analytical solution for pulsatile axial flow velocity waveforms in curved elastic tubes

864 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 8, AUGUST 2001

Analytical Solution for Pulsatile Axial Flow VelocityWaveforms in Curved Elastic Tubes

Lance Jonathan Myers* and Wayne Logan Capper

Abstract—An analytical solution for pulsatile axial flow velocitywaveforms in curved elastic tubes is presented. The result is ob-tained by exact solution of linearized Navier–Stokes and tube mo-tion equations in a torroidal coordinate system. Fourier analysis isused to divide the flow into constant and oscillatory componentswhich are separately considered. The solution is used to investi-gate the effects of curvature on volumetric axial velocity flow wave-forms, as would be measured by Doppler ultrasound techniques. Intypical human arteries, the greatest effects of curvature on the vol-umetric axial flow are exerted on the constant component and atlow values of the frequency parameter for the oscillatory compo-nents. Here, the magnitude and phase angle of oscillatory flow inthe curved tube, relative to that in the straight tube, differ by max-imum values of 1.2% and 0.15 rad, respectively. However, constantflow may vary by as much as 60% at high Dean numbers. The solu-tion is presented in a form similar to Womersley’s solution for thestraight elastic tube and may, thus, be incorporated into a trans-mission-line analog model. These models are frequently used to in-vestigate axial flow velocity variations in mamillian circulatory sys-tems and this work offers a tool which may extend these models toincorporate the effects of curvature.

Index Terms—Analytical solution, blood flow velocity waveform,curved elastic arteries, Doppler ultrasound, modeling, transmis-sion line.

I. INTRODUCTION

T HE MODELING of blood flow in straight arteries hasbeen the subject of continued investigation and improve-

ment for over half a century. Beginning with the fundamentallinearized analysis of Womersley [1], [2], and Morgan and Kiely[3] for two-dimensional (2-D) oscillatory flow of a viscous in-compressible fluid in straight elastic tubes, many subsequentresearchers have attempted to improve on their results in somemanner. These analyses have largely been performed for straightarteries, yet a significant proportion of arteries in mammalianbodies exhibit varying degrees of curvature. Common exam-ples of such arteries are the aortic arch, coronary, femoral, pul-monary and umbilical arteries.

Blood flowing through a region of curvature in an artery willbe affected by centrifugal forces which tend to set up secondaryflows in that artery. The nature of these secondary flows havebeen examined in considerable detail in many papers in the past.However, very little attention has been given to address whateffect the curvature has on the axial flow velocity. Axial flow

Manuscript received July 25, 2000; revised April 27, 2001. This work wassupported by the Medical Research Council of South Africa (MRC) and theNational Research Fund (NRF).Asterisk indicates corresponding author.

*L. J. Myers is with the University of Cape Town Medical School Observa-tory, Cape Town, South Africa (e-mail: [email protected]).

W. L. Capper is with the University of Cape Town Medical School Observa-tory, Cape Town, South Africa.

Publisher Item Identifier S 0018-9294(01)06155-9.

velocity waveforms are frequently measured using Doppler ul-trasound where either the mean or maximum values of the ve-locity profiles (or the volumetric average) are used to constructtime-varying waveforms [4], [5]. These waveforms are used forsubsequent clinical diagnosis of various pathological conditions[5]–[7]. Thus, it is of significant value to be able to quantify theexpected degree of variation of the Doppler waveform with cor-responding variations in the curvature. The focus of many pa-pers dealing with the effects of curvature has been motivated bythe effect it has on wall shear stresses in relation to atheroscle-rotic disease and the examination of time-varying volumetricaxial flow velocity waveforms have largely been neglected.

The earliest studies were conducted by Dean [8], [9], whoanalyzed steady laminar flow in a rigid curved pipe by the per-turbation method. This was limited to small values of the Deannumber, , where is the radius of the pipecross section, the radius of curvature of the pipe axis, and Reis the Reynolds number. Many further theoretical, numerical,and experimental studies on this work have been conducted andhave extended the range of solution to small, intermediate, andlarge Dean numbers [10]–[19].

