Utilizing bifurcations to separate particles in spiral inertial microﬂuidics Rahil N. Valani1aBrendan Harding2band Yvonne M. Stokes1c 1School of Mathematical Sciences University of Adelaide South Australia 5005 Australia

2025-05-06 0 0 2.77MB 6 页 10玖币

侵权投诉

Utilizing bifurcations to separate particles in spiral inertial microﬂuidics

Rahil N. Valani,1, a) Brendan Harding,2, b) and Yvonne M. Stokes1, c)

1)School of Mathematical Sciences, University of Adelaide, South Australia 5005, Australia

2)School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6012,

New Zealand

(Dated: 28 October 2022)

Particles suspended in ﬂuid ﬂow through a closed duct can focus to speciﬁc stable locations in the duct cross-section due

to hydrodynamic forces arising from the inertia of the disturbed ﬂuid. Such particle focusing is exploited in biomedical

and industrial technologies to separate particles by size. In curved ducts, the particle focusing is a result of balance

between two dominant forces on the particle: (i) inertial lift arising from small inertia of the ﬂuid, and (ii) drag arising

from cross-sectional vortices induced by the centrifugal force on the ﬂuid. Bifurcations of particle equilibria take place

as the bend radius of the curved duct varies. By using the mathematical model of Harding, Stokes, and Bertozzi 1, we

illustrate via numerical simulations that these bifurcations can be leveraged in a spiral duct to achieve large separation

between different sized particles by transiently focusing smaller particles near saddle-points. We demonstrate this by

separating similar-sized particles, as well as particles that have a large difference in size, using spiral ducts with square

cross-section. The formalism of using bifurcations to manipulate particle focusing can be applied more broadly to

different geometries in inertial microﬂuidics which may open new avenues in particle separation techniques.

The ability to separate particles of different sizes suspended

in a ﬂuid is important in many biomedical and industrial tech-

nologies2. For example, efﬁcient isolation of rare circulating

tumor cells from a large concentration of red blood cells and

white blood cells in a blood sample can revolutionize cancer

diagnostics and help in determining a likely prognosis3. An-

other example is the detection and separation of waterborne

pathogens in drinking water4–6. Microﬂuidics has become

an important tool for particle separation due to small sample

consumption, fast processing time, high spatial resolution and

high portability7. Amongst the different possible microﬂuidic

technologies aimed at particle separation, inertial microﬂu-

idics has been used widely because of its ease of operation

and high separation resolution8,9.

Segré and Silberberg10 ﬁrst reported that particles sus-

pended in ﬂuid ﬂow through a straight pipe with a circular

cross-section can migrate across streamlines and accumulate

to an annular region at approximately 60% of the pipe radius.

This deviation of particles from ﬂuid streamlines is due to the

inertial lift force acting on the particle that arises from small

but non-negligible inertia of the disturbed ﬂuid ﬂow at low to

moderate Reynolds numbers. This results in the phenomenon

of inertial migration and subsequent focusing of particles. In

straight ducts, the locations where no net force acts on the

particle in the duct cross-section, henceforth referred to as

particle equilibria, vary with the geometry of the duct cross-

section11, particle size1and the ﬂow Reynolds number12,13.

Adding curvature to the duct also inﬂuences particle equilib-

ria via the introduction of cross-sectional vortices to the ﬂow,

known as Dean vortices1,14. Curved microchannels with cir-

cular and spiral geometries are commonly used in inertial mi-

croﬂuidic devices aimed at particle separation by size2. In

these channels, the interplay between (i) the inertial lift force

a)Electronic mail: rahil.valani@adelaide.edu.au

b)Electronic mail: brendan.harding@vuw.ac.nz

c)Electronic mail: yvonne.stokes@adelaide.edu.au

arising from ﬂuid inertia, and (ii) the secondary drag force

arising from Dean vortices, determines the number, nature and

location of particle equilibria. By tuning the relative strength

of these forces via changes in the bend radius, the particle

equilibria can be manipulated to achieve particle separation

for different sized particles.

