Fluids

The term fluid applies to both liquids and gases, and indeed to some things which we think of as solids (ie., glass). The essential differences between fluids and solids can be summarized as follows:

1. Fluids are shapeless. In more technical terms, we say that fluids do not resist shear. That is, it is relatively hard to bend a solid object, but a fluid splashes under the same force.

2. When a force is applied to a fluid, the pressure increases, but whereas the force is directional, the pressure is omnidirectional. Consider the force applied to the surface of a glass of water by the atmosphere. That force is downward, but the resulting pressure is felt throughout the water. If we measure the pressure at any given depth, it is the same on all sides of the measuring instrument.

3. Fluids are viscous. While fluids do not resist shear, they do resist changes in the rate of change of shear. The more viscous the fluid, the greater its resistance to changes in the rate of shear: colloquially, we say the fluid is syrupy. Viscosity is measured in Poise (denoted P):

   1 P = 0.1 kg / (m s).

   It is most often described in terms of cgs units: units whose fundamental measures are the gram, the centimeter and the second. In cgs units, since there are 1000 grams in a kilogram and 100 centimeters in a meter, we have

   1 P = 0.1 kg / (m s) * (1000 g / kg) * (1 m / 100 cm)

   = 1 g / (cm s).

   The viscosity of water is .01 Poise, and the viscosity of blood is .04 Poise.

4. Fluids flowing past a solid surface obey the no slip condition: the velocity of the fluid at the solid surface is zero. This may seem surprising, until you recall that your dishwasher often does not clean well when, for instance, the dishes are coated with cheese. The no slip condition means that the water by itself is not as effective as scrubbing with a solid object, and so many of us wipe the dishes before putting them in a dishwasher. Since the velocity of the fluid is nonzero elsewhere, the no slip condition implies that a velocity gradient exists in the flowing fluid: the velocity depends on the position where it is measured. In a pipe (or blood vessel), the velocity profile across the diameter is essentially parabolic (depending on r²) and has axial (circular) symmetry:
   

   the velocity does not depend on the angular position on any concentric circle, but only on the distance from the center.

   The fact that the velocity is constant on each circle leads us to think of the fluid as flowing in concentric sheets, but in fact the velocity is a smooth function of distance from the center. For higher viscosity, the velocity gradient is shallower. Essentially, viscosity opposes the existence of steep velocity gradients. The larger the viscosity, the gentler is the shape of the parabolic velocity gradient, and the velocity is more nearly constant across the section.

5. Fluids are subject to turbulence. Smooth flow is called laminar, implying that layers of fluid flow past each other in a sheet-like fashion. The flow is turbulent when eddies and vortices (whirlpools) occur. Fluids lose energy through dissipation in both types of flow. During laminar flow, the viscosity causes dissipative losses between the sheets of fluid: the fluid resists changes in the velocity gradient, and that costs energy. In turbulent flow, energy is required to create the eddies and vortices, resulting in an energy loss by the fluid as a whole.

   Fluid flow in a pipe crosses the threshold from laminar to turbulent flow when a dimensionless parameter called the Reynold's Number (denoted Re) reaches about 2000. It is defined as

   Re = 2 r r v / h,

   where r denotes the density of the fluid (1.05 g / cm³ for blood), v is the fluid's velocity and h is the Greek letter eta denoting the viscosity. Re is essentially the ratio of the inertial forces (tending to keep the fluid flowing) to the viscous forces experienced by a layer of fluid. Its value indicates the relative unimportance of viscosity (ie., low Re corresponds to very viscous situations). Typical values for human arteries and veins are:

   vessel	r	v	Re
   artery	.2 cm	28 cm/s	294
   vein	.25 cm	36 cm/s	472.5

For laminar flow through a uniform straight pipe, the flow rate (volume per unit time) is given by Poiseuille's Equation:

F = DP p r ⁴ / 8 h l,

We would like to use Poiseuille's Equation to discuss the blood vessels in animal circulatory systems. While the flow is essentially laminar outside of the capillaries, it is pulsatile throughout the arterial subsystem: the pressure varies as a periodic function of time. In addition, the equation is based on the parabolic velocity gradient, but since pressure waves in arterial walls propagate more quickly than those in blood, the velocity profile is more uniform than parabolic. Beyond that, Poiseuille's Equation assumes a constant viscosity, whereas the viscosity of blood actually changes with velocity, since blood is not a uniform fluid. In fact, the viscosity is much lower in the capillaries than in the rest of the system, since the red blood cells line up in single file to pass through them. On top of everything else, the blood vessels are not straight, uniform pipes!

All of these reservations notwithstanding, we can apply Poiseuille's Equation to the circulatory system to understand the scaling relationships between the various parameters. For instance, an occluded artery is often the subject of angioplasty or even bypass, in order to increase the inner radius of the vessel. We can use Poisseuille's Equation to understand the functional dependency of flow on radius: since flow is proportional to the fourth power of radius, we see that increasing the radius by a factor of 2 results in an increase in the flow rate by a factor of 16. We say that the flow scales with the fourth power of the radius. In a similar fashion, we can relate any two of the variables, holding all of the others constant. Since flow is equal to velocity times cross sectional area, we can also relate the blood velocity and therefore the Reynold's number to the other parameters.

It is necessary to have a single equation which contains exactly two interrelated variables in order to deduce a scaling relationship. In the event that three variables are interrelated through two or more equations, one variable must be eliminated. Any single equation which contains three interrelated variables is insufficient to deduce the relation; in that case a different equation is necessary.

Note that in this applet, the answers must be exact! Answers less than one must be entered as fractions (ie., 1/2).

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The next section builds an electrical analogy to fluid systems.

• Table of Contents
• Arithmetic without a Calculator
• Symbols and Abbreviations
• Index:
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©2005, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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