1742-4682-3-31 1742-4682 Research <p>Pulsatile blood flow, shear force, energy dissipation and Murray's Law</p> Painter R Page ppainter@oehha.ca.gov Edén Patrik patrik@thep.lu.se Bengtsson Hans-Uno hans-uno.bengtsson@thep.lu.se

Office of Environmental Health Hazard Assessment, California Environmental Protection Agency, P. O. Box 4010, Sacramento, California 95812, USA

Department of Theoretical Physics, Lund University, S-223 62 Soelvegatan 14A, Lund, Sweden

 Theoretical Biology and Medical Modelling 1742-4682 2006 3 1 31 http://www.tbiomed.com/content/3/1/31 16923189 10.1186/1742-4682-3-31 14 3 2006 21 8 2006 21 8 2006 2006 Painter et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Murray's Law states that, when a parent blood vessel branches into daughter vessels, the cube of the radius of the parent vessel is equal to the sum of the cubes of the radii of daughter blood vessels. Murray derived this law by defining a cost function that is the sum of the energy cost of the blood in a vessel and the energy cost of pumping blood through the vessel. The cost is minimized when vessel radii are consistent with Murray's Law. This law has also been derived from the hypothesis that the shear force of moving blood on the inner walls of vessels is constant throughout the vascular system. However, this derivation, like Murray's earlier derivation, is based on the assumption of constant blood flow.

Methods

To determine the implications of the constant shear force hypothesis and to extend Murray's energy cost minimization to the pulsatile arterial system, a model of pulsatile flow in an elastic tube is analyzed. A new and exact solution for flow velocity, blood flow rate and shear force is derived.

Results

For medium and small arteries with pulsatile flow, Murray's energy minimization leads to Murray's Law. Furthermore, the hypothesis that the maximum shear force during the cycle of pulsatile flow is constant throughout the arterial system implies that Murray's Law is approximately true. The approximation is good for all but the largest vessels (aorta and its major branches) of the arterial system.

Conclusion

A cellular mechanism that senses shear force at the inner wall of a blood vessel and triggers remodeling that increases the circumference of the wall when a shear force threshold is exceeded would result in the observed scaling of vessel radii described by Murray's Law.

Background

In 1926, the physiologist Cecil Murray published a theoretical explanation for the relationship between the radius of an artery immediately upstream from a branch point (parent artery) and the radii of arteries immediately downstream (daughter arteries) 12. In its simplest form, i.e., when an artery of radius Rk branches into η arteries of radius Rk+1, the relationship termed Murray's Law states that Rk3=ηRk+13 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaqhaaWcbaGaem4AaSgabaGaeG4mamdaaOGaeyypa0dcciGae83TdGMaemOuai1aa0baaSqaaiabdUgaRjabgUcaRiabigdaXaqaaiabiodaZaaaaaa@389B@. Murray's derivation of this relationship assumed that there is an energy requirement for producing the blood contained in a vessel that is proportional to the volume of blood in a vessel and that there is an energy requirement for pumping blood through a vessel that is given by Poiseuille's Law for flow in a tube. When the radius of a branching artery is increased, the cost of blood in the artery increases, but the cost of pumping blood through the artery decreases. Calculation of the radius that minimizes cost, using the calculus of variations, leads to Murray's Law.

While Murray did not suggest a mechanism for the regulation of the radius of an artery, other scientists have. A recent hypothesis is that shear force at the inner surface triggers circumferential growth if the force is above a threshold 34. This is an attractive hypothesis because high values of shear in fluids can damage or destroy cells. Furthermore, turbulence, which occurs above critical values of shear in fluids, is associated with atherosclerosis of arterial walls.

Implications of constant shear force at the wall of blood vessels have been analyzed by Kassab and Fung for a constant pressure gradient model 5. They showed that, if shear force at the inner wall of vessels is constant in their model, then scaling of the radius of blood vessels is described by Murray's Law.

If the shear-force remodeling (SFR) hypothesis is correct, it should be possible to derive Murray's Law from the equations describing the fluid mechanics of pulsatile blood flow in a tubular structure. Specifically, it should be possible to derive the law from the basic work of Womersley on pulsatile flow in an elastic tube 6. However, the results of Womersley are complex, and the approximations introduced by Womersley may result in inaccurate predictions. Furthermore, the elastic tube in Womersley's model is not tethered to surrounding structures. Therefore, we analyze an elastic tube model where the inside surface can move relative to the outside surface but where the outside wall is attached to surrounding structures. We also use a general solution for the differential equation describing the pulsatile flow model that was not used directly by Womersley and that leads to an exact solution. We show that the exact solution can be closely approximated by a simpler expression and use this result to analyze the relationship between vessel radius and shear force during pulsatile blood flow.