Spiral channels with rectangular and trapezoidal cross-

sections have shown promise for efﬁcient size-based particle

separation2. In these geometries, the various parameters are

chosen such that one obtains a horizontal separation between

the particle equilibria of the chosen particle sizes to be sep-

arated. The particles focus to their respective stable equilib-

rium points or a stable limit cycle towards the end of the spiral

channel after which the channel is split into multiple chan-

nels to collect the separated particles15–18. Another separation

method commonly used with circular and spiral channels hav-

ing rectangular cross-section is known as Dean Flow Fraction-

ation (DFF)19,20. In this method, the spiral channel has two

inlets, one consisting of the sample containing particles (typi-

cally a mix of two distinct sizes) and the other through which

a sheath ﬂow is introduced. As the particles ﬂow through the

curved channel, the Dean vortices transport the smaller parti-

cles towards the outer wall, while a balance of inertial lift and

secondary drag equilibrates the larger particles near the inner

wall, thus achieving separation between the two particle sizes.

Note that with this method, the smaller particles do not reach

a stable equilibrium position, rather, their well-controlled mi-

gration due to Dean vortices drives their separation from the

larger particles. A variant of DFF has also been developed

to separate several different smaller (sub-micron) sized parti-

cles and is called High-Resolution Dean Flow Fractionation

(HiDFF)21.

Many advances in particle separation methods are primar-

ily driven by experimental trial-and-error, with the potential

of predicting and optimizing particle separation based on the-

oretical models and numerical simulations not yet being fully

exploited. Although the use of theoretical and numerical

methods has progressed our understanding of particle equi-

libria and their bifurcations in straight channels22–25, only

arXiv:2210.15197v1 [physics.flu-dyn] 27 Oct 2022

FIG. 1. Schematic of the theoretical setup. A particle of radius awith

center located at xp=x(θp,rp,zp)is suspended in an incompress-

ible ﬂuid ﬂow through an Archimedean spiral duct having a uniform

square cross section of side length l. The enlarged view of the cross-

section illustrates the local cross-sectional (r,z)co-ordinate system,

and the secondary ﬂow (gray closed curves) induced by the curvature

of the duct. The edge labeled “inner wall" is the side closer to ori-

gin (x,y,z) = (0,0,0)while the edge labeled “outer wall" is the side

further away from the origin.

recently, progress has also been made to gain a systematic

understanding of the particle equilibria in curved channels1.

This has improved the understanding of how various system

parameters, such as particle size, bend radius and aspect ratio

of the cross-section, can affect the location and nature of the

particle equilibria. Subsequently, rich bifurcations in particle

equilibria have been observed with respect to variations in the

bend radius of the curved duct26–28. Herein, we illustrate how

these bifurcations can be utilized to design spiral microchan-

nels that produce large separation between two sets of differ-

ent sized particles. Although we restrict the present study to a

square cross-section and certain particle sizes to illustrate the

separation mechanism, the general formalism is applicable to

a broad range of geometries in which similar bifurcations in

the particle equilibria take place.

With reference to Fig. 1, consider a particle of density ρ

and radius asuspended in an incompressible ﬂuid ﬂow of

the same density ρand dynamic viscosity µﬂowing through

an Archimedean spiral duct. The instantaneous radius of

the spiral varies with the azimuthal angle θaccording to

R(θ) = Rstart + (∆R/2πNturns)θwhere ∆R=Rend −Rstart is

the change in radius from the start to the end of the spiral

duct and Nturns represents the number of turns of the spiral.

The cross-section of the channel is uniform and has a square

geometry with side length l. The horizontal and vertical co-

ordinates within the square cross-section are denoted by rand

z, respectively, with the origin at the center of the square i.e.

the domain is −l/2≤r≤l/2, −l/2≤z≤l/2. These cross-

sectional co-ordinates are related to the global co-ordinates of

the three-dimensional spiral duct as follows:

x(θ,r,z) = (R(θ) + r)cos(θ)i+ (R(θ) + r)sin(θ)j+zk.

