Introduction

Fluid inertia e�ects in linear viscoelastic ¯ow have beenstudied by a number of workers for both controlledstrain and controlled stress instruments. Controlledstrain theory was developed by Walters (1960, 1961)for the concentric cylinder geometry; by Maude andWalters (1964), Nally (1965), and Walters and Kemp(1968a) for the cone- and-plate geometry; and byWalters and Kemp (1968b) for the parallel plategeometry. In the case of the controlled stress instrument,a theory which included ¯uid inertia e�ects was ®rstdeveloped by Holder (1982) for the cone- and-plate andparallel plate geometries. Jones et al. (1984) showed thatthe controlled stress rheometer could be used to producerealistic complex viscosity data. The theoretical workwas extended by Jones et al. (1987) to develop a ®rst-order ¯uid inertia correction for the concentric cylindergeometry. In this paper, experiments were also carried

out to validate the ®rst order theory for small ¯uidinertia e�ects. Golden (1990) developed second-order¯uid inertia corrections for the cone- and-plate, parallelplate and concentric cylinder geometries. In this work,an experimental programme was carried out to test therange of validity of these corrections. Golden (1990) alsoconsidered the numerical solution of the exact governingequations for these measurement systems and was ableto correct the complex viscosity data for large ¯uidinertia e�ects.

For the controlled stress and controlled straininstruments, the governing equations which describethe e�ect of ¯uid inertia on the complex viscosityfunction do not possess a closed form solution. Fur-thermore, these equations possess multiple solutionswhich give rise to realistic complex viscosity predictions.In such a situation, any numerical iterative schemethat is used to solve the governing equations will bedependent on the initial estimate of the solution. These

Rheol Acta 38: 365±374 (1999)Ó Springer-Verlag 1999 ORIGINAL CONTRIBUTION

Kevin GoldenMansel Davies

Problems of non-uniqueness wheninterpreting the effect of ¯uid inertiaon the complex viscosity function

Received: 27 July 1998Accepted: 9 April 1999

K. Golden (&)Faculty of Computer studiesand MathematicsUniversity of the West of EnglandFrenchay Campus, Coldharbour LaneBristol, BS16 1QY, UK

M. DaviesSchool of Mathematics and StatisticsUniversity of Plymouth, Drake CircusPlymouth, Devon, PL4 8AA, UKe-mail: ken.jukes@uwe.ac.uk

Abstract A theoretical investigationis carried out into the interpretationof the e�ect of ¯uid inertia on thecomplex viscosity function as mea-sured on a controlled stress rheo-meter. The problem of non-uniquesolutions to the governing equationsis considered for the parallel plategeometry. The locations of thesesolutions are investigated by con-sidering the critical points of thecomplex mapping associated withthe linear viscoelastic equations ofmotion. It is shown that these crit-ical points play an important role indetermining where convergenceproblems are likely to occur when

applying numerical methods of so-lution to the governing equations.Analytical approximations based ona series expansion about a criticalpoint are developed as an alternativeapproach to a numerical solution inthe neighbourhood of a criticalpoint. In order to verify the theo-retical predictions a numerical sim-ulation of the behaviour of a singleelement Maxwell ¯uid on a con-trolled stress rheometer is carriedout for a parallel plate geometry.

Key words Linear viscoelasticity áComplex viscosity á Fluid inertia áControlled stress rheometer

problems were encountered in the work by Golden(1990) who found that, in certain circumstances, theconvergence of the scheme to the required solution wassensitive to the starting estimate for the scheme.Convergence to the wrong solution was identi®ed by adiscontinuity in the complex viscosity data when it wasplotted against the frequency of oscillation.

The theoretical investigation carried out in this paperis concerned with the e�ect of ¯uid inertia on theprediction of the complex viscosity function as measuredon a conventional controlled stress rheometer. In ¯owsituations where ¯uid inertia e�ects are considered to besmall, the complex viscosity may be determined from theexperimental data using perturbation analysis (Holder1982; Jones et al. 1987; Golden 1990). However, when¯uid inertia e�ects are large, numerical techniques havenormally been used to predict the complex viscositybehaviour (Golden 1990). In this paper we show that,for a particular ¯uid, there are certain ranges of thefrequency of oscillation for which two realistic solutionslie close to each other in the solution domain. This willoccur in the neighbourhood of the critical points of thecomplex transformation which arises from the governingequations in the theory. It is in these regions that thenumerical problems discussed earlier are likely to occur.In this work we propose an alternative to the iterativemethod for solving the governing equations in theneighbourhood of a critical point. The governingequations may be expanded about a critical point toproduce a series solution for the complex viscosity in thisregion. The series expansions produced are able todescribe large ¯uid inertia e�ects and represent anextension of the work carried out by Golden (1990).

