role of linear viscoelasticity and rotational diffusivity...

Role of linear viscoelasticity and rotational diffusivityon the collective behavior of active particles

Yaser Bozorgi and Patrick T. Underhilla)

Department of Chemical and Biological Engineering, Rensselaer PolytechnicInstitute, 110 8th Street, Troy, New York 12180

(Received 12 July 2012; final revision received 3 December 2012;published 23 January 2013)

Synopsis

A linear dynamics scheme has been used to quantify the impact of viscoelasticity of the

suspending fluid on the collective structure of active particles, including rotational diffusivity. The

linear stability examines the response near an isotropic state using a mean-field theory including

far-field hydrodynamic interactions of the swimmers. The kinetic model uses three possible

constitutive models, the Oldroyd-B, Maxwell, and generalized linear viscoelastic models inspired

by fluids like saliva, mucus, and biological gels. The perturbation growth rate has been quantified

in terms of wavenumber, translational diffusivity, rotational diffusivity, and material properties of

the fluids. A key dimensionless group is the Deborah number, which compares the relaxation time

of the fluid with the characteristic timescale of the instability. An advantage of the model

formalism is the ability to calculate some properties analytically and others efficiently

numerically in the presence of rotational diffusion. The different constitutive equations examined

help illustrate when and why the dispersion relation can have a peak at a particular wavenumber.

The fluid properties can also change the role of rotational diffusion; diffusion always stabilizes a

system in a Newtonian fluid but can destabilize a system in a Maxwell fluid. VC 2013 The Societyof Rheology. [http://dx.doi.org/10.1122/1.4778578]

I. INTRODUCTION

The self-propulsion of micro-organisms is important for many physical processes.

Fluid mechanics plays an important role in this process. The negligible role of inertia

influences how organisms can move, as mentioned in the classic work of Purcell (1977)

and much work since [e.g., reviewed by Lauga and Powers (2009)]. Non-Newtonian flu-

ids can also have a strong influence on how a single organism moves, which has been

studied by many researchers [Chaudhury (1979); Sturges (1981); Fulford et al. (1998);

Lauga (2007); Fu et al. (2009); Fu et al. (2010)]. Finally, a fluid can influence the interac-

tions between organisms; an organism causes a flow when it moves which affects other

organisms. There has been much recent work on how these hydrodynamic interactions

(HIs) can lead to collective behaviors of groups. However, almost all of the work has

focused on interactions of groups of organisms suspended in a Newtonian fluid. Only

a)Author to whom correspondence should be addressed; electronic mail: [email protected]

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recently has the influence of a non-Newtonian fluid on group behavior generated by HIs

been examined [Bozorgi and Underhill (2011)].

A number of collective behaviors have been observed experimentally including

enhanced transport, large scale structures, and increased speed of the organisms [Pedley

and Kessler (1992); Mendelson et al. (1999); Wu and Libchaber (2000); Dombrowski

et al. (2004); Tuval et al. (2005); Sokolov et al. (2007); Cisneros et al. (2007); Leptos

et al. (2009)]. It has been speculated that HIs are important for many of these processes.

Agent-based models have been used to examine the onset and properties of collective

behavior [Vicsek et al. (1995); Toner and Tu (1995, 1998)]. Recently, many of these

have included HIs [Hernandez-Ortiz et al. (2005); Saintillan and Shelley (2007); Ishi-

kawa and Pedley (2007a, 2007b); Underhill et al. (2008); Mehandia and Nott (2008);

Saintillan and Shelley (2008b); Hernandez-Ortiz et al. (2009); Baskaran and Marchetti

(2009); Underhill and Graham (2011)]. It has been shown that HI is sufficient to produce

qualitatively similar features as some experiments. The mechanism for and impact of the

collective behavior has been examined.

Field theories have also been used to understand collective behavior [Simha and Ram-

aswamy (2002); Kruse et al. (2004); Aranson et al. (2007); Lau and Lubensky (2009);

Saintillan and Shelley (2008a); Subramanian and Koch (2009); Hohenegger and Shelley

(2010); Bozorgi and Underhill (2011)]. Some models have been based on phenomenol-

ogy, while others are derived from agent-based models using a mean-field assumption.

The theories including HI have shown that the uniform and isotropic state can be unstable

to perturbations if the organisms are pushed from behind (pushers). The mechanism

deduced from this is consistent with the agent-based models; correlations of the orienta-

tions of organisms give rise to collective behavior.

Our goal in this article is to generalize the results of Bozorgi and Underhill (2011) to

better understand how a viscoelastic suspending fluid can alter the collective behavior of

swimming micro-organisms. We do this using a mean-field theory and a linear stability

analysis on an Oldroyd-B constitutive model since it may represent the biological fluids

like mucus and saliva, as discussed later. We also find it instructive to study a Maxwell

fluid suspension since the Maxwell model is a special case of the Oldroyd-B model in the

absence of Newtonian fluid contribution (and convected derivative terms that do not play a

role in the linear stability). Finally, an attempt has been made to study the effects of viscoe-

lasticity more generally by examining the generalized linear viscoelastic (GLVE) model.

We develop a formalism to understand the stability in any fluid in the linear region.

This formalism is used to better understand when and how a non-Newtonian fluid can

lead to a peak in the dispersion relation. We also develop a formalism to include rota-

tional diffusion in an efficient way. Rotational diffusion, which is always stabilizing in a

Newtonian fluid, can be destabilizing in a Maxwell fluid, as shown later.

In Sec. II, the model and the linear stability of the uniform isotropic state are pre-

sented for an Oldroyd-B suspension. In Secs. III and IV, the solution to the model is for-

mulated and results presented. The solution and results for a Maxwell fluid are shown

in Secs. V and VI. The generalization to the GLVE model is presented in Sec. VII. In

Sec. VIII, we present a summary of key results and the conclusions. Finally, in Appen-

dix, Oldroyd-B suspension has been studied from a different perspective, using a differ-

ent nondimensionalization.

II. MODEL

Our model is an extension for the work of Saintillan and Shelley (2008b) who studied

the stability of active particles in a Newtonian fluid. The dynamics of the collective

512 Y. BOZORGI AND P. T. UNDERHILL


structure of swimmers have been modeled by a Smoluchowski equation representing the

conservation of the probability density distribution of the particles. HIs of the particles

couple this equation with the fluid flow equation, which includes an active stress term

induced by the particles, continuity equation of the fluid, and the constitutive equation.

The equations have been perturbed around the isotropic uniform state with zero flow by

doing a linear stability analysis.

Many active particles are approximately axisymmetric and therefore we assume that

its distribution function, Wðx; n; tÞ, is expressed in terms of the center of mass (x) and

unit vector along its axis (n). The time evolution of this function follows the Smoluchow-

ski equation

@W@t¼ �rx � ð _xWÞ � rn � ð _nWÞ; (1)

where _x and _n represent the effective translational and angular velocities, respectively.

