As discussed, the Poiseuille equation (see Equation 17-9) is based on solid empirical and theoretical grounds. However, the equation requires the following assumptions:
1. The fluid must be incompressible.
2. The tube must be straight, rigid, cylindrical, and unbranched, and have a constant radius.
3. The velocity of the thin fluid layer at the wall must be zero (i.e., no “slippage”). This assumption holds for aqueous solutions, but not for some “plastic” fluids.
4. The flow must be laminar. That is, the fluid must move in concentric undisturbed laminae, without the gross exchange of fluid from one concentric shell to another.
5. The flow must be steady (i.e., not pulsatile).
6. The viscosity of the fluid must be constant. First, it must be constant throughout the cross section of the cylinder. Second, it must be constant in the “newtonian” sense; that is, the viscosity must be independent of the magnitude of the shear stress (i.e., force applied) and the shear rate (i.e., velocity gradient produced). In other words, the shear stress at each point is linearly proportional to its shear rate at that point.
To what extent does the circulatory system fulfill the conditions of the Poiseuille equation? The first condition (i.e., incompressible fluid) is well satisfied by blood. If we consider only flow in a vessel segment that is of fairly fixed size (e.g., thoracic aorta), the second assumption (i.e., simple geometry) is also reasonably well satisfied. The third requirement (i.e., no slippage) is true for blood in blood vessels. Indeed, if one forms a reservoir out of a piece of vessel (e.g., aorta) and fills it with blood, a meniscus forms with the concave surface facing upward, indicating adherence of blood to the vessel wall.
The fourth and fifth assumptions, which are more complex, are the subject of the next two sections. With regard to the sixth assumption, Chapter 18 addresses the anomalous viscosity of blood.
Blood flow is laminar
From Ohm's law of hydrodynamics (ΔP = F · R), flow should increase linearly with driving pressure if resistance is constant. In cylindrical vessels, flow does indeed increase linearly with ΔP up to a certain point (Fig. 17-6A). However, at high flow rates—beyond a critical velocity—flow rises less steeply and is no longer proportional to ΔP but to roughly the square root of ΔP, because R apparently increases. Here, blood flow is no longer laminar but turbulent. Because turbulence causes substantial kinetic energy losses, it is energetically wasteful.
FIGURE 17-6 Laminar versus turbulent flow.
The critical parameter that determines when flow becomes turbulent is a dimensionless quantity called the Reynolds number (Re), named after Osborne Reynolds: N17-4
Contributed by Chevalier Emile Boulpaep
In Equation 17-13 (shown here as Equation NE 17-3),
the mean linear velocity () is expressed in centimeters · second–1, the radius (r) in centimeters, the density (ρ) in grams · centimeter–3, and the viscosity (η) in poise. When the equation is written as above—with the term 2r or diameter in the numerator—blood flow is laminar when Re is below ~2000.
You may also encounter a similar equation with r, rather than 2r, in the numerator:
When the equation is written in terms of radius rather than diameter, blood flow is laminar when Re is below ~1000. In the first and second printings of the first edition of this text, the value 1160 was used assuming the “radius” convention in Equation NE 17-4.
Regardless of which version of the equation we use, the terms in the numerator reflect disruptive forces produced by the inertial momentum in the liquid, because of both the velocity term and the product r ⋅ ρ, which is related to the mass of the moving fluid. In other words, a high inertial momentum predisposes to turbulence. The term in the denominator reflects the cohesive forces in the liquid; that is, the viscosity that tends to keep the layers of fluid together.
One way of looking at the above equations is that at low Re, flow is laminar and a tiny volume of fluid in one layer in Figure 17-5B tends to stay in that layer. When Re is sufficiently high, flow is turbulent, and that tiny volume may leave its original layer and become part of a neighboring layer—that is, it participates in eddy formation.
Another way of looking at the above equations is that a tiny volume of fluid has a certain probability of deviating course and leaving its original layer. This tendency to stray from its original layer is enhanced when velocity () or density (ρ) is high (which raises inertial momentum) but is counteracted by the viscosity (which tends to hold it in the layer). The tendency to stray is also counteracted by a small radius, which reduces the number of layers and brings the average layer closer to a constraining wall—channeling the fluid.
http://www-history.mcs.st-and.ac.uk/Mathematicians/Reynolds.html [(Accessed August 2015)].
http://www.mace.manchester.ac.uk/about-us/hall-of-fame/mechanical-engineering/osborne-reynolds/ [Accessed August 2015].
Blood flow is laminar when Re is below ~2000 and is mostly turbulent when Re exceeds ~3000. The terms in the numerator reflect disruptive forces produced by the inertial momentum in the fluid. Thus, turbulent blood flow occurs when r is large (e.g., aorta) or when is large (e.g., high cardiac output). Turbulent flow can also occur when a local decrease in vessel diameter (e.g., arterial stenosis) causes a local increase in . The term in the denominator of Equation 17-13, viscosity, reflects the cohesive forces that tend to keep the layers well organized. Therefore, a low viscosity (e.g., in anemia—a low red blood cell count) predisposes to turbulence. When turbulence arises, the parabolic profile of the linear velocity across the radius of a cylinder becomes blunted (see Fig. 17-6B).
The distinction between laminar and turbulent flow is clinically very significant. Laminar flow is silent, whereas vortex formation during turbulence sets up murmurs. These Korotkoff sounds are useful in assessing arterial blood flow in the traditional auscultatory method for determination of blood pressure. These murmurs are also important for diagnosis of vessel stenosis, vessel shunts, and cardiac valvular lesions (Box 17-1). Intense forms of turbulence may be detected not only as loud acoustic murmurs but also as mechanical vibrations or thrills that can be felt by touch.
Heart Murmurs and Arterial Bruits
Stage 1: Turbulence as blood flows across diseased heart valves creates murmurs that can be readily detected by auscultation with a stethoscope. The factors causing turbulence are the ones that increase the Reynolds number: increases in vessel diameter or blood velocity and decreases in viscosity. Before the advent of sophisticated technology, such as cardiac ultrasonography, clinicians made a fine art of detecting these murmurs in an attempt to diagnose cardiac valvular disease. In general, it was appreciated that normal blood flow across normal heart valves is silent, although murmurs can occur with increased blood flow (e.g., exercise) and are not infrequently heard in young, thin individuals with dynamic circulations. The grading of heart murmurs helps standardize the cardiac examination from observer to observer. Thus, a grade 1 heart murmur is barely audible, grade 2 is one that is slightly more easily heard, and grades 3 and 4 are progressively louder. A grade 5 murmur is the loudest murmur that still requires a stethoscope to be heard. A grade 6 murmur is so loud that it can be heard with the stethoscope off the chest and is often accompanied by a thrill. The location, duration, pitch, and quality of a murmur aid in identifying the underlying valvular disorder.
Stage 2: Blood flowing through diseased arteries can also create a murmur or a thrill. By far the most common cause is atherosclerosis, which narrows the vessel lumen and thus increases velocity. In patients with advanced disease, these murmurs can be heard in virtually every major artery, most easily in the carotid and femoral arteries. Arterial murmurs are usually referred to as bruits.
Pressure and flow oscillate with each heartbeat between maximum systolic and minimum diastolic values
Thus far we have considered blood flow to be steady and driven by a constant pressure generator. That is, we have been working with a mean blood flow and a mean driving pressure (the difference between the mean arterial and venous pressures). However, we are all aware that the heart is a pump of the “two-stroke” variety, with a filling and an emptying phase. Because both the left and right hearts perform their work in a cyclic fashion, flow is pulsatile in both the systemic and pulmonary circulations.
The mean blood pressure in the large systemic arteries is ~95 mm Hg. This is a single, time-averaged value. In reality, the blood pressure cycles between a maximal systolic arterial pressure (~120 mm Hg) that corresponds to the contraction of the ventricle and a minimal diastolic arterial pressure (~80 mm Hg) that corresponds to the relaxation of the ventricle (Fig. 17-7). The difference between the systolic pressure and the diastolic pressure is the pulse pressure. Note that the mean arterial pressure is not the arithmetic mean of systolic and diastolic values, which would be (120 + 80)/2 = 100 mm Hg in our example; rather, it is the area beneath the curve, which describes the pressure in a single cardiac cycle (see Fig. 17-7, blue area) divided by the duration of the cycle. A reasonable value for the mean arterial pressure is 95 mm Hg.
FIGURE 17-7 Time course of arterial pressure during one cardiac cycle. The area beneath the blue pressure curve, divided by the time of one cardiac cycle, is mean arterial pressure (horizontal yellow line). The yellow cross-hatched area is the same as the blue area.
Like arterial pressure, flow through arteries also oscillates with each heartbeat. Because both pressure and flow are pulsatile, and because the pressure and flow waves are not perfectly matched in time, we cannot describe the relationship between these two parameters by a simple Ohm's law–like relationship (ΔP = F · R), which is analogous to a simple DC circuit in electricity. Rather, if we were to model pressure and flow in the circulatory system, we would have to use a more complicated approach, analogous to that used to understand AC electrical circuits. N17-5
Mechanical Impedance of Blood Flow
Contributed by Ridder Emile Boulpaep
We began this chapter by drawing an analogy between the flow of blood and electrical current, as described by a Ohm's law of hydrodynamics: ΔP = F ⋅ R. We now know that there are other factors that influence pressure. In addition to the flow resistance R (electrical analogy = ohmic resistor), we must also consider the compliance C (electrical analogy = capacitance) as well as the inertiance L (electrical analogy = inductance). A similar problem is faced in electricity when dealing with alternating (as opposed to direct) currents. In Ohm's law for alternating currents, E = I ⋅ Z, where Z is a complex quantity called the impedance. Z depends on the electrical resistance R, the electrical capacitance C, and the electrical inductance L. Similarly, for blood flow, we can write ΔP = F ⋅ Z, where Z is also a complex quantity, called mechanical impedance, that includes the following:
1. Compliant impedance that opposes volume change (compliance of the vessel).
2. Viscous (or resistive) impedance that opposes flow (shearing forces in the liquid). This term is the R of Ohm's law of hydrodynamics: ΔP = F ⋅ R (Equation 17-1).
3. Inertial impedance that opposes a change of flow (kinetic energy of fluid and vessels).
Considering all these sources of pressure, we can state that the total pressure difference at any point in time, instead of being given by Ohm's law, is
The Pgravity term in the above equation is discussed on pages 418–419 in the section titled “Gravity causes a hydrostatic pressure difference when there is a difference in height.”