The problem of unsteady flow is more complicated and thesimplification of pure oscillatory flow has been considered inmany of these studies. This work was initiated by Lyne [20]who used boundary-layer approximations to solve linearizedNavier–Stokes equations for large values of the frequency pa-rameter and both very large and very smallvalues of the parameter . Here, is proportional tothe pressure gradient amplitude,is the radial frequency, and

the kinematic viscosity. This work showed the existence of afour-vortex secondary flow system which could be divided intotwo regions: a core region in which the flow is essentially in-viscid and a boundary region in which viscous effects dominate.The net effect was to shift the maximum axial velocity towardthe inner wall of the curvature. Lyne’s studies were again for arigid-walled tube.

Zalosh and Nelson [21] carried out an analysis using finiteHankel transforms and a perturbation solution to linearisethe Navier–Stokes equations and confirmed Lyne’s results.Mullin and Greated [22] carried out an essentially parallelanalysis using the same solution procedure. Difficulties in thesesolutions were encountered at high frequencies, due to the lackof convergence of the Inverse Hankel transform and Zapryanovand Matakiev [23] obtained a closed-form analytic solution forall values of . Bertelsen and Thorsen [24] experimentally ver-ified Lyne’s findings. Numerical solutions to purely oscillatoryflow have been derived by Sudoet al. [25].

Oscillatory flow with a mean was investigated by Smith [26]who again used boundary-layer approximations to derive sev-

0018–9294/01$10.00 © 2001 IEEE

MYERS AND CAPPER: PULSATILE AXIAL FLOW VELOCITY WAVEFORMS IN CURVED ELASTIC TUBES 865

eral different flow attributes ranging between the limiting casesof steady flow in curved tubes to purely oscillatory flow instraight tubes. Simonet al. [27] investigated this using similarperturbation techniques and finite Hankel transforms. Numer-ical investigations have been conducted by Rabadiet al.[28] andHamakiotes and Berger [29], [30]. Chang and Tarbell [31] pro-vided a complete numerical solution of pulsatile flow but onceagain their result was for a rigid tube.

The only nonexperimental analysis of pulsatile flow in acurved, elastic-walled tube was done by Chandranet al. [32],[33], where a thin-walled model was applied to linearizedNavier–Stokes equations and a numerical solution used to solvethe ensuing equations.

None of the cited studies have explicitly demonstrated the ef-fect of the curvature on the axial flow velocity waveform in acurved elastic tube. The purpose of this study is to provide asimple, analytical solution for this problem. This solution maybe applied to investigate relationships between variations in cur-vature with corresponding variations in axial flow velocity asmeasured by Doppler ultrasound. The form of this solution issimilar to that of Womersley [1], [2], and the many subsequentresearchers who investigated blood flow in arteries using hismethod of solution as the basis of their approach [34]–[37].Many blood flow models have employed Womersley’s solutionin order to investigate various aspects of circulatory systems[38]–[41]. A simple revision of these models to include the ef-fects of curvature is now possible. Thus, this paper will focus onan analytical solution for the effects of curvature on axial flowvelocity waveforms, as would be measured by Doppler ultra-sound. The effects of curvature on secondary flows and velocityprofiles have been well documented in many of the above pa-pers and will not be discussed in any further detail, although itmust be noted that this solution may be used to investigate theaxial velocity profiles.

II. THEORETICAL ANALYSIS

A. Background

For a given radius of curvature,, the torroidal coordinatesystem , shown in Fig. 1, is used to define the gov-erning differential equations for fully developed, time-depen-dent flow of a Newtonian fluid in a curved elastic tube. Here,

is the radius of the tube. The arterial wall is assumed to bemade of a homogenous isotropic material with a constant mod-ulus of elasticity. The continuity and Navier–Stokes equationsas well as the equations for the motion of the fluid wall werepresented by Chandran [32] for this case. These equations aregiven in Appendix I. Chandran simplified these equations byperforming an order of magnitude study on the fluid equations,where terms of and were neglected when com-pared to the leading term. Here,is the wavelength, the wavevelocity of the pulse and is the axial velocity of the fluid. Thisled to a set of quasi-linearized equations for the flow which arealso presented in Appendix I.

B. Solution

We proceeded from here by considering only those tubeswhere and then by nondimensionalizing the system of

Fig. 1. Torroidal coordinate system.

equations according to the following:

(1)

where is the kinematic viscosity.This results in the following (now nondimensional) set of re-

lations for the fluid motion.

Continuity equation:

(2)

Momentum equations:

(3)

(4)

(5)

where it was assumed that as ,and .