The rigid tube model

To develop the model, we first consider blood flow in a rigid cylindrical tube of constant radius R. We make the same assumptions as Kassab and Fung: that "blood is an incompressible viscous fluid so that flow is described by the Navier-Stokes equation" and that "each blood vessel is a straight circular cylindrical tube" 5. The distance from the central axis of the tube is denoted r, the tube length is L, and the velocity of flow is u. The Navier-Stokes equation of motion is

μ ∂ 2 u ∂ r 2 + μ 1 r ∂ u ∂ r + A = ρ ∂ u ∂ t , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF8oqBdaWcaaqaaiabgkGi2oaaCaaaleqabaGaeGOmaidaaOGaemyDauhabaGaeyOaIyRaemOCai3aaWbaaSqabeaacqaIYaGmaaaaaOGaey4kaSIae8hVd02aaSaaaeaacqaIXaqmaeaacqWGYbGCaaWaaSaaaeaacqGHciITcqWG1bqDaeaacqGHciITcqWGYbGCaaGaey4kaSIaemyqaeKaeyypa0Jae8xWdi3aaSaaaeaacqGHciITcqWG1bqDaeaacqGHciITcqWG0baDaaGaeiilaWcaaa@4C81@

where A is the pressure gradient ΔP/L, μ is viscosity and ρ is density. Substituting ξ for μ/ρ gives

ξ ∂ 2 u ∂ r 2 + ξ 1 r ∂ u ∂ r + A ρ = ∂ u ∂ t .       ( 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF+oaEdaWcaaqaaiabgkGi2oaaCaaaleqabaGaeGOmaidaaOGaemyDauhabaGaeyOaIyRaemOCai3aaWbaaSqabeaacqaIYaGmaaaaaOGaey4kaSIae8NVdG3aaSaaaeaacqaIXaqmaeaacqWGYbGCaaWaaSaaaeaacqGHciITcqWG1bqDaeaacqGHciITcqWGYbGCaaGaey4kaSYaaSaaaeaacqWGbbqqaeaacqWFbpGCaaGaeyypa0ZaaSaaaeaacqGHciITcqWG1bqDaeaacqGHciITcqWG0baDaaGaeiOla4IaaCzcaiaaxMaadaqadiqaaiabigdaXaGaayjkaiaawMcaaaaa@506E@

Let ũ be the initial-condition-independent solution of Equation (1) when the pressure gradient is a constant (denoted Ã). The solution is Poiseuille's equation

u ˜ = A ˜ ( R 2 − r 2 ) / ( 4 μ ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG1bqDgaacaiabg2da9iqbdgeabzaaiaWaaeWaceaacqWGsbGudaahaaWcbeqaaiabikdaYaaakiabgkHiTiabdkhaYnaaCaaaleqabaGaeGOmaidaaaGccaGLOaGaayzkaaGaei4la8YaaeWaceaacqaI0aaniiGacqWF8oqBaiaawIcacaGLPaaacqGGUaGlaaa@3DBA@

The rate of blood flow in the tube, Q˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaacaaaa@2DE6@, is

∫ 0 R 2 π r u ˜ d r = π A ˜ R 4 / ( 8 μ ) ,       ( 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWdXaqaaiabikdaYGGaciab=b8aWjabdkhaYjqbdwha1zaaiaGaemizaqMaemOCaiNaeyypa0Jae8hWdaNafmyqaeKbaGaacqWGsbGudaahaaWcbeqaaiabisda0aaakiabc+caVmaabmGabaGaeGioaGJae8hVd0gacaGLOaGaayzkaaaaleaacqaIWaamaeaacqWGsbGua0Gaey4kIipakiabcYcaSiaaxMaacaWLjaWaaeWaceaacqaIZaWmaiaawIcacaGLPaaaaaa@4952@

and the shear force (per unit area) of the fluid on the inner wall of the tube is

− μ [ ∂ u ˜ / ∂ r | r = R A ˜ R / 2.       ( 4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHsisliiGacqWF8oqBdaabciqaaiabcUfaBjabgkGi2kqbdwha1zaaiaGaei4la8IaeyOaIyRaemOCaihacaGLiWoadaWgaaWcbaGaemOCaiNaeyypa0JaemOuaifabeaakiqbdgeabzaaiaGaemOuaiLaei4la8IaeGOmaiJaeiOla4IaaCzcaiaaxMaadaqadiqaaiabisda0aGaayjkaiaawMcaaaaa@456B@

