Steady-state performance of an infinitely fast reaction in a three-dimensional open Stokes flow

We investigate the steady-state performance of a single irreversible mixing-controlled reaction between segregated reactant streams entering the annular gap between coaxial cylinders that can rotate independently. The three-dimensional Stokes flow results from the superposition of a pressure-driven Poiseuille flow along the mixer axis, and a cross-sectional Couette flow generated by the steady rotation of the inner and outer cylinders. Flow protocols associated with different geometries and different conditions of the rotating cylinders are considered. For each protocol, we analyze the axial behavior of reaction yield as a function of the Peclet number, quantifying the relative importance of convective vs. diffusive transport mechanisms. Following a well-established approach developed in the context of closed bounded flows, we connect the steady-state reactor performance with the spectral (eigenvalue-eigenfunction) structure of the advection-diffusion operator restricted to the system cross-section. For slim channels, this approach indicates that reaction performance is controlled by the real part, say -Λ (Λ>0), of the dominant eigenvalue of the advection-diffusion operator, and therefore the quantitative analysis of reaction performance consists of determining how A depends on the Peclet number for the assigned protocol. The presence of a nonuniform axial flow makes the spectral analysis different from previous studies focusing on transient mixing in closed two-dimensional autonomous flows in that it transforms the standard eigenvalue-eigenfunction formulation of the problem into a generalized form, where the axial velocity plays the role of weight function for the generic eigenvalue-eigenfunction. This difference is not just formal, as it can give rise to a variety of convection-enhanced regimes which cannot be observed in the standard eigenvalue formulation. Identifying the onset conditions and the range of existence of these regimes could greatly improve the rational design of geometry and operating conditions of micro and ordinary lengthscale continuous reactors operating under laminar flow conditions.