Journal of Non-Newtonian Fluid Mechanics, 26 (1987) 43-55 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

43

BOUNDARY EFFECTS ON THE DRAG OF AN OSCILLATING SPHERE: APPLICATIONS TO THE MAGNETIC SPHERE RHEOMETER

H.S. SELLERS and W.H. SCHWARZ

Department of Chemical Engineering, The Johns Hopkins University, Baltimore, MD 21218 (U.S.A.)

and

M. SAT0 and T. POLLARD

Department of Cell Biology and Anatomy, The Johns Hopkins School of Medicine, Baltimore, MD 21205 (U.S.A.)

(Received April 7, 1986; in revised form March 19, 1987)

Summary

The effect of a closed, nearby boundary on the drag of a small, oscillating sphere in an incompressible, linearly viscoelastic medium is studied. Through a perturbation procedure the functional dependence of the drag on the complex oscillatory Reynolds number is determined and shown to differ from that of the unbounded case. These results allow one to account for boundary effects in the determination of the complex viscosity using the magnetic sphere rheometer. This paper is motivated by the rheological study of materials purified in minute quantities such as intracellular proteins, thus the need for small chamber sizes for use with the rheometer and the need for boundary corrections for the drag.

1. Introduction

The increased use of a small-volume, oscillatory magnetic sphere rheome- ter [l-8] necessitates a close scrutiny of the boundary effects on the drag of an oscillating particle. Boundaries can significantly affect the drag, yet such effects are routinely ignored in applications of the magnetic sphere rheome- ter.

0377-0257/87/$03.50 0 1987 Elsevier Science Publishers B.V.

44

Our purpose in this paper is twofold: first, to determine the effect of a closed, nearby boundary on the drag of a small oscillating sphere in an incompressible, linearly viscoelastic medium, and second, to render explicit this effect on the calculation of the complex viscosity from the magnetic sphere rheometer as it is presented in [l]. This reference discusses in detail the design and operation of the rheometer.

In the small displacement, oscillatory mode of operation of the rheometer, measurements of the sphere’s displacement and phase shift relative to the applied magnetic force are used to calculate the complex viscosity given the correct expression for the drag on a sphere. The expression currently used is the Fourier transform of the classical formula derived by Stokes for the drag in an unbounded Newtonian fluid [1,9,10] *. However, the rheological study of materials purified in minute quantities, such as the intracellular proteins actin, profilin, and actinin, requires the use of small (microliter) chamber sizes in the rheometer. As a consequence of this size limitation, boundary effects can be especially important; the expression for the drag on a sphere in an unbounded medium may no longer be applicable and should be modified to take into account the presence of boundaries.

For slow, steady motion (creeping motion approximation) of a linearly viscoelastic material, a dimensional argument shows that the boundary corrections to the drag involve only dimensionless geometric factors, which can be determined experimentally through calibrations with Newtonian fluids of known viscosity [12]. In the small-amplitude, oscillatory case there is the complication that the boundary corrections depend on the complex viscosity, in addition to the geometric factors, thereby fundamentally alter- ing the result. This paper addresses the latter, important case of unsteady, oscillatory motion.

In the following, we present a simple analysis for a smooth, closed boundary. The principle assumptions are that a small, rigid sphere executes axisymmetric, small-amplitude, harmonic oscillations in a small, closed volume of incompressible, linearly viscoelastic material. Our procedure is to employ the correspondence principle to express the equations in terms of the complex viscosity, then to introduce the Stokes stream function. Next we demonstrate a perturbation procedure based on the nearness of the boundary for the solution of the stream function. These results are extended to the calculation of the drag, where we compare the classical, unbounded solution to the bounded, perturbation solution. The example of a spherical outer

* According to the correspondence principle the Fourier transform yields the constitutive equation of linear viscoeiasticity identical to that of a Newtonian fluid, but with a complex- valued viscosity that is a function of frequency [ll].

45

boundary is examined in detail since an exact solution already exists, allowing comparison with the perturbation solution. Finally, we show how these results are used to calculate the complex viscosity.

As we are dealing here with linear viscoelastic materials having a complex viscosity, it will be convenient to leave all expressions in complex form. It is to be understood that the real part is to be taken.

