Page 1

arXiv:1002.4634v1 [physics.flu-dyn] 25 Feb 2010

Pumping by flapping in a viscoelastic fluid

On Shun Pak,1Thibaud Normand,2and Eric Lauga1, ∗

1Department of Mechanical and Aerospace Engineering,

University of California San Diego, 9500 Gilman Dr., La Jolla CA 92093-0411, USA.

2D´ epartment de M´ ecanique, Ecole Polytechnique, 91128 Palaiseau Cedex, France.

(Dated: February 25, 2010)

In a world without inertia, Purcell’s scallop theorem states that in a Newtonian fluid a time-

reversible motion cannot produce any net force or net flow. Here we consider the extent to which

the nonlinear rheological behavior of viscoelastic fluids can be exploited to break the constraints

of the scallop theorem in the context of fluid pumping. By building on previous work focusing on

force generation, we consider a simple, biologically-inspired geometrical example of a flapper in a

polymeric (Oldroyd-B) fluid, and calculate asymptotically the time-average net fluid flow produced

by the reciprocal flapping motion. The net flow occurs at fourth order in the flapping amplitude, and

suggests the possibility of transporting polymeric fluids using reciprocal motion in simple geometries

even in the absence of inertia. The induced flow field and pumping performance are characterized and

optimized analytically. Our results may be useful in the design of micro-pumps handling complex

fluids.

PACS numbers: 47.57.-s, 47.15.G-, 47.63.Gd

I.INTRODUCTION

Imagining oneself attempting to swim in a pool of viscous honey, it is not hard to anticipate that, because of

the high fluid viscosity, our usual swimming strategy consisting of imparting momentum to the surrounding fluid

will not be effective. The world microorganisms inhabit is physically analogous to this situation [1]. As a result,

microorganisms have evolved strategies which exploit the only physical force available to them – namely fluid drag

– to propel themselves or generate net fluid transport. The success of these propulsion strategies is vital in many

biological processes, including bacterial infection, spermatozoa locomotion and reproduction, and ciliary transport [2].

Advances in micro- and nano- manufacturing technology have also allowed scientists to take inspiration from these

locomotion strategies and design micropumps [3] and microswimmers [4].

The fundamental physics of small-scale locomotion in simple (Newtonian) fluids is well understood [5–7]. In contrast,

and although most biological fluids are non-Newtonian, many basic questions remain unanswered regarding the

mechanics of motility in complex fluids. Since they usually include biopolymers, most biological fluids of interest

display rheological properties common to both fluids (they flow and dissipate energy) and solids (they can store

energy). Examples include the airway mucus, which acts as a renewable and transportable barrier along the airways

of the lungs to guard against inhaled particulates and toxic substances [8], as well as cervical mucus, which is important

for the survival and transport of sperm cells [9]. The influence of viscoelasticity of the fluid on cell locomotion has

been experimentally quantified by a number of studies [10–16], including the change in the waveform, structure, and

swimming path of spermatozoa in both synthetic polymer solutions and biological mucus [17]. Gastropod mucus is

another common non-Newtonian biofluid, which is useful for adhesive locomotion, and its physical and rheological

properties have been measured [18–20]. Modeling-wise, different constitutive models have been employed to study

locomotion in complex fluids (see the short review in Ref. [21]). Among these models, the Oldroyd-B constitutive

equation is the most popular, both because of its simplicity and the fact that it can be derived exactly from kinetic

theory by modeling the fluid as a dilute solutions of elastic (polymeric) dumbbells [22–25]. Recent quantitative studies

have suggested that microorganisms swimming by propagating waves along their flagella have a smaller propulsion

speed in a polymeric fluid than in a Newtonian fluid [21, 26]. Likewise, a smaller net flow is generated by the

ciliary transport of a polymeric fluid than a Newtonian fluid. Specifically, Lauga [21] considered the problem with

a prescribed beating pattern along the flagellum, while Fu and Powers [26] prescribed the internal force distribution

instead; both studies suggest that viscoelasticity tends to decrease the propulsion speed.