In practice, large ¯uid inertia e�ects are normallyassociated with the testing of mobile ¯uids at highfrequencies of oscillation in large gap concentric geom-etries. However, in order to obtain a better understand-ing of the underlying problem, this theoretical study willconcentrate on the parallel plate geometry which leadsto a less complicated form of the governing transfor-mation. The ideas developed in this paper can beextended to the concentric cylinder geometry which willbe the subject of future work. In order to validate thetheory for the parallel plate geometry, we simulate thebehaviour of a single element Maxwell ¯uid on acontrolled stress rheometer.

Theory

All physical quantities are referred to cylindrical polarco-ordinates �r; h; z�. The parallel plate geometry isde®ned by two coaxial horizontal ¯at plates of radius a,separated by a gap h, as shown in Fig. 1.

An oscillatory torque is applied to the upper platencausing it to make small-amplitude angular oscillations

of amplitude v0 and frequency x about the z-axis. Thelower platen remains stationary. The ¯uid, located in thegap between the upper and lower platens, is assumed tobe incompressible and for a generalised linear viscoelas-tic model the stress, rij, in the ¯uid is given by

rij � g� _cij ÿ pgij ; �1�where g� is the complex viscosity function, _cij is the rate-of-strain tensor, p is the hydrostatic pressure and gij isthe metric tensor. The complex viscosity function, interms of its real and imaginary parts, is given by

g� � g0 ÿ iG0

x; �2�

where g0 and G0 are the dynamic viscosity and dynamicrigidity functions of frequency only.

Following the analysis of Jones et al. (1987), weassume a velocity pro®le of the form

mh � rf �z�eixt ; �3�which identically satis®es the equation of continuity forincompressible ¯uids. Throughout this paper, physicalcomponents are implied by the `real' part of anexpression.

This velocity pro®le gives rise to one non-zerocomponent of the stress tensor, namely

rhz � rdfdz

g�eixt : �4�We introduce the complex parameter a2, de®ned byWalters (1975) to be

a2 � ÿixqg�

; �5�

where q is the ¯uid density. The parameter a2 governsthe magnitude of ¯uid inertia e�ects in the ¯ow.Substituting Eqs. (3) and (4) into the relevant ¯uidequation of motion and satisfying the no-slip boundaryconditions, we obtain the boundary value problem

d2fdz2� a2f � 0; f �0� � 0; f �h� � ixv0; 0 � z � h :

�6�

Fig. 1 Parallel plate geometry

366

The solution of Eq. (6) can now be expressed in the form

f �z� � ixv0 sin�az�sin�ah� : �7�

The equation of motion of the upper platen is given by

C ÿ Cf � I �v ; �8�where C is the applied torque, Cf is the torque acting onthe upper platen due to the motion of the ¯uid, I is themechanical inertia of the upper rotating assembly and �vis the angular acceleration of the upper platen.

The applied sinusoidal torque may be expressed as

C � C0ei�xt�/� ; �9�where C0 is the torque amplitude and / is the phase lagof the displacement of the upper platen behind the inputtorque.

The angular displacement v of the upper platen cantherefore be written as

v � v0eixt : �10�On using Eqs. (4) and (7) it can be shown that the stress®eld acting on the upper platen, due to the motion of the¯uid, produces a torque given by

Cf � ixpg�a4v0a cot�ah�2

eixt : �11�

Substituting Eqs. (9)±(11) into Eq. (8), we obtain

ixpg�a4v0a cot�ah�2

� C0ei/ � Ix2v0 : �12�Taking the limit of Eq. (12) as a! 0 yields thecommonly used expression for the complex viscosityfunction, which is valid when ¯uid inertia e�ects arenegligible in controlled stress rheometry, which is givenby

g�0 � g00 ÿ iG00x� 2h

ixpa4v0�C0ei/ � Ix2v0� : �13�

g00 and G00 represent dynamic viscosity and dynamicrigidity functions respectively, when ¯uid inertia e�ectsare ignored in the theory. It should be noted that inthis equation g�0 can be determined explicitly from theexperimental data for the displacement amplitude v0 andthe phase angle /. From Eq. (13) it can be seen that themechanical inertia of the rotating assembly in¯uencesthe dynamic rigidity prediction and must be taken intoaccount in the theory, particularly for mobile ¯uids athigh frequencies of oscillation.

