Fluid Pressure, Fluid Flow in the Body, and Motion in Fluids

100 $\upmu$ m.

In this chapter we will discuss the concept of pressure as it relates to fluids in the body. For example, the pressure of the vitreous humor in the eyeball serves several functions, including maintaining the shape of the eyeball. This pressure is similar to the stress we examined in Chap. 4, such as that in our long bones when we walk. They both describe a force per unit area. The pressure in the fluid is hydrostatic, i.e., the force per unit area is the same stress in all directions. In solids the stress is often anisotropic. We will review the basic physics of pressure and fluid flow, including the relationship of pressure and fluid flow, and diffusion [11, 20, 23, 28]. We will also examine the flow of humans in fluids, i.e., swimming, along with the possibility of human flight.

7.1 Characteristic Pressures in the Body

7.1.1 Definition and Units

The pressure of a fluid column is given by (2.48), $P=\rho gh$ , where $\rho$ is the fluid density, g is the gravitational constant, and h is the height of the column. For mercury $\rho$ is 13.6 g/cm $^{3}$ . For water $\rho = 1.00$ g/cm $^{3}$ at 4 $^{\circ }$ C. The density of whole blood is a bit higher, 1.06 g/cm $^{3}$ at 37 $^{\circ }$ C. The units of pressure are presented in Table 2.6.

So far we have been discussing absolute pressure, $P_{\mathrm {abs}}$ , which is the total force per unit area. In discussions concerning the body it is very common to cite the gauge pressure, $P_{\mathrm {gauge}}$ , which is the pressure relative to a standard, which is usually atmospheric pressure, and so $P_{\mathrm {gauge}}=P_{\mathrm {abs}}- 1$ atm. This is helpful because it is the difference in pressure that is the net force that acts on a unit area. In discussing blood pressure and the pressure of air in the lungs, it is assumed that the term pressure P refers to the gauge pressure relative to the local atmospheric pressure. During breathing in (which is called inspiration), the pressure in the lungs is lower than that outside the body and so the internal (gauge) pressure is $$<$$

. Table 7.1 gives typical pressures in the body.

Table 7.1

Typical (gauge) pressures in the body (in mmHg)

Arterial blood pressure
Maximum (systolic)	100–140
Minimum (diastolic)	60–90
Capillary blood pressure
Arterial end	30
Venous end	10
Venous blood pressure
Typical	3–7
Great veins	1
Middle ear pressure
Typical	1
Eardrum rupture threshold	120
Eye pressure
Humors	20 (12–23)
Glaucoma threshold range	$\sim$ 21–30
Cerebrospinal fluid pressure
In brain—lying down	5–12
Gastrointestinal	10–12
Skeleton
Long leg bones, standing	$\sim$ 7,600 (10 atm.)
Urinary bladder pressure
Voiding pressure	15–30 (20–40 cmH $_{2}$ O)
Momentary, up to	120 (150 cmH $_{2}$ O)
Intrathoracic
Between lung and chest wall	10

Using data from [3]

7.1.2 Measuring Pressure

One way of directly measuring pressure is with a manometer (Fig. 7.1). The measured pressure is that corresponding to the height of the fluid column plus the reference pressure, so

$\begin{aligned} P=P_{\mathrm {ref}}+\rho gh\mathrm {.} \end{aligned}$

(7.1)

The most common way to measure blood pressure is with a sphygmomanometer (sfig-muh-ma-nah’-mee-ter), which consists of a cuff, a squeeze bulb, and a meter that measures the pressure in the cuff (Fig. 7.2). The cuff is the balloon-like jacket placed about the upper arm above the elbow; this encircles the brachial artery. The cup of a stethoscope is placed on the lower arm, just below the elbow, to listen for the flow of blood. With no pressure in the cuff, there is normal blood flow and sounds are heard through the stethoscope. Gurgling sounds are heard after the cuff is pressurized with the squeeze bulb and then depressurized by releasing this pressure with a release valve in this bulb.

