^{1}and Donald L. Koch

^{2}

### Abstract

Microscale inertia is found to break the degenerate closed-streamline configuration that occurs in a shearing flow past a neutrally buoyant torque-free spherical particle in the inertialess limit. The broken symmetry at small but finite allows heat or mass to be convected away in an efficient manner in sharp contrast to the inertialess diffusion-limited scenario. Inertial forces scale with the particle Reynolds number, defined as , where is the radius of the particle, is the characteristic magnitude of the velocity gradient, and is the kinematic viscosity of the suspending fluid. The dimensionless heat or mass transfer rate is then given by when and , the constant being a function of the flow in the vicinity of the particle. Here, is the Nusselt number defined as , where is the dimensional heat/mass flux, the appropriate transport coefficient, and the driving force viz. the temperature or concentration difference between the particle and the ambient fluid; for pure diffusion, . The Peclét number is a dimensionless measure of the relative dominance of the convective and diffusive transfer mechanisms. It is shown that equals for a two-dimensional linear flow, where measures the relative magnitudes of extension and vorticity. For simple shear , knowledge of the inertial velocity field to enables one to determine the next term in the asymptotic expansion for ; one finds in the limit . It is argued that the convective enhancement at finite via symmetry-breaking streamline bifurcations will occur in generic shearing flows with nonlinear velocity profiles; the degenerate Stokes streamline pattern around a neutrally buoyant torque-free particle in a quadratic flow serves to reinforce this assertion. The above mechanism represents a possible means for heat or mass transfer enhancement from the dispersed phase in multiphase systems. Implications for particles in turbulent flows are also discussed.

This work was supported by Department of Energy Grant No. DE-FG02-03-ER46073.

I. INTRODUCTION

II. STREAMLINE CONFIGURATION: THE INERTIALESS AND FINITE SCENARIOS

A. A torque-free cylinder in a planar linear flow

B. A torque-free sphere in a simple shear flow

C. A torque-free sphere in a planar linear flow

III. BOUNDARY LAYER ANALYSIS FOR SMALL BUT FINITE

IV. EFFECT OF FLUID INERTIA ON HEAT TRANSFER FROM PARTICLES IN NONLINEAR FLOWS

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Shear flows
- 42.0
- Heat transfer
- 26.0
- Vortex dynamics
- 23.0
- Stokes flows
- 17.0
- Convection
- 15.0

## Figures

(Top) Two-dimensional fore-aft symmetric inertialess streamline pattern for a torque-free cylinder in simple shear flow; the axis of the cylinder is aligned with the vorticity direction. The bold lines indicate the pair of separatrices that enclose the region of closed streamlines. (Bottom) Two-dimensional asymmetric streamline pattern for a torque-free cylinder in simple shear flow at small but finite . The finite separatrices, again indicated by bold lines, form a pair of recirculating wakes that bracket a central region of closed streamlines in between. The ordinates of the saddle points shown are not identically zero, but they are asymptotically small compared to their flow coordinates. The resulting streamline configuration is invariant to a rotation, and thence, consistent with the antisymmetry of simple shear.

(Top) Two-dimensional fore-aft symmetric inertialess streamline pattern for a torque-free cylinder in simple shear flow; the axis of the cylinder is aligned with the vorticity direction. The bold lines indicate the pair of separatrices that enclose the region of closed streamlines. (Bottom) Two-dimensional asymmetric streamline pattern for a torque-free cylinder in simple shear flow at small but finite . The finite separatrices, again indicated by bold lines, form a pair of recirculating wakes that bracket a central region of closed streamlines in between. The ordinates of the saddle points shown are not identically zero, but they are asymptotically small compared to their flow coordinates. The resulting streamline configuration is invariant to a rotation, and thence, consistent with the antisymmetry of simple shear.

