lambda \ll 1$). Here, $Re_c = V_{wall}H/\nu$, where $H$ denotes the separation between the channel walls, $V_\text {wall}$ denotes the speed of the moving wall, and $\nu$ is the kinematic viscosity of the Newtonian suspending fluid. Also, $\lambda = a/H$, where $a$ is the sphere radius, with $Re_p=\lambda ^2 Re_c$. The channel centreline is found to be the only (stable) equilibrium below a critical $Re_c$ ($\approx 148$), consistent with the predictions of earlier small-$Re_c$ analyses. A supercritical pitchfork bifurcation at the critical $Re_c$ creates a pair of stable off-centre equilibria, located symmetrically with respect to the centreline, with the original centreline equilibrium becoming unstable simultaneously. The new equilibria migrate wall-ward with increasing $Re_c$. In contrast to the inference based on recent computations, the aforementioned bifurcation occurs for arbitrarily small $Re_p$ provided that $\lambda$ is sufficiently small. An analogous bifurcation occurs in the two-dimensional scenario, that is, for a circular cylinder suspended freely in plane Couette flow, with the critical $Re_c$ being approximately $110$.
Inertial migration of a neutrally buoyant spheroid in plane Poiseuille flow
- Prateek Anand, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 974 / 10 November 2023
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- 03 November 2023, A39
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We study the cross-stream inertial migration of a torque-free neutrally buoyant spheroid, of an arbitrary aspect ratio $\kappa$, in wall-bounded plane Poiseuille flow for small particle Reynolds numbers ($Re_p\ll 1$) and confinement ratios ($\lambda \ll 1$), with the channel Reynolds number, $Re_c = Re_p/\lambda ^2$, assumed to be arbitrary; here $\lambda =L/H$, where $L$ is the semi-major axis of the spheroid and $H$ denotes the separation between the channel walls. In the Stokes limit ($Re_p =0)$, and for $\lambda \ll 1$, a spheroid rotates along any of an infinite number of Jeffery orbits parameterized by an orbit constant $C$, while translating with a time-dependent speed along a given ambient streamline. Weak inertial effects stabilize either the spinning ($C=0$) or tumbling orbit ($C=\infty$), or both, depending on $\kappa$. The asymptotic separation of the Jeffery rotation and orbital drift time scales, from that associated with cross-stream migration, implies that migration occurs due to a Jeffery-averaged lift velocity. Although the magnitude of this averaged lift velocity depends on $\kappa$ and $C$, the shape of the lift profiles are identical to those for a sphere, regardless of $Re_c$. In particular, the equilibrium positions for a spheroid remain identical to the classical Segre–Silberberg ones for a sphere, starting off at a distance of about $0.6(H/2)$ from the channel centreline for small $Re_c$, and migrating wallward with increasing $Re_c$. For spheroids with $\kappa \sim O(1)$, the Jeffery-averaged analysis is valid for $Re_p\ll 1$; for extreme aspect ratio spheroids, the regime of validity becomes more restrictive being given by $Re_p \kappa /\ln \kappa \ll 1$ and $Re_p/\kappa \ll 1$ for $\kappa \rightarrow \infty$ (slender fibres) and $\kappa \rightarrow 0$ (flat disks), respectively.
