scholarly journals Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number

2014 ◽  
Vol 748 ◽  
pp. 241-277 ◽  
Author(s):  
D. P. G. Foures ◽  
C. P. Caulfield ◽  
P. J. Schmid
Keyword(s):  
Scalar Field ◽  
Poiseuille Flow ◽  
Peclet Number ◽  
Passive Scalar ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Péclet Number ◽  
Dimensional Plane ◽  
Scalar Variance ◽  
Optimal Mixing

AbstractWe consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the linear instability threshold. We use a nonlinear-adjoint-looping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23–46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions.

Mixing in frozen and time-periodic two-dimensional vortical flows

2001 ◽  
Vol 442 ◽  
pp. 359-385 ◽  
Author(s):  
ALEXANDER WONHAS ◽  
J. C. VASSILICOS
Keyword(s):  
Scalar Field ◽  
Passive Scalar ◽  
Time Range ◽  
Two Dimensional ◽  
Kam Tori ◽  
Advection Diffusion ◽  
Box Counting Dimension ◽  
Theoretical Predictions ◽  
Scalar Variance ◽  
Variance Decay

In the first part of this paper, we investigate passive scalar or tracer advection–diffusion in frozen, two-dimensional, non-circular symmetric vortices. We develop an asymptotic description of the scalar field in a time range 1 [Lt ] t/T [Lt ] Pe1/3, where T is the formation time of the spiral in the vortex and Pe is a Péclet number, assumed much larger than 1. We derive the leading-order decay of the scalar variance E(t) for a singular non-circular streamline geometry,The variance decay is solely determined by a geometrical parameter μ and the exponent β describing the behaviour of the closed streamline periods. We develop a method to predict, in principle, the variance decay from snapshots of the advected scalar field by reconstructing the streamlines and their period from just two snapshots of the advected scalar field.In the second part of the paper, we investigate variance decay in a periodically moving singular vortex. We identify three different regions (core, chaotic and KAM-tori). We find fast mixing in the chaotic region and investigate a conjecture about mixing in the KAM-tori region. The conjecture enables us to use the results from the first section and relates the Kolmogorov capacity, or box-counting dimension, of the advected scalar to the decay of the scalar variance. We check our theoretical predictions against a numerical simulation of advection–diffusion of scalar in such a flow.

scholarly journals Effect of temperature gradient on the cross-stream migration of a surfactant-laden droplet in Poiseuille flow

2017 ◽  
Vol 835 ◽  
pp. 170-216 ◽  
Author(s):  
Sayan Das ◽  
Shubhadeep Mandal ◽  
Suman Chakraborty
Keyword(s):  
Temperature Field ◽  
Temperature Gradient ◽  
Poiseuille Flow ◽  
Peclet Number ◽  
Marangoni Number ◽  
Stream Velocity ◽  
Small Surface ◽  
Péclet Number ◽  
Surfactant Transport ◽  
The Cross

The motion of a viscous droplet in unbounded Poiseuille flow under the combined influence of bulk-insoluble surfactant and linearly varying temperature field aligned in the direction of imposed flow is studied analytically. Neglecting fluid inertia, thermal convection and shape deformation, asymptotic analysis is performed to obtain the velocity of a force-free surfactant-laden droplet. The droplet speed and direction of motion are strongly influenced by the interfacial transport of surfactant, which is governed by surface Péclet number. The present study is focused on the following two limiting situations of surfactant transport: (i) surface-diffusion-dominated surfactant transport considering small surface Péclet number, and (ii) surface-convection-dominated surfactant transport considering high surface Péclet number. Thermocapillary-induced Marangoni stress, the strength of which relative to viscous stress is represented by the thermal Marangoni number, has a strong influence on the distribution of surfactant on the droplet surface. The present study shows that the motion of a surfactant-laden droplet in the combined presence of temperature and imposed Poiseuille flow cannot be obtained by a simple superposition of the following two independent results: migration of a surfactant-free droplet in a temperature gradient, and the motion of a surfactant-laden droplet in a Poiseuille flow. The temperature field not only affects the axial velocity of the droplet, but also has a non-trivial effect on the cross-stream velocity of the droplet in spite of the fact that the temperature gradient is aligned with the Poiseuille flow direction. When the imposed temperature increases in the direction of the Poiseuille flow, the droplet migrates towards the flow centreline. The magnitude of both axial and cross-stream velocity components increases with the thermal Marangoni number. However, when the imposed temperature decreases in the direction of the Poiseuille flow, the magnitude of both axial and cross-stream velocity components may increase or decrease with the thermal Marangoni number. Most interestingly, the droplet moves either towards the flow centreline or away from it. The present study shows a critical value of the thermal Marangoni number beyond which the droplet moves away from the flow centreline which is in sharp contrast to the motion of a surfactant-laden droplet in isothermal flow, for which the droplet always moves towards the flow centreline. Interestingly, we show that the above picture may become significantly altered in the case where the droplet is not a neutrally buoyant one. When the droplet is less dense than the suspending medium, the presence of gravity in the direction of the Poiseuille flow can lead to cross-stream motion of the droplet away from the flow centreline even when the temperature increases in the direction of the Poiseuille flow. These results may bear far-reaching consequences in various emulsification techniques in microfluidic devices, as well as in biomolecule synthesis, vesicle dynamics, single-cell analysis and nanoparticle synthesis.

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