Fig. 17.3A ). In the case of laminar flow through a tube, the fluid consists of a series of infinitesimally thin concentric tubes sliding past one another, of which the central tube has the highest velocity. The velocities of the concentric laminae decrease parabolically towards the vessel wall. Despite the differences between the vascular system (i.e., flow is pulsatile, the vessels are not rigid cylinders, and blood is not a newtonian fluid), Poiseuille’s law does provide valuable insight into the determinants of blood flow through the vascular system. In certain unusual situations, however, flow can become turbulent (see
Fig. 17.3B ), rather than laminar. Under these conditions, vortices (swirls) are present, and the distribution of flow velocities is chaotic. This condition is described in more detail later in this chapter.
Poiseuille’s law describes the laminar flow of fluids through cylindrical tubes in terms of pressure, the dimensions of the tube, and the viscosity of liquid:
Equation 17.3
Pi − Po = pressure gradient from the inlet (i) of the tube to the outlet (o) r = radius of the tube η= viscosity of the fluid l = length of the tube
As is clear from the equation, flow through the tube increases as the pressure gradient is increased, and it decreases as either the viscosity of the fluid or the length of the tube increases. The radius of the tube is a critical factor in determining flow because it is raised to the fourth power.
Resistance to Flow
In electrical theory, Ohm’s law is that the resistance (R) equals the ratio of voltage drop (E) to current flow (I).
Equation 17.4
Similarly, in fluid mechanics, hydraulic resistance (R) may be defined as the ratio of the pressure drop (Pi − Po,) to flow (Q):
Equation 17.5
For the steady, laminar flow of a newtonian fluid through a cylindrical tube, the physical components of hydraulic resistance may be appreciated by the rearranging of Poiseuille’s law to yield the hydraulic resistance equation:
Equation 17.6
Thus when Poiseuille’s law applies, the resistance to flow depends on only the dimensions of the tube and the characteristics of the fluid.
The principal determinant of resistance to blood flow through any vessel is the caliber of the vessel because resistance varies inversely as the fourth power of the radius of the tube. In
Fig. 17.4 , the resistance to flow through small blood vessels is measured, and the resistance per unit length of vessel (R/l) is plotted against the vessel diameter. As shown, resistance is highest in the capillaries (diameter of 7 µm), and it diminishes as the vessels increase in diameter 1000 100 10 1 0.1 0.01 58 5850 5042 4234 3426 2618 187 Vessel diameter (µm) R/l(mm Hg/[mm3/sec])/µm Capillaries Arterial side Venous side •Fig. 17.4 Resistance Per Unit Length (R/l) of Individual Small Blood Vessels. Thecapillaries,withadiameterof7µm,aredenotedbythevertical dashed line. Resistancesofarteriolesareplottedtotheleftoftheverticaldashedline,andresistancesofvenules,totherightoftheverticaldashedline.Forbothtypesofvessels,theresistanceperunitlengthisinverselyproportionaltothefourthpowerofthevesseldiameter.(RedrawnfromLipowskyHH,etal.Circ Res. 1978;43:738.) on the arterial and venous sides of the capillaries. Values of R/l are virtually inversely proportional to the fourth power of the diameter (or radius) of the larger vessels on both sides of the capillaries.