When ¯uid inertia e�ects are included in the theory,Eq. (12) can now be written in terms of g�0 to produce thefollowing equation which describes the e�ect of ¯uidinertia on the complex viscosity function;

ah tan�ah� � �a0h�2 ; �14�

where

�a0h�2 � ÿixqh2

g�0: �15�

It should be noted that ah and a0h are non-dimensionalquantities. Equation (14) must be solved for a which canthen be used in Eq. (6) to produce g�.

Solution behaviour of Eq. (14)

It is convenient for us to introduce the complexparameters Z and W which are de®ned to be

Z � x� iy � ah �16�and

W � u� iv � �a0h�2 : �17�Hence, Eq. (14) may be written as

W � F �Z� � Z tan Z : �18�This equation represents a complex mapping from theZ-plane to the W-plane. It should be noted that W isa known quantity which, from Eqs. (13), (15) and (17),is related to the experimental output data taken froma controlled stress rheometer by

W � x2pa4v02h�C0ei/ � Ix2v0�

: �19�

From Eqs. (6) and (16) it can be seen that the complexviscosity function is related to Z by

g� � ÿ ixqh2

Z2: �20�

Hence, the complex viscosity of the ¯uid can bedetermined from Eq. (20) where Z is the solution toEq. (18).

Since the dynamic viscosity and the dynamic rigiditymust both be positive, then it can be seen from Eq. (20)that the argument of Z must satisfy

ÿ p4� Arg�Z� � 0 : �21�

In general, a complex transformation will not provide aone-to-one mapping for any F �Z� in the neighbourhoodof a point ZP if F 0�Z� is equal to zero in thatneighbourhood, where 0 denotes di�erentiation withrespect to Z (Wunsch 1994).

We note that any value of Z which satis®es

F 0�Z� � 0 �22�is de®ned to be a critical point of the complextransformation. Critical points of the complex mappingde®ned by Eq. (18) occur whenever the equation

2Z � sin 2Z � 0 �23�is satis®ed. Clearly, one critical point occurs at

367

Z � Z0 � 0 : �24�It can be deduced that on taking the real and imaginaryparts of Eq. (23) and using the constraint imposed byEq. (21) that the real part of all other critical pointsde®ned by

Z � Zn � xn � iyn; n � 1; 2; . . . �25�must lie in a region where

�2nÿ 1�p2

� xn � �4nÿ 1�p4

: �26�

In general, the critical points of the mapping describedby Eq. (18) are obtained from a numerical solution ofEq. (23). The critical points Zn corresponding ton � 0; 1; 2 and 3 are presented in Table 1. The corre-sponding values for Wn � Zn tan Zn are also presented inthis table.

To investigate the solution behaviour of Eq. (18), it isconvenient to de®ne the function

G�Z� � Z tan Z ÿ W : �27�It should be noted that G�Z� is equal to zero whenever Zis a root of Eq. (18). It can be seen that when W is equalto Wn both G�Zn� and G0�Zn� are zero, and since it can beshown that G00�Zn� is non-zero, the equation

Wn � Z tanZ �28�has a double root when Z is equal to Zn.

In order to demonstrate how the critical pointsin¯uence the location of the roots of Eq. (18) in theZ-plane, we consider a sequence of contour plots of themagnitude of G�Z� which show the solution tendingtowards the critical point Z � Z1. From Table 1 thecorresponding value of W is W1 � ÿ1:6506ÿ 2:06i.Consider the following sequence of values of Wmoving in the direction of the imaginary axis:(a) W � ÿ1:6506ÿ 1:5i; (b) W � ÿ1:6506ÿ 1:65i;(c) W � ÿ1:6506ÿ 1:8i; and (d) W � W1. On using thesevalues of W in Eq. (27) we produced contour plots ofjG�Z�j as shown in Fig. 2.

