On the correspondence between creeping flows of viscous and viscoelastic fluids

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J. Non-Newtonian Fluid Mech. 145 (2007) 150–172

On the correspondence between creeping flowsof viscous and viscoelastic fluids

Ke Xu a,∗, M. Gregory Forest a, Isaac Klapper b

a Departments of Mathematics & Biomedical Engineering & Institute for Advanced Materials,University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, United States

b Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717, United States

Received 20 April 2006; received in revised form 23 April 2007; accepted 12 June 2007

bstract

From the wealth of exact solutions for Stokes flow of simple viscous fluids [C. Pozrikidis, Introduction to Theoretical and Computational Fluidynamics, Oxford University Press, Oxford, 1997, pp. 222–311], the classical “viscous–viscoelastic correspondence” between creeping flows ofiscous and linear viscoelastic materials yields exact viscoelastic creeping flow solutions. The correspondence is valid for an arbitrary prescribedource: of force, flow, displacement or stress; local or nonlocal; steady or oscillatory. Two special Stokes singularities, extended to viscoelasticityn this way, form the basis of modern microrheology [T.G. Mason, D.A. Weitz, Optical measurements of the linear viscoelastic moduli of complex
uids, Phys. Rev. Lett. 74 (1995) 1250–1253]: the Stokeslet (for a stationary point source of force) and the solution for a driven sphere. We amplify
hese viscoelastic creeping flow solutions with a detailed focus on experimentally measurable signatures: of elastic and viscous responses to steadynd time-periodic driving forces; and of unsteady (inertial) effects. We also assess the point force approximation for micron-size driven beads.inally, we illustrate the generality in source geometry by analyzing the linear response for a nonlocal, planar source of unsteady stress.2007 Elsevier B.V. All rights reserved.

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eywords: Creeping flow; Generalized Stokes–Einstein relation; Microrheolog

. Introduction

Linear response theory, of thermal fluctuations and their asso-iated power spectra and of driven motion from an imposedource, provides a basis for exploring viscous, elastic andompressible properties of condensed matter. For the casef a moving sphere, the viscous–viscoelastic correspondenceas developed in 1970 by Zwanzig and Bixon [34], moti-ated by numerical experiments of Alder and Wainwright1] on atomic fluctuation spectra. Zwanzig and Bixon devel-ped a quite general theory, allowing for linear viscoelasticityassuming a single mode Maxwell law), compressibility of theurrounding medium, arbitrary degree of slip of the sphere,nd inertial (unsteady) effects. They derived the generalizedtokes–Einstein drag law for viscoelastic fluids, and then the
elocity correlation function for thermal fluctuations. We noten even earlier application of linear response theory was carriedut by Thomas and Walters [30] in 1965 to model a sedimenting
∗ Corresponding author. Tel.: +1 919 962 5752; fax: +1 919 962 9345.E-mail address: [email protected] (K. Xu).

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377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2007.06.003

coelasticity; Correspondence

phere in a viscoelastic fluid. Their focus was on the transientotion and passage to terminal velocity (which they showed

epends only on the zero strain rate viscosity of the fluid).scillatory forcing of magnetic beads in viscoelastic materialsas carried out by Ziemann et al. [33] and then modeled with

orce balance arguments and spring-dashpot mechanical mod-ls to give an alternative method for storage and loss modulusharacterization.

Mason and Weitz [19] and Mason et al. [20] had the sem-nal idea to apply the generalized Stokes–Einstein drag lawnd associated power spectra of thermally fluctuating beadso rheology. The field of microrheology is now establisheds a viscoelastic characterization technique, with many vari-nts of the original Mason–Weitz protocol. Gittes et al. [10]sed laser-based microscopy techniques to measure trajecto-ies of individual spheres, together with linear response theory.rocker et al. [8] and Levine and Lubensky [16] developed thelastic–viscoelastic correspondence to relate two-point tracer
tatistics with viscoelastic (loss and storage) moduli, in someense a mirror-equivalent approach to the viscous–viscoelasticorrespondence emphasized in the present paper. More recently,iverpool and MacKintosh [17] and Atakhorrami et al. [2] high-
mailto:[email protected]

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K. Xu et al. / J. Non-Newtonia

ighted inertial (unsteady but still linear) features of the flowenerated by colloidal particles, using the exact solution of lin-ar response theory for a stationary point source of force in aiscoelastic material.

We refer the reader to various review articles in the pastew years cf. [11,28,5,18]); the present article has a reviewomponent as well. Our paper aims to place these results inunified context in which results and perspectives from vis-

ous hydrodynamics are transferred to linear viscoelasticityith relative ease, consistent with [34,19], by a straightfor-ard prescription—the viscous–viscoelastic correspondence.he analogous elastic–viscoelastic correspondence is addressed

n detail by Christensen [6].The present paper derives from the Virtual Lung Project at

NC and specifically our collaborations with R. Superfine, D.ill and J. Cribb, in order to model their driven microbead

xperiments for viscoelastic characterization of lung airway sur-ace liquids. The two special Stokes singularities that have beenpplied in microrheology thus far, the Stokeslet and the flowenerated by a driven sphere, are analyzed here in greater detailhan in the microrheology literature. These results are necessaryo model and interpret a range of active microrheology experi-

ents, including driven magnetic beads as well as bead tracersor propagating shear waves.

Why revisit these two examples at all? After all, Mason andeitz already applied the results for a single localized source orspherical source in their seminal papers [19,20], and Levine

nd Lubensky [16] and Crocker et al. [8] already analyzed andpplied special features of the displacement of one bead due tohe thermal motion of another bead, in their analysis of two-pointassive microrheology. The two-point focus is in the specialegime where the beads are separated by several bead diameters,here bead-fluid interactions are suppressed. In our colleagues’

xperiments, the beads do not all satisfy this criterion, and its of interest to know the response function in the immediateeighborhood of a driven bead. The microscope takes data in aocal plane, so it is also relevant to know whether or on whatimescale beads will stray out of the focal plane for a givenxperiment. The standard model for driven magnetic beads [33]elies on a force balance argument with an ad hoc geometricactor, and analogies with the Voigt mechanical model are oftensed to interpret creep-recovery data. More recent models [32]ave incorporated polymer network deformations in the imme-iate neighborhood of the driven bead. It is clearly of interesto derive an explicit expression for a bead driven by a magneticeld in an arbitrary linear viscoelastic material, which yields

he bead motion in time as well as the displacement and stresselds in the neighborhood of the bead. This information followsrom our analysis of a forced sphere. In shear wave experimentsith embedded bead tracers [21], normal stress generation is

apable of generating bead motion along the direction of waveropagation. Can one quantify this effect?

For these and related applications, we analyze the viscoelas-
ic creeping flow induced by a time-varying point source and ariven sphere to contrast responses from near-field to far-field,nd in different focal planes. Can we distinguish quasi-steadyersus unsteady (inertial) effects in the field surrounding a har-
tbad

d Mech. 145 (2007) 150–172 151

onically driven sphere, and if so, where? An investigation intohis question is approximated with the viscoelastic analog of atokeslet for an imposed time-varying point force by Liverpoolnd MacKintosh [17] and Atakhorrami et al. [2]. Here, we ana-yze inertia-induced vortices in both viscous and viscoelasticuids, and in particular, we show where vortices are spawned inead diameter dimensions and analyze the vortex strength rela-ive to the applied force, for time-varying point forces and drivenpheres. In each illustration, our goal is to inform experimentalrotocols as to whether and where signatures of elastic versusiscous properties are most accessible.

To formulate the viscous–viscoelastic correspondence, therst step is to cast linear response theory in parallel with the clas-ical hydrodynamic analyses of viscous creeping (Stokes) flow12,24]. Because of linearity, generalizations to richer sourceselevant for modern experiments are immediate, e.g., point orpherical sources with oscillating strength; we provide theseesults, which are straightforward, but which have not previouslyppeared in this detail in the literature. Again, our emphasis isn illuminating experimentally measurable features. The morehallenging analysis of initial-value problems, as in Thomas andalters [30], has not been introduced into microrheology thus

ar, and we do not take up the challenge here.In the viscoelastic formulation of linear response theory, the

iscosity of simple viscous liquids is replaced by the com-lex viscoelastic modulus of linear viscoelastic materials, afterhe equations have been transformed from the time domaino frequency space. This identification is possible becauseinear viscoelasticity presumes a convolution integral for thetress tensor, whose Fourier transform yields the Stokes relationith a complex (frequency-dependent) viscosity. The creep-

ng flow equations (steady or unsteady) can then be posedonsistent with point, finite or extended sources, of force, veloc-ty, strain or stress, and the correspondence remains intact forhe associated geometry and boundary-initial value problems.

henever the creeping viscous flow problem can be solved,he analogous solution of the viscoelastic flow problem fol-ows.

Thus far, the field of microrheology has exploited two suchreeping viscoelastic flow solutions, for a stationary point sourcef force and a driven sphere. In fact, only partial features ofhese solutions are typically used; we illustrate additional infor-

ation of experimental relevance in the analysis and figureselow. These are two from a large family of special solutionsrising from “Stokes singularity theory” of viscous fluids [24].y varying the geometry of the problem and the source (localr nonlocal, steady or unsteady, of force, stress, displacement orelocity), the essential calculation is that of a Green’s function,alled in this context a viscous Stokes singularity. We presentllustrations for point, spherical, and planar sources.

All Stokes singularities (and appropriate sums of them) carryver to viscoelastic media via this simple prescription. Thenferences that can be drawn from each creeping flow solu-
ion require some analysis and work. Detailed relationshipsetween force, displacement, stress and flow fields are avail-ble, which can then be applied to experimental data, or even toesign experiments. First, we provide a straightforward exten-

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where λ is the elastic relaxation time and G is the elastic

52 K. Xu et al. / J. Non-Newtonia

ion of results in the literature to a point source of force withime-dependent strength. The response functions of viscous andiscoelastic materials are contrasted for the same forcing, orlternatively, one can deduce how to achieve identical responseunctions by modifying the applied source. The very fact thatdentical responses can be achieved in viscous and viscoelastic

aterials underscores the need for careful analysis to supportxperimental design. Next, analogous results are obtained formoving sphere, where the source may be either an imposed

elocity of the sphere or an imposed force on the sphere. Foriscous fluids, these two problems are identical, but for vis-oelastic fluids, they are not. For point and spherical sources,e use the explicit relations to explore elasticity-induced con-

rasts for viscously matched fluids with respect to measurableuantities, both without and with inertial (unsteady) effects, ando compare the point source approximation of a time periodic,orce-driven sphere. Finally, to illustrate the generality of theorrespondence principle, we shift from local sources of forcer velocity to a nonlocal, planar source of unsteady, spatiallyeterogeneous stress. This third example, when restricted tohe special case of a homogeneous planar source of stress, isquivalent to the Ferry shear wave model [9], another classi-al experimental technique and a viscoelastic extension of thetokes problem for semi-infinite flow above a driven planaroundary.