2. Governing equations

The equations of motion for an incompressible material are

div u=O, (la)

p ~+U-VV = -vp+divT. (lb)

In linear viscoelasticity the stress tensor is a linear functional of the past history of deformation. For an isotropic material this constitutive relation can be expressed as

T(t) = 21’ +(t - T)D(~) d7, -CO

D = +(vu+ (vty), w where + is the stress relaxation function, giving the time-dependent stress after imposition of unit strain.

These equations must be supplemented with boundary conditions. We use the usual no-slip conditions for viscous fluids: the material adheres to the surface of the oscillating sphere and to the surface of the outer boundary. For a solid-like material, these conditions correspond to the notion of a “sticky” solid. Thus, these boundary conditions would be appropriate to the study of sol-gel transitions in the case of small displacement of the sphere.

If an oscillatory velocity field o = V, ei“” is assumed, the linearized form of eqns. (1) and (2) transform to

divv,=O, (3a)

iopva eiwt = - Vp + v*V 2va e’“‘, ( W

q*(w) = lm*(u) ePioUdu, (3c)

= q’ - i(G’/w), (3d)

where we have introduced the complex viscosity n*. It will be convenient to express the equations of motion in dimensionless form. Let R be the radius

46

of the sphere, U the magnitude of its velocity, and b a characteristic distance from the center of the sphere to the outer boundary. We introduce the following dimensionless quantities:

2=x/R, + =Rv, i?=t?/l_J,

@ = Rp/v*U, A=b/R, f= RT/q*U, (4) S2 = iapR2/q*, Re = RUp/r)*, f= iot.

Thus (lb) becomes

Three fundamental dimensionless quantities arising in this problem are a complex translational Reynolds number Re, a complex oscillatory Reynolds number S2, and the geometric factor A. Both Re and S2 are defined as complex quantities for convenience in manipulations. If

the Re is the ratio of the convective acceleration to the viscous term, and S2 is the ratio of the unsteady acceleration to the viscous term. In the un- bounded, Newtonian case, the real part of S( = i-R) has been called the Stokes number and is the ratio of particle size to the Stokes shear wave layer thickness.

Comparing relative magnitudes, the ratio of ( Re 1 to 1 S2 1 gives maxi- mum particle displacement over particle radius. Since the operating condi- tions of the rheometer are such that this ratio is small (or perhaps, since the operating conditions are restricted to those such that this ratio is small), Re is small compared to S2; hence, in this analysis the nonlinear term will be neglected. Consequently, nonlinear effects such as steady streaming are not considered [13]. This linearization is also in the spirit of the constitutive assumption of linear viscoelasticity, which is based on infinitesimal displace- ment. In addition, the upper frequency limitation of the rheometer is about 100 Hz. With the use of very small spheres (e.g. R < 0.1 cm), the magnitude of S2 will itself be small. This fact is important, since the smallness of S2 and the nearness of the outer boundary mean that the region of viscous influence can extend to the outer boundary, allowing the particle to strongly feel its presence. Here, the assumption on the nearness of the outer boundary is understood to be that

]S]h<l. (7) Thus, in our problem the equations (modified with term.

equations of motion simplify to the steady Stokes a complex viscosity) plus a small inhomogeneous

47

The assumption of axisymmetry allows the introduction of the stream function +, further reducing the equations of motion to a single scalar equation for $. With an externally applied, oscillatory magnetic force Fe’” and the linearity of the equation, the stream function can be expressed as

$J = R*Uj(f, S, geometry) eiw’, (8)

giving the equation for 4 for a viscoelastic material in dimensionless form as

where in spherical coordinates

i*=R* a2 sin 9 a 1 a

ii?-- -- ( )I r* ae sin 8 ae * To fix ideas, we envision a sphere of reduced radius i = 1 oscillating with zero mean displacement and completely enclosed in a small region of viscoelastic material by a rigid outer boundary having a well defined normal. The coordinate system is chosen such that the origin coincides with the mean center of the sphere. The boundary conditions are that the material is at rest on the outer, enclosing boundary, and oscillates with the velocity of the sphere on the surface of the sphere. Under the assumption of negligible translational Reynolds number, this latter boundary condition can be lin- earized, allowing the application of the no-slip condition to the mean location of the sphere. In terms of the stream function, the boundary conditions translate to

& = - $sin*e

a4 at i= 1, -= ai

- sin28

and

4 = a&/a4 = 0 at the outer boundary, (Ilb)

a/G being the dimensionless derivative in the direction normal to the outer boundary.

