CHAOTIC ADVECTION IN PULSED SOURCE-SINK SYSTEMS
Pankaj Kumar and Mark A. Stremler
Department of Engineering Science and Mechanics
Virginia Polytechnic Institute and State University
In a number of microﬂuidics applications, the mixing of ﬂuid between two closely spaced ﬂat plates at low Reynolds
number, known as Hele-Shaw ﬂow, plays an important role. Under the Stokes ﬂow approximation, the depth-
averaged velocity in a Hele-Shaw ﬂow can be represented by a velocity potential. Eﬀective mixing for these systems
can thus be designed by considering chaotic advection in a two-dimensional potential ﬂow. Pulsed operation of a
source and a sink, produced by injecting and extracting ﬂuid, respectively, through small holes in one of the plates,
has been shown to generate chaotic advection in an unbounded domain . Our present work examines chaotic
advection generated by sources and sinks in a bounded circular domain. In a bounded system, the sources and
sinks must operate in pairs in order to conserve ﬂuid volume . For a source-sink pair placed on the boundary
of a circular domain, particle motions can be given explicitly by a discrete mathematical mapping. In this talk,
several diﬀerent conﬁgurations of source-sink pairs, primarily the two shown in Fig.1 (a) and (b), are examined to
determine the optimal operating parameters for producing chaos in a bounded Hele-Shaw ﬂow. Chaotic particle
motion in this system is primarily controlled by two operating parameters α and θ, where α is the dimensionless
pulse area of the ﬂuid injected and extracted by sources and sinks, respectively, and θ is the angle that the source-
sink pairs make with the horizontal axis as shown in Fig.1. The chaos diagnostics and measures considered in
this work include Poincar´ sections, Lyapunov exponents, and Kolmogorov (or KS) entropy. An example Poincar´ e
section is shown in Fig.1(c). It is found that conﬁguration (a) gives better mixing than conﬁguration (b) for small
values of α and all possible values of θ. In contrast, conﬁguration (b) gives better mixing for high values of α;
however, conﬁguration (b) with large values of θ gives ineﬃcient mixing. A few other source-sink conﬁgurations
are examined for further improvements of ﬂuid mixing in bounded Hele-Shaw ﬂow.
(a) (b) (c)
Figure 1: Arrangement of two source-sink pairs on the boundary of a circular domain. Source and sink locations are indicated
by ⊕ and , respectively. Sources and sinks connected by bold dashed lines form source-sink pairs that are operated together.
(a) Crossed source-sink pair conﬁguration. (b) Parallel source-sink pair conﬁguration. (c) Poincar´ section of ﬂuid ﬂow for
conﬁguration (a) with θ = 25 and α = 20%.
 S. W. Jones and H. Aref, “Chaotic advection in pulsed source-sink systems,” Phys. Fluids, 31, 469-485, 1988.
 M. A. Stremler and B. A. Cola, “A maximum entropy approach to optimal mixing in a pulsed source-sink
ﬂow,” Phys. Fluids, 18, 011701, 2006.