138 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



which depends only on the nature and temperature 

 of the Uquid, and is defined as its "viscosity." If a 

 shearing stress of i dyne per cm- produces a rate of 

 shear of i see"', the hquid has unit viscosity; the unit 

 being called the "poise," after Poiseuille. 



Newton in his Principia considered the force neces- 

 sary to move a solid through a liquid at a given veloc- 

 ity, and assumed as an hypothesis that the one was 

 directly proportional to the other. This implies that 

 the rate of shear is directly proportional to the applied 

 shearing stress, and thus that the viscosity of the 

 liquid is a constant, independent of either. But there 

 was no experimental justification for this until 

 Poiseuille made his extensive series of measurements 

 in 1840. Although he himself did not express his 

 results in this way, they showed that the viscosities 

 of water, of a number of other liquids, and of watery 

 solutions of crystalloidal substances were independent 

 of the shearing stress applied. Such fluids are thus 

 called "ideal" or "Newtonian" in respect of their 

 behavior when made to flow. Many kinds of colloidal 

 solution, howe\cr, and suspensions of particles which 

 are large enough to be visible (with or without a 

 microscope) do not obey the Newtonian assumption; 

 they are said to be "anomalous" or "non-Newtonian." 

 They do not have a definite coefficient of viscosity, 

 and the "apparent viscosity" in any partictilar con- 

 dition of shear is said to be "shear-dependent." Blood 

 is such an anomalous fluid, and its apparent viscosity 

 depends on the shearing stress applied. 



If it is possible to find some part of the liquid in 

 which the two parallel planes that are in relative 

 motion may have a finite area, the motion of the 

 liquid is said to be "laminar" in this region; in any 

 region where such planes can have only an infinitesi- 

 mal area, the motion is "turliuleni." Put in another 

 way, in laminar flow any small element of the fluid 

 travels in a straight line, and the course that it takes 

 is parallel to that taken by any other small element of 

 the fluid. In turbulent flow, elements of tiie fluid do 

 not, in general, travel in straight lines. Flow will be 

 laminar only if the rate of shear is less than some 

 critical value, and will become turbulent when the 

 rate of shear becomes large. 



FLOW IN TUBES 



In the study of circulatory phenomena in animals, 

 we are concerned almost entirely with the relation 

 between the pressure required to dri\e the blood 

 throueh tiie blood \essels and tiie voliunc rate of 



flow produced. This will be determined by the dimen- 

 sions of the vessels and by the apparent viscosity of 

 the blood in the conditions considered. But, owing to 

 the geometry of the system, certain complications 

 may arise. 



Suppose that we ha\e a rigid tube of circular cross 

 section. It is filled with a liquid of viscosity r; and 

 connected to a reservoir at each end; the fluid in the 

 reservoir at one end is under a hydrostatic pressure 

 Pi, and that in the reservoir at the other end is under 

 a pressure P>. Then the "pressure head'' which drives 

 the liquid through the tube is Pi — P-i = P. The 

 liquid is assumed to be incompressible, and the flow 

 to be laminar and steady, so that any element of the 

 liquid moves in a straight line parallel to the axis of 

 the tube with constant velocity. The pressure is then 

 uniform over the cross section of the tube at all dis- 

 tances from its ends (if it were not, some elements of 

 the liquid would have a velocity perpendicular to 

 the axis), and the fall in pressure per unit length ol 

 tube (the "pressure gradient") is constant from one 

 end to the other and has a value given by P/l, where 

 / is the length of the tube (if it were not, the velocity 

 parallel to the axis would not be constant). When a 

 liquid flows through a tube, the surfaces of uniform 

 \elocity will not be plane, as in the ideal conditions 

 just considered, but will be cylindrical; the shear 

 occurs between concentric circular sleeves. Consider 

 a column of liquid of radius r and unit length. The 

 force acting on its circular ends and driving it down 

 the tube will be Trr--P 7; the area of its surface, where 

 it drags against the fluid outside it, will be 27rr, so 

 that the shearing stress {t^) at this surface will be 

 given by: 



Tr = P-rhl 



(I) 



Thus the shearing stress increases from zero at the 

 axis of the tube to a maximum value at the wall of 

 the tube, the average \alue (t) being given by: 



T = P-a/4 



(2) 



where a is the radius of the tube. 



We now make two further assumptions: /) the 

 \elocity of the \ery thin layer of liquid which is in 

 contact with the wall of the tube is zero. (This 

 assumption, that the liquid does not "slip" along the 

 wall of the tuiae, is justified Ijy the most careful and 

 accurate measurements that have been made with 

 Newtonian fluids in riE;id tubes.) 2) The viscosity 

 (or apparent viscosity) of the liquid is the same at all 

 distances from the axis of the tul)e. We then find that 



