OF THE CONDUITS
The most fundamental law that governs the flow of flu-ids through cylindrical tubes was derived empirically by the French physiologist Jean Poiseuille. He was pri-marily interested in the physical determinants of blood flow, but he replaced blood with simpler liquids for his measurements of flow through glass capillary tubes.
His work was so precise and important that his obser-vations have been designated Poiseuille’s law. Subse-quently, this same law has been derived theoretically.
Poiseuille’s law is applicable to the flow of fluids through cylindrical tubes only under special condi-tions, namely, in the case of steady, laminar flow of newtonian fluids. The term steady flow signifies the absence of variations of flow in time, that is, a nonpul-satile flow. Laminar flow is the type of motion in
which the fluid moves as a series of individual layers, with each stratum moving at a different velocity from its neighboring layers (Figure 6-5). In the case of flow through a tube, the fluid consists of a series of infini-tesimally thin concentric tubes sliding past one another. Laminar flow is described in greater detail later, where it is distinguished from turbulent flow.
Also, a newtonian fluid is defined more precisely. For the present discussion, it may be considered to be a homogeneous fluid, such as water, in contradistinc-tion to a suspension, such as blood.
Pressure is one of the principal determinants of the rate of flow. The pressure, P, in dynes/cm2, at a dis-tance h, in centimeters, below the surface of a liquid is:
P= hρg (4)
where ρ is the density of the liquid in g/cm3 and g is the acceleration of gravity in cm/s2. For convenience, however, pressure is frequently expressed simply in
Proximal sensor Distal sensor
Aortic
valve Mitral
valve
PLV
PAVO
100 mm Hg
LV-LV LV-AVO
PLV
PAO
LV-AO
A B C
FIGURE 6-4 n Pressures (P) recorded by two transducers in a patient with aortic stenosis.
A, Both transducers were in the left ventricle (LV-LV). B, One transducer was in the left ventricle and the other was in the aortic valve orifice (LV-AVO). C, One transducer was in the left ventricle and the other was in the ascending aorta (LV-AO). Pao, pressure in ascending aorta; Pavo, aortic valve orifice; Plv, pressure in left ventricle. (Redrawn from Pasipoularides A, Murgo JP, Bird JJ, et al: Fluid dynamics of aortic stenosis: mechanisms for the pres-ence of subvalvular pressure gradients. Am J Physiol 246:H542, 1984.)
terms of the height of the column of liquid above some arbitrary reference point.
Consider the tube connecting reservoirs R1 and R2 in Figure 6-6A. Let reservoir R1 be filled with liquid to height h1, and let reservoir R2 be empty, as in Figure 6-6A. The outflow pressure, Po, is therefore equal to the atmospheric pressure, which shall be designated as the zero, or reference, level. The inflow pressure, Pi, is then equal to the same reference level plus the height, h1, of the column of liquid in reservoir R1. Under these conditions, let the flow, Q, through the tube be 5 mL/s.
If reservoir R1 is filled to height h2, which is twice h1, and reservoir R2 is again empty (as in panel B), the flow is twice as great, that is, 10 mL/s. Thus with reser-voir R2 empty, the flow is directly proportional to the inflow pressure, Pi.
If reservoir R2 is now allowed to fill to height h1, and the fluid level in R1 is maintained at h2 (as in panel C), the flow again becomes 5 mL/s. Thus, flow is directly proportional to the difference between inflow and outflow pressures:
Q ∝ Pi − Po (5)
FIGURE 6-5 n In laminar flow, all elements of the fluid move in streamlines that are parallel to the axis of the tube;
movement does not occur in a radial or circumferential direction. The layer of fluid in contact with the wall is motionless; the fluid that moves along the axis of the tube has the maximal velocity.
FIGURE 6-6 n A to D, The flow, Q, of fluid through a tube connecting two reservoirs, R1 and R2, is proportional to the differ-ence between the pressure at the inflow end (Pi) and the pressure at the outflow end (Po) of the tube. h1 and h2, heights of liquid in the reservoir.
h2
Q = 10 mL/s
Q = 0 mL/s
R1 R2
h2 = 2h1 Pi Po
Pi Po R2
h2
h2 R1
B When the fluid level in R1 is increased twofold, the flow increases proportionately.
D When pressure in R2 rises to equal the pressure in R1, flow ceases in the connecting tube.
Q = 5 mL/s
Q = 5 mL/s
R1 R2
Pi Po Pi Po
R2
h1
h2 R1
A When R2 is empty, fluid flows from R1 to R2 at a rate
proportional to the pressure in R1.