Now consider the rigid tube model with an oscillating pressure gradient Ăeiωt. The solution for the velocity, ŭ, stated by Womersley 6, is

ŭ = [Ă/(ρiω)]eiωt[1 - J0(i3/2αr)/J0(i3/2αR)],

where α = (ω/ξ)1/2 and J0(i3/2αr) is the Bessel function of order 0,

∑ n = 0 n = ∞ ( − 1 ) n ( i 3 / 2 α r / 2 ) 2 n / ( n ! n ! ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaamaabmGabaGaeyOeI0IaeGymaedacaGLOaGaayzkaaWaaWbaaSqabeaacqWGUbGBaaGcdaqadiqaaiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaGGacOGae8xSdeMaemOCaiNaei4la8IaeGOmaidacaGLOaGaayzkaaaaleaacqWGUbGBcqGH9aqpcqaIWaamaeaacqWGUbGBcqGH9aqpcqGHEisPa0GaeyyeIuoakmaaCaaaleqabaGaeGOmaiJaemOBa4gaaOGaei4la8YaaeWaceaacqWGUbGBcqGGHaqicqWGUbGBcqGGHaqiaiaawIcacaGLPaaacqGGUaGlaaa@5006@

Womersley did not provide a derivation of the solution for the rigid tube. He cited Lambossy 7 as the source, but Lambossy did not arrive at this solution for ŭ. Therefore, we provide a derivation of the solution for the rigid tube model that can be extended to a solution for an elastic tube model. We write ŭ as a power series of the variable r:

ŭ = b0 + b1r + b2r2 + ...     (5)

where bi is a function of time, t. Equating the coefficients of r0 following substitution of this power series into Equation (1) gives

2 ξ b 2 + 2 ξ b 2 + A ⌣ e i ω t / ρ = b 0 /       ( 6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaIYaGmiiGacqWF+oaEcqWGIbGydaWgaaWcbaGaeGOmaidabeaakiabgUcaRiabikdaYiab=57a4jabdkgaInaaBaaaleaacqaIYaGmaeqaaOGaey4kaSIafmyqaeKbaqbacqWGLbqzdaahaaWcbeqaaiabdMgaPjab=L8a3jabdsha0baakiabc+caViab=f8aYjabg2da9iabdkgaInaaDaaaleaacqaIWaamaeaacqGGVaWlaaGccaWLjaGaaCzcamaabmGabaGaeGOnaydacaGLOaGaayzkaaaaaa@4AD2@

where bn/ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGIbGydaqhaaWcbaGaemOBa4gabaGaei4la8caaaaa@3071@ = dbn/dt. Equating the coefficients of rn for n>0 gives

( n + 2 ) ( n + 1 ) ξ b n + 2 + ( n + 2 ) ξ b n + 2 = b n / . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadiqaaiabd6gaUjabgUcaRiabikdaYaGaayjkaiaawMcaamaabmGabaGaemOBa4Maey4kaSIaeGymaedacaGLOaGaayzkaaacciGae8NVdGNaemOyai2aaSbaaSqaaiabd6gaUjabgUcaRiabikdaYaqabaGccqGHRaWkdaqadiqaaiabd6gaUjabgUcaRiabikdaYaGaayjkaiaawMcaaiab=57a4jabdkgaInaaBaaaleaacqWGUbGBcqGHRaWkcqaIYaGmaeqaaOGaeyypa0JaemOyai2aa0baaSqaaiabd6gaUbqaaiabc+caVaaakiabc6caUaaa@4E91@

Solving for bn+2 gives

b n + 2 = ( 1 / ξ ) b n / / ( n + 2 ) 2 .       ( 7 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGIbGydaWgaaWcbaGaemOBa4Maey4kaSIaeGOmaidabeaakiabg2da9maabmGabaGaeGymaeJaei4la8ccciGae8NVdGhacaGLOaGaayzkaaGaemOyai2aa0baaSqaaiabd6gaUbqaaiabc+caVaaakiabc+caVmaabmGabaGaemOBa4Maey4kaSIaeGOmaidacaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaGccqGGUaGlcaWLjaGaaCzcamaabmGabaGaeG4naCdacaGLOaGaayzkaaaaaa@46EA@

Because ∂ŭ/∂r = 0 at r = 0, b1 is 0 for all values of t. Consequently, for all odd values of n, bn is 0.

We now define the constant B1,0 by the equation

b2 = [(iω/ξ)/4]eiωtB1,0.     (8)

From this equation and Equations (5) and (7) it follows that the series

b2r2 + b4r4 + b6r6 + ... is

B1,0eiωt {[(iωr2/ξ)/4]/(1!)2 + [(iωr2/ξ)/4]2/(2!)2 + [(iωr2/ξ)/4]3/(3!)2 + ...}.