They are given as

_x ¼ visnþ u� Dtrxðln WÞ; (2)

_n ¼ ðd� nnÞ � ½ðc C�XÞ � n� Drrnðln WÞ�; (3)

where vis is the isolated swimming speed, d is the identity tensor, C ¼ ðruþru†Þ=2 is

the rate of strain tensor, X ¼ ðru�ru†Þ=2 is the vorticity tensor, Dt is the translational

diffusivity, Dr is the rotational diffusivity, and c ¼ ðA2 � 1Þ=ðA2 þ 1Þ where A is the as-

pect ratio. Equation (3) is known as the Jeffery equation, which quantifies the angular ve-

locity of a suspending particle.

The difference between the current study from the work of Saintillan and Shelley

(2008b) lies in the constitutive equation of the suspending fluid, which is given by Eqs.

(4)–(6) for an Oldroyd-B constitutive model. In the first part of this article, we examine

the case in which the total stress in the fluid follows the Oldroyd-B constitutive equation,

in which the total stress st is the sum of a polymeric stress sp and the stress due to a New-

tonian solvent (ss ¼ �2gsC)

st ¼ sp þ ss: (4)

The polymer stress follows the upper convected Maxwell model

sp þ ksp ¼ �2gpC; (5)

where k and gp are the relaxation time of the polymer and the polymer contribution to the

viscosity, respectively. The upper convected derivative is defined as

sp ¼@sp

@tþ u � rsp � ðru† � sp þ sp � ruÞ: (6)

In the linear stability analysis, the nonlinear terms in the convected derivative do not

contribute.

The fluid velocity is determined by the continuity equation and conservation of mo-

mentum. For an incompressible fluid, continuity is given as

513COLLECTIVE BEHAVIOR IN NON-NEWTONIAN FLUIDS


rx � u ¼ 0: (7)

At small Reynolds number (in the Stokes regime), conservation of momentum can be

expressed as

rxqþrx � st ¼ rx � R; (8)

where R and q represent the active particle stress and the fluid pressure, respectively. The

active forcing on the fluid by the particles occurs as a stress (or dipole) because regardless

of the propulsion mechanism the net force on a neutrally buoyant organism is zero. This

extra stress for an organism takes the form dðnn� d=3Þ where d is the dipole moment of

a swimmer. In our mean-field approximation, the stress at a position in the fluid is related

to the number of swimmers per unit volume at a particular location, which is the mean

(over orientations) of the single particle distribution function W times the number of

swimmers N. Therefore, this approximation gives the extra stress as

R ¼ dN

ðS

Wðx; n; tÞ nn� d

3

� �dn; (9)

where S is the surface of a unit sphere. Therefore, in a self-consistent way, the distribu-

tion function leads to the extra stress, which alters the fluid flow, which alters the distri-

bution function.

In order to do the linear stability analysis, it is useful to nondimensionalize the equa-

tions. From previous work looking at organisms suspended in a Newtonian fluid [Underhill

et al. (2008); Saintillan and Shelley (2008b); Subramanian and Koch (2009); Hohenegger

and Shelley (2010); Bozorgi and Underhill (2011)], we know that tc ¼ gc

cjdj reflects the char-

acteristic time needed for HI to rotate the particles and lead to an instability, where c is the

concentration of the bacteria and gc is a characteristic viscosity. For each constitutive equa-

tion examined, we will identify an appropriate gc. In some cases, multiple choices can be

made. In addition to using tc to nondimensionalize times, we nondimensionalize lengths by

lc ¼ vistc, which is the length an isolated organism can swim in time tc.It is important to note that jdj and vis are affected by the properties of the suspending

fluid in ways that are not fully understood. By using them in our characteristic scales, our

solution for dimensionless variables will not depend on these effects. However, when

comparing systems in terms of dimensional variables, changes in jdj and vis would need

to be taken into account. Unless otherwise stated, the remainder of this article uses

dimensionless variables with tc and lc as the characteristic time and length scales with

gc=tc ¼ jdjc as the characteristic stress scale.

The one exception to this is for the distribution function W. Instead, we choose the

volume of the system V to nondimensionalize W. This only acts to rescale the distribution

such that the uniform, isotropic state corresponds to W ¼ 1=ð4pÞ. Using this nondimen-

sionalization, the expression given for the extra stress, Eq. (9), in dimensionless form

becomes

R ¼ p

ðS

W nn� d

3

� �dn; (10)

where p is �1 for the swimmers whose propulsion force is actuated from the posterior

(pushers) and þ1 for the swimmers whose propulsion force is actuated from the anterior

(pullers).



The uniform isotropic distribution function is a steady state solution for which the

stress (for both polymers and solvent), pressure, velocity, and the extra stress all equal

zero. We perform a linear stability analysis over the isotropic state of the suspension,

with primed variables denoting the deviation from the steady state. Since the distribution

function is 1=ð4pÞ at steady state, we define the primed distribution as W ¼ 14p ð1þW0Þ.

With these definitions, the linear, dimensionless system is

rx � u0 ¼ 0; (11)

�gs

gc

r2xu0 þ rxq0 þ rx � s0p ¼ rx � R0; (12)

@W0

@t¼ �n � rxW

0 þ 3cnn : C0 þ Dtr2xW0 þ Drr2

nW0; (13)

R0 ¼ p

4p

ðS

W0 nn� d

3

� �dn; (14)

s0p þ 5De@s0p@t¼ �2HC0; (15)

where H ¼ gp=gc and De ¼ k=ð5gc=ðjdjcÞÞ. The Deborah number De represents the ratio

of the relaxation time of the polymers to a characteristic time scale of the instability. For

swimmers suspended in an Oldroyd-B fluid, there are two natural choices for gc: Either

gs or gs þ gp. In this article and following previous work [Bozorgi and Underhill (2011)],

we will choose gc ¼ gs. This has the advantage that it is easier to understand the effect of

adding polymers to a Newtonian solvent. For comparison, we show results in the Appen-

dix when gc ¼ gs þ gp. This choice is useful when comparing two systems with the same

total viscosity but with different solvent and polymer contributions to the viscosity.

When examining cases in which the fluid is not composed of a Newtonian solvent and an

additional component, it is not possible to choose gs as the characteristic viscosity.

We see that the dynamics have been represented by a Smoluchowski equation for con-

servation of the configurational distribution function (W), expressed in terms of three spa-

tial (x) and two orientational coordinates (n), the creeping flow expression of fluid flow

equation, which includes an extra stress term arising from the dipole stress of the par-

ticles, continuity equation, and the constitutive equation. In Sec. III, the systems of equa-

tions will be transferred into the Fourier space and will be expressed as an eigenvalue

problem by using a basis function expansion. The dynamics of an Oldroyd-B suspension

will be expressed not only in terms of De, H, Dr, and Dt but also in terms of wavenum-

bers (k).

III. SOLUTION OF THE MODEL FOR OLDROYD-B

In the absence of rotational diffusion, the instability in a Newtonian fluid and in an

Oldroyd-B fluid has been examined previously. The impact of rotational diffusion for the

Newtonian case has been examined numerically [Hohenegger and Shelley (2010)]. Our

goal here is to examine the unique features present with rotational diffusion in a non-

Newtonian fluid using a semi-analytical approach. We proceed by using conservation of

momentum and continuity to eliminate the fluid velocity, leaving coupled equations for

the distribution function and polymer stress.