3. Expansion procedure

It is necessary to know how a boundary modifies the functional depen- dence of 4 on 5, as it is through S that the stream function (and consequently the drag) depends upon the complex viscosity. Again, our

48

interest lies in the case of a nearby boundary and small S. Thus, in this section we seek a perturbation solution of (9) in the form

J= E j,<.?, geometry)$(S) + 0 [&(S)], j=O

(12a)

ww with the Sj to be determined. We are primarily concerned with the form of the gauge functions ai, not the Jj, as the Gj depend only upon the geometry; we discuss only one example where we explicitly determine the

#j*

The expansion (12a) is substituted into (9) with the resulting expression separated into terms of equal order in the lij and them consecutively solved at each order. The assumption on the Reynolds numbers is only that Re = o( S*), so that we may expect the dependence of the 8, on Re to become important in the higher order terms. For simplicity Re is neglected.

Implicit in the above expansion is that a regular perturbation is valid. It is well known that the example of unbounded, steady flow past a sphere is singular; the boundary conditions cannot be satisfied in a regular perturba- tion expansion of powers of Re (Whitehead’s Paradox) [14,15]. If, however, the boundary is sufficiently close such that it lies completely within the viscous region surrounding the sphere, then a regular perturbation (the inner solution of a singular perturbation) remains valid [16]. Our concern is for minute sample sizes, thus the use of a regular perturbation. As discussed in [16], the requirement that a regular perturbation be valid is (7).

With the boundary conditions (ll), one easily sees that upon substituting the expansion (12), the only choice for 6,(S) that leads to a nontrivial solution is So = 1. Hence, to zero order the problem simplifies to

i”J, = 0, 03) with the boundary conditions

Jo = - *sin*0

aJo -= a;

- sin*8 at i= 1, 04a)

and

Go = &j&M = 0 at the outer boundary. (I4b)

As was to be expected, the dominant term $,, corresponds to the solution for steady, slow motion of a sphere. This much discussed problem presents tremendous analytical difficulties, with existing solutions limited to simple boundary conditions (see e.g. [17]).

49

Similarly, the next nontrivial solution comes in only when 6,(S) = S2. Thus the S2-problem becomes

E”J, = &‘$,, (15)

with the homogeneous boundary conditions

~,=~J,/~i=Oat F=l, (lea)

and

&i = a&/a2 = 0 at the outer boundary. (16b)

Here $,, serves as a forcing function for the correction $i, in general nonzero.

Moreover, if Re = o( Si) for j > 1, then upon continuing this procedure one immediately concludes that due to the homogeneous boundary condi- tions at all higher orders, the functions sj are increasing euen powers of S up to Sj. This fact will lead to rather umque consequences for the drag. It should be remembered that any such consequence is a direct result of a regular perturbation and should not be expected as the boundary moves to infinity, since a regular perturbation is no longer valid. In conclusion, when Re = o(S2”) then the stream function 4 takes the asymptotic form

4 - J$C (17)

and the equations for the coefficients of the stream function become

JP& = 0, 34&j = E2Jj_i, j > 0. (1%

This procedure is easily illustrated with the example of a spherical outer boundary with radius i = A, X < 1 S I, which is concentric with the mean l_ocation of the oscillating sphere. Following the standard argument we write Go as

&= CX~( ?) sin20. (19)

This separation of variables results in

($ - $jl.di)=o, and the boundary conditions

(20)

a,(l)= -f and q(X)=0 a;(l) = -1 LX;(X) =o (21)

50 ’

Equation (20) is easily solved, with the general solution given by

(YJ 3) = A,P4 + II,? -t COP2 + Do/3 (22)

The constants A,, B,,, Co, and Do are obtained by inserting the boundary conditions (21) and solving a set of 4 simultaneous algebraic equations. After some algebra, one obtains

A =2(X’-h) O 2

c = L (4 + 5x3 - 9P) X’ O2 x ’

3(X6 - X) Do= (X3-0 (23)

Bo= x , x ’

where

x= -4 + 9x - 10x3 + 9x5 - 4x6. Now, turning to the S2-problem, substitution of (22) results in the

differential equation

le’l+c, ( 2BO = &Jo = lOA, - - i 1 sin28

and the boundary conditions *

J@ !$ =Oat i=l,

and A

+ !!g =0 at ?=h.