From Equation (6) and Equation (8), it follows that

b0 = [Ă/(ρiω)]eiωt + B1,0eiωt.

Consequently, the expression for blood velocity is

ŭ = [Ă/(ρiω)]eiωt + B1,0eiωt J0(i3/2αr).     (9)

The above derivation is similar to many published analyses of differential equations that have solutions containing Bessel functions. The derivation is provided to make it clear that all solutions of the model of blood flow in a rigid tube in response to the pressure gradient Ăeiωt are described by the above expression. This solution can also be derived from the assumption that we can write ŭ = νeiωt, where ν is a function of the single variable r. Substitution into Equation (1) leads to

ξ ∂ 2 ν ∂ r 2 + ξ 1 r ∂ ν ∂ r + A ⌣ ρ = i ω ν , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF+oaEdaWcaaqaaiabgkGi2oaaCaaaleqabaGaeeOmaidaaOGae8xVd4gabaGaeyOaIyRaemOCai3aaWbaaSqabeaacqaIYaGmaaaaaOGaey4kaSIae8NVdG3aaSaaaeaacqaIXaqmaeaacqWGYbGCaaWaaSaaaeaacqGHciITcqWF9oGBaeaacqGHciITcqWGYbGCaaGaey4kaSYaaSaaaeaacuWGbbqqgaafaaqaaiab=f8aYbaacqGH9aqpcqWGPbqAcqWFjpWDcqWF9oGBcqGGSaalaaa@4C55@

which can be written as

∂ 2 ν ∂ r 2 + 1 r ∂ ν ∂ r + i 3 α 2 ν = − A ⌣ μ . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaeGOmaidaaGGacOGae8xVd4gabaGaeyOaIyRaemOCai3aaWbaaSqabeaacqaIYaGmaaaaaOGaey4kaSYaaSaaaeaacqaIXaqmaeaacqWGYbGCaaWaaSaaaeaacqGHciITcqWF9oGBaeaacqGHciITcqWGYbGCaaGaey4kaSIaemyAaK2aaWbaaSqabeaacqaIZaWmaaGccqWFXoqydaahaaWcbeqaaiabikdaYaaakiab=17aUjabg2da9iabgkHiTmaalaaabaGafmyqaeKbaqbaaeaacqWF8oqBaaGaeiOla4caaa@4BED@

We note that, if ν0 is a solution of the equation ∂2ν∂r2+1r∂ν∂r+i3α2ν=0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaeGOmaidaaGGacOGae8xVd4gabaGaeyOaIyRaemOCai3aaWbaaSqabeaacqaIYaGmaaaaaOGaey4kaSYaaSaaaeaacqaIXaqmaeaacqWGYbGCaaWaaSaaaeaacqGHciITcqWF9oGBaeaacqGHciITcqWGYbGCaaGaey4kaSIaemyAaK2aaWbaaSqabeaacqaIZaWmaaGccqWFXoqydaahaaWcbeqaaiabikdaYaaakiab=17aUjabg2da9iabicdaWaaa@4823@, which is Bessel's equation of order 0, then ν0 + A/(i α2 μ) is a solution of Equation (1) for the oscillating pressure gradient. Noting that the solution of Bessel's equation is BJ0(i3/2αr), where B is a constant, completes the derivation.

A boundary condition for the rigid tube is that the function ŭ in Equation (9) is 0 at r = R:

0 = [Ă/(ρiω)]eiωt + B1,0eiωt J0(i3/2αR).

Consequently, B1,0 = -[Ă/(ρiω)]/J0(i3/2αR), and

ŭ = [Ă/(ρiω)]eiωt[1 - J0(i3/2αr)/J0(i3/2αR)],     (10)

which is the result published without derivation by Womersley 6.

The rate of blood flow Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@ is computed as