As with previous work, the solution proceeds by Fourier transforming the spatial coor-

dinate x into a wavevector k. We will use a tilde to denote variables in Fourier space.

This transform results in decoupled equations for each wavevector. Therefore, the wave-

vector becomes a parameter which we will vary and calculate the response of the system.

The continuity equation in Fourier space (ik � ~u0 ¼ 0) is used to eliminate the pressure.

This results in the following equation for conservation of momentum:

~u0 ¼ i

kðd� kkÞ � ð~R0 � ~s0pÞ � k; (16)

where overhats denote unit vectors. To simplify the notation, it is useful to define

S ¼ ðd� kkÞ � ~R0 � k and Q ¼ ðd� kkÞ � ~s0p � k, making the expression for the velocity

equal to

~u0 ¼ i

kðS� QÞ: (17)

Applying the Fourier transformation on the linearized forms of Smoluchowski equation

[Eq. (13)] and the polymer constitutive equation [Eq. (15)] yields

@ ~W0

@t¼ �i ~W

0k � n� Dtk

2 ~W0 þ Drr2

n~W0 þ 3i

2cnn : ðk~u0 þ ~u0kÞ; (18)

~s0p þ 5De@~s0p@t¼ �iHðk~u0 þ ~u0kÞ: (19)

Since the part of the polymer stress that affects the fluid velocity is written as Q, we

reformulate the constitutive equation as

ðd� kkÞ � ~s0p þ 5De@~s0p@t

� �� k ¼ �iHðd� kkÞ � ðk~u0 þ ~u0kÞ � k; (20)

Qþ 5De@Q

@t¼ �iH~u0k: (21)

Inserting the fluid velocity [Eq. (17)] into the Smoluchowski equation [Eq. (18)] and the

constitutive equation [Eq. (21)] gives two coupled equations

@ ~W0

@t¼ �i ~W

0k � n� Dtk

2 ~W0 þ Drr2

n~W0 � 3cðk � nÞðn � S� n � QÞ; (22)

ð1þ HÞn � Qþ 5De@ðn � QÞ@t

¼ Hn � S: (23)

The final step to form a closed system is to return to how the distribution function causes

the extra stress, and thereby the quantity S. Without loss of generality, we will choose the

wavevector k to point in the z-direction. We will also describe orientations in spherical

coordinates relative to this axis, and therefore n ¼ ðsin / cos h; sin / sin h; cos /Þ. The

angle-averaging necessary to obtain the extra stress is also simplified [as shown previ-

ously by Hohenegger and Shelley (2010)] by expanding the h-dependence of the distribu-

tion function as ~W0 ¼

PmAmðt;/Þeimh. Using these definitions, the extra stress is



n � S ¼ n � ðd� kkÞ � ~R0 � k ¼ p sin /

4dm;1eimh

ðp

0

sin2 /0 cos /0A1d/0: (24)

As shown in previous work, only the first azimuthal mode (m¼ 1) leads to an extra stress

and therefore a possible instability. The remaining modes yield only temporal decay or

saturation. Using this expression for the extra stress and the expansion of ~W0yields

@A1

@teih ¼ �ikA1 cos /eih � Dtk

2A1eih þ 3ðn � QÞc cos /

þDr1

sin2/A1 �

1

sin /@/ðsin /@/A1Þ

� �eih

� 3pc4

cos / sin /ðp

0

sin2 /0 cos /0A1d/0� �

eih; (25)

ð1þ HÞn � Qþ 5De@ðn � QÞ@t

¼ Hp sin /4

eihðp

0

sin2 /0 cos /0A1d/0: (26)

The presence of rotational diffusion and the resulting /-derivatives require a different

approach to solve the equations than is used when Dr ¼ 0. Note that the form of the

derivatives suggests a basis function expansion as A1 ¼P1

l¼1 ClðtÞP1l ð/Þ, where P1

l are

the associated Legendre polynomials (the / portion of the spherical harmonics). The

angle-dependence of n � Q can also be expanded. Since the angle-dependence comes

solely from n and not from Q, we can write n � Q ¼ TðtÞP11ð/Þeih. These two expansions

lead to the dynamical equations

@T

@t¼ �ð1þ HÞ

5DeT þ Hp

25DeC2; (27)

X1l¼1

dCl

dtP1

l ¼ �ikX1l¼1

Cl�1

l� 1

2l� 1P1

l � ikX1l¼1

Clþ1

lþ 2

2lþ 3P1

l þ TcP12

� pc5

C2P12 � Dtk

2X1l¼1

ClP1l � Dr

X1l¼1

lðlþ 1ÞClP1l : (28)

The two equations above may be expressed in matrix form as

d

dt

T

C1

C2

C3

�

2666664

3777775¼

�1þH

5De0

Hp

25De0 � � �

0 �2Dr �Dtk2 �3ik

50 � � �

c � ik

3

�pc5�Dtk

2� 6Dr �4ik

7� � �

0 0 �2ik

5�Dtk

2� 12Dr � � �

� � � � . ..

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

T

C1

C2

C3

�

2666664

3777775:

(29)



The time dynamics of these linear equations are found by diagonalizing the matrix, which

leads to exponential dynamics. Therefore, the dynamics will be a linear combination of

terms of the form expðrtÞ, in which the r’s are the eigenvalues of the square matrix. A

positive r illustrates instability whereas a negative one represents a stable response. For-

mally, the dimension of the matrix is infinite since an infinite number of associated

Legendre polynomials are needed as a basis. In practice, we truncate the matrix, calculate

the eigenvalues, then repeat with a larger matrix until the eigenvalues do not depend

on the size of the matrix. We found that a 100� 100 matrix was more than sufficient so

that the eigenvalues do not vary with increasing the matrix size. Eigenvalues were com-

puted efficiently using MATLAB (2008) since the matrix is sparse; the matrix is tridiagonal

except for the (3, 1) and (1, 3) elements.

If Dr ¼ 0, Eq. (25) loses its /-dependent derivative term. Because of this, it is not nec-

essary to expand the /-coordinate in associated Legendre polynomials. The dispersion

relation is the solution to

3ipc

4k 1þ H

1þ 5Der

� � 2a3 � 4

3aþ ða4 � a2Þln a� 1

aþ 1

� �� ¼ 1; (30)

where a ¼ �i rþDtk2

k . We will see later in this article that the term 1þ H1þ5Der is the Lap-

lace transform of the relaxation modulus of the Oldroyd-B model.