(24)

(254

(25b) Again, this equation can be solved by separation of variables with the solution given by

where

51

---

4. Drag relations

In this section, we extend the results on the stream function to the drag. In addition, we review the classical results on the unbounded case and compare it to the results derived below for the bounded case. These two cases, the nearby boundary and the unbounded medium, will prove im- portant because they both give simple relations between the drag and S.

The drag on a body is obtained by integrating over the surface of the body the stress in the direction of motion:

9= mTn dA, / (27) A

m being the unit vector in the direction of motion and n the unit normal to the sphere. Formula (27) can be expressed in terms of the dimensionless stream function. Generalizing the results of [17] for steady creeping motion to oscillatory creeping motion, the drag on a sphere in an incompressible, viscoelastic material and in the limit of zero translational Reynolds number becomes

9= m$RU eiot s2 a4 id(, --- 1 ?sinBai . (28)

Inserting the expansion for the stream function (17) in the above formula (28) and pulling the powers of S outside the integral results in the following asymptotic expression for the drag:

9- q*RUeei*’ t iSjS2j, (29 j=o

52

where the coefficients ZBj are functions of only dimensionless geometric factors. Thus, we see how the drag depends on the parameter S.

Returning to the example of a spherical outer boundary, we now calculate the coefficients Bj. The first two terms of the stream function have already been calculated. Substituting (23) and (26) into (28) and effecting the various integrations, we get to within terms of order S2,

g= aq*RUeiW’[8Bo + S2( -4B, - ?.A, - qB,, - 4C, + 20,)]. (30)

For S = 0 (steady motion), this result reduces to that in [17] for the drag on a sphere translating in a spherical envelope.

Due to the simple geometry the spherical envelope problem can be solved exactly, as done originally by Stokes [18]. His method involved separation of variables and the solving of a set of four complicated, simultaneous alge- braic equations. In our notation his solution becomes

g= trV*RUs2K eiot

where

(31)

I>1 *

3 K=l--. Ye% A> -Y(SL A-‘>

2s2 12s + S(S, X) + &Sx, X1)/X

and

y(S, A)=(S2+3S+3)(S2X2-3Sh+3)exp[S(A-l)],

S(S, A) = [(S2hz - 3SA + 3) - (S2 + 3s + 3)/X)] exp[ S(X -

Though exact, this formula does not permit a simple solution in closed form for q*. However, with assumption (7) this expression can be expanded in series form, resulting in (30).

We turn to the unbounded case. Letting h + cc, (31) reduces to

ZB= 6rrrq*RUe’“’ 1 + S+ 9 . (

S2

i (32)

This simple result is exact for Re + 0, S arbitrary. Separating real and imaginary parts, one observes that the drag consists of two parts: a dissipa- tive part proportional to the velocity and an inertial part proportional to the acceleration. In the bounded case we see from (31) that in general the drag cannot be so simply expressed. However for small S and a nearby boundary, (29) shows that the drag can be expressed as a series in S2. As opposed to the unbounded case, there is no term proportional to S1. Again, a regular perturbation was used in this derivation so that this result cannot go over to the unbounded expression as the boundary moves to infinity.

53

5. Complex viscosity

We investigate the consequences of (29) on the determination of the complex viscosity using the magnetic sphere rheometer. We suppose that an appropriate bias counteracts the gravitational force on a metallic sphere of density ps. If an oscillatory magnetic force J’eio’ is applied to the sphere suspended in a nonconducting viscoelastic medium, the sphere’s inertia is balanced by two forces: the applied, oscillatory magnetic force and the drag exerted by the medium. Assuming that the sphere’s motion is also oscillatory (a consequence of linearity), a force balance on the sphere yields

Fe iot = +rpsR3ioU eiwt + 9. (33)

A phase difference between the applied force and the particle velocity will make F complex. F is determined by the particular magnet used and is a function of the electric current, the number of windings, etc. Restricting ourselves to those conditions for which (7) holds, (33) becomes

F=xq*RU go+ - [ (;F ++2]

to within terms of order S2. Solving for q* yields

F

‘* = TRUED

The complex modulus G* is related to TJ* by

G* = iwg*,

so that with (30) and (31), G* becomes

G* = Fio

nR UcQo

(35)

(36)

(37)

For a spherical outer envelope, B0 and LB~ are found from (30). Finally, using (3d) the dynamic viscosity $ and the storage modulus G’ become

G’= -1m (39)

These two results in this approximation of neglecting terms of order higher than S2 demonstrate: (1) 9’ can be calculated by assuming steady motion of the sphere; (2) G’ contains an additional term arising from S2 that is proportional to (wR)~. It is common practice to assume steady motion. These results show in what sense this assumption can be justified.