A perturbation solution was used to further linearize the equa-tions. As was assumed to be small, the dependent variables

and for the flow and and for the tube motion, areexpanded in a power series in

(6)

A similar procedure was performed for the equations for thetube motion.

These expansions were substituted into the flow and tubeequations and terms of and were equated.

The ensuing equations for and , as well asand are the same as the linearized 2-D straight tube equationsand the solution to these is presented by Womersley [2]. Theseequations are not displayed in this paper and again the readermay be referred to Womersley’s paper for a full description. Asthe equations are linear we may write the solution for aconstant or steady component and for oscillatory or unsteadycomponents using Fourier analysis. It is important to note thatthe Fourier decomposition analyzes the flow on a harmonic byharmonic basis in the frequency domain. Time domain wave-forms may be reconstructed by employing an inverse Fouriertransform. Thus, when we speak of the constant or steady flowcomponent, we are referring to the zeroth frequency componentof the Fourier decomposition. Oscillatory or unsteady flow, thus,refers to any of the frequency-dependent harmonics or phasorsof flow and not the unsteady time domain flow. We assume thatthe form of the pressure pulse is a travelling wave with velocity

and for each harmonic,, is , where .The steady component solution is simply Poiseuille’s solutionand is

(7)

where is the viscosity.Writing the solution for the volumetric flow we have

(8)

The unsteady solution for the axial velocity is

(9)and for the unsteady volumetric flow we have

(10)

where

;;

,density;Poisson’s ratio;Young’s modulus;solution to the frequency equation

(11)

where;

wall thickness;.

The effect of the surrounding tissues may be incorporated intothis solution by the inclusion of an elastic constraint in the lon-gitudinal equation of motion of the wall. Womersley found thatthe only adjustment in this case to his solution was to redefine

as follows:

(12)

where is the natural frequency of the constraint.Once we have the solution to the first-order set of equations,

we may now continue to solve the second-order set of equations.The perturbation solution indicates that the effect of curvatureon the axial velocity is, thus, an additive component and thegoverning equation is obtained by equating terms of

(13)

From (13), it may be seen that the additive axial flow contri-bution is only dependent on the primary axial flow andnot on the radial and tangential components. The equations forthese are

(14)

and

(15)

Equations (14) and (15) indicate that for small radius to radiusof curvature ratios, first-order effects of curvature on the axial


flow velocity field (13) are only dependent upon the primaryaxial flow. However, the first-order effects of curvature on theradial and tangential components are nonlinear and dependentupon each other. Second-order effects on the axial flow velocityare nonlinear and dependent upon the behavior of the secondaryvelocity components. Furthermore, solutions that fall out of therange of validity given by the perturbation limitations will benonlinear and the axial flow velocity will again be dependenton the secondary velocity components as shown by Lyne [20]and Smith [26]. The solution in this paper will be limited tofirst-order effects of axial flow for small .

As the governing equation for secondary axial flow is linear(13), once again we may divide the solution up into constant andoscillatory components. The solution for the constant or steadyflow has been solved by many researchers and we chose to usethe form of solution presented by Manlapaz and Churchill [19]who investigated existing solutions for fully developed laminarflow in a coil of finite pitch and finite radius and calculated theirown correlating equation as the best overall representation forthe experimental data. This is

(16)

where 2 for 20, 1 for 40, and 0 for40. is the flow in a curved tube and is the flow in

a straight tube.To solve the unsteady or oscillatory flow components we

begin with the nondimensionalised solution

(17)

As this is independent of we may rewrite the equation foras

(18)

A solution which satisfies this is assumed to be of the form

(19)

Substituting this into (18) and solving by the method of vari-ation of parameters, yields

(20)

The solution must be finite at 0, thus, 0 and theboundary condition 0 is used as there is assumed tobe negligible longitudinal wall movement. Insertion of these

boundary conditions and simplification results in a final solu-tion

(21)

To obtain the volumetric flow we use the integral

(22)

and, thus, the solution for volumetric flow is

(23)

where and are the Struve functions shown inAppendix II.

We wish to obtain an expression similar to that of the steadysolution in terms of the ratio . To obtain this similaritywe incorporate into the expression for . Thus, taking theratio gives

(24)

III. RESULTS AND DISCUSSION

The formulae for the ratio of flow in a curved tube to that ina straight tube for constant and oscillatory Fourier componentsdiffer in which variables they are dependent upon. Equation (16)for the constant flow component indicates a dependence uponthe driving pressure gradient and is independent of heart rate.Alternatively, examination of (24) for the oscillatory flow com-ponents, indicates that it is independent of the driving pressuregradient and dependent upon heart rate.