C Flow from R1 to R2 is proportional to the difference between the pressures in R1 and R2.
h1
If the fluid level in R2 attains the same height as in R1, flow ceases (panel D).
For any given pressure difference between the two ends of a tube, the flow depends on the dimensions of the tube. Consider the tube connected to the reservoir in Figure 6-7A. With length l1 and radius r1, the flow Q1 is observed to be 10 mL/s.
The tube connected to the reservoir in panel B has the same radius but is twice as long. Under those con-ditions the flow Q2 is found to be 5 mL/s, or only half as great as Q1. Conversely, for a tube half as long as l1, the flow would be twice as great as Q1. In other words, flow is inversely proportional to the length of the tube:
Q ∝ 1 / l (6)
The tube connected to the reservoir in Figure 6-7C is the same length as l1, but the radius is twice as great.
Under these conditions, the flow Q3 is found to increase to a value of 160 mL/s, which is 16 times greater than Q1. The precise measurements of Poiseuille revealed
that flow varies directly as the fourth power of the radius:
Q ∝ r4 (7)
Because r3 = 2r1 in the example above (Figure 6-7C), Q3 will be proportional to (2r1)4, or 16r41; therefore Q3 will equal 16Q1.
Finally, for a given pressure difference and for a cylindrical tube of given dimensions, the flow varies as a function of the nature of the fluid itself. This flow-determining property of fluids is termed viscosity, η, which Newton defined as the ratio of shear stress to the shear rate of the fluid. Those fluids for which the shear rate is proportional to the shear stress are known as newtonian fluids. If the shear rate is not propor-tional to the shear stress, the fluid is nonnewtonian.
These terms may be comprehended more clearly if one considers the flow of a homogeneous fluid between parallel plates. In Figure 6-8, let the bottom plate (the bottom of a large basin) be stationary, and let the upper plate move at a constant velocity along the upper surface
FIGURE 6-7 n A to D, The flow, Q, of fluid through a tube is inversely proportional to the length, l, and the viscosity, η, and is directly proportional to the fourth power of the radius, r. h1 and h2, heights of liquid in the reservoir.
Q1 = 10 mL/s r1 h1
l1
Q ∼ r4 l3 = l1
Q3 =
r3 = 2r1
160 mL/s
A Reference condition: for a given pressure, length, radius, and viscosity, let the flow (V1) equal 10 ml/s.
C If tube radius doubles, flow increases 16-fold.
Q ∼
l2 = 2l1 r2 = r1
Q2 = 5 mL/s
η4 = 2η1
Q4 = 5 mL/s r4 = r1 Q ∼
l4 = l1
B If tube length doubles, flow decreases by 50%.
D If viscosity doubles, flow decreases by 50%.
1 I
η1
of the fluid. The shear stress, η, is defined as the ratio of F:A, where F is the force applied to the upper plate in the direction of its motion along the upper surface of the fluid, and A is the area of the upper plate in contact with the fluid. The shear rate is du/dy, where u is the velocity of a minute element of the fluid in the direction parallel to the motion of the upper plate, and y is the distance of that fluid element above the bottom, stationary plate.
For a movable plate traveling at constant velocity across the surface of a homogeneous fluid, the velocity profile of the fluid will be linear. The fluid layer in contact with the upper plate will adhere to it and therefore will move at the same velocity, U, as the plate. Each minute element of fluid between the plates will move at a velocity, u, proportional to its distance, y, from the lower plate. Therefore the shear rate will be U/Y, where Y is the total distance between the two plates. Because viscosity, η, is defined as the ratio of shear stress, τ, to the shear rate, du/dy, in the example illustrated in Figure 6-8:
η = (F / A) / (U / Y) (8)
Thus the dimensions of viscosity are dynes/cm2 divided by (cm/s)/cm, or dynes•s/cm2. In honor of Poiseuille, 1 dyne•s/cm2 has been termed a poise. The viscosity of water at 20° C is approximately 0.01 poise, or 1 centipoise.
With regard to the flow of newtonian fluids through cylindrical tubes, the flow varies inversely as the vis-cosity. Thus in the example of flow from the reservoir in Figure 6-7D, if the viscosity of the fluid in the
reser-voir were doubled, the flow would be halved (5 mL/s instead of 10 mL/s).
In summary, for the steady, laminar flow of a newto-nian fluid through a cylindrical tube, the flow, Q, varies directly as the pressure difference, Pi − Po, and the fourth power of the radius, r, of the tube, and it varies inversely as the length, l, of the tube and the viscosity, η, of the fluid. The full statement of Poiseuille’s law is
Q = π (Pi − Po)r4/ 8η l (9)
where π/8 is the constant of proportionality.