Circular tube pulsatile flow

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 ... 

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... 

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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