. Linear response theory and the viscous–viscoelasticorrespondence principle

Consider unsteady creeping (Stokes) flow [12] with velocity, defined by

ut + ∇p = ∇ · τ + f, ∇ · u = 0, (1)

here ρ is the fluid density, p the pressure, τ the “extra” stressensor, and f is a prescribed, general force. For incompressibleiscous fluids, τ = 2η0D, where D = (1/2)(∇u + ∇uT) is theate-of-strain tensor. For an incompressible viscoelastic fluid,he canonical linear constitutive law involves a scalar relaxation

odulus G(t) [9]:

(x, t) = 2∫ t

−∞G(t − t′)D(x, t′) dt′.

ssuming causality, i.e., G(τ) = 0 for τ < 0, the linear vis-oelastic model becomes:

(x, t) = 2∫ ∞

−∞G(t − t′)D(x, t′) dt′. (2)

It is straightforward to generalize this discussion to com-ressible viscoelastic fluids, requiring one additional modulusunction [9]; see Zwanzig and Bixon [34].)

Alternative constitutive laws to (2) are often used. One canxpress the stress τ in terms of the time history of the strain tensor
, essentially by integrating (2) by parts. One can also prescriben inverse constitutive law for the strain tensor in terms of theistory of the rate of the stress, where the memory kernel J(t)s called the creep compliance.
d Mech. 145 (2007) 150–172

In practice, G(t) is typically determined in Fourier space bytting rheometric response data using the transform of (2):

ˆ (x, ω) = 2G(ω)D(x, ω), (3)

here either of stress τ(x, ω) or rate-of-strain D(x, ω) is imposednd the other measured. Here z refers to the temporal Fourierransform defined in the example of G(t) as

ˆ (ω) =∫ ∞

−∞G(τ)e−iωτ dτ. (4)

he alternative constitutive law formulations are appropriatehen one experimentally controls time dependence of strain or

tress. The “inverse” formulations of linear viscoelastic consti-utive laws are borne out by the relationship between G(t) and(t) in transform space (we omit the time domain relation for

heir convolution):

ˆ (ω)G(ω) = − 1

ω2 . (5)

Contact with notation in the rheology literature: Depend-ng on the experiment a variety of equivalent linear viscoelasticroperties are presented.

The complex relaxation modulus G∗(ω) = iωG(ω) =G′(ω) + iG′′(ω), which defines the storage modulus G′(ω)and loss modulus G′′(ω). The ratio G′′(ω)/G′(ω) = tan δ(ω)is known as the loss tangent.The complex viscosity η∗ = G(ω) = η′ − iη′′, where wenote for later usage that η′(ω) = ω−1G′′(ω) and η′′(ω) =ω−1G′(ω).The following identity will prove useful in examples:

G(ω) = η′(ω)

| sin δ(ω)|ei(δ(ω)−π/2). (6)

Some familiar finite-mode approximations for G(t), whichharacterize viscous and elastic behavior with a finite numberf material parameters, will be used later in examples.

Newtonian viscous fluids (G = GV):

GV(t) = η0δ(t) (7)

where η0 is the (constant) viscosity, G(ω) = η0, G′(ω) = 0,and η′(ω) = η0.

0 0modulus:

G1−M(ω) = G0λ0

1 + iωλ0, tan δ1−M(ω) = 1

ωλ0(9)

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and

G′1−M(ω) = ω2G0λ

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, G′′1−M(ω) = ωG0λ0

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η′1−M(ω) = G0λ0

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N-mode Maxwell (linear viscoelastic fluids) (G = GN−M):

GN−M(t) =N−1∑j=0

Gje−t/λj , (11)

where λj andGj are, respectively, the elastic relaxation timeand the elastic modulus (or spectral strength) of mode j. Also

GN−M(ω) =N−1∑j=0

Gjλj

1 + iωλj,

tan δN−M(ω) =

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N−1∑j=0

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j )

, (12)

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G′′N−M(ω) =

N−1∑j=0

ωGjλj

1 + ω2λ2j

,

η′N−M(ω) =

N−1∑j=0

Gjλj

1 + ω2λ2j

. (13)

We follow Zwanzig and Bixon [34] to analyze the linearystem (1)–(3). Applying a time Fourier transform to (1) andubstituting (3):

iρωu(x, ω) + ∇p(x, ω) = G(ω)∇2u(x, ω) + f(x, ω),

∇ · u(x, ω) = 0. (14)

If boundary conditions are time-dependent then they shoulde transformed as well—see below for an example.

Note that (14) is formally equivalent to the Fourier transformf the unsteady viscous Stokes flow Eq. (1) with the exceptionhat G(ω) is generally complex and frequency dependent. It fol-ows that if f(x, ω) and the boundary conditions are prescribedndependently of velocity u, then any solution of viscous Stokesow transforms to a linear viscoelastic creeping flow solution innatural way. This correspondence between Newtonian viscoustokes flow and linear viscoelastic creeping flow can be written
chematically as follows:
viscous Stokes flow ⇔ viscoelastic Stokes flow,

f, p ⇔ f, p, η0 ⇔ G(ω)

•

d Mech. 145 (2007) 150–172 153

here quantities on the left apply to viscous Stokes and quanti-ies on the right to linear viscoelastic Stokes. In particular: givenin Fourier space) a system of boundary conditions and pre-cribed forcing, viscous Stokes flow and viscoelastic creepingow produce formally identical solutions where η0 is replacedy G(ω).

We make the following observations specific to a particularrequency:

. G(ω) is in general complex for ω �= 0 so that (3) impliesthat τ(x, ω) is out of phase with D(x, ω), u(x, ω) by afrequency dependent phase shift. It is standard to writeG(ω) = −i|G(ω)|eiδ(ω), where tan δ(ω) = G′′(ω)/G′(ω) isthe classical “loss tangent”. This observation is independentof flow regime or source, depending only on (3). For speci-fied sources, these observations are then accompanied by therelative phase shifts between τ, D, u and a time-dependentforce, and by pre-factors arising from |G(ω)|. (At the risk ofnotational confusion, we will use δ for both the loss angle andfor the Dirac delta function, as both notations are universallystandard.)

. Taking the divergence of (14), we see that ∇2p = ∇ · f, sothat pressure is independent of the linear viscoelastic consti-tutive law and is in phase with external forcing.

. In the case of quasi-steady creeping flow, i.e., dropping theinertial term iρωu from (14), suppose g is the (real) viscousGreen’s operator [23]. We can write:

ˆ (x, ω) =∫

g(x, x′, ω)η0 f(x′, ω)

G(ω)dx′

=∫

g(x, x′, ω)feff

(x′, ω) dx′, (15)

here

eff(x, ω) = η0 f(x, ω)

G(ω)= η0

|G(ω)|ei(π/2−δ(ω)) f(x, ω). (16)

This schematic representation will be exploited and illus-rated in examples to follow. Note that u is generically out ofhase with the forcing. Referring back to Observation 1 and Eq.3), for forced systems, τ and f are in phase, whereas both areut of phase with u, D by the frequency dependent phase factor/2 − δ(ω). Fourier inversion of the “effective force” (16):

eff(x, t) =∫ ∞

−∞feff

(x, ω)eiωt dω

2π, (17)

ields an explicit (and quite general) correspondence principle15) between quasi-steady viscous and viscoelastic experiments.t follows immediately that:

forcing feff(t) and a viscoelastic fluid with forcing f(t);conversely, for experiments where force is imposed, analo-gous formulas (developed below) relate the elasticity-inducedcontrast in viscous velocity fields.

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These general statements are worked out below for threeistinct “experimental controls”: on force, on velocity, and ontress, and for three distinct source geometries: a point, a spherend an infinite plane. In each illustration, we further specializeource features (e.g. steady and harmonic) and the viscoelasticodulus G(t) to those listed above (7)–(13). We then explicitly

dentify ”trade-offs” between sources and responses for viscousnd viscoelastic materials (see Sections 3.2.1, 3.4.1, 3.4.2, 3.6.1nd 3.7). These specific viscous–viscoelastic relations are thenmplemented in the figures to simulate experimental observa-ions of viscosity-matched and elasticity-contrasted materials.

. For unsteady (i.e. inertial terms retained) viscoelasticcreeping flow, the scaling argument made in the previousobservation fails so that all three of u, τ, and f may be out ofphase with each other at any given frequency ω. See Section3.6 for explicit illustration.

. More generally, in the case of a superposition of forcingfrequencies, since G(ω) is frequency dependent, there is asuperposition of these single-frequency solutions, each witha different phase shift and effective viscosity. The picture isthus more complicated but formally straightforward. For anarbitrary time-dependent source, a Fourier integral is neces-sary to describe the response. We do not explore this level ofgenerality here.

. Exact solutions of viscoelastic creeping flow for threeource geometries

Dimensional considerations lead naturally to a complex-alued viscoelastic penetration depth:

(ω) =√G(ω)

iρω, (18)

here we assume throughout that ω �= 0; |(ω)| measures theepth to which viscoelastic stress dominates over inertia. Whenhe relevant system length scale L is small relative to ||, then wean neglect the unsteady term iρωu in (14). Conversely, whenis large relative to ||, inertial effects become dominant and

iscoelastic response is negligible at these length scales. Thepshot is that experiments aimed at viscoelastic properties needo be tuned to appropriate length scales and cognizant of theelative role of inertial effects. We illustrate with three sourceypes and geometries.

.1. Viscous quasi-steady Stokeslets (without inertia)

Consider a point force located at X0 with time-dependenttrength α(t) in an infinite (boundary free) domain:

(t) = α(t)δ(x − X0). (19)

The velocity field and the stress field are found by solving the
ontinuity equation together with the singularly forced Stokesquation:
· uv(x, t) = 0, (20)

F

η

d Mech. 145 (2007) 150–172

pv(x, t) = η0∇2uv(x, t) + α(t)δ(x − X0). (21)

he superscript v denotes the viscous formulas for velocity, pres-ure, stress and force. (More generally, we can allow X0 = X0(t)nd transform away the time dependence by x → x − X0(t).)hese equations have Stokeslet solution (see Pozrikidis [24]):

vi (x, t) = Sv

ij(x, η0)αj(t), (22)

vij(x, η0) = 1

8πη0

(δij

r+ xixj

r3

), (23)

here η0 is the zero strain rate viscosity for the fluid, x = x −0, r = |x|, and Sv

ij(x, η0) is called the viscous Stokeslet tensor.In the Fourier domain, the corresponding uv

i (x, ω) is giveny

ˆ vi (x, ω) = Sv

ij(x, η0)αj(ω). (24)

The stress field for uv becomes, since τ = 2η0D:

vik(x, t) = − 3

4π

xixjxk

r5 αj(t), (25)

hich is independent of fluid viscosity. Finally, to make con-act with experimental tracking of passive tracers, we note theisplacement field associated with a velocity field u:

vi (x, t) =

∫ t

t0

uvi (x(t′), t′) dt′ + x(t0). (26)

.2. Viscoelastic analog of quasi-steady Stokeslets (withoutnertia)

Following the development of the viscous Stokeslet, we con-ider a concentrated point force located at a fixed point X0 inn infinite, boundary-free domain with time-dependent strength(t) as in Eq. (19) above, or equivalently in the Fourier domain:

(x, ω) = α(ω)δ(x − X0). (27)

This example appears in several recent microrheology appli-ations [16,2,17]. From (14):