The sequence of contour plots presented in Fig. 2illustrates a situation whereby two solutions to Eq. (18)exist in the neighbourhood of the critical point Z1. Themapping described by Eq. (18) is continuous in theneighbourhood of a critical point, and therefore these

solutions approach the critical point as W approachesW1. The existence of two physically realistic solutionssituated close to each other can cause di�culties whenemploying iterative methods of solution, as it is notpossible to guarantee convergence to the correct solu-tion. In such cases, it is important that a suitable initialestimate of the required solution is used. It should benoted that since W is known from Eq. (19), it is alwayspossible to determine when the solution to Eq. (18) liesin the neighbourhood of a critical point.

The multiple solutions shown in Fig. 2 are allphysically realistic descriptions of the behaviour of aviscoelastic ¯uid. The non-uniqueness of the solution toEq. (18) implies that it would not be possible from anexperimental test sampled at a single frequency ofoscillation, to determine which solution represents thecorrect characterisation of the ¯uid. Such discriminationbetween the di�erent solutions is only possible with aknowledge of the ¯uid behaviour over a range offrequencies, where a `wrong' solution would reveal itselfas a discontinuity in the dynamic viscosity and dynamicrigidity curves. If two solutions are very close to eachother in the Z-plane, then it may not be possible todecide which solution is correct, even after examiningthe dynamic viscosity and dynamic rigidity curves.However, this situation will not lead to large errors inthe characterisation of the ¯uid properties. To overcomethe numerical di�culties in predicting the correctsolution, we consider an alternative approach to solvingEq. (18) in the neighbourhood of a critical point.

Perturbation solutions

Consider a series expansion of the functionF �Z� � Z tan Z about a critical point Z � Zn given by

W � F �Z� �X1j�0

�Z ÿ Zn�jj!

F �j��Zn� ; �29�

where F �j��Zn� denotes the jth order derivative of F �Z�with respect to Z evaluated at Z � Zn.

Using the de®nition of a critical point, Eq. (29) maybe expressed as

W ÿ Wn �X1j�2

�Z ÿ Zn�jj!

F �j��Zn� ; �30�

where

Wn � Zn tan Zn : �31�Inverting the series de®ned by Eq. (30) produces a seriesexpansion for Z of the form

Z � Zn �X1j�1

aj�W ÿ Wn�j=2 ; �32�

Table 1 Critical points of the complex mapping W=Z tanZ forn=0, 1, 2, 3 correct to four decimal places of accuracy

n Critical point Zn Wn

0 0 01 2.1062 ± 1.1254i )1.6506 ± 2.0600i2 5.3563 ± 1.5516i )2.0579 ± 5.3347i3 8.5367 ± 1.7755i )2.2785 ± 8.5226i

368

where aj are complex coe�cients which have to bedetermined. De®ning the polar form of W ÿ Wn to berweihw and the polar form of aj to be bje

i/, weapproximate Z by considering a ®nite number of termsof the series in Eq. (32). We denote this approximationby Sm;k which is given by

Sm;k � Zn �Xm

j�1bjr

j=2w ei�jhw=2�/j�kp� ; �33�

where m is the number of terms of the series and k � 0; 1de®nes the two branches of the function �W ÿ Wn�1=2. In

this work we have chosen that the principal branch ofthe function be de®ned by

ÿp < hw � p : �34�It should be noted that it is not possible to obtain ageneral expression for the coe�cients of the seriesde®ned in Eq. (33). Hence, the radius of convergencefor this series as m!1 cannot be determined analyt-ically. However, our principal interest in this work is toestablish regions of theW-plane over which the accuracyof the truncated series is known.

The parameters �bj;/j� for the series de®ned inEq. (33)have been evaluated for the critical points presented inTable 1. The values �bj;/j� for the case when m � 5 arepresented, to four decimal places of accuracy, in Table 2.