Thus, without using actual measured pressure waveforms wemay compare the results. The following set of geometric andmaterial properties were chosen to test the effects of curvatureon the axial-flow velocity and were considered representative of


Fig. 2. Variation of the axial, constant flow component in a curved arteryrelative to a straight artery,Q =Q , with �, for Reynolds number,(Re). Theconstant component of axial flow in a curved artery(Q ) decreases relative toflow in a straight artery(Q ) for increasing Reynolds number and increasingcurvature.

common curved arteries in the human circulatory system. Thesewere:

radius of tube: 0.1, 0.3, 0.6 cm;wall thickness of tube: 0.1;fluid viscosity: 0.05 poise;density of fluid: 1.05 10 kg/mm ;heart rate: 70 bpm;ratio of radius to radius of curvature: 0.1, 0.0.05, 0.01;young’s modulus for the elastic tube: 1 10dynes/cm;Poisson’s ratio: 0.5.

The constant flow solution was evaluated for a range ofbetween 0.0 and 0.2 and for Reynolds numbers of 100, 200, 500,and 1000.

Examining the results for the constant flow solution, Fig. 2depicts a plot of the variation of versus for Re 100,200, 500, and 1000.

As may be seen from this figure, the effect of curvature onthe steady component of axial-flow, is to decrease the flow withincreasing and increasing Reynolds number and, therefore, in-creasing Dean number. Dean numbers vary considerably in thenormal human circulation, ranging from about 5 in the femoralarteries [42] to as high as 500 in the aortic arch [31], althoughoften rather low values of the Dean number are encountered.Furthermore, large values of the Dean number are often associ-ated with large pressure gradients and large radii and these in-troduce additional nonlinearities [43] which would violate theassumptions of this solution, rendering it inapplicable.

The ratios of oscillatory flow components in a curved tubeto those in a straight tube were plotted versus the frequencyparameter, , in the form of magnitude and phase plots. Figs. 3and 4, respectively, show the magnitude and phase plots for arange of curvatures.

From Figs. 3 and 4, it is clear that the effects of low valuesof and high values of curvature are the most pronounced. At

Fig. 3. Variation of axial, oscillatory flow component magnitude ratio (curved: straight tube flow),jQ =Q j, with frequency parameter� for � = 0.1, 0.05,and 0.01. The oscillatory component of axial flow magnitude in a curved artery(Q ) decreases relative to flow in a straight artery(Q ) for increasing curvatureand approachesQ at high values of the frequency parameter,�.

Fig. 4. Variation of axial, oscillatory flow component phase angle difference(curved - straight tube flow),arg (Q �Q ), with frequency parameter� for� =0.1, 0.05, and 0.01. The difference between the oscillatory component ofaxial flow phase angle in a curved artery(Q ) and flow in a straight artery(Q )diminishes for increasing frequency parameter and decreasing curvature.

high values of , the axial flow in the curved tube tends to thesame value in both magnitude and phase of axial flow in thestraight tube. The actual value of the difference in the magni-tudes is remarkably small, with a maximum variation of an in-crease of about 1.2% asapproaches zero. However, the phaseshows a more pronounced effect with a maximum variation ofa difference of 0.15 rad at small. For most large arteries inthe adult human circulation is approximately 2–3 and at thesevalues, curvature has less than a 1% effect on the volumetricaxial flow velocity in an artery. However, in the smaller arteriesor in arteries in the foetal circulatory system, it is possible toobtain values of that are less than one. Reduced heart rates orpathology which effects arterial size reduceand it is possible


Fig. 5. Variation of axial, oscillatory flow component magnitude ratio (curved: straight tube flow),jQ =Q j, with frequency! for r = 0.1,� = 0.1 andnatural frequency of oscillation of the tissue,m = 0, 10, 30, and 60 rad/s. Theeffect of the tethering is to shift the peak value of the curve ofjQ =Q j towardshigher frequencies asm increases.

