Fig. 17.5 may be derived. According to this equation, for resistances in parallel, the reciprocal of the total resistance (Rt) equals the sum of the reciprocals of the individual resistances; that is,
Equation 17.8
In a few simple illustrations, some of the fundamental properties of parallel hydraulic systems become apparent. For example, if the resistances of the three parallel elements in
Fig. 17.6 were all equal, then
Equation 17.9
Therefore, from
Eq. 17.8,
Equation 17.10
When the reciprocals of these terms, are equated,
Equation 17.11
Thus the total resistance is less than the individual resistances. For any parallel arrangement, the total resistance must be less than that of any individual component. For example, consider a system in which a tube with very high resistance is added in parallel to a low-resistance tube. The total resistance of the system must be less than that of the low-resistance component by itself because the high-resistance component affords an additional pathway, or conductance, for flow of fluid.
Consider the physiological relationship between the total peripheral resistance (TPR) of the entire systemic vascular bed and the resistance of one of its components, such as the renal vasculature. TPR is the ratio of the arteriovenous (AV) pressure difference (arterial pressure [Pa] − venous pressure [Pv]) to the flow through the entire systemic vascular bed (i.e., the cardiac output [Qt]). For example, the renal vascular resistance (Rr) would be the ratio of the same AV pressure difference (Pa − Pv) to renal blood flow (Qr).
In an individual with an Pa of 100 mm Hg, a peripheral Pv of 0 mm Hg, and a cardiac output of 5000 mL/min, TPR is 0.02 mm Hg/mL/minute, or 0.02 peripheral resistance units (PRUs). Normally, the rate of blood flow through one kidney would be approximately 600 mL/ minute. Renal resistance would therefore be 100 mm Hg ÷ 600 mL/minute, or 0.17 PRUs, which is 8.5 times greater than the TPR. In an organ such as the kidney, which weighs only approximately 1% as much as the whole body, the vascular resistance is much greater than that of the entire systemic circulation. Hence, it is not surprising that the resistance to flow would be greater for a component organ, such as the kidney, than for the entire systemic circulation because the systemic circulation has not only one kidney but also many more alternative pathways for blood to flow.
Fig. 17.3A ), a thin layer of fluid in contact with the tube wall adheres to the wall and hence is motionless. The layer of fluid just central to the external lamina must shear against this motionless layer, and therefore that layer moves slowly but with a finite velocity. Similarly, the next more central layer moves still more rapidly; the longitudinal velocity profile is that of a paraboloid (see
Fig.
17.3A ). The fluid elements in any given lamina remain in that lamina as the fluid moves longitudinally along the tube. The velocity at the center of the stream is maximal and equal to twice the mean velocity of flow across the entire cross-section of the tube.
Irregular motions of the fluid elements may develop in the flow of fluid through a tube; such flow is called turbulent. In this condition, fluid elements do not remain confined to a specific laminae; instead, rapid, radial mixing occurs (see
Fig. 17.3B ). Greater pressure is necessary to force a given flow of fluid through the same tube when the flow is turbulent than when it is laminar. In turbulent flow, the pressure drop is approximately proportional to the square of the flow rate, whereas in laminar flow, the pressure drop is proportional to the first power of the flow rate. Hence, to produce a given flow, a pump such as the heart must do considerably more work if turbulent flow develops.