It should be noted that the ®rst- and second-order¯uid inertia perturbation expansions, developed by

Fig. 2a±d Contour plots of jG�Z�j with the critical point Z1 denotedby j and the solutions to the equation W � Z tan Z denoted by �:a W � ÿ1:6506ÿ 1:5000i; b W � ÿ1:6506ÿ 1:6500i; c W �ÿ1:6506 ÿ1:8000i; d W � ÿ1:6506ÿ 2:0600i

369

Jones et al. (1987) and Golden (1990), for smallReynolds number ¯ows can be obtained from Eq. (20)using the coe�cients of the series expansion about thepoint Z � Z0 shown in Table 2. In order to investigatethe accuracy of the series de®ned in Eq. (33) forestimating solutions to Eq. (18), we consider thequantity

Em;k � 100jZ ÿ Sm;kjjZj

� �; �35�

which de®nes the relative error in the magnitude of Z asestimated by Eq. (33). To evaluate Em;k in Eq. (35), werequire the two exact values of Z in the neighbourhoodof a critical point corresponding to k � 0 and k � 1 for agiven value of W. An estimation of these exact valueshas been obtained using an iterative technique withsuitable initial estimates of Z. For convenience, weevaluate Em;k at points located on the boundary of acircular region of radius rw in the W-plane centred on apoint W � Wn. Along this boundary, the error de®ned byEq. (35) will vary with the argument hw, as demonstratedin Fig. 3 for a circular region of radius rw � 1:5 centredat W � W1.

In Fig. 3 it can be seen that a maximum error of lessthan 2% occurs when using the series E5;k to estimate thesolutions of Eq. (18) in the neighbourhood of the criticalpoint Z1.

In Fig. 4 we plot the variation in the maximum valueof E5;k around the boundary of a circular region in theW-plane as rw is increased. The regions considered arecentred at the points W0;W1;W2 and W3 in the W-plane.It should be noted that for W0 we only consider thesemicircular boundary de®ned by hw 2 �0;ÿp�, sincefrom Eq. (21) physically realistic solutions only occur inthis region. As expected, the maximum percentage errorincreases with increasing radius rw for each of the curves.For a given value of rw we observed that for regionscentred at Wn; n � 1; 2; 3; the maximum percentageerror decreases with the index n.

For completeness, in Fig. 5 we show the number ofterms of the series Sm;k that are required to estimate Z toan accuracy of 1%.

It should be noted that a further increase in thenumber of terms of the series will not necessarily lead toa 1% accuracy for larger values of rw as this is dependentupon the radius of convergence of the series.

Table 2 Showing the ®rst ®veterms of the series de®ned byEq. (33)

Z0 Z1 Z2 Z3

(b1, /1) (1, 0) (1, 0) (1, 0) (1, 0)(b2, /2) (0, 0) (0.3685, 1.3860) (0.3418, 1.4846) (0.3373, 1.5146)(b3, /3) (0.1667, p) (0.1062, )2.9087) (0.1096, )3.0500) (0.1103, )3.0838)(b2, /2) (0, 0) (0.0325, )1.1665) (0.0298, )1.4252) (0.0296, )1.4814)(b5, /5) (0.0306, 0) (0.0015, )2.1942) (0.0029, 0.1290) (0.0034, 0.09948)

Fig. 3 E5;k versus normalised hw for k � 0; 1 around the circumfer-ence of a circle of radius rw � 1:5 in the W-plane with centreW1 � ÿ1:6506ÿ 2:06i

Fig. 4 Maximum percentage error �E5;k� against rw for circularregions centred at W0;W1;W2 and W3 respectively

Fig. 5 Number of terms of the series S5;k versus radius rw required fora 1% accuracy in estimating Z for circular regions centred atW � W0;W1;W2 and W3

370

In Fig. 6 we plot circular regions in the W-plane,centred on Wn�n � 0; 1; 2; 3�; in which the maximumpercentage error incurred in using S5;k to approximate Zis less than 1%. It should be noted that all physicallyrealistic values of Z lie to the right of the line x � ÿyshown in Fig. 6b. Values of Wn together with thecorresponding radius of the circular region are shownin Table 3.

For a value of W which lies within any of the circularregions in the W-plane, shown in Fig. 6a there are twosolutions to Eq. (18) which appear in the correspondingregion for Z. If either of these solutions represents thetrue complex viscosity of the ¯uid, then S5;k may be usedto obtain an approximate value of Z which has amagnitude accurate to within 1%. Equation (20) maythen be used to obtain the complex viscosity, for whichthe complex viscosity modulus is accurate to approxi-mately 2%.