IV. RESULTS: OLDROYD-B MODEL

The growth rate of the linear stability r depends on five dimensionless parameters: k,

De, H, Dt, and Dr. We have shown the dependence on k, De, and H previously when

Dt ¼ Dr ¼ 0 [Bozorgi and Underhill (2011)]. Before showing the results including diffu-

sion, it is important to consider the typical ranges of the parameters. For bacterial systems

such as for Escherichia coli, possible collective structures occur at high bacterial concen-

trations, for example, from 109 (stationary phase) to 1011 (Zooming BioNematic) cells/

ml. Estimating jdj � 2� 10�18 J for an E. coli in a Newtonian fluid such as water using

the approach of Liao et al. (2007) and taking gc to be the water viscosity, tc ranges

between 0.5 and 0.005 s, with the time scale being inversely proportional to bacterial con-

centration. Using an isolated swimming speed of E. coli of vis � 30lm=s, the characteris-

tic length scale lc ranges from 15 to 0.15 lm, and is inversely proportional to bacterial

concentration.

The values of De and H depend on the material properties of the fluid. As an example,

we consider the properties of saliva and gastric mucus near neutral pH. For saliva, Hranges from near zero to �10 [Stokes and Davies (2007); Haward et al. (2010)]. The

relaxation time can vary widely, but typical values combined with the tc values above

give De up to �40. For gastric mucus near neutral pH [Celli et al. (2009)], H is approxi-

mately 30 and relaxation times lead to De up to �2. Note that the effect of the fluid prop-

erties on the dipole moment has not been taken into account when calculating these

estimates. At low pH, gastric mucus forms a gel with H ranging from 10 to thousands,

though the Oldroyd-B constitutive equation does not accurately capture the rheology of

gels.

Finally, the dimensionless values of the diffusivities Dt and Dr will depend on the

intrinsic diffusivities as well as the bacterial concentration and fluid properties by way of

the characteristic scales. Estimating the exact values of these parameters involves subtle

issues beyond the scope of this article with respect to connecting agent-based models of



interacting organisms to the mean-field theory used here. The values of Dt and Dr will

include the intrinsic value (present for dilute systems) plus a possible contribution from

interactions lost because of the mean-field approximation. We will estimate here the val-

ues not including contributions from interactions, which will act as a lower bound. Future

work will investigate how to accurately include any contributions from interactions.

Both wild-type and smooth swimming E. coli change their directions in time [Berg

(2004)]. Estimating that motion with a rotational diffusivity gives an effective Dr of

approximately 0:4 s�1 for wild-type and 0:08 s�1 for smooth swimming. This leads to a

dimensionless Dr that ranges from 0.2 to 0.002 for wild-type E. coli and from 0.04 to

0.0004 for smooth swimming E. coli. The combination of swimming and rotational diffu-

sion leads to an effective translational diffusivity equal to Dt ¼ v2is

6Dr. However, since both

swimming and rotational diffusion are included in the mean-field theory, it seems

unnecessary to include this diffusion mechanism a second time. Therefore, we estimate

the translational diffusivity using the value for nonmotile E. coli, 2� 10�13m2=s. This

corresponds to a dimensionless Dt that ranges from 0.04 to 0.0004.

We begin by showing in Fig. 1 the real and imaginary parts of the growth rate r for a

suspension of rod-like pushers in a relatively weak non-Newtonian fluid for which H¼ 5

and De¼ 1 and without rotational or translational diffusivity. Because Dr ¼ 0, we use

Eq. (30) to calculate the growth rates. The dispersion relation (r versus k) can be sepa-

rated into three regions. For 0 < k < k0, there are two positive and one negative r with

zero imaginary parts. For k0 < k < k1, the two positive roots split into a complex conju-

gate pair with positive real part. For larger wavenumbers (k > k1), Eq. (30) has no solu-

tions because the structure of the dynamics changes [Hohenegger and Shelley (2010)].

Other than the negative root, which does not appear for a Newtonian fluid, the dispersion

relation with an Oldroyd-B fluid is qualitatively the same as the Newtonian case for this

set of H and De. Note that only positive r’s lead to instability.

We can better understand the response for small k by expanding r in a series, and

determining the coefficients analytically from Eq. (30). The two roots that are nonzero

when k ! 0 behave as r ¼ r0 � r2k2 þ Oðk4Þ. The root which is zero when k ! 0

behaves as r ¼ r1k2 þ Oðk3Þ. The coefficient r0 is the solution of

De ¼ H

�pc� 5r0

� 1

5r0

: (31)

FIG. 1. Solutions for the growth rate for rod-like (c ¼ 1) pushers (p¼�1) with H¼ 5, De¼ 1, and Dr ¼ Dt ¼ 0

versus wavenumber k. (a) The real part of the three nonzero values of r. (b) The imaginary parts of r for the two

positive roots from part (a).



This can be easily solved to give

r0 ¼�ðpcDeþ 1þ HÞ6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpcDeþ 1þ HÞ2 � 4pcDe

q10De

: (32)

For each value of r0, the value of r2 is given by

r2 ¼ Dt þ3

7r0

� �1

1� Der0H

ð1þ 5Der0 þ HÞð1þ 5Der0Þ

: (33)

The value of r1, which determines the behavior of the root near zero, is

r1 ¼ �Dt �1þ H

pc: (34)

Note that p¼�1 for pushers, which often leads to a positive r1 as seen in Fig. 1. While

these expressions include the effect of translational diffusion, they do not include the

effect of rotational diffusion because Eq. (30) is not capable of accounting for rotational

diffusion.

There are a number of key features that we learn from these expansions. As illustrated

previously [Bozorgi and Underhill (2011)], curves of constant r0 as a function of De and

H will be straight lines. We can also see the impact of Dt on the dispersion relation. For

the Newtonian result [Saintillan and Shelley (2008b)], translational diffusion simply

shifts the solutions by �Dtk2. For the non-Newtonian case, translational diffusion damp-

ens the instability for many constitutive equations but not as a simple shift.

In order to understand the influence of rotational diffusion, we must use Eq. (29)

instead of Eq. (30). We again begin by examining the limit k ! 0. In this limit for

Dr 6¼ 0, the time dynamics of C1, C3, C4, etc. become uncoupled. In particular, the time

dynamics are Cl / expð�lðlþ 1ÞDrtÞ for l¼ 1, 3, 4,…. The time dynamics of T and C2

can be written as a 2� 2 matrix equation. Solving for the eigenvalues of this matrix tells

us the nonzero values of r at k¼ 0 (i.e., r0), which solve

De ¼

6DrH

pc=5þ 6Dr� H

5r0 þ pcþ 30Dr�

1þ 6DrH

pc=5þ 6Dr

5r0

: (35)

Again, this can be rewritten as

r0 ¼�ðpcDeþ 1þ H þ 30DrDeÞ

10De

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpcDeþ 1þ H þ 30DrDeÞ2 � 4pcDe� 120DrDeð1þ HÞ

q10De

: (36)

Figure 2 shows the level curves of the real and imaginary parts of r0 with Dr ¼ 0:001 for

slender (c ¼ 1) pushers (p¼�1). We plot the root which has the larger real part, corre-

sponding to using the plus sign in Eq. (36) because that root is the most unstable. This



figure illustrates the key features present when Dr < 1=30. Some key features are similar

to the case without rotational diffusion [Bozorgi and Underhill (2011)]. When H ! 0 or

when De!1 for any H, the growth rate r0 approaches the result if only the Newtonian

solvent was considered, which is 1=5� 6Dr . However, when De! 0, the polymers have

time to relax during the instability, and therefore the fluid acts as an effective Newtonian

fluid with a different viscosity. Therefore, the r0 approaches 1=ð5ð1þ HÞÞ � 6Dr.

