Finite Time Lyapunov Exponent for Micro Chaotic Mixer Design

2003 ◽  
Author(s):  
Xi-Ze Niu ◽  
Patrick Tabeling ◽  
Yi-Kuen Lee
Keyword(s):  
Lyapunov Exponent ◽  
Finite Time ◽  
Stokes Equations ◽  
Channel Length ◽  
Navier Stokes ◽  
Chaotic Mixing ◽  
Mixing Efficiency ◽  
Navier Stokes Equations ◽  
Finite Time Lyapunov Exponent ◽  
Mixer Design

In this paper, the Finite Time Lyapunov Exponent (FTLE) approach is used to analyze and optimize chaotic mixing in an active microchannel and a static mixer. The characteristics of FTLE related to chaotic mixing are discussed. By comparing the similarity of Poincare´ mapping and FTLE contour, it is shown that FTLE can be used to evaluate the chaotic mixing of liquid in the micromixer qualitatively and quantitatively. The minimum channel length needed for full mixing in the mixers can be estimated by the mean FTLE. The results are consistent with CFD simulations directly solving the Navier-Stokes equations coupled with the diffusion equation. More than 3 orders of CPU time can be saved by using FTLE compared with the classical infinite time Lyapunov exponent approach. Moreover, the FTLE is used to optimize the design and operation of the chaotic micromixers to improve the mixing efficiency for the first time.

Chaotic mixing analysis of a novel single-screw extruder with a perturbation baffle by the finite-time Lyapunov exponent method

2019 ◽  
Vol 39 (3) ◽  
pp. 287-299 ◽  
Author(s):  
Jian Liu ◽  
Xiangzhe Zhu
Keyword(s):  
Lyapunov Exponent ◽  
Finite Time ◽  
Polymer Processing ◽  
Chaotic Mixing ◽  
Mixing Efficiency ◽  
Screw Extruder ◽  
Screw Channel ◽  
Single Screw ◽  
Single Screw Extruder ◽  
Finite Time Lyapunov Exponent

Abstract The single-screw extruder with a perturbation baffle is a novel piece of equipment for polymer processing, in which the polymer melts undergo complex chaotic mixing. In this paper, from a new Lagrangian perspective, the fluid transporting mechanism in chaotic flow of the unwound screw channel was analyzed based on the finite element method. Firstly, two-dimensional velocity distributions in the unwound screw channel were calculated based on the mesh superposition technique. Fluid particle evolution processes in the extruder were tracked based on the fourth-order Runge-Kutta scheme. The numerical method used in this paper was validated by grid independence and experiments obtained from literature. Moreover, the finite-time Lyapunov exponent (FTLE) and Poincaré sections were adopted to discuss the chaotic mixing in the novel single-screw extruder. The effects of baffle width and height on the manifold structures in the flow dynamic system were analyzed. The results show that the homoclinic point of the manifold structure can give rise to chaotic mixing in the single-screw extruder. The height of the baffle is an important parameter to control the chaotic strength. In a way, increasing the height of the baffle can enlarge the kink scale and increase the stretching and folding actions, which results in the decrease of regular regions and an increase of the mixing efficiency in the single-screw extruder.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

scholarly journals Numerical Investigation of Liquid–Liquid Mixing in Modified T Mixer with 3D Obstacles

2021 ◽  
pp. 25-32
Author(s):  
Md. Readul Mahmud
Keyword(s):  
Numerical Investigation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Reynolds Numbers ◽  
Channel Length ◽  
Navier Stokes ◽  
Mixing Efficiency ◽  
Mixing Performance ◽  
Wide Range

The fluids inside passive micromixers are laminar in nature and mixing depends primarily on diffusion. Hence mixing efficiency is generally low, and requires a long channel length and longtime compare to active mixers. Various designs of complex channel structures with/without obstacles and three-dimensional geometries have been investigated in the past to obtain an efficient mixing in passive mixers. This work presents a design of a modified T mixer. To enhance the mixing performance, circular and hexagonal obstacles are introduced inside the modified T mixer. Numerical investigation on mixing and flow characteristics in microchannels is carried out using the computational fluid dynamics (CFD) software ANSYS 15. Mixing in the channels has been analyzed by using Navier–Stokes equations with water-water for a wide range of the Reynolds numbers from 1 to 500. The results show that the modified T mixer with circular obstacles has far better mixing performance than the modified T mixer without obstacles. The reason is that fluids' path length becomes longer due to the presence of obstacles which gives fluids more time to diffuse. For all cases, the modified T mixer with circular obstacle yields the best mixing efficiency (more than 60%) at all examined Reynolds numbers. It is also clear that efficiency increase with axial length. Efficiency can be simply improved by adding extra mixing units to provide adequate mixing. The value of the pressure drop is the lowest for the modified T mixer because there is no obstacle inside the channel. Modified T mixer and modified T mixer with circular obstacle have the lowest and highest mixing cost, respectively. Therefore, the current design of modified T with circular obstacles can act as an effective and simple passive mixing device for various micromixing applications.

