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Fluid Flowing in Tubes is Described by Ohm's Law

The equations that describe current flow in electrical circuits can also be adapted to describe fluid flow. By substituting flow for current and pressure for voltage, the analogy is almost exact. Ohm's law for electrical circuits states that the current through a device can be determined by dividing the voltage across it by the resistance of the device:

Current = A voltage/resistance

Ohm's law can be applied to the circulation as well. Note that the relationship among flow through a segment (Q), the difference in pressures (AP) across the segment, and the resistance to flow (R) across the segment given above in Eq. 7 is identical to Ohm's law, where Q is substituted for current and AP for A voltage.

A powerful feature of Ohm's law is that if one knows values for any two of the variables, the third can be calculated. Also, remember that R is taken directly from Poiseuille's equation above and includes terms for both the viscosity of the fluid and geometry of the vessel. For example, if flow is to be kept constant when the length of a tube is doubled, then the driving pressure (AP) must be twice as great (Fig. 8). It is important to remember that R (during laminar flow) is determined only by tube radius, length, and fluid viscosity and not by Q or AP. In the body, the length of the blood vessels or the viscosity of the blood cannot be easily changed from moment to moment. On the other hand, the radius of smooth muscle in the walls of the vessels is constantly changing. Because resistance varies according to the fourth power of the radius, small changes in radius have a profound effect on flow. Thus, if the radius of the tube is

FIGURE 8 Fluid is added at the same rate of flow to two containers that differ only in the lengths (l) of their outflow tubes such that the length in example 2 is twice that in example 1. The radii (r) of tubes 1 and 2 are equal. The column of water in each container will rise until the pressure exerted by that column is large enough to cause outflow to equal inflow. Because the resistance to flow in example 2 is twice that in example 1, column height will be twice as great in example 2 at steady state.

FIGURE 8 Fluid is added at the same rate of flow to two containers that differ only in the lengths (l) of their outflow tubes such that the length in example 2 is twice that in example 1. The radii (r) of tubes 1 and 2 are equal. The column of water in each container will rise until the pressure exerted by that column is large enough to cause outflow to equal inflow. Because the resistance to flow in example 2 is twice that in example 1, column height will be twice as great in example 2 at steady state.

FIGURE 9 Fluid is added to two containers that differ only in the radii of their outflow tubes. Because the radius of tube 2 is twice that of tube 1, the resistance to flow through tube 2 is 1/16 of that through tube 1. Thus, flow in tube 2 must be 16 times as great to maintain the same column height.

FIGURE 9 Fluid is added to two containers that differ only in the radii of their outflow tubes. Because the radius of tube 2 is twice that of tube 1, the resistance to flow through tube 2 is 1/16 of that through tube 1. Thus, flow in tube 2 must be 16 times as great to maintain the same column height.

doubled, Q will increase 16-fold for the same AP (Fig. 9). In the cardiovascular system, the dimensions of the tubes seldom are known well enough to calculate resistance directly. More often, both Q and AP are measured and R is calculated.

Flow is proportional to driving pressure only under laminar flow conditions. If the velocity of the fluid particles becomes too great, turbulence may develop. During turbulent flow the orderly pattern of flow is lost and the particles frequently change direction abruptly and move between lamina rather than along them. This results in additional energy loss and an increased resistance to flow. Generally, flow changes from laminar to turbulent as velocity increases. The physicist, Osborne Reynolds, empirically determined the factors that lead to turbulence and expressed them quantitatively in Reynolds equation:

where D is the diameter of the tube, v is the mean velocity, p is the density, and rç is the viscosity of the fluid. Turbulence can be expected whenever the Reynolds number, Re, exceeds a value of about 2000. It should be noted that some texts use radius in the derivation of Re rather than diameter. When that is done, the critical value of Re becomes 1000. While laminar flow is silent, turbulent flow is audible. In practical terms Reynolds numbers are highest where there is a deformity in the vessel wall, such as at a branch point or in the root of the aorta where velocity is very high, but normally turbulence does not occur in a blood vessel unless its shape has been modified by disease.

Resistance in Series and Parallel Circuits

Collectively, the arteries, arterioles, capillaries, ven-ules, and veins are arranged in series. Also, each vessel type, with the exception of the aorta and the pulmonary artery, finds itself in parallel with other vessels of the same type because of the extensive branching of the vascula-ture. Thus, resistances to blood flow are arranged in both series and parallel circuits. As will be discussed in other chapters, this arrangement allows for each region of the body to be perfused with blood in a regulated manner.

One can appreciate the consequences of whether the resistance is in series or in parallel with the circuit by again using the electrical analogy. Consider the following example of three tubes with values of 5, 25, and 100 resistance units (RU). If these tubes are arranged in series (Fig. 10), total resistance (Rt) is given as their simple sum because the flowing fluid must overcome all resistances in the circuit:

Because the same fluid must pass through all resistances in a series circuit, changing the resistance in one segment changes the flow in all segments. For example, constricting the arterioles in an organ equally reduces flow through the arteries, capillaries, and veins associated with that organ. When calculating the

Clinical Note

Bruits

Turbulence can occur in the cardiovascular system at points where the flow velocity is high, such as where a vessel is abruptly narrowed by an atherosclerotic plaque. Turbulent flow at such points can often be heard with a stethoscope; when such a sound is heard in the peripheral circulation it is referred to as a bruit. Bruits are not infrequently heard in the carotid arteries of the elderly and are usually accompanied by signs of impaired cerebration because of inadequate cerebral blood flow. The Reynolds number is also influenced by the viscosity of the fluid. When the viscosity of the blood is reduced by severe anemia, bruits may be heard at branch points in the large vessels even without any vascular disease.

FIGURE 10 Three tubes of different radii offer different resistances (R) to flow. If these tubes are arranged in series, the resistance offered by the combination of tubes is equal to the sum of their individual resistances. Note that the total resistance will always be greater than the largest single resistance.

FIGURE 10 Three tubes of different radii offer different resistances (R) to flow. If these tubes are arranged in series, the resistance offered by the combination of tubes is equal to the sum of their individual resistances. Note that the total resistance will always be greater than the largest single resistance.

equivalent of several resistances in series, the total must be greater than the greatest resistance in the circuit. If you do not find that to be the case, then a calculation error has been made.

If the same tubes are arranged in parallel (Fig. 11), the fluid is offered alternative paths that make it easier to flow through the circuit. Total resistance must be calculated by adding reciprocals:

Thus, it would take a AP 32.5 times as great to maintain the same Q through the three tubes in series compared with the tubes in parallel. Note that in parallel circuits the overall resistance must be less than the smallest individual resistance. In parallel beds, a change in an individual resistance affects only the flow through that leg. For example, flow to the intestine is in parallel with that to the kidneys. An increased resistance to flow in the intestine will not affect flow to the kidneys as long as aortic pressure does not change. The cardiac output would be reduced, however. The arterial pressure in the systemic circulation perfuses many parallel vascular beds.

The principle of hydrodynamics can be applied to understand blood flow in the cardiovascular system; however, additional complications are added because: (1) blood is a complex fluid, and (2) blood vessels are distensible and many can change their radii by contraction of the muscle in their walls. Each of these complications affects one or more of the parameters of resistance, pressure, and flow.

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