The inertial orientation dynamics of anisotropic particles in planar linear flows – CORRIGENDUM
- Navaneeth K. Marath, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 949 / 25 October 2022
- Published online by Cambridge University Press:
- 07 October 2022, E1
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Motion of a sphere in a viscous density stratified fluid
- Arun Kumar Varanasi, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 949 / 25 October 2022
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- 29 September 2022, A29
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We examine the translation of a sphere in a stratified ambient in the limit of small Reynolds numbers ($Re \ll 1$) and viscous Richardson numbers ($Ri_v \ll 1$); here, $Re = {\rho Ua}/{\mu }$ and $Ri_v = {\gamma a^3 g}/{\mu U}$, with $a$ being the sphere radius, $U$ the translation speed, $\rho$ and $\mu$ the density and viscosity of the stratified ambient, $g$ the acceleration due to gravity, and $\gamma$ the density gradient characterizing the ambient stratification. In contrast to most earlier efforts, our study considers the convection-dominant limit corresponding to $Pe = {Ua}/{D} \gg 1$, $D$ being the diffusivity of the stratifying agent. We characterize in detail the velocity and density fields around the particle in what we term the Stokes stratification regime, defined by $Re \ll Ri_v^{{1}/{3}} \ll 1$, and corresponding to the dominance of buoyancy over inertial forces. Buoyancy forces associated with the perturbed stratification fundamentally alter the viscously dominated fluid motion at large distances of order the stratification screening length that scales as $a\,Ri_v^{-{1}/{3}}$. The motion at these distances transforms from the familiar fore–aft symmetric Stokesian form to a fore–aft asymmetric pattern of recirculating cells with primarily horizontal motion within, except in the vicinity of the rear stagnation streamline. At larger distances, the motion is vanishingly small except within (a) an axisymmetric horizontal wake whose vertical extent grows as $O(r_t^{{2}/{5}})$, $r_t$ being the distance in the plane perpendicular to translation, and (b) a buoyant reverse jet behind the particle that narrows as the inverse square root of distance downstream. As a result, for $Pe = \infty$, the motion close to the rear stagnation streamline starts off pointing in the direction of translation, in the inner region, and decaying as the inverse of the downstream distance; the motion reverses beyond distance $1.15a\,Ri_v^{-{1}/{3}}$, with the eventual reverse flow in the far-field buoyant jet again decaying as the inverse of the distance downstream. For large but finite $Pe$, the narrowing jet is smeared out beyond a distance of $O(a\,Ri_v^{-{1}/{2}}\, Pe^{{1}/{2}})$, leading to an exponential decay of the aforementioned reverse flow.
Shape dynamics and rheology of dilute suspensions of elastic and viscoelastic particles
- Phani Kanth Sanagavarapu, Ganesh Subramanian, Prabhu R. Nott
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- Journal of Fluid Mechanics / Volume 949 / 25 October 2022
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- 29 September 2022, A22
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This paper examines the shape dynamics of deformable elastic and viscoelastic particles in an ambient Newtonian fluid subjected to simple shear. The particles are allowed to undergo large deformation, with the elastic stress determined using the neo-Hookean constitutive relation. We first present a method to determine the shape dynamics of initially ellipsoidal particles that is an extension of the method of Roscoe (J. Fluid Mech., vol. 28, issue 2, 1967, pp. 273–293), originally used to determine the shape at steady state of an initially spherical particle. We show that our method recovers earlier results for the in-plane trembling and tumbling dynamics of initially prolate spheroids in simple shear flow, obtained by a different approach. We then examine the in-plane dynamics of oblate spheroids and triaxial ellipsoids in simple shear flow, and show that they too, like prolate spheroids, exhibit time-periodic tumbling or trembling dynamics, depending on the initial aspect ratios of the particle and the elastic capillary number $G \equiv \mu \dot {\gamma }/\eta$, where $\mu$ is the viscosity of the fluid, $\eta$ is the elastic shear modulus of the particle and $\dot {\gamma }$ is the shear rate. In addition, we find a novel state wherein the particle extends indefinitely in time and asymptotically aligns with the flow axis. We demarcate all the dynamical regimes in the parameter space comprising $G$ and the initial particle aspect ratios. When the particles are viscoelastic, damped oscillatory dynamics is observed for initially spherical particles, and the tumbling–trembling boundary is altered for initially prolate spheroids so as to favour tumbling.
Inertio–elastic instability of a vortex column
- Anubhab Roy, Piyush Garg, Jumpal Shashikiran Reddy, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 937 / 25 April 2022
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- 28 February 2022, A27
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We analyse the instability of a vortex column in a dilute polymer solution at large ${{Re}}$ and ${{De}}$ with ${{El}} = {{De}}/{{Re}}$, the elasticity number, being finite. Here, ${{Re}} = \varOmega _0 a^2/\nu$ and ${{De}} = \varOmega _0 \tau$ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity ($\varOmega _0$), the radius of the column ($a$), the total (solvent plus polymer) kinematic viscosity ($\nu = (\mu _s +\mu _p)/\rho$ with $\mu _s$ and $\mu _p$ being the solvent and polymer contributions to the viscosity) and the polymeric relaxation time ($\tau$). The stability of small-amplitude perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by ${E} = {{El}}(1-\beta )$, $\beta$ being the ratio of the solvent to the solution viscosity. The neglect of the relaxation terms, in the said limit, implies that the polymer solution supports undamped elastic shear waves propagating relative to the base-state flow. Unlike the neutrally stable inviscid case, an instability of the vortex column arises for finite ${E}$ due to a pair of elastic shear waves being driven into a resonant interaction under the differential convection by the irrotational shearing flow outside the core. An asymptotic analysis for the Rankine profile shows the absence of an elastic threshold for this instability. The growth rate is $O(\varOmega _0)$ for order unity $E$, although it becomes transcendentally small for ${E} \ll 1$, being $O(\varOmega _0 {E}^2{\rm e}^{-1/{E}^{{1}/{2}}})$. An accompanying numerical investigation shows that the instability persists for smooth monotonically decreasing vorticity profiles, provided the radial extent of the transition region (from the rotational core to the irrotational exterior) is less than a certain ${E}$-dependent threshold.