Simulation

In order to validate the theory developed in this paper,we simulate the oscillatory shear ¯ow behaviour of asingle element Maxwell ¯uid on a controlled stressrheometer using a parallel plate geometry. For a singleelement Maxwell ¯uid, the complex viscosity is given by

g� � g01� ikx

�36�

where g0 is the zero shear viscosity, k is the relaxationtime and x is the frequency of oscillation. In Fig. 7 wepresent the normalised dynamic viscosity and dynamicrigidity curves plotted against the normalised frequencyfor the Maxwell ¯uid.

We shall ®rst consider the complex viscosity predic-tion obtained from Eq. (13) where ¯uid inertia e�ects arenot included in the theory. It is convenient to introducethe non-dimensional parameters

R � qh2

kg0; K � kx ; �37�

where R is a measure of the magnitude of the ¯uidinertia e�ects for the single element Maxwell model. Itcan be shown from Eqs. (6), (14), (15) and (36) that theuncorrected complex viscosity g�0 may be expressed as

g�0g0� f �R;K� �

�����������RK

iÿ K

rcot

���������������������RK�Kÿ i�

ph i: �38�

Fig. 6a, b Regions of the map-ping W � Z tan Z where E5;k isless than 1%. The circularregions of the W-plane are eachcentred on a value Wn forn � 0; 1; 2; 3. The correspond-ing position of the critical pointZn is marked on the Z-plane(�). a W-plane, b Z-plane

Table 3 The radius rw and the centre of each of the circular regionsof the W-plane shown in Fig. 6(a)

n Centre Wn Radius rw

0 0 1.331 )1.65)2.06i 1.302 )2.06)5.33i 1.963 )2.28)8.52i 2.22

Fig. 7 Normalised dynamic viscosity �g0=g0� and normalised dynamicrigidity �G0k=g0� for a single element Maxwell ¯uid

371

In Figs. 8 and 9 we use Eq. (38) to show the e�ects ofneglecting ¯uid inertia on the complex viscosity predic-tion. The e�ects shown in these ®gures have beendocumented in previous work by Jones et al. (1987) andby Golden (1990). As noted in this previous work, thedynamic rigidity prediction is more sensitive to ¯uidinertia e�ects than the dynamic viscosity prediction.This can be explained by the fact that the leading order¯uid inertia correction to the dynamic viscosity is secondorder whereas the leading order ¯uid inertia correctionto the dynamic rigidity is ®rst order. The exactprediction of the complex viscosity function correspondsto a value R being equal to 0.

We now investigate the validity of the series S5;k whencorrecting the complex viscosity prediction for thee�ects of ¯uid inertia. In this paper we shall onlyconsider the case when R � 1. In Fig. 10 we show thepath of the Maxwell ¯uid in the W-plane together withits corresponding path in the Z-plane as the normalisedfrequency is varied from 0 to 3. In this ®gure Z isobtained by substituting g� from Eq. (36) into Eq. (20)and W is obtained by substituting this value of Z into

Eq. (18). In this ®gure we also show two regions fromFig. 6 in which the series S5;k will correct the magnitudeof Z to within a 1% accuracy and hence will estimate themagnitude of g� to approximately a 2% accuracy.

We observe in Fig. 10a that the path of the Maxwell¯uid between the points A and B lies outside of thecircular regions. Therefore, it cannot be guaranteed thatthe error in the magnitude of Z as estimated by S5;k isless than 1%. The point P shown in Fig. 10a is a branchpoint of the series S5;k and corresponds to the value ofthe argument of W ÿ W1 being equal to ÿp.

Using Eq. (20), where Z is estimated by S5;k; weinvestigate the accuracy of S5;k in describing the e�ect of¯uid inertia on the complex viscosity function over thenormalised frequency range from 0 to 3. The graphspresented in Figs. 11 and 12 show normalised dynamicviscosity and dynamic rigidity predictions plottedagainst the normalised frequency kx. The series S5;kused in these ®gures has been expanded about the criticalpoints Z0 and Z1 corresponding to n � 0 and n � 1,respectively, as presented in Table 1. The parameter kde®nes the di�erent branches of the series. The seriesexpansion about Z0 gives rise to the same correction fork � 0 and k � 1. This is not the case for seriesexpansions about other critical points. It should benoted that the ¯uid inertia correction obtained for n � 0is only valid for small ¯uid inertia e�ects since thisregion of the Z-plane corresponds to small values of ah.This ¯uid inertia correction di�ers from the ¯uid inertiacorrections obtained by Golden (1990) in that thisprevious work developed a series expansion for g� asopposed to ah. One series can be obtained from theother by using Eq. (6). We denote the frequencies ofoscillation corresponding to the points A and B by xA