∫ 0 R 2 π r u ⌣ d r = ( π A ⌣ e i ω t ) / ( i ω ρ ) [ R 2 − 2 R 2 J 1 ( i 3 / 2 α R ) / ( i 3 / 2 α R ) / J 0 ( i 3 / 2 α R ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWdXaqaaiabikdaYGGaciab=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f7aHjabdkfasbGaayjkaiaawMcaaiabc+caViabdQeaknaaBaaaleaacqaIWaamaeqaaOWaaeWaceaacqWGPbqAdaahaaWcbeqaaiabiodaZiabc+caViabikdaYaaakiab=f7aHjabdkfasbGaayjkaiaawMcaaaGaay5waiaaw2faaiabcYcaSaaa@7604@

where J1(i3/2αR) is the Bessel function of order 1,

∑ n = 0 n = ∞ ( − 1 ) n ( i 3 / 2 α R / 2 ) 2 n + 1 / [ ( n + 1 ) ! n ! ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaamaabmGabaGaeyOeI0IaeGymaedacaGLOaGaayzkaaWaaWbaaSqabeaacqWGUbGBaaaabaGaemOBa4Maeyypa0JaeGimaadabaGaemOBa4Maeyypa0JaeyOhIukaniabggHiLdGcdaqadiqaaiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaGGacOGae8xSdeMaemOuaiLaei4la8IaeGOmaidacaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmcqWGUbGBcqGHRaWkcqaIXaqmaaGccqGGVaWldaWadiqaamaabmGabaGaemOBa4Maey4kaSIaeGymaedacaGLOaGaayzkaaGaeiyiaeIaemOBa4MaeiyiaecacaGLBbGaayzxaaaaaa@5465@

We simplify this expression using the identity

(i3/2αR)2J0(i3/2αR) - 2(i3/2αR)J1(i3/2αR) = -(i3/2αR)2J2(i3/2αR), where J2(i3/2αR) is the Bessel function of order 2,

∑ n = 0 n = ∞ ( − 1 ) n ( i 3 / 2 α R / 2 ) 2 n + 2 / [ ( n + 2 ) ! n ! ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaamaabmGabaGaeyOeI0IaeGymaedacaGLOaGaayzkaaWaaWbaaSqabeaacqWGUbGBaaGcdaqadiqaaiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaGGacOGae8xSdeMaemOuaiLaei4la8IaeGOmaidacaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmcqWGUbGBcqGHRaWkcqaIYaGmaaaabaGaemOBa4Maeyypa0JaeGimaadabaGaemOBa4Maeyypa0JaeyOhIukaniabggHiLdGccqGGVaWldaWadiqaamaabmGabaGaemOBa4Maey4kaSIaeGOmaidacaGLOaGaayzkaaGaeiyiaeIaemOBa4MaeiyiaecacaGLBbGaayzxaaGaeiOla4caaa@554D@

Division of both sides by (i3/2αR)2 gives

J0(i3/2αR) - 2J1(i3/2αR)/(i3/2αR) = -J2(i3/2αR), and substitution gives

Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@ = [πĂR2eiωt/(iωρ)][-J2(i3/2αR)/J0(i3/2αR)].     (11)

We define PQ(i3/2αR) as J2(i3/2αR)/[(i3/2αR)2/8] and note that as R goes to zero the imaginary part of PQ(i3/2αR) vanishes and |PQ(i3/2αR)| goes to 1. Substitution now gives Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@ = [πĂR4/(8μ)]eiωtPQ(i3/2αR)/J0(i3/2αR). This expression is further simplified to

Q ⌣ = [ π A ⌣ R 4 / ( 8 μ ) ] e i ω t + i θ P Q − i θ J 0 | P Q ( i 3 / 2 α R ) | / | J 0 ( i 3 / 2 α R ) | ,       ( 12 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaiabg2da9maadmGabaacciGae8hWdaNafmyqaeKbaqbacqWGsbGudaahaaWcbeqaaiabisda0aaakiabc+caVmaabmGabaGaeGioaGJae8hVd0gacaGLOaGaayzkaaaacaGLBbGaayzxaaGaemyzau2aaWbaaSqabeaacqWGPbqAcqWFjpWDcqWG0baDcqGHRaWkcqWGPbqAcqWF4oqCdaWgaaadbaGaemiuaaLaemyuaefabeaaliabgkHiTiabdMgaPjab=H7aXnaaBaaameaacqWGkbGscqaIWaamaeqaaaaakmaaemGabaGaemiuaa1aaSbaaSqaaiabdgfarbqabaGcdaqadiqaaiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaOGae8xSdeMaemOuaifacaGLOaGaayzkaaaacaGLhWUaayjcSdGaei4la8YaaqWaceaacqWGkbGsdaWgaaWcbaGaeGimaadabeaakmaabmGabaGaemyAaK2aaWbaaSqabeaacqaIZaWmcqGGVaWlcqaIYaGmaaGccqWFXoqycqWGsbGuaiaawIcacaGLPaaaaiaawEa7caGLiWoacqGGSaalcaWLjaGaaCzcamaabmGabaGaeGymaeJaeGOmaidacaGLOaGaayzkaaaaaa@7103@

where θPQ is the argument of PQ(i3/2αR) and θj0 is the argument of J0(i3/2αR).