Fig. 7.1

Manometer

To understand when these sounds occur and their significance, we need to understand how the pressure in the main arteries varies with time. (This will be detailed in Chap. 8.) In every heart beat cycle (roughly 1/s), the blood pressure in the major arteries, such as the brachial artery, varies between the systolic pressure ( $\sim$ 120 mmHg) and the diastolic pressure ( $\sim$ 80 mmHg), as is depicted in Fig. 7.3. (The units of these cited gauge pressures are in mmHg—see (7.1) and Chap. 2.) When the pressure in the cuff exceeds the systolic pressure, there is no blood flow to the lower arm and consequently there are no sounds. When the pressure in the cuff is lowered with the release bulb to just below the systolic pressure, there is intermittent flow. During the part of the cycle when the arterial blood pressure is lower than the cuff pressure there is no flow; when it is greater, there is flow. This intermittent flow is turbulent and produces gurgling sounds. These sounds, the Korotkoff orK sounds, are transmitted by the stethoscope. As the cuff pressure is lowered further, the K sounds get louder and then lower, and are heard until the cuff pressure decreases to the diastolic pressure. Blood flow is not interrupted when the cuff pressure is less than the diastolic pressure and the K sounds cease because the blood flow is no longer turbulent. Therefore, the onset and end of the K sounds, respectively, denote the systolic and diastolic blood pressures. (This auscultatory method of Korotkoff was introduced by Russian army physician Korotkoff [24] who discovered a century ago that sound can be heard distally from a partially occluded limb [7].)

Fig. 7.2

Measuring blood pressure with a sphygmomanometer, listening to Korotkoff sounds (of varying levels during the turbulent flow shown in a–c). (Listening to sounds is called auscultation) (From [29])

Fig. 7.3

Variation of blood pressure with time, for blood leaving the left heart for the systemic system, with the systolic and diastolic pressures shown

7.2 Basic Physics of Pressure and Flow of Fluids

In this section we overview the basics of fluids. Some of this will be a review for most. Some of the more advanced results are derived, while others are merely presented. These basics will be used in subsequent chapters.

Both gas and liquid fluids are important in the body. Gases will be treated by the ideal gas law

$\begin{aligned} P=nRT, \end{aligned}$

(7.2)

where P is the pressure, n is the gas density, R is the gas constant (

8.31 J/mol-K), and T is the temperature (in K). The gas density

, where N is the total number of molecules in a volume V. The gas constant $R=N_{\mathrm {A}}k_{\mathrm {B}}$ , where $N_{\mathrm {A}}$ is Avogadro’s number, $6.02 \times 10^{23}$ , and $k_{\mathrm {B}}$ is Boltzmann’s constant, $1.381 \times 10^{-23}$ J/K.

One guiding principle is Pascal’s Principle : the pressure applied to a confined fluid increases the pressure throughout by the same amount. Also quite important is Archimedes’ Principle: the buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by that object. Another important relation is the Law of Laplace, which relates the difference of pressures inside and outside a thin-walled object—of a given shape—to the tension in the walls of the object. We will also need to understand the properties of flowing fluids to be able to analyze the physics of the circulatory system.

7.2.1 Law of Laplace

The pressure inside blood vessel walls, P, exceeds that outside, $P_{\mathrm { ext}}$ , by $\Delta P=P-P_{\mathrm {ext}}$ . How large of a tension should the vessel walls be able to withstand to support this positive pressure difference in equilibrium? The answer is provided by the Law of Laplace for hollow cylinders. It is derived here and then used in Chap. 8.

Consider a tube of radius R and length L. Figure 7.4a shows a section of this tube with angle $\theta \ll 1$ . The outward force (upward in the diagram) on this area is the pressure difference, $\Delta P$ , times the area, $(R\theta )L$ . The circumferential tension T is the force per unit length (along the tube length). (Note that this use of the word “tension” has a different meaning than in earlier chapters, where it meant a force, often used to pull things apart.) This film tension has units of force/length or energy/area. It is equal to a circumferential stress $\sigma =T/w$ , where w is the cylinder thickness (with $w\ll R$ ). These forces can be those within the blood vessel walls (Chap. 8). The horizontal components of the film tension to the left and right cancel. The vertical components are inward and each equal to $T\sin (\theta /2)\simeq T(\theta /2)$ for small angles. With both of these tension components multiplied by L, in static equilibrium force balance gives

$\begin{aligned} \Delta P(R\theta )L=2\left( T\frac{\theta }{2}\right) L. \end{aligned}$

(7.3)

This means

$\begin{aligned} \Delta P=\frac{T}{R}\mathrm {\qquad or\qquad }T=R(\Delta P). \end{aligned}$

(7.4)