Fig. 6. Variation of axial, oscillatory flow component phase angle difference(curved - straight tube flow),arg (Q �Q ), with frequency! for r = 0.1,� = 0.1, and natural frequency of oscillation of the tissue,m = 0, 10, 30, and60 rad/s. The effect of the tethering is to shift the peak value of the curve ofarg (Q �Q ) towards higher frequencies asm increases.

to obtain very small values of in other mammals of small sizeand low heart rates. Under these circumstances, the effects ofcurvature become relevant and should not be neglected.

It is interesting to note that the propagation properties of theflow waveforms are not affected by the curvature as the valueof the wavespeed is calculated from the frequency equationfrom the first-order solution and is the same value used in thesecond-order solution. We investigated the effects of tetheringon curvature and this is illustrated in Figs. 5 and 6 for increasingmass constants.

From Figs. 5 and 6, we may see that the net effect of tetheringis to reduce the phase difference caused by curvature and to shiftthe peaks of the phase and magnitude maximums forward in fre-

Fig. 7. Normalised axial velocity distribution across tube center- line fora =0.6 cm and!t = 0. The maximum axial velocity has shifted towards the innerwall of curvature.

quency, with the peak occurring at approximately the frequencyof the natural oscillation.

The normalized profile of the axial-velocity for a single har-monic, across the center-line of the tube is illustrated in Fig. 7,for 0.6 and 0. From this curve we may see that themaximum axial velocity is shifted away from the center of thetube (where it occurs in straight tube flow) toward the inner wallof curvature. Chandran [32], [33], also observed this result forhis numerical solution on thin-walled curved elastic tubes. Lyne[20] first showed that in rigid-walled tubes, for oscillatory flow,the maximum velocity is shifted toward the inner wall of curva-ture. This is in contrast to the velocity profiles of steady flow,where the maximum velocity is shifted toward the outer wall ofcurvature.

The full solution to any wave propagation problem consistsof forward and reflected travelling waves. Often investigatorsintroduce full transmission line analogs as a tool to further ex-plore these effects [38]–[41], [44]. Our solution may easily beincorporated into such an analog with a simple adjustment to theflow. This will offer a more accurate picture of the net effect ofcurvature on the axial flow velocity waveform.

IV. CONCLUSION

A linearized set of equations was solved to calculate the effectthat a curved artery would have on the volumetric axial velocityflow waveform, as well as on the axial-velocity profile. The re-sultant equation appears in a simple and convenient form foreasy evaluation, (24). The solution is accurate to first-order andvalid for values of . The solution may be used to furtherunderstand and predict effects that arterial curvature will haveon blood flow velocities as measured by Doppler ultrasound.These measurements are frequently used in clinical diagnosisand the influence of curvature should be considered when inter-preting such measurements.

When considering a rigid-walled tube, Kang and Tarbell [45]experimentally investigated the impedance (pressure drop/flowrate) of curved arteries, considering only sinusoidal flowswith a nonzero mean and found that the impedance of themean component was always higher than that of a steadyflow at the mean flow rate. Chang and Tarbell [31] using


a complete numerical solution, found that in contrast themean impedance of pulsatile flow is lower than that of steadyflow at the mean Dean number. This may be explained byexamining the geometric properties and pressure gradients ofthe arteries under consideration. Our analysis would indicatethat the experimental results were indeed the correct ones.However, conditions which exist in the aortic-arch are thatof small radii of curvature , large radius and largepressure gradients. Thus, not only is our perturbation schemeinvalid for these conditions, but one could expect furthernonlinearities to arise when the centripetal acceleration termsin the Navier–Stokes equations are not neglected [43].

Kang and Tarbell [45] also found that the impedance to os-cillatory flow Fourier components in a rigid, curved tube wasalmost identical to that in the straight tube for all but low valuesof . The impedance to the constant flow component differedaccording to the amount of curvature present. This is in agree-ment with the results we presented for the curved elastic tube.Typical values of encountered in commonly investigated ar-teries in the adult human circulation are greater than two andthe effects of curvature on the oscillatory flow components arenegligibly small at these values. Thus, for many applications ofthis theory, one would need only account for curvature on theconstant flow component. This is often highly significant as themean value of the volumetric axial flow (the time domain equiv-alent of the constant- or zero-frequency Fourier component) isa determining factor in many indices which quantify Dopplermeasurements, such as the pulsatility index [46]. The pulsatilityindex is defined as the maximum minus the minimum dividedby the mean value.