Numerical Investigation of Liquids Mixing in Pin-Finned Microchannels

2012 ◽  
Author(s):  
Haleh Shafeie ◽  
Omid Abouali ◽  
Khosrow Jafarpur ◽  
Goodarz Ahmadi
Keyword(s):  
Differential Equation ◽  
Mass Transport ◽  
Computational Model ◽  
Numerical Investigation ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Mixing Efficiency ◽  
Navier Stokes Equations ◽  
Pin Fins

In the present work, the performance of pin-finned microchannels as the micromixers is investigated. Different patterns for distribution of pin-fins were examined (staggered and oblique distribution of fins). A 3-D computational model was developed and the Navier-Stokes equations were solved and the corresponding flow fields were evaluated. The mass transport differential equation was also solved and the concentration of liquids in the mixture was evaluated. The results for the mixing efficiency were compared between the simple and pin-finned microchannels. The results suggest that the finned microchannels with staggered distribution of pins perform very well in mixing of liquids. The mixing efficiency reaches to 100 percent for the Reynolds numbers in which the mixing efficiency is less than 10 percent for the simple microchannels.

scholarly journals Towards a finite-time singularity of the Navier–Stokes equations Part 1. Derivation and analysis of dynamical system

2018 ◽  
Vol 861 ◽  
pp. 930-967 ◽  
Author(s):  
H. K. Moffatt ◽  
Yoshifumi Kimura
Keyword(s):  
Dynamical System ◽  
Finite Time ◽  
Stokes Equations ◽  
Rate Of Change ◽  
Navier Stokes ◽  
Tipping Points ◽  
Dimensionless Time ◽  
Time Singularity ◽  
Navier Stokes Equations ◽  
Finite Time Singularity

The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity$\unicode[STIX]{x1D708}$is studied, starting from a finite-energy configuration of two vortex rings of circulation$\pm \unicode[STIX]{x1D6E4}$and radius$R$, symmetrically placed on two planes at angles$\pm \unicode[STIX]{x1D6FC}$to a plane of symmetry$x=0$. The minimum separation of the vortices,$2s$, and the scale of the core cross-section,$\unicode[STIX]{x1D6FF}$, are supposed to satisfy the initial inequalities$\unicode[STIX]{x1D6FF}\ll s\ll R$, and the vortex Reynolds number$R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the ‘tipping points’) is determined solely by the curvature$\unicode[STIX]{x1D705}(\unicode[STIX]{x1D70F})$at the tipping points and by$s(\unicode[STIX]{x1D70F})$and$\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$, where$\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D6E4}/R^{2})t$is a dimensionless time variable. The Biot–Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot–Savart law breaks down just before this singularity is realised, when$\unicode[STIX]{x1D705}s$and$\unicode[STIX]{x1D6FF}/\!s$become of order unity. The dynamical system admits ‘partial Leray scaling’ of just$s$and$\unicode[STIX]{x1D705}$, and ultimately full Leray scaling of$s,\unicode[STIX]{x1D705}$and$\unicode[STIX]{x1D6FF}$, conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter$\unicode[STIX]{x1D705}$; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot–Savart approach just before the incipient singularity is realised. The Euler flow situation ($\unicode[STIX]{x1D708}=0$) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.

scholarly journals Maximum palinstrophy amplification in the two-dimensional Navier–Stokes equations

2018 ◽  
Vol 837 ◽  
pp. 839-857 ◽  
Author(s):  
Diego Ayala ◽  
Charles R. Doering ◽  
Thilo M. Simon
Keyword(s):  
Initial Data ◽  
Finite Time ◽  
Stokes Equations ◽  
Optimization Problems ◽  
Navier Stokes ◽  
Instantaneous Growth Rate ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Wide Range

We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the two-dimensional incompressible Navier–Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number $Re$ and palinstrophy ${\mathcal{P}}$ which maximize $\text{d}{\mathcal{P}}/\text{d}t$ is constructed by numerically solving suitable optimization problems for a wide range of $Re$ and ${\mathcal{P}}$, providing numerical evidence for the sharpness of the analytic estimate $\text{d}{\mathcal{P}}/\text{d}t\leqslant (a+b\sqrt{\ln Re+c}){\mathcal{P}}^{3/2}$ with respect to both $Re$ and ${\mathcal{P}}$. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations, and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e. $\sup _{t>0}{\mathcal{P}}(t)$, are observed to scale with the magnitude of the initial palinstrophy ${\mathcal{P}}(0)$ in accord with the corresponding a priori estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier–Stokes equation are discussed.

scholarly journals Shape optimization of stirring rods for mixing binary fluids

2020 ◽  
Vol 85 (5) ◽  
pp. 762-789 ◽  
Author(s):  
Maximilian F Eggl ◽  
Peter J Schmid
Keyword(s):  
Stokes Equations ◽  
Passive Scalar ◽  
Industrial Applications ◽  
Adjoint Equations ◽  
Navier Stokes ◽  
Mixing Efficiency ◽  
Mixing Processes ◽  
Navier Stokes Equations ◽  
Wide Range ◽  
Time And Energy

Abstract Mixing is an omnipresent process in a wide range of industrial applications, which supports scientific efforts to devise techniques for optimizing mixing processes under time and energy constraints. In this endeavour, 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 nonlinear Navier–Stokes equations, complemented by an evolution equation for a passive scalar. Immersed and moving stirrers are treated by a Brinkman penalization 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 utilize a Fourier-based approach for parameterizing and optimizing the embedded stirrers and consider a variety of geometries to achieve enhanced mixing efficiency. We consider a restricted optimization 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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