M

an jpvn

According to the no slip condition, velocity at the wall should be zero. That location is also defined by r = R, since r = 0 corresponds to the centerline of the vessel. In order to solve for the constant Cx use the boundary condition of u = 0 at r = R.

an jpvn

jpwnJ0(AR)

The total solution to the differential equation, for harmonic n, now becomes fn(r)

an jpvn

This is still the complex form that is time independent. If we take the pressure gradient as the real part of anejvnt and substitute u from Eq. (7.35), the corresponding velocity as a function of r and t becomes

1 f eJvnt

Recall that this solution applies to the results of each harmonic. Now to find the velocity as a function of radius r and time t for the entire driving pressure, add together the results from all harmonics.

Figure 7.3 shows an example plot of the total velocity u(r) for two different r values in a mathematical model of an artery under pulsatile driving pressure. To get the flow resulting from these velocity profile, a n

Figure 7.3 A graph of axial velocity as a function of radius and time, where r is the radius variable and R is the radius of the artery.

Figure 7.3 A graph of axial velocity as a function of radius and time, where r is the radius variable and R is the radius of the artery.

one needs only to integrate the velocities multiplied by the differential area, over the entire cross-section. The differential area as a function of r is 2prdr, so the flow term becomes

Wormersley (1955) published his solution when it was not so easy to make up a computer model of the flow. He integrated the velocity terms, solved for flow and published the following equation for the flow component resulting from each of the driving pressure gradient harmonics:

The total flow Q, which can be compared to Q from Eq. (7.55), is

where the pressure gradient associated with each harmonic is

The magnitude of the driving pressure is given by

The angle of the phase shift is given by depending on the quadrant fn tan

Wormersley compiled the values of the constants for M10/a2 and for eio as a function of the alpha parameter. As alpha approaches zero, M10/a2 approaches 1/8 and eio approaches 90° and the solution becomes Poiseuille's law. Recall from Chap. 1 that a is the Wormersley number, or alpha parameter, which is a ratio of transient to viscous forces, and is defined by

Figure 7.4 shows a velocity waveform plotted in terms of time and vessel radius for a pulsatile flow condition in the uterine artery of a cow.

7.7 Pulsatile Flow in Rigid Tubes: Fry Solution

This solution was published by Greenfield and Fry (1965) for axisym-metric, uniform, fully developed, horizontal, newtonian, pulsatile flow and is comparable to the Wormersley solution in the sense that they are

25 2

Figure 7.4 A three-dimensional plot that shows a flow waveform plotted in terms of time and vessel radius for a pulsatile flow condition. The curve was generated using Wormersley's version of the Navier-Stokes equation solution. The flow, Q, is shown in mm3/s, the time t is shown in seconds, the radius r in millimeters.

25 2

Figure 7.4 A three-dimensional plot that shows a flow waveform plotted in terms of time and vessel radius for a pulsatile flow condition. The curve was generated using Wormersley's version of the Navier-Stokes equation solution. The flow, Q, is shown in mm3/s, the time t is shown in seconds, the radius r in millimeters.

both solutions relating flow rate to pressure gradient for the pulsatile flow condition. Since flow in arteries is pulsatile, this is an important case for human medicine. We will find the Fry solution particularly useful in Chap. 8 in modeling the behavior of an extravascular catheter/transducer pressure measuring system. This solution lends itself nicely to the development of an electrical analog to the pulsatile flow behavior.

In this solution one may begin from the Navier-Stokes equations where u is velocity in the x-direction (down the tube centerline), r is the radial direction variable (where r = 0 on the centerline), vr is radial velocity, Vq is velocity in the transverse direction and n is m/p or the kinematic viscosity. Recall from Eq. (7.34) the differential equation for axisymmetic, uniform, fully developed horizontal, newtonian pulsatile flow.

Equation (7.34) can be rewritten in the form of Eq. (7.59).

dr2 r dr v dt m dx Two of the terms in Eq. (7.59) can be combined into

d2U 1 1 du dr2 r dr and be written from the chain rule as in Eq. (7.60).