∗Electronic address: elauga@ucsd.edu

Page 2

2

In a Newtonian fluid, Purcell’s scallop theorem states that swimming and pumping in the absence of inertia can

only be achieved by motions or body deformations which are not identical under a time-reversal symmetry (so-

called “non-reciprocal” motion) [1]. This poses of course an interesting challenge in designing artificial swimmers

and pumps in simple fluids, which has recently been addressed theoretically and experimentally (see the review in

Ref. [7]). The question we are addressing in this paper is the extent to which the scallop theorem holds in complex

fluids. Because polymeric fluids display nonlinear rheological properties such as shear-dependance or normal-stress

differences [23, 24], in general reciprocal motions are effective in polymeric fluids [27]. New propulsion and transport

methods can therefore be designed on small scales to specifically take advantage of the intrinsic nonlinearities of the

fluid. The goal of this paper is to study such a system in the context of fluid pumping with a simplified geometrical

setup where the pumping performance can be characterized analytically.

For simple flow geometries, it is not obvious a priori whether a simple oscillatory forcing of a nonlinear fluid leads

to a net (rectified) flow. For example, for all Oldroyd-like fluids, a sinusoidally-forced Couette flow leads to zero time-

averaged flow [23]. In previous work [28], we considered a biologically-inspired geometric example of a semi-infinite

flapper performing reciprocal (sinusoidal) motion in a viscoelastic (Oldroyd-B) fluid in the absence of inertia. We

showed explicitly that the reciprocal motion generates a net force on the flapper occurring at second order in the

flapping amplitude, and disappearing in the Newtonian limit as dictated by the scallop theorem. However, there was

no time-average flow accompanying the net force generation at second order [28]. Here, we report on the discovery

of a net fluid flow produced by the reciprocal flapping motion in an Oldroyd-B fluid. The net flow transport is

seen to occur at fourth order in the flapping amplitude, and is due to normal-stress differences. The dependence of

the pumping performance on the actuation and material parameters is characterized analytically, and the optimal

pumping rate is determined numerically. Through this example, we therefore demonstrate explicitly the breakdown

of the scallop theorem in complex fluids in the context of fluid pumping, and suggest the possibility of exploiting

intrinsic viscoelastic properties of the medium for fluid transport on small scales.

The geometric setup in this paper is motivated by the motion of cilia-like biological appendages. Cilia are short

flagella beating collaboratively to produce locomotion or fluid transport [5, 29]. For example, cilia cover the outer

surface of microorganisms such as Paramecium for self-propulsion. They are also present along our respiratory tracts

to sweep up dirt and mucus and along the oviduct to transport the ova. Our setup is also relevant to the rigid

projections attached to the flagellum of Ochromonas, known as mastigonemes, which protrude from the flagellum into

the fluid [5]. As waves propagate along the flagellum, the mastigonemes are flapped back-and-forth passively through

the fluid, a process which can lead to a change in the direction of propulsion of the organism [30–32].

Our study is related to the phenomenon known as steady (or “acoustic”) streaming in the inertial realm [33–46],

which has a history of almost two centuries after being first discovered by Faraday [33]. Under oscillatory boundary

conditions, as in the presence of an acoustic wave or the periodic actuation of a solid body in a fluid, migration of fluid

particles occur in an apparently purely oscillating flow, manifesting the presence of nonlinear inertial terms in the

equation of motion. This phenomenon occurs in both Newtonian and non-Newtonian fluids [37–46]. In particular, it

was found that the elasticity of a polymeric fluid can lead to a reversal of the net flow direction [37–40]. As expected

from the scallop theorem, no steady streaming phenomenon can occur in a Newtonian fluid in the absence of inertia.

However, as will be shown in this paper, the nonlinear rheological properties of viscoelastic fluids alone can lead to

steady streaming. In other words, we consider here a steady streaming motion arising purely from the viscoelastic

effects of the fluid, ignoring any influence of inertia.

Recently, polymeric solutions have been shown to be useful in constructing microfluidic devices such as flux stabi-

lizers, flip-flops and rectifiers [47, 48]. By exploiting the nonlinear rheological properties of the fluid and geometrical

asymmetries in the micro-channel, microfluidic memory and control have been demonstrated without the use of ex-

ternal electronics and interfaces, opening the possibility of more complex integrated microfluidic circuit and other

medical applications [47]. In the setup we study here, we do not introduce any geometrical asymmetries and exploit

solely the non-Newtonian rheological properties of the polymeric fluid for microscopic fluid transport.