The shear force (per unit area) at the inner wall of the tube is -μ[∂u/∂r|r=R = -[Ăμ/(ρiω)]eiωti3/2αJ-1(i3/2αR)/J0(i3/2αR), where i3/2αJ-1(i3/2αR) = dJ0(i3/2αr)/dr|r=R. We note that -J-1(i3/2αR) = J1(i3/2αR), which is written as (i3/2αR/2)PS(i3/2αR). Substitution now gives the expression for the shear force (per unit area), [ĂeiωtR/2]PS(i3/2αR)/J0(i3/2αR). This expression is further simplified to

− μ [ ∂ u / ∂ r ] r = R = [ A ⌣ R / 2 ] e i ω t + i θ P S − i θ J 0 | P S ( i 3 / 2 α R ) | / | J 0 ( i 3 / 2 α R ) | ,       ( 13 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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f7aHjabdkfasbGaayjkaiaawMcaaaGaay5bSlaawIa7aiabc+caVmaaemGabaGaemOsaO0aaSbaaSqaaiabicdaWaqabaGcdaqadiqaaiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaOGae8xSdeMaemOuaifacaGLOaGaayzkaaaacaGLhWUaayjcSdGaeiilaWIaaCzcaiaaxMaadaqadiqaaiabigdaXiabiodaZaGaayjkaiaawMcaaaaa@7894@

where θPS is the argument of PS(i3/2αR). Note that PS(i3/2αR) is a function with an imaginary part that vanishes and a modulus that approaches 1 as R approaches 0.

The final step in the description of the arterial pressure gradient is to express it as a sum of a forward-pumping gradient and a purely oscillatory gradient. In large human arteries under normal physiological conditions, pressure cycles from approximately 120 mm Hg (systolic) to approximately 80 mm Hg (diastolic). Pressure in the terminal arterioles is much lower than 80 mm Hg. These pressures suggest a model where there is a forward-pumping pressure gradient, Ã, plus an oscillating pressure gradient, Ăeiωt. The flow velocity of this forward-pumping, pulsatile model is the sum of the solutions for the constant gradient model and the oscillating gradient model. Similarly, the flow rate in the tube and the shear force at the inner wall of the tube are the sums of the flow rates and the shear forces, respectively, of the constant gradient model and the oscillating gradient model.

Flow in an elastic tube

Now consider flow in an elastic tube where the radius is constant but the inside surface moves in response to the pull of adjacent fluid. The thickness of the wall is denoted h, and wall tissue density is denoted ρw. The displacement of a point on the inside wall of the tube from the locus when the oscillatory component of force is identically 0 is denoted Z, and the coefficient of deformation relating Z to the force per unit area along the inside wall is K.

The model considered in this section differs from the Womersley model in how the elastic tube responds to shear force on the inner wall. In the Womersley model, the outer wall is not connected to surrounding structures. The full thickness of the wall moves in response to shear force, and regions of relatively high force stretch upstream regions of the wall and compress downstream regions. In the model analyzed below, the outer wall is tethered by branching arteries, and its movement is further restricted by contact with adjacent tissues. The tube matrix between the inner and outer wall is modeled as elastic tissue.

We first consider the pressure gradient Ăeiωt. From Equation (9), the expression for ŭ is again [Ă/(ρiω)]eiωt + B1,0eiωtJ0(i3/2αr), where B1,0 is determined by the boundary condition requiring the velocity of fluid at the wall surface to equal the velocity of the wall surface. The solution for wall displacement, Z, is periodic with period 2π/ω. Therefore, the position of the inner wall of the tube is described by a Fourier series

Z = ∑ n = − ∞ n = + ∞ C n e i n ω t . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGAbGwcqGH9aqpdaaeWaqaaiabdoeadnaaBaaaleaacqWGUbGBaeqaaOGaemyzau2aaWbaaSqabeaacqWGPbqAcqWGUbGBiiGacqWFjpWDcqWG0baDaaGccqGGUaGlaSqaaiabd6gaUjabg2da9iabgkHiTiabg6HiLcqaaiabd6gaUjabg2da9iabgUcaRiabg6HiLcqdcqGHris5aaaa@4595@

The condition requiring the velocity of the wall to equal the velocity of the adjacent fluid gives