∇p(x, ω) = G(ω)∇2u(x, ω) + α(ω)δ(x − X0),

∇ · u(x, ω) = 0. (28)

hese equations have a Stokeslet-like solution (22)–(26), con-tructed simply by replacing η0 with G(ω):

ˆ i(x, ω) = Svij(x, η0)

η0αj(ω)

G(ω)= Sv

ij(x, η0)αeffj (ω), (29)

here

ˆ effj (ω) = η0αj(ω)

G(ω), (30)

nd factors ofη0 (an arbitrary real viscosity scalar) are inserted toake explicit contact with the viscous Stokeslet (23). Following
erry [9], a standard choice for η0 is
′(ω0) = G′′(ω0)

ω0, (31)

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The inverse transform of (29) yields the viscoelastic analogf the Stokeslet:

i(x, t) = Svij(x, η0)αeff

j (t), (32)

here the renormalized source strength αeff(t) is defined from(t) and the shear modulus G(t) by

effj (t) = η′(ω0)

∫ ∞

−∞αj(ω)

G(ω)eiωt dω

2π. (33)

Note: the viscous limit, where G(ω) = η′(ω0) = η0 is con-tant, yields αeff

j (t) = αj(t), and recovers (22).From here, we can make explicit connections between the

iscoelastic analog of a Stokeslet (32) and the correspondingiscous Stokeslet, (23). Namely, a purely viscous fluid of vis-osity η′(ω0) forced by feff(x, t) = αeff(t)δ(x − X0) will havedentical Stokes singularity solution (i.e., velocity field u(x, t))s a viscoelastic fluid with modulus G(t) forced by f(x, t) =(t)δ(x − X0). For special forcing strength α(t) and modulus(t), this correspondence can be made exact; see Section 3.2.1

elow. This relationship underscores the fact that the viscous andiscoelastic response functions for a point source are not fun-amentally different, so that care has to be taken in identifyingignatures of viscoelasticity.

Finally, we observe that the stress field for (32) is

ik(x, t) = − 3

4π

xixjxk

r5 αj(t), (34)

hich is independent of G; hence, the non-inertial stress is iden-ical for viscous and linear viscoelastic fluids. That is, onceransients and inertial effects have died out, there is no dis-inguishable stress signature of elasticity nor viscosity.

.2.1. Quasi-steady creeping flows for single frequency,armonic point forces: αj(t) = δ1jF0Im[eiω0t]

To make explicit predictions and comparisons, we restrict theime-dependent source strength to a single frequency ω0, suchs one might impose with an oscillatory magnetic field strengthn a stationary microscopic “point” source. Thus we impose:

(t) = F Im[eiω0t], (35)

here

= F0[1, 0, 0] = F0δ1j, (36)

nd recall the Fourier transform of eiω0t is 2πδ(ω − ω0). Select-ng the viscosity parameter consistent with the modulus G at themposed frequency, η0 = η′(ω0) = G′′(ω0)/ω0, we obtain, with
eff(t) = (αeff
1 (t), 0, 0),

ˆ eff1 (ω) = η′(ω0)α1(ω)

G(ω)= 2πF0

η′(ω0)δ(ω − ω0)

G(ω). (37)

rsλ

r

d Mech. 145 (2007) 150–172 155

This characterization in transform space can be explicitlynverted back to the time domain:

eff1 (t) = F0η

′(ω0)Im

[eiω0t

G(ω0)

], (38)

= F0η′(ω0)

|G(ω0)| sin(ω0t + π/2 − δ0), (39)

here

an(δ0) = G′′(ω0)

G′(ω0). (40)

n explicit amplitude and phase renormalization of the sourcetrength follows:

eff1 (t) = η′(ω0)

|G(ω0)| α1

(t + π/2 − δ0

ω0

). (41)

Furthermore, by virtue of the separable form of the vis-oelastic creeping flow solution (32), the amplitude and phaseenormalization due to viscoelasticity transfers directly to theelocity field in the real time domain with no effects on thetress field:

The viscoelastic creeping flow and viscous Stokeslet veloci-ies, u and uv respectively, for the same harmonic point source19), (35), (36), are related by a transparent relation:

i(x, t) = A(δ0)uvi (x, t + φ(δ0)), (42)

ik(x, t) = τvik(x, t), (43)

here the phase shift φ and attenuation factor A are obtainedxplicitly from the loss tangent δ(ω0) at the frequency ω0 of theource:

(δ0) = | sin(δ0)| ≤ 1, (44)

(δ0) = π/2 − δ0

ω0. (45)

f we consider the single-mode Maxwell constitutive law,8)–(10), the attenuation and phase formulae become especiallyransparent:

= 1√1 + ω2

0λ20

and φ = 1

ω0tan−1(ω0λ0). (46)

We now use formulae (42)–(46) to illustrate quasi-steady,on-inertial viscoelastic creeping flow and viscous Stokeslet fea-ures in Figs. 1–7, with a single forcing frequency. A benefit ofhese relations is that real-time predictions of an experimentan be made, as opposed to standard matching in transformpace. From the velocity formulae above, we generate displace-ent fields of passive markers in the vicinity of the source. The

arameters we use are listed in Table 1 and are representative ofirway mucus material properties and cilia forcing frequency.he characteristic shear relaxation times for mucus span the

ange 10−2 s to 102 s. In addition to a viscous fluid, we con-ider two single-mode Maxwell fluids with relaxation times0 = 0.01 s and λ0 = 0.1 s, respectively. (Predictions for longerelaxation times follow immediately from (46) but since they

156 K. Xu et al. / J. Non-Newtonian Fluid Mech. 145 (2007) 150–172

Fig. 1. A snapshot of planar displacement fields produced by a quasi-steady viscous Stokeslet and two analogous viscoelastic creeping flow solutions due to astationary point source at the origin, directed parallel to the x axis, with harmonic strength α(t) = 100 sin(ω0t) pN, and frequency ω0 = 10 Hz. Each fluid hasidentical viscosity, η0 = 50 cP. The length scale is 1 �m. In (a)–(c), a field of markers in the z = 0 plane is labeled at an initial time when their effective force is zero,tracked for a half-period (t = 0.05 s) of positive force along the positive x-axis. By symmetry, all markers remain in the z = 0 plane. (a) Purely viscous fluid (red).(b) Viscoelastic 1-mode Maxwell fluid with relaxation time λ0 = 0.01 s (blue). (c) Viscoelastic 1-mode Maxwell fluid with λ0 = 0.1 s (green). (d) Superposition ofthe displacements of two lines of particles from (a), (b) and (c), one line starting along x = −4 �m and the other line along x = −2 �m. The yellow dots are thestarting positions for each line, while the triangle-arrow represents the position and direction of the point force. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of the article.)

Fig. 2. Comparison of displacement fields vs. time of a line of red viscous(λ0 = 0 s) and blue viscoelastic (λ0 = 0.01 s) markers. The markers begin atthe same locations at t = 0, then are tracked for the same full cycle of the forcestrength α(t), 0 ≤ t ≤ 0.1 s, revealing apparent recoil due to the elastic phaselag. (For interpretation of the references to colour in this figure legend, the readeris referred to the web version of the article.)

Fig. 3. Comparison of displacement fields vs. time of a line of red viscous andgreen viscoelastic (λ0 = 0.1 s) markers, as in Fig. 2. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web versionof the article.)

K. Xu et al. / J. Non-Newtonian Fluid Mech. 145 (2007) 150–172 157

Fig. 4. A snapshot of 3D displacement fields produced by non-inertial viscous and viscoelastic analogs of Stokeslets for the same solutions but different markersthan Fig. 1. In (a)–(c), a field of markers in the z = 1 �m plane is labeled at an initial time when their effective force is zero, tracked for a half-period (t = 0.05 s)of positive force along the positive x-axis. Each marker executes planar motion in the plane of r and F, which is not the z = 1plane. In (d), we extract two differentinitial lines of tracers, along x = −3.5 �m and x = −1.5 �m, and their deformations ffluid λ0 = 0 (red) and two viscoelastic fluids of Maxwell type with relaxation times λby 90◦. (For interpretation of the references to colour in this figure legend, the reader

Fig. 5. A snapshot of planar displacement fields for a line of markers startingfrom z = 1 �m, y = 0 in three different fluids. The displacements are producedby a stationary point source at the origin, directed parallel to the x axis, withharmonic strength α(t) = 100 sin(ω0t) pN, with ω0 = 10 Hz. These markers arelabeled at an initial time when their effective force is zero, tracked for a half-period (t = 0.05 s) of positive force along the positive x-axis. By symmetry, allmarkers remain in the y = 0 plane. Each fluid has identical viscosity, η0 = 50 cPbut different relaxation times λ0 = 0 (red), 0.01 s (blue) and 0.1 s (green). Thelength scale is 1 �m. The yellow dots are the markers’ original positions. (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of the article.)

gemifi(1

F(p−sltv

rom (a) to (c) are superimposed to illustrate the elasticity contrasts for a viscous

0 = 0.01 s (blue) and 0.1 s (green). Note, figure (d) is rotated around the z-axisis referred to the web version of the article.)

enerate highly attenuated displacement fields for this modelxperiment, they are omitted.) The viscosity of “healthy” humanucus is about 50 times that of water in the frequency range of

nterest. In order to highlight elasticity contrasts in displacementelds for viscosity-matched liquids, we fix the viscosity η′(ω0)31) to be 50 cP for each of the three sample fluids, and vary the-mode Maxwell elasticity parameter λ0. Healthy cilia beat in

ig. 6. Comparison of displacement fields vs. time of a line of viscous (red)λ0 = 0 s) and viscoelastic (blue) (λ0 = 0.01 s) markers in the z = 1 � m focallane. The markers begin at the same locations (x = −1 �m, y ranging from10 to 10 �m) for each fluid, then are tracked for the same full cycle of the force

trength α(t), 0 ≤ t ≤ 0.1 s, revealing apparent recoil due to the elastic phaseag. The yellow dots are the markers’ original positions. (For interpretation ofhe references to colour in this figure legend, the reader is referred to the webersion of the article.)

158 K. Xu et al. / J. Non-Newtonian Flui

Fig. 7. Comparison of displacement fields vs. time of a line of viscous (red) andvmfi

tli

mfateptFmez

ity

F

pS

TP

P

ω

η

λ

G

G

δ

A

tfotvleit

iaifiaflivλ

s(aaFty

d

awscsfteoap0tw

iscoelastic (green) (λ0 = 0.1 s) markers, as in Fig. 6. The black dots are thearkers’ original positions. (For interpretation of the references to colour in thisgure legend, the reader is referred to the web version of the article.)

he range of 10 ∼ 15 Hz; we choose ω0 to be 10 Hz in the fol-owing simulations. A reasonable estimate of forces in the lungs 100 pN picoNewtons, which we use for F0.