Figure 2 also illustrates a key feature found for a non-Newtonian fluid when rotational

diffusion is present that the growth rate at k¼ 0 can have an imaginary part. This imagi-

nary (i.e., oscillatory) response at k¼ 0 only occurs when H � 1=ð30DrÞ � 1 and when

De � 1=ð30DrÞ. Each of these has a physical significance associated with them. The con-

dition on H is equivalent to saying that the value of r0 when De¼ 0 (the “effective New-

tonian” response) must be negative in order to see an oscillatory response at larger De.

The condition on De is related to a comparison in times scales between the polymer

relaxation time and the rotational diffusivity. In order to see an oscillatory behavior at

k¼ 0, it is necessary for the relaxation time to be larger than the time scale for relaxation

by rotational diffusion. While these conditions are necessary, they are not sufficient to

see oscillatory behavior. The oscillatory behavior occurs when the two real solutions

from Eq. (36) join into a complex conjugate pair. This occurs when the quantity inside

the square root passes from positive to negative, which is when

ð�Deþ 1þ H þ 30DrDeÞ2 þ 4De� 120DrDeð1þ HÞ ¼ 0: (37)

This equation is shown as a dashed line in Fig. 2(a) and forms the boundary of the non-

zero imaginary part in Fig. 2(b). The oscillatory response is particularly important for

this non-Newtonian fluid. It was found previously [Bozorgi and Underhill (2011)] when

Dr ¼ 0 that the oscillatory response that occurs for k > 0 could lead to a peak in the dis-

persion relation. We see here that rotational diffusion can lead to an oscillatory response

even at k¼ 0.

The inclusion of rotational diffusion for k 6¼ 0 must be done numerically. As discussed

previously, we numerically calculate the eigenvalues of the matrix in Eq. (29). We then

make the matrix larger and larger until the eigenvalues do not depend on the size of the

matrix. Figure 3 shows the results for rod-like pushers (c ¼ 1 and p¼�1) for the weakly

non-Newtonian case used in Fig. 1, H¼ 5 and De¼ 1. In order to focus on the role of

FIG. 2. Maximal value of the real part (a) and absolute value of the imaginary part (b) of the growth rate of

instability at k! 0 ðr0Þ for pushers with c ¼ 1 and Dr ¼ 0:001 in an Oldroyd-B suspension. The values result

from using the plus sign in Eq. (36). The white regions represent Reðr0Þ < 0 and Imðr0Þ ¼ 0. The dashed line

represents the region with complex values, and is given by Eq. (37).



rotational diffusion, we set Dt ¼ 0. For Dr 6¼ 0 and k 6¼ 0, the number of eigenvalues

from Eq. (29) equals the size of the matrix (which is tending to infinity). Many of these

eigenvalues are large negative numbers (stable modes), so we focus on the largest few

eigenvalues. For these parameters of H¼ 5 and De¼ 1, most features are similar to previ-

ous results for a Newtonian fluid. Recall that without rotational diffusion, the solution

method fails at a particular value of k ¼ k1. However, with rotational diffusion there are

solutions for all values of k; the limit Dr ! 0 is a singular limit in which the rotational

diffusion smooths out the singularity that led to the failure of the solution. As the rota-

tional diffusivity increases, the growth rate of the instability decreases. The key differen-

ces from the Newtonian result arise at high enough values of Dr. At high enough Dr, the

solution changes character by becoming oscillatory even at k¼ 0. For each set of H and

De, there is a value of Dr such that the system lies within the envelope of oscillatory solu-

tions in Fig. 2. Therefore, we see that even at a moderate H and De, the rotational diffu-

sion enhances the oscillatory nature of the dynamics versus the Newtonian result.

Having seen the impact of rotational diffusion on a “weakly non-Newtonian fluid,” we

next examine the impact on a non-Newtonian fluid with H¼ 40. At this value of the ratio

of polymer to solvent contributions to the viscosity, it was shown previously [Bozorgi and

Underhill (2011)] that the oscillatory nature of the instability can lead to a peak in the dis-

persion relation at a particular k. Although rotational diffusion typically dampens the insta-

bility, since it also enhances the oscillatory nature of the instability, we expect that the peak

in the dispersion relation may also be enhanced. Figure 4 shows the impact of Dr on the

instability of rod-like pushers (c ¼ 1, p¼�1) in an Oldroyd-B fluid with H¼ 40 at differ-

ent De. At any value of k, the rotational diffusion seems to reduce the maximal value of r(making it more stable). At the extreme values of Deborah number (De ¼ 1 and De¼ 0),

the response is that of a Newtonian fluid in which the rotational diffusion reduces the

growth rate almost uniformly across the range of k. One key difference between De ¼ 1and De¼ 0 is the effective viscosity of the solution; the system with De¼ 0 has a viscosity

(1þH) times that of the De¼ 0 case, and therefore a smaller growth rate.

At intermediate De, we find that the rotational diffusion can enhance the peak in the

dispersion relation. The reason behind the formation of a peak at large H and De � H

FIG. 3. The effect of increasing Dr on the collective response of rod-like pushers in a weak Oldroyd-B suspend-

ing fluid with H¼ 5, De¼ 1, and Dt ¼ 0. The curves correspond to the largest few eigenvalues for Dr ¼ 0 (solid

black), Dr ¼ 0:005 (dotted red), and Dr ¼ 0:01 (dashed green).



was discussed elsewhere [Bozorgi and Underhill (2011)]. We also have given another ex-

planation in Sec. VI in which the results using the Maxwell model illustrate the limiting

effects of the Newtonian solvent part in the Oldroyd-B model. Figure 4 shows the disper-

sion relation for De¼ 100 and De¼ 30. In the case of De¼ 100, the dispersion relation

for Dr ¼ 0 has a maximal growth rate at k¼ 0. As Dr increases, the growth at small k is

reduced more than at higher k, leading to a peak at nonzero k for Dr ¼ 0:01. This occurs

when the growth process has an oscillatory nature at k¼ 0. At higher Dr, the growth

becomes stable for all k. The case of De¼ 30 illustrates the response when there is a peak

in the dispersion relation when Dr ¼ 0. We again see that rotational diffusion reduces the

growth at small k more than at larger k. At Dr ¼ 0:001, this leads to a stable (negative) rat both small and large k, while it is unstable at intermediate k. The dynamics of pullers

have not been shown since they are stable in Oldroyd-B suspensions for all values of H,

De, Dr, and k. Rotational diffusivity acts to further stabilize the puller suspensions (which

are stable when Dr ¼ 0).