The location of the particle’s center is given by xp=

x(θp,rp,zp). In the absence of the particle, the incompressible

steady ﬂuid ﬂow in a curved duct driven by a steady pressure

gradient is referred to as Dean ﬂow14,29,30. The presence of

a particle disturbs this background Dean ﬂow and the parti-

cle responds to this disturbed ﬂow. Harding et al.1developed

a general model for the leading order forces that govern the

motion of such a particle in ﬂow through a curved duct (con-

stant R(θ)) at sufﬁciently small ﬂow rates. The forces on the

particle from the ﬂuid are calculated and used to construct a

ﬁrst order model for the trajectory of the particle giving the

following dynamical equations of motion1:

drp

dt=−Rep

Fs,r

,dzp

dt=−Rep

Fs,z

and dθp

dt=¯ua

R/a+rp

where ¯uais the axial component of the background ﬂuid ﬂow

velocity, Fs,r=Fs·erand Fs,z=Fs·ezare the radial and

the vertical components of the cross-sectional force, respec-

tively, with corresponding drag coefﬁcients Crand Czthat vary

with the particle’s position in the cross-section. The particle

Reynolds number is Rep=Re(a/l)2, where Re =ρUml/µ

is the channel Reynolds number with Umthe characteristic

axial velocity of the background ﬂuid ﬂow. Here, we use a

quasistatic approximation for the background ﬂuid ﬂow and

extend this particle dynamics model to investigate particle dy-

namics in spiral ducts. This approximation is reasonable for

spiral ducts with slowly changing curvature where the ﬂow

locally does not differ signiﬁcantly from Dean ﬂow in a con-

stant curvature duct with the same curvature31. Numerical

implementation of this model involves using a ﬁnite element

method to compute the forces acting on the particle (see Hard-

ing et al.1for more details). Once the forces are pre-computed

at numerous points in the cross-section and for numerous sys-

tem parameter values, interpolants of Cr,Cz,Fs,r,Fs,zare con-

structed and the particle dynamics are then simulated using

the MATLAB solver ode45. For simulations of particle dy-

namics in a spiral duct, we ﬁx the channel Reynolds number

to Re =25. All the results presented herein are in dimen-

sionless units with the dimensionless variables denoted by an

overhead tilde and the lengths scaled by l/2.

Consider particles of three different sizes ˜a1=0.05, ˜a2=

0.10 and ˜a3=0.15. For these particle sizes in a curved duct of

constant bend radius, nine different particle equilibria exist in-

side the square cross-section for large bend radius (see Fig. 2

and also Fig. 2(a,b) of Valani, Harding, and Stokes 27 ); four

stable nodes (green) near the center of the four edges, four

saddle points (yellow) near the corners and an unstable node

(red) near the center of the square cross-section. A slow mani-

fold is formed along a closed curve that connects all the stable

nodes and saddle points due to a large disparity in the magni-

tude of the two eigenvalues for these equilibria. As the bend

radius is decreased progressively, a number of bifurcations

take place. Firstly, saddle-node bifurcations take place where

the stable nodes near the center of the top and bottom walls of

the square collide and annihilate with the saddle points located

near the inner wall. Further decreasing the bend radius results

in the stable node located near the outer wall undergoing a

subcritical pitchfork bifurcation with the two saddle points lo-

cated above and below, and the three equilibria merge into a

single saddle point. As the bend radius is yet further reduced,

the unstable node near the center of the duct migrates towards

the stable node located near the inner wall and undergoes a

series of bifurcations28. Finally, at small bend radius, we get

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

UtilizingbifurcationstoseparateparticlesinspiralinertialmicrouidicsRahilN.Valani,1,a)BrendanHarding,2,b)andYvonneM.Stokes1,c)1)SchoolofMathematicalSciences,UniversityofAdelaide,SouthAustralia5005,Australia2)SchoolofMathematicsandStatistics,VictoriaUniversityofWellington,Wellington6012,NewZealand(Da...

展开>> 收起<<

Utilizing bifurcations to separate particles in spiral inertial microﬂuidics Rahil N. Valani1aBrendan Harding2band Yvonne M. Stokes1c 1School of Mathematical Sciences University of Adelaide South Australia 5005 Australia.pdf

共6页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Utilizing bifurcations to separate particles in spiral inertial microﬂuidics Rahil N. Valani1aBrendan Harding2band Yvonne M. Stokes1c 1School of Mathematical Sciences University of Adelaide South Australia 5005 Australia

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: