HANDBOOK OF PHYSIOLOGY -^ CIRCULATION I 



4. VISCOSITY OF FILLING LIQ^UID AND ITS INFLUENCE 

 ON DAMPING AND SPEED OF PROPAGATION 



In the previous section we ha\e introduced into 

 the theoretical treatment the visco-elastic behavior 

 of tiie tube walls and ignored the energy loss in the 

 viscous filling fluid. The fact that the theory fits in 

 general, even at this stage of development, with our 

 experience and with the earlier findings of A. Miiller 

 [(12) who made determinations in 1951 on propaga- 

 tion and damping of pressure waves in rubber tubes 

 of similar dimensions with fluids of different viscosities], 

 indicates that liquid viscosity cannot have very much 

 influence, at least on the speed of propagation. 

 Nevertheless, we shall complete the theory in this 

 direction and compare the results with experience. 



For this purpose, we complete equation i .9 with a 

 term which accounts for a resistance to flow in the 

 .^-direction and obtain 



dp 



dz 



d-Z ^ d^ 



p 77 + -"^ :r 

 di- at 



(4-0 



The procedure of calculation is much the same as in 

 the previous section, and we shall give merely the 

 result of it. For the quantities /3 and a we have the 

 following equations 





+ a>=C2) 



j2C2) 



(4-2) 



where C stands for t)a'(2/p). Solving for a and (i 

 we obtain 



0)2 



a' = — = a,'-(;V + oj-C')- 



-(--77')1"-(-7) 



(4-3) 



In order to discuss the influence of the newly arising 

 terms Rovd'/ipu) and CRJ p we must first determine 

 the frictional constant R^ . According to equation 

 4.1 the quantity 7?2 </f/(// = iTJs/Q, is equal to a certain 

 gradient of pressure —dp'/dz- If we suppose that 

 Poiseuille's law holds, even for pulsating current, we 

 have / = —{clp'/dz)Trr*/(8rjj), where rji stands for the 



viscositv of the liquid. It follows from this with 

 0,= r-T 



8,1 



R, = 



(44) 



However, as we shall see later, Poiseuille's law does 

 not hold for pulsating flow, and formula 4.4 can only 

 be used for very low frequencies. Let us take as an 

 example the thick-walled tube with r = 1.02 cm, 

 a = 0.193 cm, and vu = 880 cm sec. For rji , we 

 take the value 0.078 poises, which we obtain for a 

 glycerine-water mixture of density 1.14. With this we 

 obtain the limiting value R-2 = 0.6. From experiments 

 and calculations, which we shall describe later on, 

 we obtain instead of 4.4 the semi-empirical formula 



R2 = 0.6 -|- a)o.o6 = 0.6 4- J/-0.377 



14-5) 



For a frequency of 10 cps, which corresponds to the 

 highest harmonic which might be safely evaluated 

 by harmonic analysis of pulse curves, R2 would be 

 4.37. The term R^vo'/in^p) would then be 4.37-880-/ 

 (62.8- 1. 14) = 4.27-10''. The quantity C = wr\a/2rp 

 becomes then 14.0- 10^-0.193/(20.4- 1. 1 4) = 11. 6-10* 

 if we take for cot; the value 14-10^ previously used. 

 The additional term RiVo'/iup) is therefore about 

 half as great as wC. With the small frequency of 2 

 cps (pulse frequency of a normal dog) R-i = 0.6 -|- 

 47r-o.o6 = 0.6 + 0.725 = 1.352, and R-iVa-fioip) 

 becomes i .35-88o'/(i2.56- 1.14) = 7.32 ■ lo'', whereas 

 oiC becomes 2.32-10^. The additional term is there- 

 fore about three times larger than oiC. It remains 

 for us to consider the term C-R-> p. Because uit) is 

 approximately constant, we write it in the form 

 uCRi/iwp) and obtain for it: 1 1.6- lo^- 1.352/ (12.56- 

 1.14) = i.og-io'' for the small frequency of 2 cps. 

 It is, therefore, small even for this frequency, and 

 certainly for higher frequencies it is small compared 

 to Vo- = 77.5- lo''. We do not know much about the 

 behavior of toC at very small frequencies below i cps. 

 We might assume that it approaches o because 

 otherwise a = 01/v w-ould approach infinity and v 

 would approach zero, which is certainly not the case. 

 Furthermore, we might be justified in supposing that 

 damping also vanishes when the frequency approaches 

 zero. Table i shows the values for v and calculated 

 from equations 4.3. It can be seen from the table that 

 liquid damping at a frequency of 2 cps has only a 

 slight influence, and at a frequency of 10 cps prac- 

 tically no influence, on the velocity of propagation. 

 This is in agreement with the results obtained by A. 

 Miiller (12) using a glycerine-water mixture with a 

 density of 1.1B8 and a viscosity of 0.256 poises and a 