A few remarks help to motivate the features illustrated. Anyarker remains in the plane of its position vector and the applied

orce, for all time. It follows that every plane passing through thexis of the point force is an “invariant plane” of this flow; here,he force is at the origin and along the x-axis. In microscopyxperiments, the stage is set so that one can adjust the focallane vertically, which in our choice of coordinates correspondso heights z =constant. Thus, z = 0 is an invariant plane, andigs. 1–3 track markers in this plane, contrasting the displace-ent fields for materials with identical viscosity and varying

lasticity (parametrized by λ0). Note that any other focal plane,=constant, which does not contain the point source, is not an

nvariant plane, and markers will move vertically as well as inhe plane. Figs. 4, 6 and 7 will address this feature. Note that= 0 is also an invariant plane. We will illustrate this feature in
ig. 5.
In Fig. 1, we present a snapshot of the displacement fieldsroduced by viscous (a) and viscoelastic analogs (b and c) oftokes singularities over a time interval in which each respec-

able 1arameters

arameters Viscous Stokeslet V–E analog ofStokeslet 1

V–E analog ofStokeslet 2

0 20π s−1(10 Hz) 20π s−1(10 Hz) 20π s−1(10 Hz)′ 50 cP 50 cP 50 cP

0 0 s 0.01 s 0.1 s′(ω0) 0 Pa 1.97 Pa 19.74 Pa′′(ω0) 3.14 Pa 3.14 Pa 3.14 Pa

0 π/2 0.32 π 0.05 π(δ0) 1 0.85 0.16

nEv(Ccvs

ae(cIto

d Mech. 145 (2007) 150–172

ive material is exposed to a half-cycle of unidirectional, positiveorce. The formulas (42) and (46) allow us to remove the effectsf the viscoelastic phase shift, by a simple translation of theime interval, and focus on elasticity-induced attenuation foriscosity-matched materials exposed to identical forcing. Theength scale in each figure is 1 �m; typical suspended bead diam-ters are 200 nm to 1 �m. A field of markers in the z = 0 planes labeled at t = 0 in (a), when α(t) = 0, then phase shifted to0 = −φ in (b and c), when αeff(t) = 0. Recall the period of αs 0.1 s (ω0 = 10 Hz). The field of markers is tracked for .05 s,half-period of forcing in which each respective force strength

s non-negative, causing motion along the positive x-axis. Thesegures thus correspond to the maximum displacements observ-ble in each material under these conditions. The purely viscousuid, withλ0 = 0, clearly experiences the largest displacements,

n the direction of the force. There is no storage of stress. Theiscoelastic fluids, one with λ0 = 0.01 s (b) and another with= 0.1 s (c), experience smaller displacement fields, since they

tore a fraction of the stress in each cycle. The contrasts betweena–c) are explained from the velocity attenuation contrasts (46)nd formula (42): A = 1 for the viscous fluid, A = 0.85 in (b)nd A = 0.16 in (c). The attenuation contrast is highlighted inig. 1 d, which superposes the final positions corresponding

o two initial lines of particles (positions x = −4 or −2 �m,= −10 to 10 �m with spacing 0.5 �m), for each of the three

ifferent fluids.Figs. 2 and 3 further illustrate the predictions of formulas (42)

nd (46), through time traces of particle paths. As in Fig. 1d,e start with a line of markers in the z = 0 plane, with the

ame x-coordinate x = −3.5 �m, and with 40 equally spaced y-oordinates, −10 �m to 10 �m. (Note that we omit the markertarting from [−3.5, 0, 0] �m to prevent contact with the pointorce.) Since the solutions are non-inertial, and periodic in time,hey preserve the Stokes property of time-reversibility. Namely,ach solution will trace out a curve, that is retraced each periodf the forcing function. However, the relative phase of the forcemplitude α matters. Here we evaluate all solutions over oneeriod of the forcing function, starting at t = 0 and through t =.1 s. We do not phase shift as in Fig. 1 to generate the response ofhe viscoelastic materials. Thus, the viscoelastic markers beginith a partial cycle of positive force, followed by a 0.5 cycle ofegative force, then the remainder of the positive force cycle.ach figure contrasts the viscous displacement data (blue) andiscoelastic data, with λ0 = 0.01 s (red) in Fig. 2 and λ0 = 0.1 sgreen) in Fig. 3. The length scale in each figure is 0.5 �m.learly, the viscous markers move to the right under the half-ycle of positive force, then retrace back to their origin. Theiscoelastic markers exhibit an “apparent recoil” due to the phasehift expressed in αeff(t), (41).

In Figs. 4–7, we address the issue of the paths of tracers thatre not in the focal plane z = 0 of the source in a microscopyxperiment, and which thereby execute apparent 3D motion.Each particle of course lies on a closed segment of a planar
urve, but the microscope will not be focused along that path!)n Fig. 4 we illustrate the displacement fields of an array ofracers initialized in the z = 1 �m plane. The markers all moveut of the z = 1 �m plane, displacing vertically and transversely.

n Flui

Ttafro1F(r

wtyst

ttλ

lla

cpdoueerss

3

o

U

ooeFtafi

u

(w

af

τ

w

T

T

S

F

(s

amttsbdcrr

3l

3a

tT(i

u

τ

F


he attenuation versus increased λ0 is apparent, and one findshe maximum vertical displacement occurs at the marker justbove the position of the force, with the maximum height rangingrom 1.63 �m to 1.55 �m to 1.13 �m for λ0 = 0, 0.01, 0.1 s,espectively. Since each marker traces a closed path in the planef the force (x-axis) and its initial position, the line (x, y = 0, z =�m) remains in the y = 0 plane and we illustrate this feature inig. 5. These particle paths clearly show the relative translationalalong x) and transverse (along y) displacements versus elasticelaxation contrast, for viscosity-matched liquids.

Figs. 6 and 7 are analogs of Figs. 2 and 3. In these two figures,e start with a line of markers in the z = 1 �m plane, with

he same x-coordinate x = −1 �m, and with 21 equally spaced-coordinates, −10 �m to 10 �m. As before, we evaluate allolutions over one period of the forcing function, starting at= 0 s and through t = 0.1 s. We do not phase shift to generate

he response of the viscoelastic materials. Each figure contrastshe viscous displacement data (blue) and viscoelastic data, with0 = 0.01 s (red) in Fig. 6 and λ0 = 0.1 s (green) in Fig. 7. The

ength scale in each figure is 0.5 �m. The attenuation and phaseag due to elasticity of viscosity-matched fluids are apparent,nd each effect is important for interpretation of data.

If multiple frequencies are present, the velocity and stressorrespondences are more complicated; for example, the super-osition in (32) results in a sum of the above solutions withifferent viscosity scalings and phase shifts. Even in the casef single frequency forcing, if the unsteady term is present then

ˆ (x, ω0) is no longer linear in G(ω0). As a further remark, resultsxtend to multiple point forces by superposition, and can bextended to viscoelastic dipoles and other singularities, all byeplacing η with G(ω) and carrying out the appropriate analy-is. See Pozrikidis [24] for a complete catalog of viscous Stokesingularities.

.3. Forced spheres in viscous fluids (the quasi-steady limit)

When the sphere velocity is prescribed as a periodic functionf period P of the form:

(t) = U0

M−1∑k=0

Im[eiωk(t+ξk)], ωk = 2πk

P, (47)

ne can construct the fluid velocity for Stokes flow past a spheref radius a in a fluid with viscosity η0. Because of the lin-arity of the equations, the Stokes equation can be solved inourier space for each fixed frequency ωk; each of these solu-

ions is constructed by combining two singular solutions, namelyStokeslet and a potential dipole [24]. By summation, the floweld generated by the imposed sphere motion (47) is then:

vi (x, t) = 1

4

a

r

(3 + a2

r2

)Ui(t)

( 2)
+3
4

a

r1 − a

r2

xixj

r2 Uj(t). (48)

Note that the velocity field is independent of fluid viscosityhen the sphere motion is controlled, and the viscosity then

w

U

d Mech. 145 (2007) 150–172 159

rises in the measured stress field, or equivalently, in the dragorce Fv

drag(t) on the sphere due to the velocity U(t).)The corresponding stress tensor is given by [24]:

vik(t) = 3aη0

4T SijkUj(t) − a3η0

4TDijkUj(t), (49)

here

Sijk = −6

xixjxk

r5 ,

Dijk = 6

δijxk + δikxj + δjkxi

r5 − 30xixjxk

r7 .

The drag force Fvdrag(t) on the sphere is given by the viscous

tokes drag law,

vdrag(t) = −6πaη0U(t) (50)

found by integrating the fluid stress over the surface of thephere).

Classical Stokes formulas (48)–(50) convey that u, τ and Fre easily related for viscous fluids, all in terms of the boundaryotion U(t) on the sphere. For viscoelastic fluids, these rela-

ions are non-trivial. If one controls either U or F on a sphere,hen linear response theory gives explicit formulas for the corre-ponding F or U. These formulas are simple in transform space,ut non-trivial in the time domain. Once again, we specialize theriving conditions and viscoelastic modulus to arrive at exactorrespondences in the time domain. (Generalizations of theseesults for slip and partial-slip boundary conditions follow fromesults in Zwanzig and Bixon [34].)

.4. Forced spheres in viscoelastic fluids (the quasi-steadyimit)

.4.1. Imposed periodic velocity on spheres in fluids withrbitrary relaxation modulus

It is straightforward to solve the quasi-steady restriction ofhe creeping viscoelastic flow Eq. (14) in the Fourier domain.he viscoelastic solution corresponds to the viscous counterpart

48)–(50), with η0 replaced by G(ω). One obtains the general-zed Stokes drag law [34,19,20] in transform space:

ˆ i(x, ω) = 1

4

a

r

(3 + a2

r2

)Ui(ω) + 3

4

a

r

(1 − a2

r2

)

× xixj

r2 Uj(ω), (51)

ˆik(ω) = τvik

G(ω)

η0(52)

= 3aη0

4T SijkU

effj (ω) − a3η0

4TDijkU

effj (ω), (53)

ˆ drag(ω) = −6πaG(ω)U(ω) = −6πaη0Ueff

(ω), (54)

here we have introduced an “effective sphere velocity”:

ˆ eff(ω) = G(ω)

η0U(ω). (55)

1 n Flui

Rr

U

Ad

u

τ

F

w

U

I

U

(tsqt

•

•

•

•

U

pp

rh

U

F

U

waiav

F

τ

wvsa

chi

12

tft(

3q

svms

U

a


ecalling the identity (6) for G(ω), we have an alternative rep-esentation:

ˆ eff(ω) = η′(ω)

η0

1

| sin(δ(ω))|ei(δ(ω)−π/2)U(ω). (56)

nalogous formulas for the viscoelastic u, τ,F in the real timeomain are:

i(t) = uvi (t), (57)

ik(t) = 3aη0

4T SijkU

effj (t) − a3η0

4TDijkU

effj (t), (58)

drag(t) = −6πaη0Ueff(t), (59)

here

eff(t) =∫ ∞

−∞U(ω)

G(ω)

η0eiωt dω

2π. (60)

f we specify U(t) as a finite Fourier sum (47):

eff(t) = −U0

{M−1∑k=0

η′(ωk)η0

1

| sin(δ(ωk))| cos[δ(ωk)

+ωk(t + ξk)]

}. (61)

Note: The insertion of a viscosity parameter η0 in (54) and

59), and thereby also in the definition of Ueff

(ω), (55), is forhe sole purpose of identifying explicit relations for the imposedphere velocity boundary-value-problem, (51)–(56) in the fre-uency domain, and (57)–(60) in the time domain. We make thehe following observations:

The viscous limit, G(ω) = η0, yields Ueff(t) = U(t), while(58) and (59) recover (49) and (50).If one prescribes U(t), then the non-inertial quasi-steadyvelocity fields for viscous and viscoelastic fluids are identical,(51) and (57).To achieve U(t) on the boundary of the sphere, therequired viscous drag force Fv

drag(t) is immediate: Fvdrag(t) =

−6πaη0U(t). For viscoelastic fluids, however, the drag forceFdrag(t) is nontrivially dependent on the modulus G(t), from(54)–(56), (59) and (60). There is a trade-off due to elasticity

captured by the scaled sphere velocity, Ueff

(ω) or Ueff(t), (55)or (60). We return to applications of these formulas below.(Realistically, an experiment controls the applied force F(t)on the sphere, and the induced drag force Fdrag = −F.)Whereas the velocity fields for prescribed sphere motion areidentical, the stress fields responsible for the drag forcesare clearly elasticity-dependent. The chosen representations,τ(ω) in (52), (53) and τ(t) in (58), are efficient for elucidatingelasticity contrasts in viscosity-matched liquids, analogous to
the point force illustrations above.
For illustration, we turn to specialized imposed sphere motion(t) where the above correspondence formulas yield exact trans-

gvft

d Mech. 145 (2007) 150–172

arent predictions, i.e. exact force specification to achieve therescribed sphere motion.

