SUMMARYThe periodic boundary displacement protocol leading to the optimum wall-to-fluid heat-transfer rate, or to the most efficient mixing rate, in 2-D annular Stokes flows is determined by calculating the steady periodic velocity and temperature fields. To obtain the steady periodic state one usually solves the dynamical system obtained after the spatial coordinates have been discretized. Here, we calculate the steady periodic state using an implicit method based on the discretization of the time coordinate over a period and the asymptotic regime is enforced by the periodicity condition in the computed temperature field. The obtained system of equations is solved using a Newton-type iterative algorithm with invariant Jacobian. At each iteration step, the sparse linearized system is solved using a multi-grid algebraic technique of rapid convergence. From a computational point of view and for the problem considered here, this method is an order of magnitude faster than the one based on a spatial discretization.
Mixing in a special class of three-dimensional, non-inertial periodic flows is studied numerically. In the type of flow considered here, the cross-sectional velocity components are independent of the axial flow and the axial flow is independent of the axial coordinate. Using the eccentric helical annular mixer as a prototype, we consider the counter-rotating case with steady rotation of the outer cylinder and sinusoidal modulation of the inner one. Apart from the mixer geometry, the behavior of the system is governed by two dimensionless parameters obtained by scaling the cross-sectional stirring protocol with respect to the characteristic residence time of the fluid in the mixer. The first parameter is related to the average number of turns of the outer cylinder and the second one is related to the average number of modulation periods of the inner cylinder. The convection-diffusion equation is solved numerically, with temperature as a passive scalar, at high Péclet number. For a given three-dimensional mixer geometry and axial flow rate we show that there is an optimum modulation frequency for which the exit standard deviation of the temperature field is a minimum. Lagrangian simulations at infinite Péclet number and the use of other tools to study mixing, such as stretching calculations and tracer tracking methods, confirm that the optimized protocol does result in very effective mixing.
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