Axial velocity profiles for curved tube flow in elastic, thin-walled arteries show that the maximum axial-velocity is shiftedtoward the inner wall of curvature, which is in agreement withpreviously reported results by Chandran [32]. Chandran inves-tigated these profiles in detail and it was not the purpose of thisstudy to reproduce his findings. Chandran used a numerical fi-nite difference scheme to determine these results and an advan-tage of our solution is that it offers a simplified means of calcu-lating the same results.

Many arteries do not have thin elastic walls and,thus, one would prefer to use a thick-walled model to investigatethem. However, the main effect of changing the wall thicknessis to change the phase velocity and damping factors. We haveshown that these are determined by the first-order set of equa-tions and for which curvature does not have any effect. Thus,one could expect to use the adjustment for curvature presentedhere for both thin- and thick-walled arteries. The same wouldapply for viscoelastic tubes by introducing a complex Young’smodulus.

The full effect of curvature may only be assessed by consid-ering both forward and reflected travelling waves. This has notbeen examined but due to the linearity of the solution it is asimple extension of the tools presented in this paper.

APPENDIX I

The governing equations for the fluid and tube motion are[32] as follows:


(25)


(25)

Momentum equations:

(26)

(27)


(28)In the above, is the axial velocity, is the radial

velocity, and is the tangential velocity. The axial pressuregradient is and is assumed independent of and

in fully developed flow. is the viscosity and the density.The equations of motion for the tube wall are

(29)

(30)

(31)

In these equations, is the undisturbed radius of the tube,is the tube thickness, and are the and tube dis-placement components, respectively.is the modulus of elas-ticity, is Poisson’s ratio, is the spring constant and is thedensity of the tube material.

Chandran performed an order of magnitude study to eliminatethe nonlinear terms. The quasi-linearized and simplified equa-tions for the fluid are as follows:


(32)

Momentum equations:

(33)


(34)

(35)

For the tube equations, Morgan and Kiely showed in an orderof magnitude study that the inertial stress terms may be ne-glected for small values of and . Where is the wave-length.

APPENDIX II

Struve’s function may be determined according to

(36)

For small and these usually converge quite rapidly.is the gamma function and is

(37)

This may be evaluated easily for the Struve function, usingthe formula

(38)

REFERENCES

[1] J. R. Womersley, “Oscillatory motion of a viscous liquid in a thin-walledelastic tube—I: The linear approximation for long waves,”Phil. Mag.,vol. 46, pp. 199–219, 1955.

[2] , “Oscillatory flow in arteries: The constrained tube as a modelof arterial flow and pulse transmission,”Phys. Med. Biol., vol. 2, pp.178–187, 1957.

[3] G. W. Morgan and J. P. Kiely, “Wave propagation in a viscous liquidcontained in a flexible tube,”J. Acoust. Soc. Amer., vol. 26, pp. 323–328,1954.

[4] N. T. Ursem, H. J. Brinkman, P. C. Struijk, W. C. Hop, M. H. Kempski,B. B. Keller, and J. W. Wladimiroff, “Umbilical artery waveform anal-ysis based on maximum, mean and mode velocity in early human preg-nancy,”Ultrasound Med. Biol., vol. 24, pp. 1–7, 1998.

[5] D. Griffin, T. Cohen-Overbeek, and S. Campbell, “Fetal and utero-pla-cental blood flow,”Clin. Obstet. Gynaecol., vol. 10, pp. 565–602, 1983.

[6] D. S. Macpherson, D. H. Evans, and P. R. F. Bell, “Common femoralartery Doppler wave-forms: A comparison of three methods of objectiveanalysis with direct pressure measurements,”Br. J. Surg., vol. 71, pp.46–49, 1984.

[7] T. R. P. Martin, D. C. Barber, S. B. Sherriff, and D. R. Prichard, “Objec-tive feature extraction applied to the diagnosis of carotid artery diseaseusing a Doppler ultrasound technique,”Clin. Phys. Physiol. Meas., vol.1, pp. 71–81, 1980.

[8] W. R. Dean, “Note on the motion of fluid in a curved pipe,”Phil. Mag.,vol. 4, pp. 208–223, 1927.

[9] , “The stream line motion of fluid in curved pipes,”Phil. Mag., vol.5, pp. 673–693, 1928.