The major points of this section are as follows:

• The inclusion of rotational diffusion required a different formalism for which the solu-

tion for the growth rate is numerically calculated as the eigenvalues of a nearly tridiag-

onal matrix.• In order to improve our understanding, the response at k ! 0 has been examined with

and without Dr. We showed that the translational diffusivity Dt dampens the instability

in an Oldroyd-B suspension, but differently from a Newtonian one. We also showed

that rotational diffusion can change the character of the instability at k¼ 0 to be

oscillatory.• Rotational diffusion has been shown to enhance the peak that can occur in the disper-

sion relation when H is large and De � H.

V. SOLUTION OF THE MODEL FOR MAXWELL FLUID

In Secs. II–IV, we have examined the linear stability of a suspension of swimming

organisms in an Oldroyd-B fluid. Because of the linear dynamics, the convected deriva-

tive in the Oldroyd-B model does not contribute. Therefore, the results are the same as if

the Jeffreys model had been used. The Jeffreys model is the sum of a Newtonian fluid

FIG. 4. The effect of Dr on the collective response of rod-like pushers in Oldroyd-B suspending fluids with

H¼ 40 at two different De. For each k, the largest two solutions are plotted. If only line is shown, it represents

a complex conjugate pair. (a) Response for De¼ 100 for Dr ¼ 0 (solid black), Dr ¼ 0:01 (dotted red), and

Dr ¼ 0:025 (dashed green). (b) Response for De¼ 30 for Dr ¼ 0 (solid black), Dr ¼ 0:001 (dotted red), and

Dr ¼ 0:01 (dashed green).



part and a Maxwell model part. We found that particularly interesting features occur in

the dynamics when the Maxwell contribution to the viscosity was much larger than the

Newtonian fluid contribution. This motivates looking at the dynamics when the suspend-

ing fluid follows the Maxwell model only.

For this case, the only change in Eqs. (11)–(15) is that the “solvent viscosity” in Eq.

(12) is zero. Therefore, it is not appropriate to use gs as the characteristic viscosity gc.

Instead, we choose gc ¼ gp. This results in H¼ 1 in Eq. (15). It also means that the pa-

rameters De, k, Dr, and Dt are now nondimensionalized using a different characteristic

viscosity. These changes require that a slightly different approach is taken to solve the

linear dynamics. Previously, conservation of mass and momentum were used to solve for

the fluid velocity, which was inserted into the Smoluchowski equation and the Maxwell

constitutive equation. However, Eq. (12) no longer contains the fluid velocity explicitly.

Instead, combining momentum and mass conservation leads to the condition that S¼Q.

This is combined with the Smoluchowski equation and constitutive equation which are

still given by

@ ~W0

@t¼ �i ~W

0k � n� Dtk

2 ~W0 þ Drr2

n~W0 þ 3icðn � kÞðn � ~u0Þ; (38)

Sþ 5De@S

@t¼ �i~u0k; (39)

where we have used that S¼Q. Solving Eq. (39) for the velocity and substituting into

Eq. (38) give the one final equation that the distribution function satisfies

@ ~W0

@t¼ �i ~W

0k � n� Dtk

2 ~W0 þ Drr2

n~W0 � 3cðn � kÞ n � Sþ 5De

@S

@t

� �� : (40)

In the absence of Dr, this equation can be solved in exactly the same way as for the New-

tonian case or the Oldroyd-B case, for which the dispersion relation must satisfy

3ipc

4k1

1þ 5Der

� � 2a3 � 4


aþ 1

� �� ¼ 1; (41)

where 1=ð1þ 5DerÞ is the Laplace transform of the relaxation modulus of the Maxwell

fluid.

When the rotational diffusivity is nonzero, we again expand the distribution in spher-

ical harmonics by first writing ~W0 ¼

PmAmðt;/Þeimh. Because of the integral involved

in S, only the m¼ 1 term contains unstable modes [see Eq. (24)]. As we did for the

Oldroyd-B case, we expand the m¼ 1 coefficient as A1 ¼P1

l¼1 ClðtÞP1l ð/Þ, where P1

l

are the associated Legendre polynomials. After some algebra, we obtain

pcDe@C2

@tP1

2 þX1l¼1

@Cl

@tP1

l ¼ �ikX1l¼2

Cl�1

l� 1

2l� 1P1

l � ikX1l¼0

Clþ1

lþ 2

2lþ 3P1

l

�Dtk2X1l¼1

ClP1l � Dr

X1l¼1

lðlþ 1ÞClP1l �

pc5

C2P12: (42)



Since the P1l are linearly independent, we can identify a set of coupled linear differential

equations that can be represented in matrix form as

@

@t

C1

C2

C3

�

2664

3775 ¼

�2Dr � Dtk2 �3ik

50 � � �

�ik

3ð1þ pcDeÞ�Dtk

2 � 6Dr � pc=5

1þ pcDe

�4ik

7ð1þ pcDeÞ � � �

0�2ik

5�Dtk

2 � 12Dr � � �

� � � . ..

0BBBBBBBBBBB@

1CCCCCCCCCCCA

C1

C2

C3

�

2664

3775:

(43)

VI. RESULTS: MAXWELL MODEL

The first key feature to note about the results using a Maxwell fluid model is that

when De¼ 0, we obtain the Newtonian result. This can be seen explicitly in both Eqs.

(41) and (43). This is expected based on the insight gained about the role of the Deborah

number from the Oldroyd-B results. In this limit, the fluid is able to relax during the

instability and therefore acts like an effective Newtonian fluid. The new features occur

for larger De, when the fluid does not have time to relax during the instability. For the

Oldroyd-B fluid, as De!1, the Maxwell part does not contribute at all; the dispersion

relation grows until it reaches the Newtonian “solvent” result. However, for a Maxwell

model, there is no Newtonian solvent that bounds the instability.

As an example of the response, Fig. 5 shows the real part of the dispersion relation for

rodlike (c ¼ 1) pushers (p¼�1) without including the effects of rotational or transla-

tional diffusion [from solving Eq. (41)]. For De < 1, the behavior is qualitatively the

FIG. 5. The real part of the growth rate of instability of pushers versus k at De¼ 0.35 (dashed black), De¼ 0.9

(solid black), De¼ 1.1 (dotted red), and De¼ 2.5 (dotted-dashed red) in a Maxwell fluid with Dr ¼ Dt ¼ 0. The

negative r’s are not shown.



same as for the Newtonian result. However, for De > 1, the response changes dramati-

cally. The nonzero positive growth rate at k¼ 0 disappears, and the growth rate increases

approximately linear in k, with a decreasing slope as De increases. This shifts the type of

instability from one that is most unstable at k ! 0 to one that is most unstable as k!1.

This occurs when De ¼ kjdjc=ð5gpÞ > 1, when the growth rate that would have occurred

in a Newtonian fluid with viscosity gp is faster than the fluid can relax.