Use the simplifying assumption that the pressure gradient dP

where / represents the length between points 1 and 2, and the governing differential equation from Eq. (7.59) can be rewritten:

If we multiply both sides of Eq. (7.61) by 2prdr and integrate from r = 0 to R where R is the tube radius, we get Eq. (7.62).

The integral of derivative in the left-hand side of Eq. (7.62) returns the term r(du/dr) so we can rewrite Eq. (7.62) as shown in Eq. (7.63).

2P 2

Notice that the integral from zero to R of u(2pr)dr is the integral of the velocity multiplied by the differential area so that the term is simply the flow through the tube. It is possible now to replace the first term on the right-hand side of the equation with the symbol Q, representing flow. After evaluating the last term between 0 and the vessel radius R we arrive at Eq. (7.64).

1 dQ

y dt

By dividing both sides of Eq. (7.64) by pR2 we arrive at Eq. (7.65)

Recall that for a newtonian fluid, the shear stress at the wall is equal to the viscosity times the velocity gradient as shown in Eq. (7.66).

Substitute Eq. (7.66) into Eq. (7.65) to obtain Eq. (7.67)

Next solve for the pressure gradient.

If we solved the same equation for Poiseuille flow, we could use Eq. to substitute for the shear stress at the wall. In that case the shear can be estimated from the flow rate as shown in Eq. (7.69).

8mQ 2pR3

After the replacement of the shear stress term, Eq. (7.68) for the case of Poiseuille flow would be

2 8mQ

Now we have an ordinary, first-order differential equation relating the flow rate and the time rate of change of flow to the pressure gradient. Recall from Eq. (7.69) that shear stress in Poiseuille flow depends on the vessel's hydraulic resistance and the flow rate. For the pulsatile flow case, assume that the shear stress depends on both the flow rate and the first derivative of the flow rate as shown by Eq. (7.72).

The values for Rv and Ljin Eq. (7.72) are then given by Eqs. (7.68) and (7.74), respectively.

Rv Rviscous CvttR4 {1-1S)

eq 7.72

Next, substituting the expressions for the viscous resistance term, Rv and the inertia term, Lj, yields the following equation. The two constants ci and cv are constants, which may be solved for empirically or predicted by the Wormersley equation.

Finally, it is possible to simplify the Fry solution to the following first order ordinary differential equation with terms that represent fluid inertance and fluid resistance as shown in Eq. (7.78).

where

cv8m p R4

It is possible to solve empirically for cu and cv or to predict those values from the Wormersley solution. For an alpha parameter of 7, cu = 1.1 and cv = 1.6.

7.8 Instability in Pulsatile Flow

It is helpful here to point out some useful information concerning stability and instability under pulsatile flow conditions. As a rule, flow in large arteries in humans is highly pulsatile. By the time the flow reaches small diameter arteries and arterioles the pulsatile elements drop out. By the time the flow reaches the capillaries, the flow becomes steady. Flow in large diameter veins becomes pulsatile once again.

The Reynolds number for flow in humans, even in large diameter arteries, is generally much less than 2000 and is therefore typically laminar. However, the presence of branching and other fluid wall interactions can sometimes result in local flow instabilities.

When there are disturbances throughout the flow field and throughout the entire oscillatory cycle, this condition constitutes turbulent flow. The presence of vortices at a specific location, or during a specific time in the pulsatile cycle, indicate a local instability only and not turbulent flow. For example, an instability associated with a valvular stenosis is an example of flow that is considered local flow instability as opposed to turbulent flow.

Bibliography

Munson BR, Young DF, Okiishi T. Fundamental of Fluid Mechanics. Wiley; 1994. Greenfield JC, Fry DL. Relationship between instantaneous aortic flow and the pressure gradient. Circ. Res. October 1965; Vol. XVII:340-348.

Rainville ED, Bedient PE. Elementary Differential Equations, 5th ed. New York: Macmillan; 1974.

Del Toro V. Principles of Electrical Engineering, 2nd ed. Englewood Cliffs, NJ Prentice-Hall; 1972.

Wormersley JR. Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 1955, 127, 553-563.

Chapter 8

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