Linear stability of a rotating liquid column revisited
- Pulkit Dubey, Anubhab Roy, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 933 / 25 February 2022
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- 06 January 2022, A55
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We revisit the somewhat classical problem of the linear stability of a rigidly rotating liquid column in this article. Although the literature pertaining to this problem dates back to 1959, the relation between inviscid and viscous stability criteria has not yet been clarified. While the viscous criterion for stability, given by $We < n^2 + k^2 -1$, is both necessary and sufficient, this relation has only been shown to be sufficient in the inviscid case. Here, $We = \rho \varOmega ^2 a^3 / \gamma$ is the Weber number and measures the relative magnitudes of the centrifugal and surface tension forces, with $\varOmega$ being the angular velocity of the rigidly rotating column, $a$ the column radius, $\rho$ the density of the fluid and $\gamma$ the surface tension coefficient; $k$ and $n$ denote the axial and azimuthal wavenumbers of the imposed perturbation. We show that the subtle difference between the inviscid and viscous criteria arises from the surprisingly complicated picture of inviscid stability in the $We$–$k$ plane. For all $n > 1$, the viscously unstable region, corresponding to $We > n^2 + k^2-1$, contains an infinite hierarchy of inviscidly stable islands ending in cusps, with a dominant leading island. Only the dominant island, now infinite in extent along the $We$ axis, persists for $n=1$. This picture may be understood, based on the underlying eigenspectrum, as arising from the cascade of coalescences between a retrograde mode, that is the continuation of the cograde surface-tension-driven mode across the zero Doppler frequency point, and successive retrograde Coriolis modes constituting an infinite hierarchy.
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 of Fluid Mechanics / Volume 933 / 25 February 2022
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- 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 centre-mode instability of viscoelastic plane Poiseuille flow
- Mohammad Khalid, Indresh Chaudhary, Piyush Garg, V. Shankar, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 915 / 25 May 2021
- Published online by Cambridge University Press:
- 12 March 2021, A43
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A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a ‘centre mode’ with phase speed close to the maximum base-flow velocity, $U_{max}$. The governing dimensionless groups are the Reynolds number $Re = \rho U_{max} H/\eta$, the elasticity number $E = \lambda \eta /(H^2 \rho )$ and the ratio of solvent to solution viscosity $\beta = \eta _s/\eta$; here, $\lambda$ is the polymer relaxation time, $H$ is the channel half-width and $\rho$ is the fluid density. For experimentally relevant values (e.g. $E \sim 0.1$ and $\beta \sim 0.9$), the critical Reynolds number, $Re_c$, is around $200$, with the associated eigenmodes being spread out across the channel. For $E(1-\beta ) \ll 1$, with $E$ fixed, corresponding to strongly elastic dilute polymer solutions, $Re_c \propto (E(1-\beta ))^{-3/2}$ and the critical wavenumber $k_c \propto (E(1-\beta ))^{-1/2}$. The unstable eigenmode in this limit is confined in a thin layer near the channel centreline. These features are largely analogous to the centre-mode instability in viscoelastic pipe flow (Garg et al., Phys. Rev. Lett., vol. 121, 2018, 024502), and suggest a universal linear mechanism underlying the onset of turbulence in both channel and pipe flows of sufficiently elastic dilute polymer solutions. Although the centre-mode instability continues down to $\beta \sim 10^{-2}$ for pipe flow, it ceases to exist for $\beta < 0.5$ in channels. Whereas inertia, elasticity and solvent viscous effects are simultaneously required for this instability, a higher viscous threshold is required for channel flow. Further, in the opposite limit of $\beta \rightarrow 1$, the centre-mode instability in channel flow continues to exist at $Re \approx 5$, again in contrast to pipe flow where the instability ceases to exist below $Re \approx 63$, regardless of $E$ or $\beta$. Our predictions are in reasonable agreement with experimental observations for the onset of turbulence in the flow of polymer solutions through microchannels.