We can better understand the dramatic change at De¼ 1 by examining the nonzero

values of the growth rate at k¼ 0, r0. Without rotational diffusion, this is found by

expanding Eq. (41). With rotational diffusion, we examine the eigenvalues of the matrix

in Eq. (43) when k¼ 0. When k¼ 0, the matrix is diagonal, and the most relevant eigen-

value is the (2, 2) element. Therefore, we find that

r0 ¼ �6Dr þ

pc5

1þ pcDe: (44)

We can explicitly see that for c ¼ 1 and p¼�1, a singularity occurs when De¼ 1. There-

fore for Dr ¼ 0, the mechanism that leads to an instability for pushers actually leads to a

stable decay. Instead, it is the solution which is zero at k¼ 0 that gives the unstable

growth. Note that for pullers (p¼ 1), the growth rate remains negative (stable).

The impact of rotational diffusion on pushers is illustrated in Fig. 6. For De < 1, the

response is qualitatively the same as the Newtonian case, in which rotational diffusion

stabilizes the system (reduces r). For De > 1, the response can be different. In particular,

when the rotational diffusivity is large enough (Dr > 1=30), the growth rate actually

increases with increasing Dr. This is also in sharp contrast to all other cases that have

been examined previously, that more rotational diffusion can cause the system to be

more unstable. We do not show the results for pullers, which are always stable and for

which rotational diffusion always stabilizes the system. The matrix in Eq. (43) shows that

the impact of translational diffusion is also different from the Newtonian case. The (2, 2)

FIG. 6. The effect of Dr on the collective response of rod-like pushers in a Maxwell suspending fluid at two dif-

ferent De. For De¼ 0.9, rotational diffusion stabilizes the system from Dr ¼ 0 (solid black) to Dr ¼ 0:004

(dashed black). For De¼ 2.5, rotational diffusion destabilizes the system from Dr ¼ 0 (dotted-dashed red) to

Dr ¼ 0:4 (dotted red).



element of the matrix shows that translational diffusion can cause the system to be more

unstable if De > 1 for pushers (p¼�1).

By examining the response in a Maxwell fluid, in comparison to the Oldroyd-B fluid,

we can see the importance of the “Newtonian solvent” even when the “polymer” contri-

bution to the viscosity is very large. We can also understand better the peak that occurs in

the dispersion relation. The peak occurs in the Oldroyd-B fluid when H is very large and

De � H. For large H, we would expect the Maxwell model to be accurate. For the

Oldroyd-B fluid, the ratio De=H ¼ kjdjc=ð5gpÞ. Therefore, it is the change in the disper-

sion relation for the Maxwell case for De > 1 that leads to the increasing side of the peak

for the Oldroyd-B case. Simultaneously, the Newtonian solvent part in the Oldroyd-B

case tries to limit the growth. The balance of these competing features leads to the peak.

To summarize the section, the key results for the Maxwell fluid case are as follows:

• As with the Oldroyd-B case, for Dr ¼ 0 the dispersion relation can be found from an

algebraic equation. When including rotational diffusion, the values of the growth rate

are found as the eigenvalues of a matrix.• The response at k¼ 0 was used to show how a dramatic change in response occurs

between De < 1 and De > 1.• For pushers when De > 1, large rotational or translational diffusion can act to more

strongly destabilize the system. We do not currently have a physical interpretation for

how this counterintuitive process occurs.• The increase of r with k for large De gives more insight into the mechanism for the

peak seen in the Oldroyd-B case.

VII. MODEL AND RESULTS: GLVE MODEL

Thus far, we have examined the response in a suspending fluid that obeys either the

Oldroyd-B or Maxwell constitutive equation. For the linear stability analysis, the results

depend only on the response of these fluids in the linear regime. We have pointed out in

the derivations that the Laplace transform of the relaxation modulus plays a key role. We

will examine here a system suspended in a GLVE fluid, and show explicitly that for any

model, the results depend on this Laplace transform.

Following the previous steps for other constitutive equations, we will identify a char-

acteristic viscosity that is used to construct the characteristic time scale. We will focus on

systems for which there exists a zero shear rate viscosity, which corresponds to the inte-

gral of the relaxation modulus. That is, we will choose gc ¼ gG ¼Ð1

0GðtÞdt. In dimen-

sionless quantities, the constitutive equation is

s0 ¼ �ðt

�1Gðt� t0ÞC0dt0; (45)

where the primes on s and C denote that they are the perturbations from the steady state

in the linear stability analysis. Similar to the Maxwell model case, since the velocity does

not explicitly appear in momentum conservation, we find that S¼Q and the constitutive

equation becomes

S ¼ �ik

ðt

�1Gðt� t0Þ~u0dt0: (46)

Following the method used for the Maxwell fluid, we would try to solve this equation for

the fluid velocity, which is then inserted into Eq. (38). This is possible if the constitutive



equation has a differential form, but is not possible for a fluid with arbitrary relaxation

modulus. However, if we assume that all variables including the fluid velocity are propor-

tional to ert, then we can reformulate Eq. (46) as

S ¼ �ik~u0ð1

0

GðsÞe�rsds ¼ �ik ~GðrÞ~u0; (47)

where ~GðrÞ is the Laplace transform of the relaxation modulus. This proportionality

between S and ~u0 is the same form found in the case of a Maxwell fluid. Following the

same steps as for the Maxwell case, we find

3ipc

4k ~GðrÞ2a3 � 4


aþ 1

� �� ¼ 1; (48)

when there is no rotational diffusion. Calculating the influence of rotational diffusion

cannot proceed in exactly the same way as previously since we do not in general have a

differential form of the constitutive equation. Therefore, we cannot obtain a set of

coupled, linear differential equations. Since we have assumed dynamics of the form ert,

we find a set of linear, coupled equations that must be satisfied if the time dynamics are

of this exponential form. Therefore, we find that r must take the values such that the de-

terminant of the following matrix is zero:

det

�2Dr � Dtk2 � r

�3ik

50 � � �

�ik

3�Dtk

2 � 6Dr �pc

5 ~GðrÞ� r

�4ik

7� � �

0�2ik

5�Dtk

2 � 12Dr � r � � �

� � � . ..

0BBBBBBBBBBB@

1CCCCCCCCCCCA

¼ 0:

(49)

These two expressions (48) and (49) are significant since they describe the linear stability

within any material that has a nonzero value of the zero shear rate viscosity, because in

the linear region any material can be expressed in terms of the GLVE model.

For k ! 0, we can solve the dispersion relation exactly since the matrix in Eq. (49)

becomes diagonal. The values of r include the values �2Dr; �12Dr; �20Dr, etc. The

growth rate which is nonzero even when Dr ¼ 0 has previously been called r0 and must

solve

ðr0 þ 6DrÞ ~Gðr0Þ ¼ �pc5: (50)

This equation reduces to the results for an Oldroyd-B or Maxwell fluid [Eqs. (31) and

(44)] when the corresponding relaxation modulus is used.