The simplest example of this class of exact solutions is toestrict to a 1-mode Maxwell fluid with single frequency (ω0)armonic oscillation of a sphere, where

(t) = U0 sin(ω0(t + ξ0)). (62)

rom (60), it is straightforward to obtain:

eff(t) = U01

| sin(δ0)| sin[ω0(t + ξ0) + δ0 − π

2

], (63)

here δ0 is defined as tan(δ0) = 1/(ω0λ0). Then the drag forcend corresponding stress field for the viscoelastic fluid aremmediate from (58) and (59). This result can be expresseds an explicit relation between viscosity-matched viscous andiscoelastic fluids for the oscillating sphere experiment:

drag(t) = Fvdrag

(t + δ0 − π/2

ω0

)1

A(δ0), (64)

ik(t) = τvik

(t + δ0 − π/2

ω0

)1

A(δ0), (65)

here Fvdrag(t) and τv

ik(t) are the viscous drag force (50) and theiscous stress tensor (49) with U(t) given in (62). RecallA(δ0) =in(δ0) is the attenuation factor in the point force example (46)nd η0 is the common viscosity.

These two explicit applications of the correspondence prin-iple, for a simple harmonic stationary point force and singlearmonic oscillatory sphere, (42)–(46) and (64)–(65), share sim-lar solution features:

. Equal but opposite-signed phase shifts φ(δ0) and −φ(δ0).

. Reciprocal multiplicative factors, A(δ0) and 1/A(δ0).

We note that forω � λ (orω � max(λj) in the N-mode case)he viscoelastic Stokes drag law reduces to a viscous one, i.e.,or time scales long compared to the longest elastic relaxationime, the drag converges to the viscous law with viscosity λG0or∑λjGj in the N-mode version).

.4.2. Imposed periodic force (stress) on a sphere in theuasi-steady limit

Suppose the experimental control is the force F(t) on thephere where F(t) is a periodic function in either a viscous oriscoelastic fluid; this is the appropriate formulation for drivenagnetic beads. From the Stokes drag law (50), the viscous

phere motion is

v(t) = F(t)

6πaη0, (66)

nd the velocity and stress fields of the surrounding fluid are
iven by (49)–(50). To get the velocity and stress fields of aiscoelastic fluid, one proceeds as follows. In transform space,rom the Stokes formula (54), the sphere velocity in a viscoelas-ic fluid with modulus G(t) (or equivalently compliance J(t))

n Flui

i

U

a

U

v

u

τ

u

τ

t(

p

F

T

U

3tf

U

=

wia

•

•

3tWpsS

F

i

U

wt

3et

fwp

F

o

F

F

F

at

x

F


s

ve(ω) = F(ω)

6πaG(ω)= − ω2

6πaF(ω)J(ω), (67)

nd in the real time domain:

ve(t) =∫ ∞

−∞F(ω)

6πaG(ω)eiωt dω

2π

= −∫ ∞

−∞ω2

6πaF(ω)J(ω)eiωt dω

2π. (68)

Following the procedure outlined above, we then obtain theelocity and stress fields:

ˆ i(x, ω) = 1

4

a

r

(3 + a2

r2

)Uvei (ω) + 3

4

a

r

(1 − a2

r2

)

× xixj

r2 Uvej (ω), (69)

ˆik(ω) = 1

−24π(3T S

ijkFj(ω) − a2TDijkFj(ω)), (70)

i(x, t) = 1

4

a

r

(3 + a2

r2

)Uvei (t) + 3

4

a

r

(1 − a2

r2

)

× xixj

r2 Uvej (t), (71)

ik(t) = 1

−24π(3T S

ijkFj(t) − a2TDijkFj(t)). (72)

Note: It is apparent that for an imposed force on the sphere,he induced stress field is independent of the material propertiesviscous or viscoelastic).

Analogous to the finite-mode assumption (47), suppose theeriodic force has the form:

(t) = F0

M−1∑k=0

Im[eiωk(t+ξk)].

hen the sphere motion is given by

ve(t) = F0

6πa

{M−1∑k=0

1

η′(ωk)| sin(δ(ωk))| cos[δ(ωk)

−ωk(t + ξk)]

}. (73)

.4.2.1. Specialized formulas for a Maxwell fluid. As before,he simplest illustration is a 1-mode Maxwell fluid with singlerequency harmonic forcing: F(t) = F0 sin(ω0t). Then

ve(t) = Im

[∫ ∞

−∞F (ω)

6πaG(ω)eiωt dω

2π

]

= | sin(δ0)|Uv(t + π/2 − δ0

ω0

)(74)

A(δ0)Uv(t + φ), (75)

pd

x

d Mech. 145 (2007) 150–172 161

here recall tan(δ0) = G′′(ω0)/G′(ω0). Now from Uve(t), thenduced flow is immediate from (71), with the same attenuationnd phase shift features.

In summary,

if an identical force is imposed on a viscous and viscoelas-tic liquid, the stress fields match while the viscoelasticvelocity field is attenuated by A(δ0) = | sin(δ0)| and phasedshifted by some angle φ = π/2 − δ0/ω0 where tan(δ0) =G′′(ω0)/G′(ω0);conversely, if identical sphere motion is imposed, the induceddrag force and stress field on the viscoelastic liquid are ampli-fied by 1/A(δ0) and phase shifted by −φ relative to the viscousliquid.

.4.2.2. Special case of a constant force due to gravity. Fromhe above solutions, we easily recover the result of Thomas and

alters [30]: the terminal velocity of a falling sphere is inde-endent of elasticity. We specialize to a constant force on thephere, F = F0, in a viscoelastic medium with modulus G(t).ince

ˆ (ω) = 2πF0δ(ω),

t is straightforward to calculate:

ve(t) =∫ ∞

−∞F(ω)

6πaG(ω)eiωt dω

2π= F0

6πaG(0), (76)

here G(0) = η0 is the zero strain rate viscosity. Thus, theerminal velocity does not depend onG(t) except through G(0).

.4.2.3. Special case of a step force. In driven magnetic beadxperiments, a step force is applied at t = 0 for a period of time0 and bead displacement data are registered during both theorce-on and force-off phase. To model the bead dynamics, weill prescribe the force as a Heaviside function (for long forceulses) or the sum of two Heaviside functions for finite t0,

step(t) = F0H(t) (77)

r

pulse(t) = F0(H(t) −H(t − t0)). (78)

irst we recall (54) and (55),

ˆ (ω) = 6πaG(ω)U(ω),

nd note the relation in transform space between the bead posi-ion x(t) and velocity U(t) = x(t):

ˆ(ω) = U(ω)

−iω .

urther recalling the relationship between the modulus and com-ˆ ˆ 2
liance in transform space, G(ω)J(ω) = −(1/ω ), Eq. (5), we
etermine the bead position in transform space:

ˆ = − iω

6πaFJ . (79)

1 n Fluid Mech. 145 (2007) 150–172

xw

x

w

x

F

x

Waac

3p

bvnama

orhTpao

α

F

u

S

flv

u

w

b

Fig. 8. Stokeslets and their viscoelastic analogs. Snapshots are given at t =0.01 s for four distinct normalized velocity amplitudes |u|/U0 vs. distance rmeasured in a where U0=0/6πaη04, f0 is the force amplitude, η0 = 50 cP anda = 1 �m is the bead radius, illustrating contrasts due to inertia and/or elas-ticity on the baseline of a non-inertial viscous Stokeslet (red dotted curve).Each response u(r, t) is due to the same stationary point force at x = 0, F =100δ1j cos(ω0t) pN with frequency ω0 = 10 Hz, and each fluid has identicalzero strain rate viscosity η0 = 50 cP. The viscoelastic responses are illustratedfor a 1-mode Maxwell fluid with relaxation time λ0 = 0.01 s. The constant slope−1 and −3 lines are superimposed to identify r−1 and r−3 scaling. (For inter-pt

d

It

U

W

a

ticta

3


The bead position will be labeled xstep when F = Fstep and

pulse when F = Fpulse. Further note Fstep = 2πF0(δ(ω) + 1iω

),hich yields:

ˆstep = −F0

3a(iωδ(ω) + 1)J , (80)

hich inverts back to the time domain as

step(t) = − F0

6πaJ(t). (81)

or the pulse force, we deduce:

pulse(t) = − F0

6πa(J(t) − J(t − t0)). (82)

e thereby rigorously recover the formulas derived in [33] byd hoc arguments, and indeed we can now draw an explicitnalogy between driven magnetic microbead motion and a bulkreep-recovery shear model.

.5. Finite size effects: quasi-steady driven spheres versusoint sources

Here we quantify the approximation of driven sphere motiony point source response functions (the Stokeslet and itsiscoelastic analog). (We continue with the quasi-steady oron-inertial limit here, and couple inertial effects below.) Asnticipated, the accuracy of the approximation depends on whereeasurements are made relative to the location of the source. Forpoint source, the natural lengthscale is the penetration depth

(ω); for a sphere, we can also measure response on the scalef the bead diameter. All figures below depict (on a log scale)esponse features at one micron from the source, and then eachorizontal unit corresponds to a factor of 10 bead diameters.his allows one to easily visualize the comparisons betweenoint and sphere sources, for viscous and viscoelastic fluids,nd for quasi-steady and inertial (unsteady) responses, in unitsf bead diameter.

The velocity field around a stationary point force of strength(t) in a viscoelastic fluid with relaxation modulus G(t), inourier space, is given from (29):

ˆ pointi (x, ω) = Sv

ij(x, η0)η0αj(ω)

G(ω), (83)

vij(x, η0) = 1

8πη0

(δij

r+ xixj

r3

). (84)

Meanwhile, the fluid velocity (denoted usphere) for creepingow past a sphere of radius a moving at velocity U(t) in the sameiscoelastic fluid is given from (71):

ˆ spherei (x, ω) = 1

8π

(δij

r+ xixj

r3

)bj(ω)

+ 1(

−δij + 3xixj

)d (ω), (85)

4π r3 r5 j

here

ˆ = 6πG(ω)aU, (86)

iρ

e

retation of the references to colour in this figure legend, the reader is referredo the web version of the article.)

