Pulsatile flow, tube

In addition, the effects of pulsatile flow cannot be ignored. One measure of the impact of oscillary flow is the Wcmersley parameter (a) a= h/2tt f/v where r is the tube radius, f the frequency of oscillation and v is the kinematic viscosity of the fluid (Wcmersley, 1955). The degree of departure from parabolic flow increases with and frequency effects may become important in straight tubes when a > 1 (Ultman, 1985). For conditions of these experiments, a exceeds one to beyond the third generation. 

Laminar pulsatile flow in a tube Flow in a tube is in one direction, parallel to the electrode surface, (taken as the jr-direction). The time-dependent convective-diffusion equation for this geometry is given by equation (10.12). Mass transport to the surface of the electrode is thus determined both by the gradient perpendicular to the surface of the tangential flow, dujdy and the concentration gradient perpendicular to the surface ... 

Polystyrene can be easily prepared by emulsion or suspension techniques. Harkins (1 ), Smith and Ewart(2) and Garden ( ) have described the mechanisms of emulsTon polymerization in batch reactors, and the results have been extended to a series of continuous stirred tank reactors (CSTR)( o Much information on continuous emulsion reactors Ts documented in the patent literature, with such innovations as use of a seed latex (5), use of pulsatile flow to reduce plugging of the tube ( ), and turbulent flow to reduce plugging (7 ). Feldon (8) discusses the tubular polymerization of SBR rubber wTth laminar flow (at Reynolds numbers of 660). There have been recent studies on continuous stirred tank reactors utilizing Smith-Ewart kinetics in a single CSTR ( ) as well as predictions of particle size distribution (10). Continuous tubular reactors have been examined for non-polymeric reactions (1 1 ) and polymeric reactions (12.1 31 The objective of this study was to develop a model for the continuous emulsion polymerization of styrene in a tubular reactor, and to verify the model with experimental data. 

The velocity profile across the tube lumen with pulsatile flow is not of the same parabolic form as that found in a steady laminar flow. The velocity profiles oscillate sinusoidally as discussed in detail by Hale et al. [44]. For example. Figure 8.26 shows the velocity profiles, at intervals of 15°, resulting from a simple sinusoidal pressure gradient (cos[mf]) during the half cycle (0°-180°) as for a simple harmonic motion, the second half is the same. 

Many researchers have assessed the effect of pulsatile flow on different membrane processes with wide range of feeds. One of the first studies was by Kennedy et al. [48] who showed that flux in the RO of sucrose solution could increase by 70% by pulsatile flow at 1 Hz. Gupta et al. [49] reported a 45% enhancement of flux in MF of raw apple juice with a pressure waveform provided by a fast piston return followed by a fast forward stroke at 1 Hz. Jaffrin [50], using hollow fiber filters, demonstrated a 45% enhancement in flux in plasma filtration. Using the collapsible-tube oscillation generator described above, Bertram et al. [47] demonstrated that pulsation resulted in a 60% increase in permeate flux in the filtration of silica suspensions. 

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, .

Pulsatile flow in an elastic vessel is very complex, since the tube is able to undergo local deformations in both longitudinal and circumferential directions. The unsteady component of the pulsatile flow is assumed to be induced by propagation of small waves in a pressurized elastic tube. The mathematical approach is based on the classical model for the fluid-structure interaction problem, which describes the dynamic equilibrium between the fluid and the tube thin wall (Womersley, 1955b Atabek and Lew, 1966). The dynamic equilibrium is expressed by the hydrodynamic equations (Navier-Stokes) for the incompressible fluid flow and the equations of motion for the wall of an elastic tube, which are coupled together by the boundary conditions at the fluid-wall interface. The motion of the liquid is described in a fixed laboratory coordinate system (f , 6, f), and the dynamic... 

Young DF, Rogge TR, Gray TA, Rooz E (1981) Indiret evaluation of system parameters of pulsatile flow in flexible tubes. J Biomech 14 339-347... 

Skalank, KM., Wang, C.Y., 1977. Pulsatile flow in tube with wall injection and suction. Appl. Sci. Res. 33, 269-307. 

Because of the mathematical complexity of analysis of pulsatile flows in elastic tubes, and the variety of physical phenomena associated with them, space does not permit a full description of this topic. The reader is instead referred to a number of excellent references for a more complete treatment [ 18-20]. Here we only briefly summarize the most important features of these flows, to give the reader a sense of the richness of the physics underlying them. 

FIGURE 7.3 Local bulging of the tube wall at regions of high pressure in pulsatile flow in an elastic tube. 

Qiu, Y. and TarbeU, J., Numerical simulation of pulsatile flow in a comphant curved tube model of a coronary artery, /. Biomech. Eng., 122,77-85,2000. 

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