[10] H. C. Topakoglu, “Steady laminar flows of an incompressible viscousfluid in curved pipes,”J. Math. Mech., vol. 16, pp. 1321–1328, 1967.

[11] M. Sankaraiah and Y. V. M. Rao, “Analysis of steady laminar flow ofan incompressible Newtonian fluid through curved pipes of small cur-vature,”Trans. ASME I: J. Fluids Eng.., vol. 95, pp. 75–80, 1973.

[12] C. H. Li, “A note in comment on ‘Analysis of steady laminar flow of anincompressible fluid through curved pipes of small curvature’,”Trans.ASME I: J. Fluids Eng.., vol. 98, pp. 323–325, 1976.

[13] M. Van Dyke, “Extended stokes series: Laminar flow through a looselycoiled pipe,”J. Fluid. Mech., vol. 86, pp. 129–145, 1978.

[14] S. M. Barua, “On secondary flow in stationary curved pipes,”Q. J. Mech.Appl. Maths, vol. 15, pp. 61–77, 1963.

[15] H. Ito, “Laminar flow in curved pipes,”Z. Angew. Math. Mech., vol. 11,pp. 653–663, 1969.

[16] L. C. Truesdell and R. J. Adler, “Numerical treatment of fully devel-oped laminar flow in helically coiled tubes,”AIChE J., vol. 16, pp.1010–1015, 1970.

[17] L. R. Austin and J. D. Seader, “Fully developed viscous flow in coiledcircular pipes,”AIChE J., vol. 19, pp. 85–94, 1973.

[18] J. M. Tarbell and M. R. Samuels, “Momentum and heat transfer in helicalcoils,” Chem. Eng. J., vol. 5, pp. 117–127, 1973.

[19] R. L. Manlapaz and S. W. Churchill, “Fully developed laminar flow ina helically coiled tube of finite pitch,”Chem. Eng. Commun., vol. 7, pp.57–78, 1980.

[20] W. H. Lyne, “Unsteady viscous flow in a curved pipe,”J. Fluid. Mech.,vol. 45, pp. 13–31, 1970.

[21] R. G. Zalosh and W. G. Nelson, “Pulsating flow in a curved tube,”J.Fluid. Mech., vol. 59, pp. 693–705, 1973.

[22] T. Mullin and C. A. Greated, “Oscillatory flow in curved pipes. Part 2,”J. Fluid. Mech., vol. 98, pp. 397–416, 1980.

[23] Z. Zapryanov and V. Matakiev, “An exact solution of the problem ofunsteady fully-developed viscous flow in slightly curved porous tube,”Arch. Mech. (Poland), vol. 32, pp. 461–474, 1980.

[24] A. F. Bertelsen and L. K. Thorsen, “An experimental investigation ofoscillatory flow in pipe bends,”J. Fluid. Mech., vol. 118, pp. 269–284,1982.

[25] K. Sudo, M. Sumida, and R. Yamane, “Secondary motion of fully de-veloped oscillatory flow in a curved pipe,”J. Fluid. Mech., vol. 237, pp.189–208, 1992.

[26] F. T. Smith, “Pulsatile flow in curved pipes,”J. Fluid. Mech., vol. 71,pp. 15–42, 1975.

[27] H. A. Simon, M. H. Chang, and J. C. F. Chow, “Heat transfer in curvedtubes with pulsatile, fully developed, laminar flows,”J. Heat Trans., vol.99, pp. 590–595, 1977.

[28] N. J. Rabadi, H. A. Simon, and J. C. F. Chow, “Numerical solution forfully developed, laminar pulsating flow in curved tubes,”Num. HeatTrans., vol. 3, pp. 225–239, 1980.

[29] C. C. Hamakiotes and S. A. Berger, “Fully developed pulsatile flow in acurved pipe,”J. Fluid. Mech., vol. 195, pp. 23–55, 1988.

[30] , “Periodic flows through curved tubes: The effect of the frequencyparameter,”J. Fluid. Mech., vol. 210, pp. 353–370, 1990.

[31] L. J. Chang and J. M. Tarbell, “Numerical simulation of fully developedsinusoidal and pulsatile (physiological) flow in curved tubes,”J. Fluid.Mech., vol. 161, pp. 175–198, 1985.