ˆ = −πa3U. (87)

f we assume that the applied force on the sphere is the same ashe point force, then the sphere motion is

ˆi(ω) = αi(ω)

6πaG(ω). (88)

e now observe the difference in velocity fields is

upointi (x, ω) − u

spherei (x, ω)

= a2

3r2

1

8πG(ω)

(−δijr

+ 3xixj

r3

)αj(ω). (89)

It is clear that if a passive tracer is close to the sphere,/r ∼ O(1), then this difference is comparable in magnitudeo the point force solution (29). We return to this comparisonn the discussion below and Figs. 8–15, where we present vis-ous and viscoelastic, quasi-steady and unsteady, solutions onhe same plots. But first, we extend the above explicit solutionsnd analysis to the unsteady case.

.6. Unsteady (inertial) viscoelastic analog of Stokeslets

Comparing the magnitudes of the unsteady and viscous termsn (14), we see that the unsteady term becomes important whenL2ω/|G(ω)| = L2/||2 is order one or larger. Here L is the rel-vant characteristic length scale. Thus the unsteady term iρωu

K. Xu et al. / J. Non-Newtonian Fluid Mech. 145 (2007) 150–172 163

Fig. 9. Stokeslets and their viscoelastic analogs, continued. The t = 0.03 s snap-shots corresponding to Fig. 8. (For interpretation of the references to colour int

m

id

pst

Fsi(r

Fig. 11. Stokeslets and their viscoelastic analogs, continued. The t = 0.03 ssnapshots corresponding to Fig. 10. (For interpretation of the references to colouri

wtv

u

his figure legend, the reader is referred to the web version of the article.)

ust be retained for either L large (relative to the viscous damp-

ng length (|G(ω)|/ρω)1/2

) or ω large (relative to the inverseamping time |G(ω)|/ρL2).

We reconsider in this light the example of a concentratedoint force located at a stationary point X0 with time-dependenttrength in an infinite (boundary free) domain, now satisfyinghe unsteady creeping flow equations:

iρωu(x, ω) + ∇P(x, ω) = G(ω)∇2u(x, ω) + α(ω)δ(x − X0),

∇ · u(x, ω) = 0, (90)

ig. 10. Stokeslets and their viscoelastic analogs, continued. Snapshots of theame solutions at t = 0.01 s of Fig. 8, except the distance r from the point sources along the y-axis, transverse to the direction of the force in the z = 0 focal plane.For interpretation of the references to colour in this figure legend, the reader iseferred to the web version of the article.)

w

A

B

i

u

Wtiα

ad

n this figure legend, the reader is referred to the web version of the article.)

here P is a modified pressure. These equations have iner-ial (or unsteady) viscous Stokeslet (see [24]) and analogousiscoelastic solution:

ˆ i(x, ω) = 1

8πη0

(δij

rA(r, (ω)) + xixj

r3 B(r, (ω))

)

× η0αj(ω)

G(ω), (91)

= Sij(x, η0, ω)αeffj (ω), (92)

here

(r, (ω)) = 2e−r/(ω)(

1 + (ω)

r+ 2(ω)

r2

)− 2

2(ω)

r2 ,

(93)

(r, (ω)) = 2e−r/(ω)(

1 + 3(ω)

r+ 3

2(ω)

r2

)+ 6

2(ω)

r2 ,

(94)

(ω) = (G(ω)/iρω)1/2. (95)

Note that the viscous limit corresponds to G(ω) = η0. Recall-ng (30), one finds

i(x, t) = 1

2π

∫ ∞

−∞Sij(x, η0, ω)αeff

j (ω)eiωt dω. (96)

e call (96) an inertial (or unsteady) viscoelastic analog ofhe inertial viscous Stokeslet. From (96), there is generally an
nertial coupling between Sij(x, η0, ω) and the forcing strengthˆ effj (ω). Aspects of this solution have been explored by Liverpoolnd MacKintosh [17]. In particular, they focus on the frequency-ependent prefactor Sij(x, η0, ω) in frequency space. We remark


Fig. 12. Inertial Stokeslets (red) and their viscoelastic analogs (green). Two-dimensional representations (in the plane z = 0) of the inertial velocity fields att = 0.01 s from Figs. 8 and 11. The respective coordinate frames are shifted tocapture the center of the vortices, whose 1-d signatures are the oscillations in theearlier figures. Both of x and y are measured in viscous/viscoelastic dampinglength: δv = 0.89 × 10−3 m, δve = 0.97 × 10−3 m. (For interpretation of thero

tupisf

τ

Fig. 13. Inertial Stokeslets and their viscoelastic analogs, continued. The t =0ov

w

T

2 G(ω) η′(ω) i(δ(ω)−π/2)

eferences to colour in this figure legend, the reader is referred to the web versionf the article.)

hat conclusions about the spatial fields associated with Sij relypon special forcing functionsα(t) for which the inertial solutionreserves the time-space separability of the quasi-steady, non-nertial formula. We return to explicit illustrations below, whereignificant fluctuations in the spatial flow field arise for a periodicorcing function typical of an AC magnetic field.

The corresponding stress tensor is given by

ik(x, t) = 1

16π2

∫ ∞

−∞Tijkαj(ω)eiωt dω, (97)

O

.03 s snapshot of the 2D velocity fields depicted in Fig. 12. (For interpretationf the references to colour in this figure legend, the reader is referred to the webersion of the article.)

here

ijk(x, ω) = − 2

r3 (δijxk + δkjxi)[e−r/(ω)(r−1(ω) + 1)

−B(r, (ω))] − 2

r3 δikxj(1 − B(r, (ω)))

−2xixjxk

r5 [5B(r, (ω)) − 2e−r/(ω)(r−1(ω)

+1)]. (98)

Recall from (55), 2(ω) admits the representation:

(ω) =iρω

=iρω| sin(δ(ω))|e . (99)

bserve then that

K. Xu et al. / J. Non-Newtonian Flui

Fig. 14. Viscous and viscoelastic vortex tracking in the z = 0 focal plane fromthe inertial solutions in Fig. 8. The center of the nearest vortices to the pointsource in the half-plane y > 0 are tracked for viscous (red) and viscoelastic(green) point-force-driven media, for two periods of the periodic force. Thepro

•

•

Fpoeadi

•

pcbu

pB

l

osition data is measured in bead radii a = 1 �m. (For interpretation of theeferences to colour in this figure legend, the reader is referred to the web versionf the article.)

There is an inertial stress signature, in contrast to the quasi-steady limit. Note that (97)–(99) with 2

V(ω) = η0/iρω is thestress tensor for a purely viscous fluid with inertia.In the case of single frequency forcing (35) and (36), recalling:

αj(ω) = δ1j2πF0δ(ω − ω0), (100)

and selecting η0 = η′(ω0), the stress integral (97) can again beexplicitly computed. This stress tensor in the real time domain

ig. 15. Stress-driven spheres: contrasts due to inertia and/or elasticity. Theoint-force solutions of Fig. 9 are extended here to a driven sphere whose centerscillates about x = 0, with the identical force, zero strain rate viscosity, andlastic relaxation time. The snapshot is again taken at t = 0.03 s with velocitymplitudes measured along the x-axis of the force to distinguish identicallyriven spheres and point sources. (For interpretation of the references to colourn this figure legend, the reader is referred to the web version of the article.)

•

•

•

d Mech. 145 (2007) 150–172 165

is

τik(x, t) = F0

8π

{− 2

r3 (δi1xk + δk1xi)[e−r/0 (r−1

0 + 1)

−B(r, 0)] − 2

r3 δikx1(1 − B(r, 0))

−2xix1xk

r5 [5B(r, 0) − 2e−r/0 (r−10 + 1)]

},

(101)

where

20 = 2(ω0) = η0

iρω0| sin(δ0)|ei(δ0−π/2)

= 2V(ω0)

1

| sin(δ0)|ei(δ0−π/2). (102)

Thus we see that the inertial viscoelastic “Stokeslet” stresstensor acquires an amplified penetration depth.In addition, note that A(r, (ω)) and B(r, (ω)) are non-separable.

Again, one can extend these previous formulas to multipleoint forces by superposition, and build other unsteady vis-oelastic singularities, e.g. unsteady viscoelastic doublets, ally replacing η0 with G(ω). See Pozrikidis [24] for a catalog ofnsteady viscous Stokes singularities.

Note that (90)–(98) differ from their quasi-steady counter-arts (28)–(34) through the perturbative factors A(r, (ω)) and(r, (ω)), which are frequency-dependent. We make the fol-

owing observations for the respective velocity fields.

For r � |(ω)|, i.e., for distances smaller than the vis-cous damping length, A(r, (ω)) and B(r, (ω)) are 1 +O(r|(ω)|−1) and so at this length scale, quasi-steady andinertial viscoelastic analogs of Stokeslets are comparable.For r > |(ω)|, the first terms in the formulas forA(r, (ω)) and B(r, (ω)) are exponentially suppressedas e−rRe (ω)/|(ω)|2 , leaving A(r, (ω)) ≈ −22(ω)/r2 =−2G(ω)/(iρωr2), B(r, (ω)) ≈ 62(ω)/r2 = 6G(ω)/(iρωr2). Thus (96) becomes

ui(x, t) ≈ 1

16π2ρ

(2δij

r3 − 6xixj

r5

)∫ ∞

−∞iαj(ω)

ωeiωt dω.

(103)

Note that: (i) inertial velocity decay goes as r−3 rather thanr−1 (as for the quasi-steady case), and, importantly, (ii)u is independent of G(ω) to an exponentially small cor-rection. In other words, at distances beyond the viscousdamping length |(ω)|, the fluid velocity decays much morerapidly with r and in any case cannot be used to determineconstitutive information about viscosity or viscoelasticity.Exponential damping beyond the viscous damping lengthis characteristic of oscillating fluids, viscous or viscoelas-
tic; see the example of planar oscillatory stress sourcesbelow.There is an overlap region r = O(|(ω)|) where differencesbetween quasi-steady and unsteady viscoelastic analogs of

1 n Flui

3c

it

u

w

C

A

B

u

w

α

α

α

tacsattotwatl1d

tottvt(1f

ia

•


Stokeslets are significant, and where information about G(ω)is available. This is the appropriate range for experimentalinterrogation.