Fig. 7.4

Derivation of the Law of Laplace for the cylinder in (a), with the force diagram for a section of a cylinder in (b), leading to the force diagram in (c), and the resolution of pressures for analysis of a half cylinder in (d)

This is a differential method. Alternatively we could integrate the forces over a half cylinder, as shown in Fig. 7.4d. The total downward force is the area of the walls, 2wL, times the stress, $\sigma$ , or $2wL\sigma$ . The total upward force is the cross-sectional area, 2RL, times the pressure difference, $\Delta P$ , or $2RL(\Delta P)$ . In equilibrium

$\begin{aligned} 2RL(\Delta P)= & {} 2wL\sigma \end{aligned}$

(7.5)

$\begin{aligned} \Delta P= & {} \frac{w\sigma }{R}=\frac{T}{R}, \end{aligned}$

(7.6)

which is the same as (7.4). (Figure 7.4d shows that the total upward force is really the integral of the upward force component, $\Delta P \cos \theta$ , times the area element, $RL\mathrm{d}\theta$ , integrated from $-90^{\circ }$ to $90^{\circ }$ or

$\begin{aligned} (\Delta P)RL\int _{-90^{\circ }}^{90^{\circ }}\cos \theta \mathrm{d}\theta =(\Delta P)RL(\sin (90^{\circ })-\sin (-90^{\circ }))=2(\Delta P)RL, \end{aligned}$

(7.7)

which turns out to be the same as expected.)

The Law of Laplace is also important in spheres, such as soap bubbles and the alveoli in the lungs. For a sphere of radius R and wall thickness w, we can balance the forces in the half sphere. The total downward force is the area of the walls, $2\pi Rw$ , times the stress, $\sigma$ , or $2\pi Rw\sigma$ . The total upward force is the cross-sectional area, $\pi R^{2}$ , times the pressure difference, $\Delta P$ , or $\pi R^{2}(\Delta P)$ . In equilibrium

$\begin{aligned} \pi R^{2}(\Delta P)= & {} 2\pi Rw\sigma \end{aligned}$

(7.8)

$\begin{aligned} \Delta P= & {} \frac{2w\sigma }{R}=\frac{2T}{R}. \end{aligned}$

(7.9)

This is the Law of Laplace for a sphere. We will use it in Chap. 9. (It is derived in more detail in Problem 7.12.)

For a spheroid with different radii of curvature, $R_{1}$ and $R_{2}$ , (7.4) and (7.9) generalize to

$\begin{aligned} \Delta P=\frac{T}{R_{1}}+\frac{T}{R_{2}}. \end{aligned}$

(7.10)

For a cylinder, $R_{1}=R$ and $R_{2}=\infty$ and this reduces to (7.4). For a sphere, $R_{1}=R$ and $R_{2}=R$ and it reduces to (7.9).

Our force balance arguments have made a direct connection between this tension, or really surface tension, and its units of force/length. Surface tension also has the same units as energy/area. This is reasonable because it is also the energy “cost” of making a unit area of a surface (or interface). Representative values of surface tension are given in Table 7.2.

Table 7.2

Surface tension ( $\gamma$ ) for several liquids

Liquid	T ( $^{\circ }$ C)	$\gamma$ (10 $^{-4}$ N/m)
Water	0	7.56
	20	7.28
	60	6.62
	100	5.89
Whole blood	20	5.5–6.1
Blood plasma	20	5.0–5.6
Lung surfactant	20	0.1
Cerebrospinal fluid	20	6.0–6.3
Saliva	20	1.5–2.1
Benzene	20	2.89
Mercury	20	46.4

Using data from [9, 15, 26]

7.2.2 Fluids in Motion

There are five attributes of the flow of fluids:

Flow can be laminar/streamline/steady or turbulent/unsteady . In laminar flow, a particle in the flow moves in a smooth manner along well-defined streamlines. In contrast, the motion is very random locally in turbulent flow. The Reynolds number Re is a dimensionless figure of merit that crudely divides the regimes of laminar and turbulent flow. It is the ratio between inertial force ( $\rho u^{2}/2$ ; $\rho u^{2}$ is used here) and viscous force $(\eta u/d)$ per unit volume on the fluid, where $\rho$ is the fluid density, u is the average speed of flow, d is the tube diameter, and $\eta$ is the fluid coefficient of viscosity or the dynamic or absolute viscosity, which is defined later. This gives

$\begin{aligned} Re=\frac{\rho u^{2}}{\eta u/d}=\frac{\rho ud}{\eta }=\frac{ud}{ \upsilon }, \end{aligned}$

(7.11)

where $\upsilon =\eta /\rho$ is the coefficient of kinematic viscosity.