An anisotropic particle in a simple shear flow: an instance of chaotic scattering
- Mahan Raj Banerjee, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 913 / 25 April 2021
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- 19 February 2021, A2
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In the Stokesian limit, the streamline topology around a single neutrally buoyant sphere is identical to the topology of pair-sphere pathlines, both in an ambient simple shear flow. In both cases there are fore–aft symmetric open and closed trajectories spatially demarcated by an axisymmetric separatrix surface. We show that the topology of the fluid pathlines around a neutrally buoyant freely rotating spheroid, in simple shear flow, is profoundly different, and will have a crucial bearing on transport from such particles in shearing flows. An inertialess non-Brownian spheroid in a simple shear flow rotates indefinitely in any one of a one-parameter family of Jeffery orbits. The parameter is the orbit constant $C$, with $C = 0$ and $C = \infty$ denoting the limiting cases of a spinning (log-rolling) spheroid, and a spheroid tumbling in the flow–gradient plane, respectively. The streamline pattern around a spinning spheroid is qualitatively identical to that around a sphere regardless of its aspect ratio. For a spheroid in any orbit other than the spinning one ($C >0$), the velocity field being time dependent in all such cases, the fluid pathlines may be divided into two categories. Pathlines in the first category extend from upstream to downstream infinity without ever crossing the flow axis; unlike the spinning case, these pathlines are fore–aft asymmetric, suffering a net displacement in both the gradient and vorticity directions. The second category includes primarily those pathlines that loop around the spheroid, and to a lesser extent those that cross the flow axis, without looping around the spheroid, reversing direction in the process. The residence time, in the neighbourhood of the spheroid, is a smooth function of upstream conditions for pathlines belonging to the first category. In contrast, the number of loops, and thence, the residence time associated with pathlines in the second category, is extremely sensitive to upstream conditions. Plots reveal a fractal structure with singularities distributed on a Cantor-like set, suggesting the existence of a chaotic saddle in the vicinity of the spheroid.
Linear instability of viscoelastic pipe flow
- Indresh Chaudhary, Piyush Garg, Ganesh Subramanian, V. Shankar
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- Journal of Fluid Mechanics / Volume 908 / 10 February 2021
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- 03 December 2020, A11
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A modal stability analysis shows that pressure-driven pipe flow of an Oldroyd-B fluid is linearly unstable to axisymmetric perturbations, in stark contrast to its Newtonian counterpart which is linearly stable at all Reynolds numbers. The dimensionless groups that govern stability are the Reynolds number $Re = \rho U_{max} R /\eta$, the elasticity number $E = \lambda \eta /(R^2 \rho )$ and the ratio of solvent to solution viscosity $\beta = \eta _s/\eta$; here, $R$ is the pipe radius, $U_{max}$ is the maximum velocity of the base flow, $\rho$ is the fluid density and $\lambda$ is the microstructural relaxation time. The unstable mode has a phase speed close to $U_{max}$ over the entire unstable region in ($Re$, $E$, $\beta$) space. In the asymptotic limit $E (1-\beta ) \ll 1$, the critical Reynolds number for instability diverges as $Re_c \sim (E (1-\beta ))^{-3/2}$, the critical wavenumber increases as $k_c \sim (E (1-\beta ))^{-1/2}$, and the unstable eigenfunction is localized near the centreline, implying that the unstable mode belongs to a class of viscoelastic centre modes. In contrast, for $\beta \rightarrow 1$ and $E \sim 0.1$, $Re_c$ can be as low as $O(100)$, with the unstable eigenfunction no longer being localized near the centreline. Unlike the Newtonian transition which is dominated by nonlinear processes, the linear instability discussed in this study could be very relevant to the onset of turbulence in viscoelastic pipe flows. The prediction of a linear instability is, in fact, consistent with several experimental studies on pipe flow of polymer solutions, ranging from reports of ‘early turbulence’ in the 1970s to the more recent discovery of ‘elasto-inertial turbulence’ (Samanta et al., Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 10557–10562). The instability identified in this study comprehensively dispels the prevailing notion of pipe flow of viscoelastic fluids being linearly stable in the $Re$–$W$ plane ($W = Re \, E$ being the Weissenberg number), marking a possible paradigm shift in our understanding of transition in rectilinear viscoelastic shearing flows. The predicted unstable eigenfunction should form a template in the search for novel nonlinear elasto-inertial states, and could provide an alternate route to the maximal drag-reduced state in polymer solutions. The latter has thus far been explained in terms of a viscoelastic modification of the nonlinear Newtonian coherent structures.