.6.1. Specialized point-source formulas: 1-mode Maxwellonstitutive laws with single-frequency harmonic forcing

As a concrete example, consider a single frequency forcing,.e., α(ω) = 2πF0 δ(ω − ω0), and a 1-mode Maxwell constitu-ive law (8)–(10). Then (96) becomes

i(x, t) = 1

8πη0Im

[(δij

rA(r, ω0) + xixj

r3 B(r, ω0)

)

× η0(1 + iω0λ0)

G0λ0Fje

iω0t

](104)

= | sin(δ0)|8πη0

Im

[(δij

rA(r, ω0) + xixj

r3 B(r, ω0)

)

× Fjei(ω0t+π/2−δ0)] , (105)

ithA(r, ω0) andB(r, ω0) evaluated using = (ω0) as follows:

(ω0) = (G(ω0)/iρω0)1/2 = C0ei(−π/2+δ0/2),

0 =√√√√ G0λ0

ρω0

√1 + ω2

0λ20

,

(r, ω0) = 2e−rC0

ei(π/2−δ0/2)

(1 + C0

rei(δ0/2−π/2) + C2

0

r2 ei(δ0−π)

)

−2C20

r2 ei(δ0−π),

(r, ω0) = −2e−rC0

ei(π/2−δ0/2)(

1 + 3C0

rei(δ0/2−π/2)

+3C20

r2 ei(δ0−π)

)+ 6C2

0

r2 ei(δ0−π)

Manipulating further,

| sin(δ0)| [δijFj i(ω0t+π/2−δ0)

]
i(x, t) =
8πη0Im

rA(r, ω0)e

+| sin(δ0)|8πη0

Im

[xixjFj

r3 B(r, ω0)ei(ω0t+π/2−δ0)]

(106)

= | sin(δ0)|8πη0

δijFj

r

[2e

−rC0

(sin(δ0/2))(sin(α1) + C0

rsin(α2)

+c20

r2 sin(α3)) + 2C2

0

r2 cos(ω0t)

]

•

d Mech. 145 (2007) 150–172

+| sin(δ0)|8πη0

xixjFj

r3

[−2e

−rC0

(sin(δ0/2))(sin(α1) + 3C0

rsin(α2)

+3c2

0

r2 sin(α3)) − 6C2

0

r2 cos(ω0t)

], (107)

here

1 = ω0t + π/2 − δ0 − r cos(δ0/2)

C0, (108)

2 = ω0t − δ0/2 − r cos(δ0/2)

C0, (109)

3 = ω0t − π/2 − r cos(δ0/2)

C0. (110)

The general observations made for viscous and viscoelas-ic response functions due to point sources of force (Stokesletsnd their viscoelastic analogs) are now illustrated for the spe-ial case of a 1-mode Maxwell liquid (with the same zerotrain rate viscosity as the viscous fluid it is compared with)nd a single harmonic force amplitude. In these and all figureso follow, we adopt physical parameters that are representa-ive of pulmonary mucus: zero strain rate viscosity η0 = 50 cP,r about 50 times as viscous as water, a single relaxationimeλ0 = 0.01 s, a uni-directional force F(t) = 100 cos(ω0t) pNith frequency ω0 = 10 Hz consistent with cilia beat cycles,

nd the density ρ for both fluids is 103 kg/m3. With respecto the viscoelastic and viscous penetration depths noted ear-ier, for these specified fluid properties we find = 9.7 ×0−4 m, v = 8.9 × 10−4 m, each on the order of 103 beadiameters.

In the figures to follow, we superimpose all four solu-ions, quasi-steady and inertial, viscous and viscoelastic, inrder to amplify the respective effects of elasticity and iner-ia for otherwise controlled conditions. Figs. 8 and 9 depictwo different snapshots, at 0.01 s and 0.03 s, of the spatiallyarying velocity amplitudes on a log-log scale, where dis-ance r is along the direction (x-axis) of the imposed force.Note: all the velocity amplitudes are normalized by U0 =.06 × 10−4 m/s which is defined as U0 = f0/6πaη0) where0 = 100 pN is the force amplitude and η0 = 50 cP is the match-

ng viscosity, while the distance r is measured in bead radii= 1 �m.

For near-field observations close to the point force (r/a ∼1 − 10), or more generally for |r| � |(ω)|, the effects ofinertia are negligible for either viscous or viscoelastic fluids.Quasi-steady solutions exhibit uniform r−1 decay at all timesand time-space separability. However, the relative amplitudesof the viscous and viscoelastic velocities fluctuate about oneanother due to the elastic phase shift, in effect disguisingthe quasi-steady elastic attenuation factor; the t = 0.01 snap-
shot indicates a viscoelastic attenuation whereas the t = 0.03snapshot indicates amplification. The post-transient time his-tory over a full period of the driving force has to be monitoredto detect elastic attenuation.

n Flui

•

•

•

•

tFetFisIb

•

•

•

z

FFfisviatFvoha

Tdmdcifnfis

3

esc(g

u

w

c

d

t


Unsteady (inertial) solutions clearly show non-uniformity inr and t and non-separability, with |uve(r, t)| and |uv(r, t)| fluc-tuating about one another in both position and time.In the far-field, r � |(ω)|, inertial solutions decay as r−3

while quasi-steady solutions decay as r−1.On intermediate lengthscales, r ∼ |(ω)|, the distinctionbetween viscous and viscoelastic response is most pro-nounced, but again phase shifts are evident in any snapshot,and the amplitude of the signal has to be considered rela-tive to experimental noise. In this spatial range along thedirection of the force, Fig. 8 shows a slight inertial ampli-fication for viscoelastic fluids which then falls off toward ther−3 scaling; in contrast, there is an inertial decay for vis-cous fluids which continues to grow with r toward the r−3

scaling.On intermediate and far-field lengthscales, inertia-inducedoscillations in the velocity amplitudes emerge for viscous andviscoelastic fluids. This feature is highlighted in recent articles[17,2]. Again due to non-separability, the inertial signaturesof what appear to be vortices are strongly time-dependent.Since we are showing velocity amplitudes along the direc-tion of the point force, one might infer stronger vorticitysignals transverse to this direction; this is the focus of the nextfigures.

Next, we extract the scaling behavior of these same four solu-ions along any axis orthogonal to the x-axis of the point force.or relevance with respect to the focal plane of a microscopyxperiment, we remain in the z = 0 plane and measure dis-ance r along y at x = 0. The same snapshots are presented inigs. 10 and 11. As anticipated, since the oscillatory point force

nduces oscillatory shear gradients with respect to y, both snap-hots detect multiple velocity fluctuations, or apparent vortices.n addition to the features highlighted above from the x-directionehavior, we note:

The number of apparent vortices is greater moving outtransverse to the forcing direction, suggesting that vorticesare spawned at some distance from the point force andthat they propagate away from the force axis as well asaway from the point source. Both snapshots suggest vis-coelastic vortices initially propagate faster than their viscousanalogs.To an observer sitting at any fixed distance beyondthe spawning location of velocity fluctuations, the rela-tive effects of elasticity are also fluctuating; the inertialviscous and viscoelastic amplitudes cross one anotherrepeatedly.For the vortices to be observable, the observer has to sit nearly103 bead radii away from the force source. Also, the velocityamplitude has dropped to nearly 10−4 of the sphere velocity,which supports the statement in [17] that no direct experi-mental observation of this vortex-induced back-flow has been
made.
Next, we provide 2d flow field snapshots in the focal plane= 0 of the unsteady (inertial) solutions depicted in the above

terv

d Mech. 145 (2007) 150–172 167

igs. 8–11. Fig. 12 corresponds to the t = 0.01 snapshot andig. 13 to the t = 0.03 snapshot. We only show the velocityeld surrounding the nearest vortex in the half-plane y > 0,ince nearer to the source the velocity amplitudes swamp theseortex features. Clearly there are vortices spawned in both flu-ds at some distance away from the point source with centerslso at some distance away from the axis of the force. The vor-ices then propagate transverse to the axis of the point force. Inig. 14, we track the centers of the vortices in the z = 0 planeersus time. Since all solutions are periodic with the frequencyf the driving force, vortices are spawned apparently one peralf period, and then they propagate away from the point sourcend x-axis.

Of course, solutions due to a point force are idealized.ypically, such solutions are good approximations to a stress-riven sphere if one is sufficiently far away from the center ofass of the sphere. While this comparison is extremely well-

ocumented for viscous fluids, we do not find such a carefulomparison in the microrheology literature for viscoelastic flu-ds. Thus our motivation is to extend the Figures just presented toorce-driven spheres of a prescribed micron diameter, to see theear-field discrepancies between point force and sphere velocityelds, and to see the corresponding inertial signatures for drivenpheres.

.6.2. Spherical forcing with inertiaAs before, singular solutions can be combined to produce an

xact solution of unsteady creeping flow around an oscillatingphere of radius a. In this case, the fluid velocity u for unsteadyreeping flow past a sphere of radius a moving at velocity U(t)an arbitrary periodic function) in a fluid with viscosity η0 isiven by

i(x, t) = 1

8πη0

(δij

rA(r) + xixj

r3 B(r)

)cj

+ 1

4π

[−δijr3 e−r/(1 + r−1 + (r−1)

2)

+3xixj

r5 e−r/(

1 + r−1 + (r−1)2

3

)]dj, (111)

here

= 6πη0a(1 + a−1 + (1/3)(a−1)2)U, (112)

= −πa3(6/(a−1)2)(ea/ − 1 − a−1 − (1/3)(a−1)

2)U.

(113)

As usual, this same velocity applies after Fourier transformo the linear viscoelastic case with η0 replaced by G. Note also
hat the correction terms to the unsteady Stokeslet velocity arexponentially damped beyond the viscous damping length. Thusemarks made previously about Stokeslet signals beyond theiscous damping length continue to apply.


Fig. 16. Stress-driven spheres: contrasts due to inertia and/or elasticity, contin-urft

c

F

wmid

wtrFfbfii

3tsigi

F

as

c

fs

Fig. 17. Finite-size source effects: near-field errors. The percent error in velocityvs. distance from the source (measured in bead radius units in the z = 0 focalplane along the axis of the applied force) between responses due to a point source(tv

(

u

pdFstcvtcato

3

ed. Snapshot of the same solutions at t = 0.03 s of Fig. 15, except the distancefrom the point source is along the y-axis (transverse to the direction of the

orce) in the z = 0 focal plane. (For interpretation of the references to colour inhis figure legend, the reader is referred to the web version of the article.)

The unsteady or inertial generalized Stokes drag law can bealculated to be [34]:

ˆ (ω) = −6πG(ω)a

(1 + a−1 + 1

9(a−1)

2)

U(ω)

+iω(ρs − ρ)a3U(ω), (114)

here ρs is the density of the sphere. If ρs ∼ ρ then ω/G(ω)ust be O(a−2) or larger before corrections to (54) become

mportant. Note that the unsteady correction terms alter both therag amplitude and phase lag.

The driven-sphere solutions are illustrated in Figs. 15 and 16,hich are snapshots taken at t = 0.03 s for points located in

he z = 0 focal plane, along the x-direction and y-directionespectively, analogous to the driven point source solutions inigs. 8 and 9. In Figs. 15 and 16, the velocity curves fall offrom the line with slope −1, showing a measurable differenceetween the point force and driven sphere solutions in the near-eld. Next we continue the discussion on finite size effects with

nertial effects incorporated.

