# 511 does not take into account the viscous coupling

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• rjgcabrera
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defined by Eq.5.1.1 does not take into account viscous coupling between fluid and wall movements. Hence it is referred to as the inviscid wave speed, since the viscous shear force exerted by the moving fluid is being neglected, as if the fluid were inviscid. The wave speed c in an elastic tube, as defined by Eq.5.7.21, does take viscous effect into account since it is obtained by solving the coupled equations. In fact, as may be expected on physical grounds, viscosity reduces the speed with which the wave propagates in an elastic tube, as can be observed from Fig.5.7.1. 11. For n = 1, we read from Appendix A = -0.7071 + 0.7071i Jo(A) = 0.9844 + 0.2496i J1 (A) = -0.3959 + 0.3076i G(A) = 1.0049 -0.0014i Using Eqso4.7.7, 5.7.20, 5.9.9, we 9 = 0.9798 -0.1216i I :: I (rigid) = 0.9860 I :: I (elastic) = 0.9963 difference ~ 1 Repeating for n = 3 we 9 = 004990 -0.3599i I :: I (rigid) = 0.5484 I :: I (elastic) = 0.6252
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difference ~ 14% and for n = 10 we find 9 = 0.1416 -0.1312i I ~: I (rigid) = 0.0695 I ~: I (elastic) = 0.0721 difference ~ 4% Appendix B: Solutions to Problems 211 The results agree visually with those in Fig.5.9.9. Chapter 6 1. Wave reflections in a tube require the presence of a propagating along the tube. In the case of a rigid tube there is nO wave propaga-tion along the tube, as was seen in Chapter 4, therefore there wave reflections under the conditions studied in that chapter. A way in which wave propagation and wave reflections can occur in a rigid is when the fluid within the tube is compressible. In that case compress-ibility of the fluid takes the place of elasticity of the tube wall phenomenon unfolds in essentially the same way. 2. The ultimate purpose of pulsatile flow analysis is to determine lationship between flow and pressure gradient, as was done for steady flow in Chapter 3. The same was done for pulsatile flow in Chapters and 5, but the results there are only valid in the absence of wave reflec-tions. When wave reflections are present, the relation between flow pressure gradient can no longer be obtained from a single solution of governing equations. All sources of wave reflections may have an effect On that relation and must be considered along with a solution of governing equations. From the standpoint of physics, and stated ply, the flow in a tube is then no longer determined by only the pressure gradient. 3. While the final form of the one-dimensional wave equations (Eqs.6.2.18, 19) does not contain the radial coordinate r, the equations contain wave speed Co, which depends On the cross sectional area of (Eq.6.2.14) and hence On the tube radius. Furthermore, the wave equa-tion for the pressure contains the time rate of change of pressure, which in turn depends On the time rate of change of cross-sectional hence of tube radius. Thus while the radial coordinate r has been inated, the tube radius and its time rate of change remain implicitly in the equations.
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212 Appendix B: Solutions to Problems 4. From Eq.6.3.6, by differentiation, we have dpx = _ iw Be-iwx/co + iw Ceiwx/co dx Co d2p w2 w2 __ x = __ Be-iwx/co __ dx2 c~ Substituting in Eq.6.3.5 we have d2px w2 w2 -d 2 + 2""Px(x) --c2
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