The Physics of Coronory Blood Flow - M. Zamir

270 8 Elements of Unlumped-Model Analysis

streamwise distance X (normalized)

Fig. 8.4.3. Pressure distributions within the three vessel segments forming an arterial bifurcation, under the idealized conditions of steady Poiseuille flow and on the assumption of a power law relation between the radius of each vessel and the flow rate through it. If the power law index is more than 3, as it is here, the pressure drop in the branch with the smaller radius (branch-2) is lower than that in the other branch.

where r is radial coordinate within the vessel and a is its radius. Using the results for flow in a tube and resistance to flow in sections 2.3,4, this equation for the shear stress can be expressed in terms of the flow rate, to give

or in terms of the pressure drop, to give

The first of these results indicates that if a power law relation as in Eq. 8.4.22 exists between the radius of a vessel and the flow rate which it carries, then the shear stress in vessels of di erent radii varies as

If the value of γ is less than 3, the shear stress will be higher in vessels of smaller radii, which clearly cannot be supported on physiological grounds. If the value of γ is more than 3, the shear stress will be lower in vessels of smaller

streamwise distance X (normalized)

Fig. 8.4.4. Pressure distributions within the three vessel segments forming an arterial bifurcation, under the idealized conditions of steady Poiseuille flow and on the assumption of a power law relation between the radius of each vessel and the flow rate through it. If the power law index is equal to 3, as it is here, the pressure drop is the same along both branches.

radii, which is more plausible on physiological grounds. But if the value of γ is equal to 3, the shear stress in Eq. 8.4.32 will be altogether independent of the radius a, which means that the shear stress will be the same in vessels of di erent radii. With this value of γ, the flow rate is proportional to the third power of the radius, or conversely, the radius is proportional to the cube root of the flow rate. This relation between the radius and flow rate is widely known as the “cube law” and is of particular interest because it was first derived by showing that it actually provides an optimum compromise between the pumping power required to drive the flow through the bifurcation, which is lower when the vessel radii are large, and the metabolic power required to maintain the volume of blood contained within the three vessels forming the bifucation, which is lower when the vessel radii are small [147, 212, 213].

Of interest in the present context is the second of the above results, namely that in Eq. 8.4.33, which indicates that if the length of a vessel is assumed to be proportional to its radius then the pressure drop becomes proportional to the shear stress, and hence everything that has been said above about the shear stress now applies equally to the pressure drop. In particular, for the two branches at a bifurcation, if the value of γ is more than 3, the pressure drop along the branch with the smaller radius is lower than that along the

272 8 Elements of Unlumped-Model Analysis

branch with the larger radius. This is somewhat unlikely on physiological or fluid dynamic grounds. On the other hand, if the value of γ is less than 3, the reverse is true, which is more plausible on both grounds. If the value of γ is equal to 3, the pressure drop is the same along both branches. These results are shown in Figs. 8.4.2–4.

The interesting conclusion from this discussion is that from the point of view of the shear stress acting on endothelial tissue, the more likely values of γ are 3 or higher, but from the point of view of pressure drop the more likely values are 3 or lower. The only possible compromise between these two conflicting requirements is clearly γ = 3, and this lends further theoretical support to the cube law. As stated earlier, values of γ based on actual measurements from the cardiovascular system have shown much scatter not only within the range of 2−4 but also outside this range [220]. The scatter, however, is generally found to center around the value γ = 3.

8.5 Pulsatile Flow in a Rigid Tube

We have seen in Sections 2.3 and 8.3 that in steady flow through a tube the pressure varies linearly along the tube, dropping from a high at entrance to a low at exit, the di erence between the two being the pressure drop driving the flow. All other properties of the flow field such as the flow rate, the shear stress at the tube wall, or velocity at any point within the tube are constant in the sense that they do not depend on the streamwise coordinate x. Only the pressure varies along the tube, but the pressure drop between the two ends of the tube is constant. These features of the flow are embodied by the equation for Poiseuille flow presented in Section 2.3 which we reproduce here using a subscript s to indicate that the flow properties are for steady flow only, as distinct from flow properties in pulsatile flow which we consider below. Thus, from Eq. 2.3.1 we write

where us is the streamwise velocity within the tube, μ is viscosity of the fluid, a is the tube radius, r is radial coordinate measured from the tube axis, and ks is pressure gradient defined by (Eq. 2.3.2)

where l is the length of the tube and Δps is the pressure di erence between the two ends of the tube.