and xB respectively. As expected, at normalised fre-quencies of oscillation below kxA, the ¯uid inertiacorrection obtained using S5;k, when expanded about thepoint Z0, provides an accurate prediction of the dynamicproperties of the ¯uid. At normalised frequencies ofoscillation between kxA and kxB there is disagreementbetween the prediction provided by the series, expandedabout either Z0 or Z1, and the exact complex viscosityfunction. This is to be expected as the path AB lies in aregion where the error in the magnitude of Z cannot beguaranteed to be less than 1%. The results presented inFig. 5 suggest that greater accuracy could be obtainedby using more terms of the series. However, care musttaken when using the series for larger values of rw sinceconvergence of the series cannot be guaranteed. Alter-natively, a numerical method of solution could be usedover this range since the solution lies outside of theneighbourhood of a critical point where numericaldi�culties are less likely occur. It should be noted thatover the range of normalised frequencies between kxA

and kxB the series expansion about Z1, with a branchde®ned by k � 1, provides a more accurate representa-

Fig. 8 E�ect of neglecting ¯uid inertia on the dynamic viscosityfunction

Fig. 9 E�ect of neglecting ¯uid inertia on the dynamic rigidityfunction

372

tion of the complex viscosity function than the seriesexpanded about Z0.

For normalised frequencies lying in the range kxB tokxP the series expanded about Z1, with the branchde®ned by k � 1, now provides an accurate representa-tion of the complex viscosity function. For normalisedfrequencies greater than kxP we see that the k � 0branch of the series provides the appropriate ¯uid inertiacorrection. We note that at x � xP the path of the

Maxwell ¯uid in the W-plane crosses the branch cut ofthe series as shown in Fig. 10a.

Conclusions

In this work we have investigated the di�culties thatarise when interpreting the e�ect of ¯uid inertia on thelinear dynamic properties of a viscoelastic ¯uid. Iterativemethods of solution of the governing equations arein¯uenced by the property that these equations aremultivalued. Hence, it is not possible to guaranteeconvergence to the correct value of the complex viscosityfunction. The work has shown that there exist a numberof regions in the complex Z-plane in which numericaldi�culties occur. These regions correspond to theneighbourhoods surrounding each critical point. Thework has indicated that it is possible to determine, fromthe uncorrected experimental data, where numericaldi�culties associated with non-unique solutions arelikely to occur.

To overcome these numerical di�culties, we havedeveloped analytical approximations based on seriesexpansions about a critical point which can then be usedto correct the complex viscosity function for the e�ect of¯uid inertia, to the required degree of accuracy, in theneighbourhood of a critical point. For solutions whichlie outside of these neighbourhoods, numerical methodsprovide an appropriate method of solution for thegoverning equations.

It is well known that, except for mobile ¯uids testedat high frequencies of oscillation, ¯uid inertia e�ects arenot large for a parallel plate measurement system. Thisimplies that for most situations encountered, the ®rst-order correction developed by Jones et al. (1987) and thesecond-order correction developed by Golden (1990)would be adequate for the purpose of correcting theexperimental data for the e�ects of ¯uid inertia.The theory developed in this paper can be used tocorrect the complex viscosity data for the e�ects of ¯uid

Fig. 10a, b Path of a Maxwell ¯uid in the W-plane (a) and theZ-plane (b) for R � 1:0 over a normalised frequency range of (0±3)

Fig. 11 Dynamic viscosity behaviour obtained from the series S5;k

Fig. 12 Dynamic rigidity behaviour obtained from the series S5;k

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inertia in the case of mobile ¯uids tested at highfrequencies of oscillation.

Fluid inertia e�ects are known to bemore important inthe concentric cylinder geometry than in the parallel plategeometry and similar numerical di�culties to thosediscussed in this paper are likely to occur. The workcarried out in this paper would suggest that a similar

approach could be used to develop series expansionsabout critical points of the governing equations to correctcomplex viscosity data for the e�ects of ¯uid inertia in aconcentric geometry. This will be considered at the nextstage of this investigation where the theoretical expres-sions developed for the concentric cylinder geometry willbe used to correct experimental data.

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