We will examine another choice of the relaxation modulus here, which is relevant to

mucus gels. For a “critical gel,” the relaxation modulus becomes a power law decay and

the zero shear rate viscosity diverges. Friedrich et al. (1989) presented a relaxation modu-

lus that is a product of a power law and an exponential. This form reduces to the Maxwell



fluid when the exponential part dominates and to the critical gel when the exponential

part disappears. Using a relaxation modulus GðtÞ / t�K expð�t=kÞ and using the integral

of G(t) as the characteristic viscosity gives a dimensionless relaxation modulus of

GðtÞ ¼ t�Ke�t=5De

ð5DeÞ1�KCð1� KÞ; (51)

where De ¼ k5gGjdjc

. Note that with this definition, De!1 corresponds to a critical gel. We

also see that K¼ 0 corresponds to the Maxwell fluid. The storage and loss modulus for

the GLVE fluid with this G(t) can be fit well to the experimental data for gastric mucus at

both acidic and neutral pH [Celli et al. (2009)].

Using the Laplace transform of Eq. (51) in Eq. (50), we can calculate how r0 depends

on De and K. This r0 solves

r0 þ 6Dr

ð1þ 5Der0Þ1�K¼ � pc

5: (52)

Similar to previous cases, when De! 0, the fluid has time to relax during the instability

and the system behaves like a Newtonian fluid. As De increases from zero, the response

depends on K. When K¼ 0 (the Maxwell case), the r0 diverges when De! 1, and

becomes negative when De > 1. However, for any 0 < K < 1, the growth rate only

approaches infinity when De!1.

As a summary, the key results of this section are as follows:

• The solution formalism was expanded to the GLVE provided that the fluid has a finite

zero shear rate viscosity, which is used for nondimensionalization. This fluid contains

the previous results from this article as special cases.• The influence of rotational diffusion must be treated differently than the Oldroyd-B

and Maxwell cases if there is no differential form of the constitutive relation.• The growth rate has been quantified at k¼ 0 for a fluid with relaxation modulus rele-

vant for mucus gels.

VIII. SUMMARY AND CONCLUSION

In summary, the influence of viscoelasticity and rotational diffusion on the dynamics

of active particles has been analyzed by using a nonequilibrium continuum model. Three

different constitutive functions have been considered, the Oldroyd-B, Maxwell, and

GLVE models. The formalism using spherical harmonics allows for analytical calcula-

tions for small wavevectors and is convenient for numerical calculations for arbitrary

wavevectors. These results illustrate key new features about the role of viscoelasticity

TABLE I. Summary of the key equations and non-Newtonian dimensionless numbers.

Oldroyd-B Maxwell GLVE Remarks

Dr ¼ 0 Eq. (30) Eq. (41) Eq. (48) Algebraic form

Dr 6¼ 0 Eq. (29) Eq. (43) Eq. (49) Matrix form

Non-Newtonian

dimensionless

numbers

De ¼ k=ð5gs=ðjdjcÞÞH ¼ gp=gs

De ¼ k=ð5gp=ðjdjcÞÞ De ¼ k=ð5gG=ðjdjcÞÞ



and how rotational diffusion has a different role in viscoelastic fluids compared with

Newtonian fluids.

The key equations of this paper and the non-Newtonian dimensionless numbers for

each suspension have been tabulated in Table I. A summary of the results for each sus-

pension is given at the end of Secs. IV, VI, and VII.

Previously Bozorgi and Underhill (2011) showed that swimmers in a Oldroyd-B fluid

have a peak in the dispersion relation at a finite wavenumber for large H and when

De � H. The extension shown here to a Maxwell fluid helps to understand the mecha-

nism for this peak. The differences between the results in the two fluids are important for

future experimental studies, in which it will be important to understand the rheological

properties of the suspending fluid. For example, it is possible that the Oldroyd-B and

Maxwell models can match experimental storage and loss moduli over a certain fre-

quency range, but give different collective behavior of swimming organisms. By extend-

ing the linear stability to the GLVE model, we have developed the formalism to

understand the linear dynamics in any fluid, since any fluid can be represented by this

model in the linear regime.

The suspending fluid also impacts the role of rotational diffusion. In general, for a

Newtonian fluid, increasing the rotational diffusivity causes the system to become more

stable. For a non-Newtonian fluid, this is not necessarily the case. In a Maxwell fluid,

when De > 1 and Dr > 1=30, increasing the rotational diffusivity causes the system to

become more unstable for pushers only. In an Oldroyd-B fluid, rotational diffusion

does stabilize the system, but can change the character of the instability. Rotational dif-

fusion can change the instability to be oscillatory in nature even at k¼ 0. This can lead

to a dispersion relation that has a peak with rotational diffusion that would not have

one without it.

The linear stability analysis presented is an important step in understanding when col-

lective behavior might result. The exact structure of that collective behavior will be de-

pendent on the nonlinear dynamics, which has not been included but is currently being

studied. In addition to nonlinear terms in the Smoluchowski equation, there can be non-

linear terms in the fluid constitutive equation. Key nonlinear effects that do not influence

FIG. 7. Contours of the growth rate of instability at k ! 0 for rodlike pushers with Dr ¼ 0 in an Oldroyd-B sus-

pension with characteristic viscosity gc ¼ gs þ gp.



the linear analysis but could influence the nonlinear dynamics include shear-thinning,

elongational thickening, and normal stress differences.

ACKNOWLEDGMENT

This work was supported by the National Science Foundation Grant No. CBET-

0954445.

APPENDIX: A DIFFERENT CHARACTERISTIC VISCOSITY FOR OLDROYD-B

The nondimensionalization for the Oldroyd-B fluid in Secs. III and IV used the viscos-

ity of the Newtonian solvent as the characteristic viscosity. This is convenient when try-

ing to understand how adding polymers to a Newtonian will affect the growth rate.

Figure 4 confirms that adding polymers to a Newtonian fluid suspension reduces the

growth rate. An alternative choice is to choose the zero shear viscosity as the characteris-

tic viscosity, which we show in this appendix. This is a useful choice when comparing

the Oldroyd-B fluid with a Newtonian fluid with the same viscosity.

With the choice tc ¼ ðgs þ gpÞ=ðjdjcÞ, all nondimensionalized variables are corre-

spondingly changed, including r, k, Dt, and Dr. We also change the definition of the

Deborah number as De ¼ k=ð5tcÞ ¼ kjdjc=ð5ðgs þ gpÞÞ. Finally, it is useful to use the

variable f ¼ gp

gpþgsinstead of H to quantify how much of the fluid is non-Newtonian. The

limit f ! 1 will correspond to the Maxwell fluid presented in Secs. V and VI.

Figure 7 shows the level curves of the maximal growth rate at k¼ 0 for rodlike

pushers with Dr ¼ 0 versus De and f. The lines of constant growth rate follow

De ¼ pcþ 5r0

5r0ð�pc� 5r0ð1� f ÞÞ : (A1)

The growth rate is always greater than the Newtonian result. The Newtonian result is

obtained at f ! 0 for all De and at De! 0 for all f. For 0 < f < 1, as De!1, the

growth rate approaches �pc=ð5ð1� f ÞÞ. In the limit f¼ 1, the system is a Maxwell fluid,

discussed previously.

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role of linear viscoelasticity and rotational diffusivity...

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