Pulsatile Flow in a Circular Tube

We can now use the Taylor dispersion equation in either of the forms (3-244) or (3-245) to show that the cross-sectionally averaged temperature profile is a Gaussian in the z direction, with the peak concentration remaining at z = 0 (i.e., converting downstream relative to fixed coordinates at the mean velocity U). 

The reader who wishes to learn more about Taylor dispersion, and a large number of generalizations to other systems in which dispersion is produced by a coupling between convection and diffusion, may wish to consult the excellent book by Brenner and Edwards.22 

In the preceding sections of this chapter, we have considered several examples of transient unidirectional flows. In each case, it was assumed that the flow started from rest with the abrupt imposition of either a finite pressure gradient or a finite boundary velocity, and we saw that the flow evolved toward steady state by means of diffusion of momentum with a time scale tc = i2Jv. Here we consider a final example of a transient unidirectional flow problem in which time-dependent motion is produced in a circular tube by the sudden imposition of a periodic, time-dependent pressure gradient  

This problem of pulsatile flow in a circular tube has been studied extensively in the context of model studies of blood flow in the arteries,23 though it is considerably simpler than the 

We begin in this section by deriving the exact solution. Later, in the next chapter, we will reexamine the problem by using asymptotic methods of analysis. The governing DE is just (3-12), with the pressure-gradient function given by (3-249)  

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In this chapter, we discuss general concepts about asymptotic methods and illustrate a number of different types of asymptotic methods by considering relatively simple transport or flow problems. We do this by first considering pulsatile flow in a circular tube, for which we have already obtained a formal exact solution in Chap. 3, and show that we can obtain useful information about the high- and low-frequency limits more easily and with more physical insight by using asymptotic methods. Included in this is the concept of a boundary layer in the high-frequency limit. We then go on to consider problems for which no exact solution is available. The problems are chosen to illustrate important physical ideas and also to allow different types of asymptotic methods to be introduced ... 

A. PULSATILE FLOW IN A CIRCULAR TUBE REVISITED - ASYMPTOTIC SOLUTIONS FOR HIGH AND LOW FREQUENCIES... 

Figure 4-1. Velocity profiles for pulsatile flow in a circular tube for three different values of Rm all plotted at t = jr/2.

Figure 4-2. A schematic of the flow domain for pulsatile flow in a circular tube at very large values of Rm. In the core region, the velocity field is characterized by a length scale lc = R and the velocity field is dominated by inertia (acceleration) effects that are due to the time-dependent pressure gradient. In the near-wall region, on the other hand, the characteristic length scale for changes in velocity is much shorter, 0(RRZ1 2), and viscous effects remain important even for very large values of R, .

In regard to nonsteady tube flows. Mason et al. have observed both inward and outward radial migration of rigid, neutrally buoyant spheres in oscillatory (S9b, G9b) and pulsatile (Tl) flows in circular tubes at frequencies up to 3 cps, at which frequencies inertial effects are likely to be important. We refer here to inertial effects arising from the local acceleration terms in the Navier-Stokes equations, rather than from the convective acceleration terms. In the oscillatory case the spheres (a/R 0.10) attained equilibrium positions at about P = 0.85. Important Reynolds numbers here are those based upon mean tube velocity for one-half cycle and upon frequency. Nonneutrally buoyant spheres in oscillatory flow migrate permanently to the tube axis, irrespective of whether they are denser or lighter than the fluid (K4a). 

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