114 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



has been considerably enlarged, and it has been 

 found that no one of the simple mechanical models 

 made up of springs and dashpots can account for all 

 the properties of elastomers in a satisfactory manner. 

 This is especially the case for the dependence on 

 frequency. The best way to overcome this difficulty, 

 therefore, is to use the simplest model of a spring 

 damped by a dashpot connected in parallel to it, and 

 to introduce an eventual frequency dependence of 

 the restoring force (spring constant) and the friction 

 at any suitable step from experimental data. In this 

 v^fay we do not obtain a theory in the strict sense of 

 the word, but merely a set of semi-empirical formulas 

 of the kind used in engineering. 



To obtain these formulas, we have to complete 

 the equilibrium condition in the radial direction 



,-,„ = .:..(i-;) 



(3-0 



which is the integral form of equation 1.14, with a 

 frictional term Ri dr/dt, as we did for the spring 

 model in section 2 ; and we obtain 



p-p, = E-a{---] + R,% (3.2)' 



where pa and ro denote the equilibrium values of p 

 and r for the tube at rest. Differentiation with respect 

 to / leads to 



dp E-a dr dh 



- = h «i — 



dt r"- dl dfl 



(3-3) 



Introducing the radial current dir = 27rr {dr/dt)-dz 

 we obtain 



dp E-adi, Ri d'-ir 

 dl 2irr' dz ^itr dzdl 



or with the relation di^ = —di. 



dp 

 'dl 



E-a 



dz 



R^ d% 

 ■2in dzdt 



(3.4) 



(3.5) 



DifTerentiating again Newton's equation i.ii with 

 respect to / and equation 3.5 with respect to z, we 

 can eliminate the pressure and obtain with Q, = r% 



dH Ea dH Rj^ dH 

 dl- 2rp dz' 2p dz-dt 



(3.6) 



2 To avoid confusion, we mention that R, in equation 3.2 

 has a somewhat different meaning from R in equation 2.3 

 because /fi dr/dt has the dimension of a pressure, whereas 

 R dx/dt in equation 2.3 has the dimension of a force. 



We again dropped the index z for the current in the 

 ^-direction. For Ri = o, that is to say in the absence 

 of damping, equation 3.6 is, of course, identical with 

 equation 1.19. Equation 3.6 may be more concisely 

 written in the form 



d^i dH dH 



— = K„2 h C • — - 



dfi dz^ dz^dt 



(3-7) 



where vn is the speed of propagation for the undamped 

 case and C stands for Rir'{2p). For the pressure we 

 obtain the analogous equation 



dfi "" ' dz'' ' dz-di 



(3-8) 



For this equation it will be natural to try the solution 

 p = Pof-f'sinoill - -] (3.9) 



Much more convenient, however, in this case is the 

 use of complex numbers. Instead of equation 3.9 

 we write, then. 



p = Po exp 



_-'-(' -;)-'^^_ = 



Poe-y-e'" 



(3.10) 



With this we obtain from equation 3.8 



-o.'- = (6= - a- + 2ja0)(l',f + j'^C) (3.1 I) 



This complex equation may be split into two real 

 equations : 



u' + y(?(/52 - a-) - 2a;3a;C = O 

 a)C(/S- — a-) — 2al3i;,- = O 



and we obtain finally 



(32 _ „2 = -aiW/iv,.' + ^-C-} 

 2al3 = o>^-C/W + ofcr-) 



Elimination of a leads to 



(3.12) 



l.3-'3) 



{3>4) 



The choice of the right sign can be made using the 

 following reasoning: for vanishing resistance, that 

 is to say, for i? = o and therefore also for C = o, 

 the damping and therefore /3 must also vanish. This 

 means that we have to use the plus sign and we ob- 

 tain finally 



