^{1,a)}

### Abstract

The efficiency of a micromixing device may be quantified by the time taken for a given initial state of separated fluids to reach a desired level of homogenization. In the physically relevant case of high Peclet number the accurate prediction of the mixing time is a challenging problem, even in simple two-dimensional flows within bounded domains. In this paper a closed-form solution for the time dependence of mixing in an annular micromixer is derived and verified by numerical simulation. The mixing time is found to scale with Peclet number as a power law, but the power-law exponent depends on the level of homogeneity desired in the final state. Numerical simulation of a recent model of chaotic mixing reveals a vortexlike stirring effect in quasiperiodic islands of the Poincaré map of the flow, which strongly influences the mixing time. This stirring effect is identified with an exponential decrease in solute variance on an intermediate time scale, being subdominant to the asymptotic long-time decay, but sensitive to the initial loading of fluids in the mixer. The subdominant decay rate is calculated to scale with Peclet number as the square root of the dominant decay rate.

This work was supported by a Science Foundation Ireland Investigator Award, under Program No. 02/IN.1/IM062.

I. INTRODUCTION

II. MATHEMATICAL FORMULATION

III. LAMINAR MIXING

A. Annular micromixer

B. Persistent structures in vortexmixing

IV. CHAOTIC MIXING

V. CONCLUSIONS

### Key Topics

- Micromixing
- 54.0
- Diffusion
- 22.0
- Solution processes
- 14.0
- Interface diffusion
- 11.0
- Rotating flows
- 10.0

## Figures

Operation of an idealized annular micromixer at three times.

Operation of an idealized annular micromixer at three times.

Annular geometry showing the centerline radius and the channel half-width .

Annular geometry showing the centerline radius and the channel half-width .

Mixing measure as a function of nondimensional time, calculated in numerical simulations with and various Peclet numbers as shown. The asymptotic result (20) for is also shown with a dotted curve, almost indistinguishable from the numerical result.

Mixing measure as a function of nondimensional time, calculated in numerical simulations with and various Peclet numbers as shown. The asymptotic result (20) for is also shown with a dotted curve, almost indistinguishable from the numerical result.

Nondimensional mixing times as a function of Peclet number for . Asymptotic results are shown as lines, and numerical results as symbols for values of the mixing measure: (dashed line, squares), (solid line, points), and (dotted line, triangles). The asymptotic result (23) is applied to the and cases, while the long-time version (25) is needed in the well-mixed situation.

Nondimensional mixing times as a function of Peclet number for . Asymptotic results are shown as lines, and numerical results as symbols for values of the mixing measure: (dashed line, squares), (solid line, points), and (dotted line, triangles). The asymptotic result (23) is applied to the and cases, while the long-time version (25) is needed in the well-mixed situation.

Poincaré section of the standard map (26) . Note the -periodic variable is shown over the range to highlight the central island. Initial condition 1 of the mixing problem has the solute distributed uniformly inside the dashed rectangle, with the solid rectangle bounding initial condition 2.

Poincaré section of the standard map (26) . Note the -periodic variable is shown over the range to highlight the central island. Initial condition 1 of the mixing problem has the solute distributed uniformly inside the dashed rectangle, with the solid rectangle bounding initial condition 2.

Concentration density for initial condition 1 (left column) and condition 2 (right column). (a) Initial conditions, (b) , (c) , (d) , (e) . The highest concentration values in each picture are white, and lowest are black. Note the stirring effect within the island in the left column of (c) and (d).

Concentration density for initial condition 1 (left column) and condition 2 (right column). (a) Initial conditions, (b) , (c) , (d) , (e) . The highest concentration values in each picture are white, and lowest are black. Note the stirring effect within the island in the left column of (c) and (d).

Mixing measure for the noisy standard map. Filled symbols are for initial condition 1 (which intersects the quasiperiodic island); open circles are for initial condition 2. Note the exponential decay of condition 1 between and —the rate of decay is faster than (subdominant to) the final exponential decay, but nevertheless has a strong impact on the mixing measure.

Mixing measure for the noisy standard map. Filled symbols are for initial condition 1 (which intersects the quasiperiodic island); open circles are for initial condition 2. Note the exponential decay of condition 1 between and —the rate of decay is faster than (subdominant to) the final exponential decay, but nevertheless has a strong impact on the mixing measure.

Mixing measures for initial condition 1, for various Peclet numbers from 555 to . The Peclet number increases moving upwards among the curves. Exponential fits within the stirring interval are shown, the scaling of these fits is shown in Fig. 9 .

Mixing measures for initial condition 1, for various Peclet numbers from 555 to . The Peclet number increases moving upwards among the curves. Exponential fits within the stirring interval are shown, the scaling of these fits is shown in Fig. 9 .

Scaling of the exponential fits to the curves. The triangles correspond to the stirring interval; the squares show the scaling of the long-time exponential decay rate. The fitted lines correspond to scalings of (triangles) and (squares).

Scaling of the exponential fits to the curves. The triangles correspond to the stirring interval; the squares show the scaling of the long-time exponential decay rate. The fitted lines correspond to scalings of (triangles) and (squares).

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