It must be emphasized that the reference here is to fully developed flow in a tube as described in Section 2.3. While in practice flow in a tube is rarely fully developed along its full length, we continue to make this assumption here

0

Because the equations governing fully developed flow in a tube are linear, these two parts of the driving pressure can be dealt with separately, each producing a flow field as if it were the only driving pressure, and the two resulting flow fields are then added to produce the complete flow field. The situation is precisely the same as that in which the harmonic components of a composite pressure wave and the flows which they produce can be treated separately and the results are then added, as discussed in Section 6.2. Indeed, here too, the oscillatory part of the pressure, pφ(t), may itself be a composite wave which is then decomposed into harmonic components that are dealt with as in Chapter 6.

As a matter of terminology, when Δp(t) is a periodic function of time we shall refer to the flow which it produces as “pulsatile flow”, to the flow which Δps produces as “steady flow”, and to the flow which Δpφ(t) produces as “oscillatory flow”. Because the steady and the oscillatory parts of pulsatile flow can be dealt with separately as discussed above, and because steady flow in a tube has already been dealt with in Section 2.3, it remains to deal with only the oscillatory part of the flow.

A classical solution of the equations governing oscillatory flow in a rigid tube exists for an oscillatory pressure gradient of the form

In other words, the driving pressure consists of a simple sine or cosine wave with amplitude k0. We recall that, in this complex exponential formulation, the solution obtained is also in complex form where the real part describes the flow when the driving pressure is of the form k0 cos ωt, and the imaginary part describes the flow when the driving pressure is of the form k0 sin ωt.

For the oscillatory flow velocity field within the tube, the solution gives [177, 208, 194, 221]

where J0 is Bessel function of order zero of the first kind, and

To interpret uφ(r, t) in relation to the corresponding Poiseuille flow velocity in steady flow, namely us(r), it is convenient to use as a reference the maximum velocity in Poiseuille flow (Eq. 8.5.1) which occurs on the axis of the tube (r = 0), namely

Also, comparison between the steady and the oscillatory velocity profiles is made easier by taking the amplitude of the oscillatory pressure gradient to be the same as the pressure gradient in steady flow, that is, we take

Nondimensional forms of the steady and the oscillatory velocity profiles are then defined by (Eq. 8.5.1)

Evaluation of the last expression requires values of the Bessel functions, which are available numerically in math tables and in most computational programs [137, 198]. Because both us(r) and uφ(r, t) have been normalized in the same way, they can be compared graphically on the same scale, as shown in Fig. 8.5.1. The figure compares the oscillatory velocity profile with the corresponding profile in Poiseuille flow at di erent times within the oscillatory cycle. The comparison is made when the oscillatory flow is driven by the real part of the oscillatory pressure gradient, namely k0 cos ωt (k0 = ks). Thus, at t = 0 the oscillatory flow and the Poisuille flow are momentarily driven by the same pressure gradient and, in the absence of other factors, we would expect the two velocity profiles to be the same at that point in time within the oscillatory cycle.

The failure of peak velocity in oscillatory flow to reach the same value as peak velocity in Poiseuille flow (normalized at 1.0), is due clearly to inertial e ects. More accurately the di erence between the two is a function of the nondimensional frequency parameter Ω as defined in Eq. 8.5.12, also known as the Womersley number, which combines the e ects of frequency of oscillation, tube radius, and both the density and viscosity of the fluid. The di erences observed in Fig. 8.5.1 occur with Ω = 2.0. In particular, the velocity profiles at peak pressure gradient, which occurs at ωt = 0, is short of maximum Poiseuille flow velocity at this value of Ω. Results at higher and lower values of Ω are shown in Fig. 8.5.2.

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Fig. 8.5.1. Velocity profiles of oscillatory flow in a rigid tube, driven by a pressure gradient of the form ks cos ωt where ks is a constant, ω is frequency and t is time. The profiles are shown at di erent times within the oscillatory cycle, as indicated by the value of the phase angle ωt (in degrees) on each curve. For comparison, the bold curve represents the velocity profile in Poiseuille flow driven by a pressure gradient ks. Velocities are normalized so that their values are comparable on the same scale. The di erences between the oscillatory profiles at di erent points in time within the oscillatory cycle and the Poiseuille flow profile indicate that peak velocity in oscillatory flow is lower than that in Poiseuille flow and occurs at a later time than peak pressure gradient, which occurs at ωt = 0.

and normalizing in terms of the corresponding flow rate in Poiseuille flow

we have