∑ n = − ∞ n = + ∞ C n i n ω e i n ω t = [ A ⌣ / ( ρ i ω ) ] e i ω t + B 1 , 0 e i ω t J 0 ( i 3 / 2 α R ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdoeadnaaBaaaleaacqWGUbGBaeqaaOGaemyAaKMaemOBa4gcciGae8xYdCNaemyzau2aaWbaaSqabeaacqWGPbqAcqWGUbGBcqWFjpWDcqWG0baDaaGccqGH9aqpdaWadiqaaiqbdgeabzaauaGaei4la8YaaeWaceaacqWFbpGCcqWGPbqAcqWFjpWDaiaawIcacaGLPaaaaiaawUfacaGLDbaacqWGLbqzdaahaaWcbeqaaiabdMgaPjab=L8a3jabdsha0baakiabgUcaRiabdkeacnaaBaaaleaacqaIXaqmcqGGSaalcqaIWaamaeqaaOGaemyzau2aaWbaaSqabeaacqWGPbqAcqWFjpWDcqWG0baDaaGccqWGkbGsdaWgaaWcbaGaeGimaadabeaakmaabmGabaGaemyAaK2aaWbaaSqabeaacqaIZaWmcqGGVaWlcqaIYaGmaaGccqWFXoqycqWGsbGuaiaawIcacaGLPaaacqGGUaGlaSqaaiabd6gaUjabg2da9iabgkHiTiabg6HiLcqaaiabd6gaUjabg2da9iabgUcaRiabg6HiLcqdcqGHris5aaaa@6F59@

Equating the coefficients of einωt leads to C1= (Ă/ρ)/(iω)2 + B1,0J0(i3/2αR)/(iω), and from the definition of Z, it follows that C0 = 0. Furthermore, for n > 1 and for n < 0 Cn = 0. Consequently, we have

Z = {(Ă/ρ)/(iω)2 + B1,0J0(i3/2αR)/(iω)}einωt.     (14)

If the outer wall is assumed to be stationary, the average velocity of the wall is described by iω C1eiωt/2, and the requirement that the force (per unit surface area) on a point on the wall, -μ[∂v/∂r|r=R-KC1eiωt, equals the rate of change of wall momentum (per unit wall surface area), (hρw/2)(iω)2C1eiωt, gives

-μ[∂B1,0eiωtJ0(i3/2αr)/∂r|r=R] - KC1eiωt = (hρw/2)(iω)2C1eiωt.

Substitution for C1 gives

μB1,0eiωti3/2αJ-1(i3/2αR)eiωt/[-K/(iω) - (hρw/2)(iω)]

= [Ă/(iω)]eiωt + B1,0eiωtJ0(i3/2αR),

where i3/2αJ-1(i3/2αR) = ∂J0(i3/2αr)/∂r|r=R. Consequently,

B1,0 = - [Ă/(ρiω)]/{J0(i3/2αR) - μ[i3/2αJ-1(i3/2αR)]/(-K/(iω) - (iω)hρw/2)},     (15)

and

ŭ = [Ă/(ρiω)]eiωt{1 - J0(i3/2αr)/[J0(i3/2αR)

+μi3/2αJ1(i3/2αR)/(K/(iω) + (iω)hρw/2)]}.     (16)

Now, consider the relationship between the thickness h and the elastic coefficient K of the wall. Using the analogy of a sheet of rubber with thickness h, we can write K = κ/h, where κ is a constant. Substituting κ/h for K and -J1(i3/2αR) for J-1(i3/2αR) in Equation (15) gives

ŭ = [Ă/(ρiω)]eiωt{1 - J0(i3/2αr)/[J0(i3/2αR) + DPS(i3/2αR)]},     (17)

where

D = (μ/κ)(h/R)(iω)(i3/2αR)2/2/(1 + (h/κ)(iω)2hρw/2).     (18)

Integration over the cross-sectional area of the tube now gives

Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@= [πĂR2/(ρiω)]eiωt{1 - PS(i3/2αR)/[J0(i3/2αR) + DPS(i3/2αR)]}.     (19)

From Equation (17), the shear force (per unit area) at the wall of the elastic tube is

-μ[∂u/∂r|r=R = [Ăμ/(ρiω)]eiωt(-i3/2α)(i3/2αR/2)PS(i3/2αR)/[J0(i3/2αR) + DPS(i3/2αR)].

This expression is further simplified to

-μ[∂u/∂r|r=R = [ĂR/2]eiωtPS(i3/2αR)/[J0(i3/2αR) + DPS(i3/2αR)].     (20)

We note that, as D goes to 0, the above expressions for flow velocity, flow rate and shear force in the elastic tube model approach the values given by Equation (10), Equation (11) and Equation (13), respectively, derived for the rigid tube model. A bound on D can be derived from the expression for Z, Equation (14), which is simplified by substituting the value of B1,0 from Equation (15) and the definition of D from Equation (18) to give

Z = [Ă/(ρω2)]eiωtDPS(i3/2αR)/[J0(i3/2αR) + DPS(i3/2αR)].

