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Register a new userBounds on the mixing enhancement for a stirred binary fluid
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Added by
Jean-Luc Thiffeault
Publication date
2007
fields
Physics
and research's language is
English
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The Cahn-Hilliard equation describes phase separation in binary liquids. Here we study this equation with spatially-varying sources and stirring, or advection. We specialize to symmetric mixtures and time-independent sources and discuss stirring strategies that homogenize the binary fluid. By measuring fluctuations of the composition away from its mean value, we quantify the amount of homogenization achievable. We find upper and lower bounds on our measure of homogenization using only the Cahn-Hilliard equation and the incompressibility of the advecting flow. We compare these theoretical bounds with numerical simulations for two model flows: the constant flow, and the random-phase sine flow. Using the sine flow as an example, we show how our bounds on composition fluctuations provide a measure of the effectiveness of a given stirring protocol in homogenizing a phase-separating binary fluid.
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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.
This paper presents some novel contributions to the theory of inviscid flow regarding the forces exerted on a body moving through such a fluid in two dimensions. It is argued that acceleration of the body corresponds to vorticity generation that is independent of the instantaneous velocity of the body and thus the boundary condition on the normal velocity. The strength of the vortex sheet representing the body retains a degree of freedom that represents the net effect of the tangential boundary condition associated with the viscous flow governed by the higher-order Navier-Stokes equations. This degree of freedom is the circulation of the vorticity generated by the acceleration of the body. Equivalently, it is the net circulation around a contour enclosing the body and any shed vorticity. In accordance with Kelvins circulation theorem, a non-zero value of the circulation around this system is necessarily communicated to infinity. This contrasts with the usual acceptance of the theorem as requiring this circulation to be zero at all times; a condition that is incapable of capturing the effect of newly generated vorticity on the body surface when it accelerates. Additionally, the usual boundary condition of continuity of normal velocity is relaxed to allow for fluid entrainment into surfaces of discontinuity that represent the mass contained within the viscous layers of the physical problem. The generalized force calculation is presented in detail. The importance of the vorticity generation due to body acceleration is demonstrated on some modeled problems relevant to biological propulsion.
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.
Truncated Taylor expansions of smooth flow maps are used in Hamiltons principle to derive a multiscale Lagrangian particle representation of ideal fluid dynamics. Numerical simulations for scattering of solutions at one level of truncation are found to produce solutions at higher levels. These scattering events to higher levels in the Taylor expansion are interpreted as modeling a cascade to smaller scales.
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