Enhanced velocity fluctuations in interacting swimmer suspensions
- Sankalp Nambiar, Piyush Garg, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 907 / 25 January 2021
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- 25 November 2020, A26
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This paper characterizes the nature of velocity fluctuations in swimmer suspensions by determining the fluid velocity variance and the diffusivity of immersed passive tracers in dilute suspensions of hydrodynamically interacting slender microswimmers. The swimmers considered include straight-swimmers whose orientations change only on account of hydrodynamic interactions, and run-and-tumble particles (RTPs) whose orientations change in addition due to tumble events obeying Poisson statistics. In a dilute non-interacting swimmer suspension, the fluid velocity variance is finite and the covariance is short ranged, decaying for distances larger than the swimmer length. In contrast, we show, for a suspension of interacting straight-swimmers, that pair interactions lead to a non-decaying velocity covariance, and a variance that diverges logarithmically with system size. For suspensions of RTPs, the aforementioned divergence is arrested due to tumbling. While the variance remains finite, and the covariance short ranged, for suspensions of interacting rapid tumbling RTPs (short run lengths), the underlying straight-swimmer divergence manifests as a logarithmic increase of the variance with the swimmer run length for persistent RTPs (long run lengths), with a correspondingly long-ranged covariance. The tracer mean squared displacement undergoes an increasingly broad crossover from the ballistic to the diffusive regime for persistent RTPs, with the tracer diffusivity exhibiting a stronger linear increase with the swimmer run length. Our analysis explains the bifurcation of the velocity variance and tracer diffusivities between pusher and puller suspensions, as well as numerous observations of a volume-fraction-dependent crossover time for passive tracer dynamics.
Concentration banding instability of a sheared bacterial suspension
- Laxminarsimharao Vennamneni, Piyush Garg, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 904 / 10 December 2020
- Published online by Cambridge University Press:
- 06 October 2020, A7
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We demonstrate a novel shear-induced mechanism for growth of concentration fluctuations in a bacterial suspension. Using a linear stability analysis, a homogeneous bacterial suspension, subject to a simple shear flow, is shown to be susceptible to exponentially growing layering perturbations in the shear rate and bacterial concentration. A semi-analytical expression for the growth rate of concentration perturbations is first obtained using the method of multiple scales, in the limit where the time scales characterizing the positional and orientation degrees of freedom are well separated. Next, the eigenspectrum obtained numerically from a full linear stability analysis is used to validate and extend the multiple scales result, and draw a contrast with the known orientation-shear instability. Finally, fully nonlinear simulations, but restricted to one-dimensional variations of the relevant fields (velocity, concentration and swimmer orientation distribution) show that the initial instability leads to gradient-banded velocity profiles, with a local depletion of bacteria at the interface between the homogeneous shear bands. Our results demonstrate that long-ranged hydrodynamic interactions serve as an alternate explanation for recent observations of shear bands in bacterial suspensions.
Shear-induced migration of microswimmers in pressure-driven channel flow
- Laxminarsimharao Vennamneni, Sankalp Nambiar, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 890 / 10 May 2020
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- 12 March 2020, A15
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We study shear-induced migration in a dilute suspension of microswimmers (modelled as active Brownian particles or ABPs) subject to plane Poiseuille flow. For wide channels characterized by $U_{s}/HD_{r}\ll 1$, the separation between time scales characterizing the swimmer orientation dynamics (of $O(D_{r}^{-1})$) and those that characterize migration across the channel (of $O(H^{2}D_{r}/U_{s}^{2})$), allows for use of the method of multiple scales to derive a drift-diffusion equation for the swimmer concentration profile; here, $U_{s}$ is the swimming speed, $H$ is the channel half-width and $D_{r}$ is the swimmer rotary diffusivity. The steady state concentration profile is a function of the Péclet number, $Pe=U_{f}/(D_{r}H)$ ($U_{f}$ being the channel centreline velocity), and the swimmer aspect ratio $\unicode[STIX]{x1D705}$. Swimmers with $\unicode[STIX]{x1D705}\gg 1$ (with $\unicode[STIX]{x1D705}\sim O(1)$), in the regime $1\ll \text{Pe}\ll \unicode[STIX]{x1D705}^{3}$ ($Pe\sim O(1)$), migrate towards the channel walls, corresponding to a high-shear trapping behaviour. For $Pe\gg \unicode[STIX]{x1D705}^{3}$ ($Pe\gg 1$ for $\unicode[STIX]{x1D705}\sim O(1)$), however, swimmers migrate towards the centreline, corresponding to a low-shear trapping behaviour. Interestingly, within the low-shear trapping regime, swimmers with $\unicode[STIX]{x1D705}<2$ asymptote to a $Pe$-independent concentration profile for large $Pe$, while those with $\unicode[STIX]{x1D705}\geqslant 2$ exhibit a ‘centreline collapse’ for $Pe\rightarrow \infty$. The prediction of low-shear trapping, validated by Langevin simulations, is the first explanation of recent experimental observations (Barry et al., J. R. Soc. Interface, vol. 12 (112), 2015, 20150791). We organize the high-shear and low-shear trapping regimes on a $Pe{-}\unicode[STIX]{x1D705}$ plane, thereby highlighting the singular behaviour of infinite-aspect-ratio swimmers.
Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow
- Indresh Chaudhary, Piyush Garg, V. Shankar, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 881 / 25 December 2019
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- 24 October 2019, pp. 119-163
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A linear stability analysis of plane Poiseuille flow of an upper-convected Maxwell (UCM) fluid, bounded between rigid plates separated by a distance $2L$, has been carried out to investigate the interplay of elasticity and inertia on flow stability. The stability is governed by the following dimensionless groups: the Reynolds number $Re=\unicode[STIX]{x1D70C}U_{max}L/\unicode[STIX]{x1D702}$ and the elasticity number $E\equiv W/Re=\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}/(\unicode[STIX]{x1D70C}L^{2})$, where $W=\unicode[STIX]{x1D706}U_{max}/L$ is the Weissenberg number. Here, $\unicode[STIX]{x1D70C}$ is the fluid density, $\unicode[STIX]{x1D702}$ is the fluid viscosity, $\unicode[STIX]{x1D706}$ is the micro-structural relaxation time and $U_{max}$ is the maximum base-flow velocity. The stability is analysed for two-dimensional perturbations using both pseudo-spectral and shooting methods. We also analyse the linear stability of plane Couette flow which, along with the results for plane Poiseuille flow, yields insight into the structure of the complete elasto-inertial eigenspectrum. While the general features of the spectrum for both flows remain similar, plane Couette flow is found to be stable over the range of parameters examined ($Re\leqslant 10^{4},E\leqslant 0.01$). On the other hand, plane Poiseuille flow appears to be susceptible to an infinite hierarchy of elasto-inertial instabilities. Over the range of parameters examined, there are up to seven distinct neutral stability curves in the $Re$–$k$ plane (here $k$ is the perturbation wavenumber in the flow direction). Based on the symmetry of the eigenfunctions for the streamwise velocity about the centreline, four of these instabilities are antisymmetric, while the other three are symmetric. The neutral stability curve corresponding to the first antisymmetric mode is shown to be a continuation (to finite $E$) of the Tollmien–Schlichting (TS) instability already present for Newtonian channel flow. As $E$ is increased beyond $0.0016$, a new elastic mode appears at $Re\sim 10^{4}$, which coalesces with the continuation of the TS mode for a range of $Re$, thereby yielding a single unstable mode in this range. This trend persists until $E\sim 0.0021$, beyond which this neutral curve splits into two separate ones in the $Re$–$k$ plane. The new elastic mode which arises out of this splitting has been found to be the most unstable, with the lowest critical Reynolds number $Re_{c}\approx 1210.9$ for $E=0.0066$. The neutral curves for both the continuation of the original TS mode, and the new elastic antisymmetric mode, form closed loops upon further increase in $E$, which eventually vanish at sufficiently high $E$. For $E\ll 1$, the critical Reynolds number and wavenumber scale as $Re_{c}\sim E^{-1}$ and $k_{c}\sim E^{-1/2}$ for the first two of the symmetric modal families, and as $Re_{c}\sim E^{-5/4}$ for first two of the antisymmetric modal families; $k_{c}\sim E^{-1/4}$ for the third antisymmetric family. The critical wave speed for all of these unstable eigenmodes scales as $c_{r,c}\sim E^{1/2}$ for $E\ll 1$, implying that the modes belong to a class of ‘wall modes’ in viscoelastic flows with disturbances being confined in a thin region near the wall. The present study shows that, surprisingly, even in plane shear flows, elasticity acting along with inertia can drive novel instabilities absent in the Newtonian limit.
