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Register a new userShape optimisation of stirring rods in mixing binary fluids
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Mixing is an omnipresent process in a wide-range of industrial applications, which supports scientific efforts to devise techniques for optimising mixing processes under time and energy constraints. In this endeavor, we present a computational framework based on nonlinear direct-adjoint looping for the enhancement of mixing efficiency in a binary fluid system. The governing equations consist of the non-linear Navier-Stokes equations, complemented by an evolution equation for a passive scalar. Immersed and moving stirrers are treated by a Brinkman-penalisation technique, and the full system of equations is solved using a Fourier-based pseudospectral approach. The adjoint equations provide gradient and sensitivity information which is in turn used to improve an initial mixing strategy, based on shape, rotational and path modifications. We utilise a Fourier-based approach for parameterising and optimising the embedded stirrers and consider a variety of geometries to achieve enhanced mixing efficiency. We consider a restricted optimisation space by limiting the time for mixing and the rotational velocities of all stirrers. In all cases, non-intuitive shapes are found which produce significantly enhanced mixing efficiency.
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Mixing of binary fluids by moving stirrers is a commonplace process in many industrial applications, where even modest improvements in mixing efficiency could translate into considerable power savings or enhanced product quality. We propose a gradient-based nonlinear optimization scheme to minimize the mix-norm of a passive scalar. The velocities of two cylindrical stirrers, moving on concentric circular paths inside a circular container, represent the control variables, and an iterative direct-adjoint algorithm is employed to arrive at enhanced mixing results. The associated stirring protocol is characterized by a complex interplay of vortical structures, generated and promoted by the stirrers action. Full convergence of the optimization process requires constraints that penalize the acceleration of the moving bodies. Under these conditions, considerable mixing enhancement can be accomplished, even though an optimum cannot be guaranteed due to the non-convex nature of the optimization problem. Various challenges and extensions of our approach are discussed.
A computational framework based on nonlinear direct-adjoint looping is presented for optimizing mixing strategies for binary fluid systems. The governing equations are the nonlinear Navier-Stokes equations, augmented by an evolution equation for a passive scalar, which are solved by a spectral Fourier-based method. The stirrers are embedded in the computational domain by a Brinkman-penalization technique, and shape and path gradients for the stirrers are computed from the adjoint solution. Four cases of increasing complexity are considered, which demonstrate the efficiency and effectiveness of the computational approach and algorithm. Significant improvements in mixing efficiency, within the externally imposed bounds, are achieved in all cases.
The mixing of binary fluids by stirrers is a commonplace procedure in many industrial and natural settings, and mixing efficiency directly translates into more homogeneous final products, more enriched compounds, and often substantial economic savings in energy and input ingredients. Enhancements in mixing efficiency can be accomplished by unorthodox stirring protocols as well as modified stirrer shapes that utilize unsteady hydrodynamics and vortex-shedding features to instigate the formation of fluid filaments which ultimately succumb to diffusion and produce a homogeneous mixture. We propose a PDE-constrained optimization approach to address the problem of mixing enhancement for binary fluids. Within a gradient-based framework, we target the stirring strategy as well as the cross-sectional shape of the stirrers to achieve improved mixedness over a given time horizon and within a prescribed energy budget. The optimization produces a significant enhancement in homogeneity in the initially separated fluids, suggesting promising modifications to traditional stirring protocols.
Particles transported in fluid flows, such as cells, polymers, or nanorods, are rarely spherical. In this study, we numerically and theoretically investigate the dispersion of an initially localized patch of passive elongated Brownian particles constrained to one degree of rotational freedom in a two-dimensional Poiseuille flow, demonstrating that elongated particles exhibit an enhanced longitudinal dispersion. In a shear flow, the rods translate due to advection and diffusion and rotate due to rotational diffusion and their classical Jefferys orbit. The magnitude of the enhanced dispersion depends on the particles aspect ratio and the relative importance of its shear-induced rotational advection and rotational diffusivity. When rotational diffusion dominates, we recover the classical Taylor dispersion result for the longitudinal spreading rate using an orientationally averaged translational diffusivity for the rods. However, in the high-shear limit, the rods tend to align with the flow and ultimately disperse more due to their anisotropic diffusivities. Results from our Monte Carlo simulations of the particle dispersion are captured remarkably well by a simple theory inspired by Taylors original work. For long times and large Peclet numbers, an effective one-dimensional transport equation is derived with integral expressions for the particles longitudinal transport speed and dispersion coefficient. The enhanced dispersion coefficient can be collapsed along a single curve for particles of high aspect ratio, representing a simple correction factor that extends Taylors original prediction to elongated particles.
We study the dynamics of simple reactions where the chemical species are confined on a general, time-modulated surface, and subjected to externally-imposed stirring. The study of these inhomogeneous effects requires a model based on a reaction-advection-diffusion equation, which we derive. We use homogenization methods to show that up to second order in a small scaling parameter, the modulation effects on the concentration field are asymptotically equivalent for systems with or without stirring. This justifies our consideration of the simpler reaction-diffusion model, where we find that by modulating the substrate, we can modify the reaction rate, the total yield from the reaction, and the speed of front propagation. These observations are confirmed in three numerical case studies involving the autocatalytic and bistable reactions on the torus and a sinusoidally-modulated substrate
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