(Top) Fore-aft symmetric inertialess streamline pattern in the plane of shear for a torque-free sphere in simple shear flow. (Bottom) Corresponding three-dimensional streamline topology. Note the axisymmetric separatrix envelope that separates the closed from the open streamlines; here, the , , and axes correspond to the flow, gradient and vorticity directions of the ambient simple shear.

(Top) Fore-aft symmetric inertialess streamline pattern in the plane of shear for a torque-free sphere in simple shear flow. (Bottom) Corresponding three-dimensional streamline topology. Note the axisymmetric separatrix envelope that separates the closed from the open streamlines; here, the , , and axes correspond to the flow, gradient and vorticity directions of the ambient simple shear.

(Top) Streamline pattern in the plane of shear for a torque-free sphere in simple shear flow for small but finite . (Bottom) Finite three-dimensional streamline topology is depicted.

(Top) Streamline pattern in the plane of shear for a torque-free sphere in simple shear flow for small but finite . (Bottom) Finite three-dimensional streamline topology is depicted.

In-plane trajectory for in an ambient simple shear flow. The trajectory originates from , spirals outward, heading off downstream.

In-plane trajectory for in an ambient simple shear flow. The trajectory originates from , spirals outward, heading off downstream.

Two orthogonal projections of an off-plane trajectory originating from for in an ambient simple shear flow. The trajectory approaches the plane of shear while spiraling outward, and eventually escapes downstream at approximately .

Two orthogonal projections of an off-plane trajectory originating from for in an ambient simple shear flow. The trajectory approaches the plane of shear while spiraling outward, and eventually escapes downstream at approximately .

Degenerate inertialess streamline configuration in a planar hyperbolic linear flow . (Top) Streamlines in the plane of symmetry, the separatrices being denoted by dashed lines and (bottom) three-dimensional axisymmetric envelope containing closed streamlines; the inset shows the streamlines of the undisturbed ambient linear flow.

Degenerate inertialess streamline configuration in a planar hyperbolic linear flow . (Top) Streamlines in the plane of symmetry, the separatrices being denoted by dashed lines and (bottom) three-dimensional axisymmetric envelope containing closed streamlines; the inset shows the streamlines of the undisturbed ambient linear flow.

Characterization of the nature of the bifurcation at for a hyperbolic planar linear flow. (Top) Corresponds to the flow pattern at small but finite , and for a coordinate closer to the plane of symmetry, where the near-field trajectories spiral out; heat is carried away via the shaded convective channels. (Bottom) Flow pattern at a larger coordinate, where the finite streamline topology is altered in a reversed sense to yield inward-flowing convective channels in opposing quadrants.

Characterization of the nature of the bifurcation at for a hyperbolic planar linear flow. (Top) Corresponds to the flow pattern at small but finite , and for a coordinate closer to the plane of symmetry, where the near-field trajectories spiral out; heat is carried away via the shaded convective channels. (Bottom) Flow pattern at a larger coordinate, where the finite streamline topology is altered in a reversed sense to yield inward-flowing convective channels in opposing quadrants.

In-plane trajectory for in an ambient hyperbolic planar linear flow with . The trajectory originates from a point on , close to the sphere and exits via a convective channel that opens up for finite .

In-plane trajectory for in an ambient hyperbolic planar linear flow with . The trajectory originates from a point on , close to the sphere and exits via a convective channel that opens up for finite .

Two orthogonal projections of an off-plane trajectory originating from for in an ambient hyperbolic planar linear flow with . (Left) projection as the trajectory spirals outward, and toward the plane of shear, eventually escaping via a convective channel close to the plane of symmetry and (right) projection shows the approach of the trajectory toward the plane of shear.

Two orthogonal projections of an off-plane trajectory originating from for in an ambient hyperbolic planar linear flow with . (Left) projection as the trajectory spirals outward, and toward the plane of shear, eventually escaping via a convective channel close to the plane of symmetry and (right) projection shows the approach of the trajectory toward the plane of shear.

