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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.
Collective motion in a suspension of micro-swimmers that run-and-tumble and rotary diffuse
- Deepak Krishnamurthy, Ganesh Subramanian
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- Journal:
- Journal of Fluid Mechanics / Volume 781 / 25 October 2015
- Published online by Cambridge University Press:
- 28 September 2015, pp. 422-466
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Recent experiments have shown that suspensions of swimming micro-organisms are characterized by complex dynamics involving enhanced swimming speeds, large-scale correlated motions and enhanced diffusivities of embedded tracer particles. Understanding this dynamics is of fundamental interest and also has relevance to biological systems. The observed collective dynamics has been interpreted as the onset of a hydrodynamic instability, of the quiescent isotropic state of pushers, swimmers with extensile force dipoles, above a critical threshold proportional to the swimmer concentration. In this work, we develop a particle-based model to simulate a suspension of hydrodynamically interacting rod-like swimmers to estimate this threshold. Unlike earlier simulations, the velocity disturbance field due to each swimmer is specified in terms of the intrinsic swimmer stress alone, as per viscous slender-body theory. This allows for a computationally efficient kinematic simulation where the interaction law between swimmers is known a priori. The neglect of induced stresses is of secondary importance since the aforementioned instability arises solely due to the intrinsic swimmer force dipoles.
Our kinematic simulations include, for the first time, intrinsic decorrelation mechanisms found in bacteria, such as tumbling and rotary diffusion. To begin with, we simulate so-called straight swimmers that lack intrinsic orientation decorrelation mechanisms, and a comparison with earlier results serves as a proof of principle. Next, we simulate suspensions of swimmers that tumble and undergo rotary diffusion, as a function of the swimmer number density $(n)$, and the intrinsic decorrelation time (the average duration between tumbles, ${\it\tau}$, for tumblers, and the inverse of the rotary diffusivity, $D_{r}^{-1}$, for rotary diffusers). The simulations, as a function of the decorrelation time, are carried out with hydrodynamic interactions (between swimmers) turned off and on, and for both pushers and pullers (swimmers with contractile force dipoles). The ‘interactions-off’ simulations allow for a validation based on analytical expressions for the tracer diffusivity in the stable regime, and reveal a non-trivial box size dependence that arises with varying strength of the hydrodynamic interactions. The ‘interactions-on’ simulations lead us to our main finding: the existence of a box-size-independent parameter that characterizes the onset of instability in a pusher suspension, and is given by $nUL^{2}{\it\tau}$ for tumblers and $nUL^{2}/D_{r}$ for rotary diffusers; here, $U$ and $L$ are the swimming speed and swimmer length, respectively. The instability manifests as a bifurcation of the tracer diffusivity curves, in pusher and puller suspensions, for values of the above dimensionless parameters exceeding a critical threshold.