The structure of the paper is the following.In §II, the flapping problem is formulated with the geometrical

setup, governing equations, nondimensionalization and the boundary conditions. In §III, we present the asymptotic

calculations up to the fourth order (in flapping amplitude), where the time-average flow is obtained.

characterize analytically the net flow in terms of the streamline pattern, directionality and vorticity distribution

(§IV). Next, we study the optimization of the flow with respect to the Deborah number (§V). Our results are finally

discussed in §VI.

We then

Page 3

3

ǫ

?t?

?n?

θ

eθ(θ)

er(θ)

π/2

r

FIG. 1: Geometrical setup and notations for the flapping calculation. A semi-infinite plane flaps sinusoidally with small

amplitude ǫ around an average position at right angle with an infinite wall.

II. FORMULATION

A. Geometrical setup

In this paper, we consider a semi-infinite two-dimensional plane flapping sinusoidally with small amplitude in a

viscoelastic fluid. The average position of the flapper is situated perpendicularly to a flat wall with its hinge point

fixed in space (see Fig. 1). The flapper is therefore able to perform reciprocal motion with only one degree of freedom

by flapping back-and-forth. Such a setup is directly relevant to the unsteady motion of cilia-like biological appendages

(see §I).

It is convenient to adopt planar polar coordinates system for this geometrical setup. The instantaneous position of

the flapper is described by the azimuthal angle θ(t) = π/2+ ǫΘ(t), where Θ(t) is an order one oscillatory function of

time and ǫ is a parameter characterizing the amplitude of the flapping motion. The polar vectors er(θ) and eθ(θ) are

functions of the azimuthal angle, and the velocity field u is expressed as u = urer+uθeθ. In this work, we derive the

velocity field in the the domain (0 ≤ θ ≤ π/2) in the asymptotic limit of small flapping amplitude, i.e. ǫ ≪ 1; the

time-averaged flow in the domain (π/2 ≤ θ ≤ π) can then be deduced by symmetry.

B. Governing equations

We assume the flow to be incompressible and the Reynolds number of the fluid motion to be small, i.e. we neglect

any inertial effects. Denoting the pressure field as p and the deviatoric stress tensor as τ, the continuity equation and

Cauchy’s equation of motion are respectively

∇ · u = 0, (1)

∇p = ∇ · τ. (2)

We also require constitutive equations, which relate stresses and kinematics of the flow, to close the system of

equations. For polymeric fluids, the deviatoric stress may be decomposed into two components, τ = τs+ τp, where

τsis the Newtonian contribution from the solvent and τpis the polymeric stress contribution. For the Newtonian

contribution, the constitutive equation is simply given by τs= ηs˙ γ, where ηsis the solvent contribution to the viscosity

and ˙ γ = ∇u+t∇u. For the polymeric contribution, many models have been proposed to relate the polymeric stress

to kinematics of the flow [22–25]. We consider here the classical Oldroyd-B model, where the polymeric stress, τp,

satisfies the upper-convected Maxwell equation

τp+ λ

▽

τp= ηp˙ γ, (3)

where ηpis the polymer contribution to the viscosity and λ is the polymeric relaxation time. The upper-convected

Page 4

4

derivative for a tensor A is defined as

▽

A=∂A

∂t

+ u · ∇A −?t∇u · A + A · ∇u?, (4)

and represents the rate of change of A in the frame of translating and deforming with the fluid. From Eq. (3), we

can obtain the Oldroyd-B constitutive equation for the total stress, τ, as given by

τ + λ1

▽τ= η

?

˙ γ + λ2

▽

˙ γ

?

, (5)

where η = ηs+ ηp, λ1= λ, and λ2= ηsλ/η. Here, λ1and λ2are the relaxation and retardation times of the fluid

respectively. The relaxation time is the typical decay rate of stress when the fluid is at rest, and the retardation time

measures the decay rate of residual rate of strain when the fluid is stress-free [23, 24]. It can be noted that λ2< λ1,

and both are zero for a Newtonian fluid.

C. Nondimensionalization

Periodic flapping motion with angular frequency ω is considered in this paper. Therefore, we nondimensionalize

shear rates by ω and stresses by ηω. Lengths are nondimensionalized by some arbitrary length scale along the flapper.

Under these nondimensionalizations, the dimensionless equations are given by

∇ · u = 0, (6a)

∇p = ∇ · τ, (6b)

τ + De1

▽τ = ˙ γ + De2

▽

˙ γ, (6c)

where De1= λ1ω and De2= λ2ω are defined as the two Deborah numbers and we have adopted the same symbols

for convenience.