.6.2.1. Finite size effects for inertial solutions. In Section 3.5,he difference between quasi-steady solutions for a fixed pointource and an oscillating sphere are given in (89). As shownn Section 3.6.2, the Stokes drag force in the Fourier domain isiven by (114), which, if ρ ∼ ρs and a � ||, can be simplifiednto

ˆ (ω) = −6πaG(ω)U(ω) (115)

s in the quasi-steady case. Also, c and d in (111) can also beimplified as

∼ 6πaG(ω)U, d ∼ −πa3U

or the viscoelastic analog in Fourier space. Once again, if theame force is applied on the sphere as with the point force, from

vsiT

Fig. 9) and a driven sphere of micron radius (Fig. 15). (For interpretation ofhe references to colour in this figure legend, the reader is referred to the webersion of the article.)

91) and (111),

upointi (x, ω) − u

spherei (x, ω)

= a2

3r2

αj(ω)e−r−1

8πG(ω)

{− δij

r[1 + r−1 + (r−1)

2]

+3xixj

r3

[1 + r−1 + (r−1)

2

3

]}.

It is clear that in the near-field |r/| � 1, upointi (x, ω) −

ˆ spherei (x, ω) decays as r−3, which we now amplify in Fig. 17. Welot the relative velocity differences (i.e. percent error) for theriven sphere solutions of Fig. 15 and the point force solutions ofig. 9. The percent error in the point source approximation due touppression of the finite bead size is given versus distance fromhe source, in bead radius units. For all 4 solutions (inertial vis-ous, quasi-steady viscous, inertial viscoelastic and quasi-steadyiscoelastic), errors of 10–50% occur within 2 bead radii, buthen errors fall to below 1% at 10 radii. Fig. 18 addresses the per-ent error on a log scale again in bead radius units. Clearly, therere intermediate and far-field absolute velocity differences dueo finite bead radius, but they become vanishingly small beyondne to two decades of bead radii.

.7. Forced flow in a half-plane

We have emphasized that linear response theory and the
iscous–viscoelastic correspondence provide a method for con-tructing viscoelastic solutions from Newtonian flow solutionsn far more generality, e.g., with nonlocalized sources of stress.o illustrate this point, consider an infinite flat plane with oscil-

K. Xu et al. / J. Non-Newtonian Flui

Fig. 18. Finite-size source effects: intermediate to far-field errors. FollowingFig. 17, the percent error in velocity between responses due to a point source(Fig. 9) and a driven sphere of micron radius (Fig. 15), shown here on a loga-rfi

lo

τ

d

(

G

wt

i

h

w

γ

(pNpttv

In the time domain,

Fω

(

ithmic scale in bead radii. (For interpretation of the references to colour in thisgure legend, the reader is referred to the web version of the article.)

ating space-dependent stress boundary condition which consistsf the real part of

xy(x, y = 0, t) = Mei(ω0t+kx). (116)

Here x refers to a direction parallel to the plate and y to theirection perpendicular to the plate. The Fourier transform of

ig. 19. Responses due to oscillatory stress imposed on an infinite plane. Instantan. A single-mode Maxwell constitutive law is assumed with unit values of all para

instantaneous) stagnation points.

d Mech. 145 (2007) 150–172 169

116) (recall (3)) is

ˆ (ω) (x, y = 0, ω) = 2πMeikxδ(ω − ω0), (117)

here ∇ × u = z is the vorticity (in the ez = ex × ey direc-ion). Taking the curl of (14) (with f = 0) we obtain:

ρω (x, ω) = G(ω)∇2 .

This equation, together with plate boundary condition (117),as solution:

ˆ (x, ω) = 2πMeikx−γyδ(ω − ω0)

G(ω),

here γ is a complex reciprocal penetration depth:

(ω) =√k2 + i

ρω

G(ω)=√k2 + −2 (118)

with the square root chosen so that Re γ is positive). Vorticityenetration is determined by the decay length scale [Re γ(ω)]−1.ote that presence of elasticity, i.e. G′ > 0, increases vorticityenetration in some sense. More precisely, if we were to comparewo materials with G′′

1 = G′′2 but G′

1(ω) > G′′2(ω), then we find

hat Re γ1(ω) < Re γ2(ω), i.e., material 1 would have a deeperorticity penetration depth than material 2.

(x, t) = M Re

[eikx−γ(ω0)yeiω0t

G(ω0)

]. (119)

eous snapshots of the stream function (120) (real part) for various values ofmeters M, k, ρ, λ, η. Open streamlines are ψ = 0 level sets emanating from


F ed. Insa ell cr )| at e

∇

ψ

f

u

absi

fa

u

wws

(

sTws0op

4

ZepvfoitiltSttsp

ig. 20. Responses due to oscillatory stress imposed on an infinite plane, continut cross-section {x = 0.1, y ≥ 0}. For the viscoelastic fluid a single-mode Maxwesulting modulus G(ω)). For the viscous fluid, the viscosity is set to η = |G(ω

The stream function ψ(x, t) for this problem (which satisfies2ψ = − with ψ = 0 on y = 0) is given by

(x, t) = M

ρω0Re[i(e−ky − e−γ(ω0)y)ei(ω0t+kx)

], (120)

rom which we obtain the velocity

(x, t) = −Mk

ρω0Re

[ei(kx+ω0t)

(i

(e−ky − γ(ω0)

ke−γ(ω0)y

),

−(e−ky − e−γ(ω0)y)

)]. (121)

Note that the viscoelastic flow is not just a phase-shiftednd attenuated version of purely viscous flow. The relationshipetween forcing and velocity (and hence also stress) is not soimple as can be seen by comparing (117) and (121); note theres a y-dependent phase shift.

Notice also that if we write γ(ω0) as γR0 + iγI0 = |γ(ω0)|eiβ,or k = 0, we recover the shear wave propagation result [9] withphase shift π/2 + β/ω0. The translational velocity u1(x, t) is

1(x, t) = −M|γ(ω0)|ρω0

e−γR0 y sin[ω0t + β − γI0y], (122)

here 2π/γI0 is the wavelength and 1/γR0 is the distance withinhich the amplitude falls off by a factor of 1/e. For k �= 0, the

olution (121) reveals:

(i) a coupling of the wavelength-dependent boundary stress tothe translational velocity (u1(x, t)) wave structure;

ii) a vertical flow (u2(x, t)) with penetration depth given by themaximum of k−1 and [Re(γ(ω0))]−1.

ctr

tantaneous comparisons of the viscoelasticψve and viscousψv stream functionsonstitutive law is assumed with unit values of all parameters M, k, ρ, λ, η (andach value of ω.

As a particular example, we illustrate the real part of thetream function (120) for a single-mode Maxwell fluid in Fig. 19.he penetration depth is O(1) for these parameters. In Fig. 20e show a comparison of instantaneous viscoelastic and viscous

tream functions along a cross-section of the half plane {y ≥}. Velocity amplitude differences are largest (i.e., most easilybserved) near frequencies whereG′(ω) = G′′(ω) which, for thearticular choices of parameters made here, occurs at ω = 1.

. Conclusion

We have formulated linear response theory, followingwanzig and Bixon [34] and Mason and Weitz [19], to yield anxplicit correspondence in the governing equations of incom-ressible creeping flow between a viscous fluid and any lineariscoelastic material, valid for an arbitrary prescribed source:orce, flow, displacement or stress; local or nonlocal; steady orscillatory. Upon specification of the geometry and source, non-nertial and inertial viscous Stokes singularities [12,24] transfero exact solutions for linear viscoelastic fluids. This formulations not new; it is our intention that such a transparent formu-ation of the viscous–viscoelastic correspondence will facilitatehe transfer of detailed knowledge of special solutions of viscoustokes flow to the analogous, creeping flow limit, of viscoelas-

ic soft matter. The three examples presented here illustratehis perspective by applying the viscoelastic analog of Stokesingularities to simulate experimental features and to highlighthenomena associated with elastic and inertial effects.

The explicit correspondences imply how elasticity contrastsan be measured for viscosity-matched materials, and howo modify sources for different materials to yield identicalesponses. The length scales at which the effects due to iner-

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ia, viscosity, and elasticity are most distinguishable have beenlarified and illustrated. Additionally, variability in experimen-al snapshots for time-varying sources has been illustrated. Twoxamples that have been central in microrheology, namely aoint source of force and a spherical source of velocity or stress,ave been revisited and developed in detail for oscillatory or stepulse forcing strength. These illustrations clarify the approx-mation of forced spheres by point sources, and in particular,here in the response field the errors are negligible. Follow-

ng the recent attention on inertial effects in microrheology,e have amplified the mechanism of vortex generation due to

ime-varying point sources [17,2] and forced spheres.From a modeling perspective, explicit formulas are produced

or Maxwell fluid laws, where it is straightforward to carry theorrespondence back to the time domain. This step is valuableor simulating passive tracers in the response field surroundingctively driven microspheres. The results presented serve as aoadmap for extension to more complex linear constitutive laws,hich are perhaps more tedious but do not require any concep-

ual advance. In these special examples, we have developed aariety of illustrations of simulated experiments, focusing onhe role of elastic properties with respect to measurable quasi-teady and inertial features. Finally, we illustrated the generalityn source type and geometry of this viscous–viscoelastic creep-ng flow correspondence by analyzing the linear response for aonlocal, planar source of unsteady stress.

A take-home message of our paper is that many additionalpplications of linear response theory await implementation,sing the extensive literature on viscous Stokes flow. Ouriscous–viscoelastic approach is the mirror symmetry of theevine, Lubensky and Crocker et al. [16,8] analysis whichegins with linear elasticity and extends the formalism toiscoelasticity by complexifying the elastic moduli. Indeed,ombining both correspondences leads to a transfer betweenlastic and viscous classical exact solutions [13]. Looking aheado more sophisticated applications, by summing appropriate sin-ularities, and exploiting linearity of the equations of motion,olutions of complex boundary value problems for viscous flu-ds have been developed; examples include slender body theory14], the viscous flow around a rotating rod [3,15], and numer-cal methods for fluid-structure interactions (cf. the immersedoundary method [23], the blob projection method [7], many-article codes [4,29,27,22,25], and flows of slender filaments26,31]). The formulation given here is a step toward extensionf these concepts and tools to viscoelastic materials, with pro-isos on the validity of Stokes approximations. The issue ofonlinearity is, of course, a major challenge [29].

cknowledgements

This research problem and results have resulted from the Vir-ual Lung Project at the University of North Carolina, involving

any colleagues across the basic and medical sciences. The
uthors acknowledge motivation and input from faculty and stu-ents (R. Boucher, R. Camassa, J. Cribb, R. Cortez, W. Davis,. Elston, D. Hill, C. Hohenegger, T-J Leiterman, R. McLaugh-
in, M. Minion, S. Mitran, P. Mucha, M. Rubinstein, J. Sheehan,

[

d Mech. 145 (2007) 150–172 171

nd R. Superfine). Experiments done at UNC, and discussionsf their design and interpretation, are the inspiration for thisaper. The authors likewise thank the anonymous referees andhe managing editor for helpful suggestions. I.K. was supportedy NIH award 5R01GM067245-02 and by NIH award P20R-16455-05 from the INBRE-BRIN Program of the Nationalenter for Research Resources. M.G.F. and K.X. acknowledge

upport from the National Science Foundation (DMS-0554501),IH (R01-HL077546-01A2) and the CISMM (P41-EB002025-1A1).

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On the correspondence between creeping flows of viscous and viscoelastic fluids

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Transcript of On the correspondence between creeping flows of viscous and viscoelastic fluids