140 HANDBOOK OF PHYSIOLOGY -^' CIRCULATION I 



Turbulence 



The assumption that the flow is laminar is justified 

 onlv so long as the mean linear velocity of the liquid 

 is not too large. The critical quantity concerned is 

 the "Reynolds Number" {Re), which is defined as: 



Re = 



uap 



(7) 



where the quantity t]/ p is the "kinematic viscosity." 

 (Some authors use the diameter of the tube in place 

 of the radius, a; their values of Re are then twice as 

 large as those given here.) \i Re is small, the flow will 

 be laminar, and if Re is large, the flow will be turbu- 

 lent; but it is not possible to state some precise value 

 of Re below which the flow will certainly be laminar 

 and above which it will certainly be turbulent. The 

 smooth rectilinear motion of the liquid in the tube is 

 likely to be disturbed if the liquid in the reservoir is 

 in motion before it enters the tube; if the end of the 

 tube is cut off sharply and does not open out gradually 

 in a bell-mouth; and if there are discontinuities, 

 sharp bends, or branches in the tube. The greater the 

 value of Re, the more likely it is that turbulence will 

 start. If Re is greater than about 200, turbulence may 

 occur at branches of a system of tubes, for example; 

 but if it is less than 1000, this turbulence will not 

 persist but will die away as the liquid flows along the 

 tube, and the flow will be approximately laminar. If 

 Re is greater than 1000, turbulence is likely to occur 

 even in a single straight tube; although if special 

 precautions are taken to eliminate all the factors 

 which are likely to initiate turbulence, laminar flow 

 may persist up to values of Re of 10,000 or more. 



More complete treatments of the nature of fluid 

 flow and of the techniques of viscosity measurement 

 will be found in appropriate monographs, such as 

 that by Barr (2). 



TURBULENCE IN BLOOD. The Conditions determining 

 the onset of turbulence in blood are sensibly the same 

 as those which determine its onset in water. In a 

 smooth straight tube the critical value of Re is about 

 1000 (12). This value is not ordinarily attained in 

 any part of the vascular system of an animal except 

 the aorta. Values of several hundred may occur in 

 the larger arteries, so that transient turbulence may 

 be expected to occur at branches and sharp bends 

 as, indeed, has been observed (32). The greater part 

 of the fall in pressure, however, occurs in the smaller 

 arterioles, and it is most unlikely that turbulence will 

 occur in the.se. 



VISCOSITY OF SUSPENSIONS 



The effect on the viscosity of a liquid of inserting a 

 number of spherical particles, so as to make a .suspen- 

 sion, was studied theoretically by Einstein (14, 15). 

 Provided the particles are so far apart that the 

 motion of any one of them does not affect that of any 

 other, the viscosity of the suspension is directly 

 proportional to the total volume of particles in unit 

 volume of suspension, and is independent of their 

 size. Thus, if the viscosity of the suspending fluid 

 (which we will suppose to be w'ater) is ?)„,, a suspension 

 in which the volume of the particles in unit volume of 

 suspension (the volume fraction) is a will have a 

 viscosity i), gisen by: 



ijAu = 7;, = I + 2-5- 



(8) 



where rjs is the "relative viscosity" of the suspension. 



If the particles are not spherical, theoretical studies 

 by Jeffery (28) indicate that the coefficient 2.5 should 

 be replaced by a smaller figure; in the limiting case 

 of flat discs (to which we may approximate the red 

 blood cells) the expected figure is 2.061 (this assumes 

 that the flow takes place with a minimum dissipation 

 of energy). 



The simple Einstein relation between relative 

 viscosity and volume concentration is valid, not only 

 for suspensions but also for solutions, crystalloidal and 

 colloidal. But it ceases to hold even approximately if 

 the volume fraction is greater than about o.i (10%). 

 If the concentration is greater than this, the observed 

 relative viscosity is greater than that calculated, 

 owing to the interactions between the particles. If 

 one particle moves past another, each will exert a 

 drag on the other even if they do not actually come 

 into contact. If they collide, they may remain in 

 contact for varying periods of time, and this will have 

 two consequences: /) A group of 2, 3, or more particles 

 will enclose and "immobilize" a certain volume of the 

 suspending fluid; the effective \olume fraction of the 

 particles is thus greater than the measured volume 

 fraction, and the viscosity of the suspension is corre- 

 spondingly increased; and 2) a certain force may 

 have to be applied in order to drag the particles 

 apart again; the suspension will then have non- 

 Newtonian properties, as will be discussed in more 

 detail later. Theoretical studies by Guth & Simha 

 (2 1 ) and by Vand (39) show that the effect of these 

 interactions between the particles on the relative 

 viscosity of the suspension may be expressed by adding 

 to the Einstein equation terms in a-, a.^, etc. in an 

 infinite series. The coefficients of these higher powers 