D. Boundary conditions

The boundary condition in this problem can be simply stated; on the flat wall (θ = 0), we have the no-slip and the

no-penetration boundary conditions. In vector notation, we have therefore

u(r,θ = 0) = 0

(7)

along the flat wall.

Along the flapper, we also have the no-slip condition, ur(r,θ = π/2 + ǫΘ(t)) = 0. The other boundary condition

imposed on the fluid along the flapper is given by the rotation of the flapper, uθ(r,θ = π/2 + ǫΘ(t)) = rΩ(t), where

Ω(t) = ǫ˙Θ. In vector notation, we have then

u(r,θ = π/2 + ǫΘ(t)) = rΩ(t)eθ. (8)

III.ANALYSIS

Noting that a two-dimensional setup is considered, the continuity equation, ∇ · u = 0, is satisfied by introducing

the streamfunction Ψ(r,θ) such that ur= (∂Ψ/∂θ)/r and uθ= −∂Ψ/∂r. The instantaneous position of the flapper is

described by the function θ = π/2+ǫΘ(t), and we consider here a simple reciprocal flapping motion with Θ(t) = cost.

Since small amplitude flapping motion (ǫ ≪ 1) is considered, we will perform the calculations perturbatively in the

flapping amplitude and seek perturbation expansions of the form

{u,Ψ,τ,p,σ} = ǫ{u1,Ψ1,τ1,p1,σ1} + ǫ2{u2,Ψ2,τ2,p2,σ2} + ...,(9)

where σ = −p1+τ is the total stress tensor and all the variables in Eq. (9) are defined in the time-averaged domain

0 ≤ θ ≤ π/2. Since a domain-perturbation expansion is performed, careful attention has to be paid on the distinction

Page 5

5

between instantaneous and average geometry. Recall that the polar vectors er(θ(t)) and eθ(θ(t)) are functions of

the azimuthal angle which oscillates in time. To distinguish the average geometry, we denote ?t? = er(π/2) and

?n? = eθ(π/2) as the average directions along and perpendicular to the flapper axis (See Fig. 1). In this paper, ?...?

denotes time-averaging.

In addition, we employ Fourier notation to facilitate the calculations. In Fourier notation, the actuation becomes

Θ = Re{eit} and˙Θ = Re{ieit}. Because of the quadratic nonlinearities arising from boundary conditions and the

constitutive modeling, the velocity field can be Fourier decomposed into the anticipated form

u1= Re{˜ u1eit},

u2= Re{˜ u(0)

u3= Re{˜ u(1)

u4= Re{˜ u(0)

(10a)

2

3eit+ ˜ u(3)

+ ˜ u(2)

+ ˜ u(2)

2e2it}, (10b)

3e3it},

4e2it+ ˜ u(4)

(10c)

4

4e4it}, (10d)

with similar decomposition and notation for all other vector and scalar fields.

We now proceed to analyze Eq. (6) order by order, up to the fourth order, where the time-average fluid flow occurs.

The boundary conditions, Eqs. (7) and (8), are also expanded order by order about the average flapper position using

Taylor expansions.

A. First-order solution

1. Governing equation

The first-order Oldroyd-B relation is given by

τ1+ De1∂τ1

∂t

= ˙ γ1+ De2∂ ˙ γ1

∂t,

(11)

which in Fourier space becomes

˜ τ1=1 + iDe2

1 + iDe1

˜˙ γ1.(12)

We then note that we have ∇×∇·τ = 0 by taking the curl of Eq. (6b). Therefore, we take the divergence and then

curl of Eq. (12) to eliminate the stress and obtain the equation for the first-order streamfunction

∇4˜Ψ1= 0. (13)

2.Boundary conditions

At θ = π/2, the boundary condition at this order is given by

u1= r˙Θ?n?, (14)

which becomes

˜ u1= ir?n?, (15)

upon Fourier transformation. We also have the no-slip and no-penetration boundary condition at θ = 0.

3.Solution

The solution satisfying the above equation and boundary conditions is given by

˜Ψ1=ir2

4

(cos2θ − 1), (16a)

˜ u1r= −ir

2sin2θ,(16b)

˜ u1θ=ir

2(1 − cos2θ). (16c)