Stress relaxation in a dilute bacterial suspension: the active–passive transition
- Sankalp Nambiar, Phanikanth S., P. R. Nott, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 870 / 10 July 2019
- Published online by Cambridge University Press:
- 15 May 2019, pp. 1072-1104
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This paper follows a recent article of Nambiar et al. (J. Fluid Mech., vol. 812, 2017, pp. 41–64) on the linear rheological response of a dilute bacterial suspension (e.g. E. coli) to impulsive starting and stopping of simple shear flow. Here, we analyse the time dependent nonlinear rheology for a pair of linear flows – simple shear (a canonical weak flow) and uniaxial extension (a canonical strong flow), again in response to impulsive initiation and cessation. The rheology is governed by the bacterium orientation distribution which satisfies a kinetic equation that includes rotation by the imposed flow, and relaxation to isotropy via rotary diffusion and tumbling. The relevant dimensionless parameters are the Péclet number $Pe\equiv \dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D70F}$, which dictates the importance of flow-induced orientation anisotropy, and $\unicode[STIX]{x1D70F}D_{r}$, which quantifies the relative importance of the two intrinsic orientation decorrelation mechanisms (tumbling and rotary diffusion). Here, $\unicode[STIX]{x1D70F}$ is the mean run duration of a bacterium that exhibits a run-and-tumble dynamics, $D_{r}$ is the intrinsic rotary diffusivity of the bacterium and $\dot{\unicode[STIX]{x1D6FE}}$ is the characteristic magnitude of the imposed velocity gradient. The solution of the kinetic equation is obtained numerically using a spectral Galerkin method, that yields the rheological properties (the shear viscosity, the first and second normal stress differences for simple shear, and the extensional viscosity for uniaxial extension) over the entire range of $Pe$. For simple shear, we find that the stress relaxation predicted by our analysis at small $Pe$ is in good agreement with the experimental observations of Lopez et al. (Phys. Rev. Lett., vol. 115, 2015, 028301). However, the analysis at large $Pe$ yields relaxations that are qualitatively different. Upon step initiation of shear, the rheological response in the experiments corresponds to a transition from a nearly isotropic suspension of active swimmers at small $Pe$, to an apparently (nearly) isotropic suspension of passive rods at large $Pe$. In contrast, the computations yield the expected transition to a nearly flow-aligned suspension of passive rigid rods at high $Pe$. We probe this active–passive transition systematically, complementing the numerical solution with analytical solutions obtained from perturbation expansions about appropriate base states. Our study suggests courses for future experimental and analytical studies that will help understand relaxation phenomena in active suspensions.
Heat or mass transport from drops in shearing flows. Part 2. Inertial effects on transport
- Deepak Krishnamurthy, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 850 / 10 September 2018
- Published online by Cambridge University Press:
- 06 July 2018, pp. 484-524
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We analyse the singular effects of weak inertia on the heat (or equivalently mass) transport problem from drops in linear shearing flows. For small spherical drops embedded in hyperbolic planar linear flows, which constitute a one-parameter family (the parameter being $\unicode[STIX]{x1D6FC}$ with $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 1$, and whose extremal members are simple shear ($\unicode[STIX]{x1D6FC}=0$) and planar extension ($\unicode[STIX]{x1D6FC}=1$)), there are two distinct regimes for scalar (heat or mass) transport at large Péclet numbers ($Pe$) depending on the exterior streamline topology (Krishnamurthy & Subramanian, J. Fluid Mech., vol. 850, 2018, pp. 439–483). When the drop-to-medium viscosity ratio ($\unicode[STIX]{x1D706}$) is larger than a critical value, $\unicode[STIX]{x1D706}_{c}=2\unicode[STIX]{x1D6FC}/(1-\unicode[STIX]{x1D6FC})$, the drop is surrounded by a region of closed streamlines in the inertialess limit ($Re=0$, $Re$ being the drop Reynolds number). Convection is incapable of transporting heat away on account of the near-field closed streamline topology, and the transport remains diffusion limited even for $Pe\rightarrow \infty$. However, weak inertia breaks open the closed streamline region, giving way to finite-$Re$ spiralling streamlines and convectively enhanced transport. For $Re=0$ the closed streamlines on the drop surface, for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$, are Jeffery orbits, a terminology originally used to describe the trajectories of an axisymmetric rigid particle in a simple shear flow. Based on this identification, a novel boundary layer analysis that employs a surface-flow-aligned non-orthogonal coordinate system, is used to solve the transport problem in the dual asymptotic limit $Re\ll 1$, $RePe\gg 1$, corresponding to the regime where inertial convection balances diffusion in an $O(RePe)^{-1/2}$ boundary layer. Further, the separation of time scales in the aforementioned limit, between rapid convection due to the Stokesian velocity field and the slower convection by the $O(Re)$ inertial velocity field, allows one to average the convection–diffusion equation over the phase of the Stokesian surface streamlines (Jeffery orbits), allowing a simplification of the original three-dimensional non-axisymmetric transport problem to a form resembling a much simpler axisymmetric one. A self-similar ansatz then leads to the boundary layer temperature field, and the resulting Nusselt number is given by $Nu={\mathcal{H}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})(RePe)^{1/2}$ with ${\mathcal{H}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ given in terms of a one-dimensional integral; the prefactor ${\mathcal{H}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ diverges for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{+}$ due to assumptions underlying the Jeffery-orbit-averaged analysis breaking down. Although the separation of time scales necessary for the validity of the analysis no longer exists in the transition regime ($\unicode[STIX]{x1D706}$ in the neighbourhood of $\unicode[STIX]{x1D706}_{c}$), scaling arguments nevertheless highlight the manner in which the Nusselt number function connects smoothly across the open and closed streamline regimes for any finite $Pe$.