Two orthogonal projections of an off-plane trajectory originating in an incoming convective channel, and initially heading straight toward the vorticity axis. Sufficiently close to the vorticity axis, it starts to spiral around it, gradually approaching the plane.

Two orthogonal projections of an off-plane trajectory originating in an incoming convective channel, and initially heading straight toward the vorticity axis. Sufficiently close to the vorticity axis, it starts to spiral around it, gradually approaching the plane.

Plot of the azimuthal correction to the coordinates of the fixed points in the plane of symmetry; is plotted as a function of for planar hyperbolic linear flows.

Plot of the azimuthal correction to the coordinates of the fixed points in the plane of symmetry; is plotted as a function of for planar hyperbolic linear flows.

Plot of , defined in the text, as a function of for hyperbolic planar linear flows.

Plot of , defined in the text, as a function of for hyperbolic planar linear flows.

Differing nature of the invariant manifolds associated with the saddle points and in the plane of symmetry, as changes across ; the abbreviations and denote the unstable and stable manifolds, respectively.

Differing nature of the invariant manifolds associated with the saddle points and in the plane of symmetry, as changes across ; the abbreviations and denote the unstable and stable manifolds, respectively.

(Left) Inertialess closed streamlines in the plane of symmetry of an elliptic linear flow with . (Right) Spiraling in-plane trajectory for in the same linear flow; as the spiraling is very tight for small , a slightly larger value is chosen for an exaggerated depiction.

(Left) Inertialess closed streamlines in the plane of symmetry of an elliptic linear flow with . (Right) Spiraling in-plane trajectory for in the same linear flow; as the spiraling is very tight for small , a slightly larger value is chosen for an exaggerated depiction.

Two orthogonal projections of an off-plane trajectory in an elliptic linear flow with and as it spirals toward the plane.

Two orthogonal projections of an off-plane trajectory in an elliptic linear flow with and as it spirals toward the plane.

Streamline pattern in the plane for a torque-free spherical particle in a pure quadratic flow— in (49) . In a reference frame moving with the particle, the fluid elements close to move faster than the particle, whereas those at greater ordinate values lag behind. The symmetry of the pattern implies a zero torque, and thence, a zero rate of rotation.

Streamline pattern in the plane for a torque-free spherical particle in a pure quadratic flow— in (49) . In a reference frame moving with the particle, the fluid elements close to move faster than the particle, whereas those at greater ordinate values lag behind. The symmetry of the pattern implies a zero torque, and thence, a zero rate of rotation.

Streamline pattern in the plane for a torque-free spherical particle in a general quadratic flow for small values of in (49) . The degeneracy of the pattern is twofold—the presence of a homoclinic point whose stable and unstable manifolds are coincident, forming a loop, that encloses closed streamlines within; the presence of a heteroclinic connection between the two saddle points and .

Streamline pattern in the plane for a torque-free spherical particle in a general quadratic flow for small values of in (49) . The degeneracy of the pattern is twofold—the presence of a homoclinic point whose stable and unstable manifolds are coincident, forming a loop, that encloses closed streamlines within; the presence of a heteroclinic connection between the two saddle points and .

Streamline pattern in the plane for a torque-free spherical particle in a general quadratic flow for larger values of in (49) ; the velocity field arising from the linear part of the flow is now dominant. The up-down asymmetry of the streamline pattern arises because the linear and quadratic parts of the flow reinforce (oppose) each other for . The invariant manifolds of a saddle point separate far-field regions where the quadratic portion drives flow from right to left from regions just above the particle where the fluid flows in the opposite direction.

Streamline pattern in the plane for a torque-free spherical particle in a general quadratic flow for larger values of in (49) ; the velocity field arising from the linear part of the flow is now dominant. The up-down asymmetry of the streamline pattern arises because the linear and quadratic parts of the flow reinforce (oppose) each other for . The invariant manifolds of a saddle point separate far-field regions where the quadratic portion drives flow from right to left from regions just above the particle where the fluid flows in the opposite direction.

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