[32] K. B. Chandran, W. M. Swanson, D. N. Ghista, and H. W. Yao, “Oscil-latory flow in thin-walled curved elastic tubes,”Ann. Biomed. Eng., vol.2, pp. 392–412, 1974.

[33] K. B. Chandran, R. R. Hosey, D. N. Ghista, and V. W. Yao, “Analysis offully developed unsteady viscous flow in a curved elastic tube model toprovide fluid mechanical data for some circulatory path-physiologicalsituations and assist devices,”J. Biomech. Eng., vol. 101, pp. 114–123,1979.

[34] H. B. Atabek, “Wave propagation through a viscous fluid contained in atethered initially stressed, orthotropic elastic fluid,”Biophys. J., vol. 8,pp. 626–649, 1968.

[35] H. B. Atabek and H. S. Lew, “Wave propagation through a viscous in-compressible fluid contained in an initially stressed elastic tube,”Bio-phys. J., vol. 6, pp. 481–503, 1966.


[36] R. H. Cox, “Wave propagation through a Newtonian fluid containedwithin a thick-walled, viscoelastic tube,”Biophys. J., vol. 8, pp.691–709, 1968.

[37] I. Mirsky, “Wave propagation in a viscous fluid contained in an or-thotropic elastic tube,”Biophys. J., vol. 7, pp. 165–186, 1968.

[38] A. P. Avolio, “Multi-branched model of the human arterial system,”Med. Biol. Eng. Comput., vol. 18, pp. 709–718, 1980.

[39] M. B. McIlroy, W. S. Seitz, and R. C. Targett, “A transmission line modelof the normal aorta and its branches,”Cardiovasc. Res., vol. 20, pp.581–587, 1986.

[40] A. A. Hill, D. R. Surat, R. S. Cobbold, B. L. Langille, L. Y. Mo, andS. L. Adamson, “A wave transmission model of the umbilicoplacentalcirculation based on hemodynamic measurements in sheep,”Amer. J.Physiol., vol. 269, pp. R1267–R1278, 1995.

[41] L. Y. Mo, P. A. Bascom, K. Ritchie, and L. M. McCowan, “A trans-mission line modeling approach to the interpretation of uterine Dopplerwaveforms,”Ultrasound Med. Biol., vol. 14, pp. 365–376, 1988.

[42] L. H. Back, E. Y. Kwack, and D. W. Crawford, “Flow measurementsin an athersclerotic curved, tapered femoral artery model of man,”J.Biomech. Eng., vol. 110, pp. 310–319, 1988.

[43] D. M. Wang and J. M. Tarbell, “Nonlinear analysis of oscillatory flow,with a nonzero mean, in an elastic tube (artery),”J. Biomech. Eng., vol.117, pp. 127–135, 1995.

[44] W. R. Milnor, Hemodynamics, 2nd ed. Baltimore, MD: Williams andWilkins, 1989, ch. 8, pp. 204–224.

[45] S. G. Kang and J. M. Tarbell, “The impedance of curved artery models,”J. Biomech. Eng., vol. 105, pp. 275–282, 1983.

[46] R. G. Gosling and D. H. King,Ultrasound Angiology in Arteries andVeins, A.W. Marcus and L. Adamson, Eds. Edinburgh, U.K.: ChurchillLivingstone, 1975, pp. 61–98.

Lance Jonathan Myers received the B.Sc. (en-gineering) and M.Sc. (engineering) degrees inelectrical engineering from the University of theWitwatersrand, South Africa, in 1994 and 1996,respectively. He is currently a final-year Ph.D.degree candidate in biomedical engineering at theUniversity of Cape Town, South Africa.

His research interests include the mathematicalmodeling of vascular systems, specifically the foetalcardiovascular system, neural networks, imageprocessing, and statistical pattern recognition.

Wayne Logan Capper received the B.Sc. Eng.degree (electronic) from the University of Natal,Durban, South Africa, in 1981, and the M.Sc. (med),Ph.D. and the MBA degrees from the Universityof Cape Town, Cape Town, South Africa, in 1984,1989, and 2001, respectively.

He joined the Department of Biomedical Engi-neering at the same university in 1991, and currentlyheads the Flow Studies laboratory. His researchinterests include noninvasive lower limb vascularassessment, the study of airway dynamics, fetal

blood flow modeling, and fetal assessment using Doppler ultrasound.

analytical solution for pulsatile axial flow velocity waveforms in curved elastic tubes

Documents