We now divide Equation (20) by -iωπR2Z to give

D = {[J0(i3/2αR) - PS(i3/2αR)]/PS(i3/2αR)}/[Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@/(iωπR2Z) - 1],

which implies

D ≤ [|J0(i3/2αR)|/|PS(i3/2αR)|+1]/|Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@/(iωπR2Z) - 1|.

We can set an upper bound on D by setting an upper bound on Z. For example, a reasonable assumption is that the maximum value of Z is bounded by h/2. This condition limits the movement of the inner surface of the tube during a cycle of pulsatile flow to a distance no greater than the thickness of the vessel wall. In small muscular arteries and arterioles, h is approximately equal to or slightly less than R 8. As R increases, h/R decreases to a value of approximately 0.2 for the aorta 9. The viscosity of blood at a shear gradient of 100/s is approximately 0.033 dyne-s/cm2 10, and the density is approximately 1.06 g/cm3. Therefore, at a heart rate of 1/s, α2 is approximately 32/cm2.

The rate of arterial blood flow in the proximal aorta is equal to the cardiac output, which in humans is approximately 70 ml/s. The velocity ranges from approximately 100 cm/s in early systole to 0 or a slightly negative value in diastole. In our oscillatory, forward-pumping model, total blood flow in the human aorta is described by Q = Q˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaacaaaa@2DE6@ + Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@, where Q˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaacaaaa@2DE6@ = 70 ml/s and Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@ = (70 ml/s)eiωt. From the value of R, approximately 1 cm, αR is approximately 5, and h is approximately 0.2 cm. Therefore, from Table 1, |J0(i3/2αR)|/|PS(i3/2αR)| is approximately equal to (but less than) 3. From the bounding of Z assumption, the quantity |Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@/(iωπR2Z)| is greater than |2Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@/(iωπR2h)|, which is in turn greater than 201. Therefore, 1/|Q⌣ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGrbqugaafaaaa@2DF2@/(iωπR2Z) - 1| is less than 1/200, and D is less than 2 × 10-2.

<p>Table 1</p>

Values of |J0(i3/2αR)|, |PS(i3/2αR)|/|J0(i3/2αR)|, |PQ(i3/2αR)|/|J0(i3/2αR)| and θd = -θPQ + θJ0 (in degrees).

αR

|J0(i3/2αR)|

|PS(i3/2αR)|

| P S ( i 3 / 2 α R ) | | J 0 ( i 3 / 2 α R ) | MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaamaaemGabaGaemiuaa1aaSbaaSqaaiabdofatbqabaGcdaqadiqaaiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaGGacOGae8xSdeMaemOuaifacaGLOaGaayzkaaaacaGLhWUaayjcSdaabaWaaqWaceaacqWGkbGsdaWgaaWcbaGaeGimaadabeaakmaabmGabaGaemyAaK2aaWbaaSqabeaacqaIZaWmcqGGVaWlcqaIYaGmaaGccqWFXoqycqWGsbGuaiaawIcacaGLPaaaaiaawEa7caGLiWoaaaaaaa@493F@

| P Q ( i 3 / 2 α R ) | | J 0 ( i 3 / 2 α R ) | MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaamaaemGabaGaemiuaa1aaSbaaSqaaiabdgfarbqabaGcdaqadiqaaiabdMgaPnaaCaaaleqabaGaeG4mamJaei4la8IaeGOmaidaaGGacOGae8xSdeMaemOuaifacaGLOaGaayzkaaaacaGLhWUaayjcSdaabaWaaqWaceaacqWGkbGsdaWgaaWcbaGaeGimaadabeaakmaabmGabaGaemyAaK2aaWbaaSqabeaacqaIZaWmcqGGVaWlcqaIYaGmaaGccqWFXoqycqWGsbGuaiaawIcacaGLPaaaaiaawEa7caGLiWoaaaaaaa@493B@

-θPQ + θJ0

0.00

1.000000

1.000000

1.000000

1.000000

0.000000

0.25

1.000061

1.00001

0.999949

0.999942

0.596807

0.50

1.000976

1.000163

0.999187

0.999079

2.385789

0.75

1.004934

1.000824

0.99591

0.995364

5.354071

1.00

1.015525

1.002602

0.987275

0.985567

9.452664

1.25

1.037563

1.006346

0.969913

0.965839

14.56122

1.50

1.076683

1.013132

0.940975

0.932856

20.45769

1.75

1.138718

1.02425

0.899476

0.885319

26.81781

2.00

1.229006

1.041167

</