Heat or mass transport from drops in shearing flows. Part 1. The open-streamline regime
- Deepak Krishnamurthy, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 850 / 10 September 2018
- Published online by Cambridge University Press:
- 06 July 2018, pp. 439-483
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We study the heat or mass transfer from a neutrally buoyant spherical drop embedded in an ambient Newtonian medium, undergoing a general shearing flow, in the strong convection limit. The latter limit corresponds to the drop Péclet number being large ($Pe\gg 1$). We consider two families of ambient linear flows: (i) planar linear flows with open streamlines (parametrized by $\unicode[STIX]{x1D6FC}$ with $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 1$, the extremal members being simple shear flow ($\unicode[STIX]{x1D6FC}=0$) and planar extension ($\unicode[STIX]{x1D6FC}=1$)) and (ii) three-dimensional extensional flows (parameterized by $\unicode[STIX]{x1D716}$, with $0\leqslant \unicode[STIX]{x1D716}\leqslant 1$, the extremal members being planar ($\unicode[STIX]{x1D716}=0$) and axisymmetric extension ($\unicode[STIX]{x1D716}=1$)). For the first family, an analysis of the exterior flow field in the inertialess limit (the drop Reynolds number, $Re$, being vanishingly small) shows that there exist two distinct streamline topologies separated by a critical drop-to-medium viscosity ratio ($\unicode[STIX]{x1D706}$) given by $\unicode[STIX]{x1D706}_{c}=2\unicode[STIX]{x1D6FC}/(1-\unicode[STIX]{x1D6FC})$. For $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ all streamlines are open, while the near-field streamlines are closed for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$. For the second family, the exterior streamlines remain open regardless of $\unicode[STIX]{x1D706}$. The two streamline topologies lead to qualitatively different mechanisms of transport for large $Pe$. The transport in the open streamline regime is enhanced in the usual manner via the formation of a boundary layer. In sharp contrast, the closed-streamline regime displays diffusion-limited transport, so there is only a finite enhancement even as $Pe\rightarrow \infty$. For $Re=0$, the drop surface streamlines in a planar linear flow may be regarded as generalized Jeffery orbits with a flow and viscosity dependent aspect ratio Jeffery orbits denote the aspect-ratio-dependent inertialess trajectories of a rigid axisymmetric particle in a simple shear flow; see Jeffery (Proc. R. Soc. Lond. A, vol. 102 (715), 1922, pp. 161–179). A Jeffery-orbit-based non-orthogonal coordinate system thus serves as a natural candidate to tackle the transport problem from a drop, in a planar linear flow, in the limit $Pe\gg 1$. Use of this system allows one to derive a closed-form expression for the dimensionless rate of transport (the Nusselt number $Nu$) from a drop in the open-streamline regime ($\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$). Symmetry arguments point to a Jeffery-orbit-based coordinate system for any linear flow, and a variant of this coordinate system is therefore used to derive the Nusselt number for the family of three-dimensional extensional flows. For both classes of flows considered, the boundary-layer-enhanced transport implies that the Nusselt number takes the form $Nu={\mathcal{F}}(P,\unicode[STIX]{x1D706})Pe^{1/2}$, with the parameter $P$ being $\unicode[STIX]{x1D6FC}$ or $\unicode[STIX]{x1D716}$, and ${\mathcal{F}}(P,\unicode[STIX]{x1D706})$ given as a one and two-dimensional integral, respectively, which is readily evaluated numerically.
The inertial orientation dynamics of anisotropic particles in planar linear flows
- Navaneeth K. Marath, Ganesh Subramanian
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- Journal of Fluid Mechanics / Volume 844 / 10 June 2018
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- 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 of Fluid Mechanics / Volume 830 / 10 November 2017
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- 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.