54

6. Conclusions

For oscillatory motion of a sphere the drag is in general a complicated function of the geometric factors and the dimensionless parameter S. We have shown that for small S and a nearby boundary, the drag can be expanded in a power series in S2, allowing a simple formula for the complex viscosity. This result differs from that of the unbounded case in that there is no dependence on S’. Thus for applications to the magnetic sphere rheome- ter involving minute sample sizes, the unbounded drag formula should be replaced by (29). With this expression for the drag it is shown how the complex viscosity is calculated.

Acknowledgements

We gratefully thank Mr Roy Tishler for help in understanding the operation of the magnetic sphere rheometer and Professor M. Smith for discussions on this work. This work was supported by National Institute of Health Research Grants GM 26338 and GM 26132 (TDP) and a Postdoc- toral Fellowship from the Muscular Dystrophy Association (MS).

References

1 R.J. Lutz, M. Litt and L.W. Chakrin, Physical-chemical factors in mucus rheology, in: H.L. Gabelmck and M. Litt (Eds.) Rheology of Biological Systems, Charles C. Thomas, Springfield, 1973.

2 M. Litt, M.A. Khan and D.P. Wolf, Mucus rheology: Relation to structure and function, Biorheology, 13 (1976) 37.

3 M. Litt, M.A. Khan, L.W. Chakrin, J.R. Wardell, Jr. and P. Christian, The viscoelasticity of fractionated canine tracheal mucus, Biorheology, 11 (1974) 111.

4 M. King, The effect of mucus cross-linking on the clearance of mucus, Mod. Probl. Paediat., 19 (1977) 182.

5 K. Hirata, Y. Majima and Y. Sakakura, Viscoelastic measurement of microliter quantities of mucus using a new type of oscillating sphere magnetic rheometer, Biorheology, 22 (1985) 360.

6 Y. Majima, K. Hirata and Y. Sakakura, Dynamic viscoelasticity of middle ear effusions and its role on mucociliary transport, Biorheology, 22 (1985) 363.

7 K. Nakamura and I. Kaneda, The study of mechanical properties of the white of a hen’s egg by the magnetic microrheometer, Biorheology, 22 (1985) 364.

8 S.L. James and C. Marriott, A modified oscillating sphere magnetic microrheometer for use with biological secretions, J. Phys. E: Sci. Instrum., 15 (1982) 179.

9 S.C. Hunter, The motion of a rigid sphere embedded in an adhering elastic or viscoelastic medium, J. Edinburgh Math. Sot., 16 (II : 1) (1968) 55.

10 B. Gauthier-Manuel, The sphere rheometer: II. Dynamic measurements, J. Phys. E: Sci. Instrum., 17 (1984) 1183.

11 K. Walters, Rheometry, John Wiley and Sons, New York, 1975.

55

12 A. Ambari, B. Gauthier-Manuel and E. Guyon, Effect of a plane wall on a sphere moving parallel to it, J. Phys. Lett., 44 (1983) L143.

13 K.R. Frater, Secondary flow in an elastico-viscous fluid caused by rotational oscillations of a sphere. Part 1, J. Fluid Mech., 20 (3) (1964) 369.

14 A.N. Whitehead, Second approximations to viscous fluid motions, Q. J. Pure App. Math., 23 (1889) 143.

15 I. Proudman and J.R.A. Pearson, Expansions at small Reynolds number for the flow past a sphere and a circular cylinder, J. Fluid Mech., 2 (3) (1957) 237.

16 L.G. Leal, Particle motions in a viscous fluid, Ann. Rev. Fluid Mech., 12 (1980) 435. 17 J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Noordhoff, Leyden,

1973. 18 G.G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums,

Trans. Cambridge Philos. Sot., 9 (II) (2851) 8.