Although this dividing line is not hard and fast, generally, flow in a rigid tube with $$Re < 2$$

,000 is laminar and that with $$Re > 2$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_7_Chapter_IEq72.gif”></SPAN>,000 is turbulent. This dividing region is often cited as being between 1,200–2,500, and in the higher part of this range for smoother-walled tubes. Figure <SPAN class=InternalRef><A href=

$$Re > 2$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_7_Chapter_IEq72.gif”></SPAN>,000 is turbulent. This dividing region is often cited as being between 1,200–2,500, and in the higher part of this range for smoother-walled tubes. Figure <SPAN class=InternalRef><A href=

7.5 shows flow in the laminar and turbulent regimes, and in the transition region between them.

Flow can be compressible or incompressible . Gases, such as air, are very compressible. Liquids are less compressible, and are often approximated as being incompressible.

Flow can be viscous or nonviscous. Fluids (other than superfluids) always have some viscosity, but in some cases it can be ignored totally, or first ignored and then considered as a perturbation.

Flow can be rotational or irrotational. In the cases we will consider there is no local rotation (such as vortices), so the flow will be irrotational.

Flow can be steady (constant in time) or pulsatile (with pulsing changes). Blood flow in the body is pulsatile, but is commonly treated as being in steady state in simple models. We will use both steady and pulsatile models in Chap. 8.

Fig. 7.5

Motion of a filament of dye in a straight pipe, showing (a) steady, laminar flow at low Re, (b) short bursts of turbulence for Re above the critical value, and (c) fully turbulent flow with random motion of the dye streak for higher Re (From [4]. Used with permission of Oxford University Press)

7.2.3 Equation of Continuity

The equation of continuity is a statement of the conservation of mass during flow. As seen in Fig. 7.6, when a fluid of a given mass density $\rho$ moves with average speed u in a tube of cross-sectional area A, the product $\rho Au$ is constant (i.e., it is conserved). Because the speed is a longitudinal distance per unit time, Au is the volume flow per unit time (because $A \; \times$ distance $$=$$

volume). Consequently, $\rho Au$ is the mass per unit time. In steady state, the same mass flows into a volume and leaves it. For the regions marked 1 and 2 in Fig. 7.6, this means that

$\begin{aligned} \rho _{1}A_{1}u_{1}=\rho _{2}A_{2}u_{2}. \end{aligned}$

(7.12)

Fig. 7.6

Continuity of flow when the tube cross-sectional area changes

Fig. 7.7

For irrotational and nonviscous flow, the pressure, flow speed, and height are related by Bernoulli’s equation along any streamline

If the fluid is incompressible, the densities in this equation do not depend on pressure and are the same everywhere. With $\rho _{1}=\rho _{2}$ , we follow the volume or volumetric flow rate Q, which is now a constant. This means $Q_{1}=A_{1}u_{1}$ and $Q_{2}=A_{2}u_{2}$ , and so the continuity equation becomes $Q=Q_{1}=Q_{2}$ with

$\begin{aligned} Q=A_{1}u_{1}=A_{2}u_{2}. \end{aligned}$

(7.13)

7.2.4 Bernoulli’s Equation

Bernoulli’s Principle (or equation) relates the average flow speed u, pressure P, and height y of an incompressible, nonviscous fluid in laminar, irrotational flow (Fig. 7.7). At any two points

$\begin{aligned} P_{1}+\frac{1}{2}\rho u_{1}^{2}+\rho gy_{1}=P_{2}+\frac{1}{2}\rho u_{2}^{2}+\rho gy_{2}. \end{aligned}$

(7.14)

The densities $\rho _{1}=\rho _{2}=\rho$ for this incompressible fluid. (Bernoulli’s equation actually applies to any two points along a streamline.)

There are three special cases of Bernoulli flow. (1) For static fluids $$ (u=0) $$

, and Bernoulli equation’s reduces to $P_{1}+\rho gy_{1}=P_{2}+\rho gy_{2}$ . (2) It reduces to Torricelli’s theorem when $P_{1}=P_{2}$ , namely $\rho u_{1}^{2}/2+\rho gy_{1}=\rho u_{2}^{2}/2+\rho gy_{2}$ . (3) It reduces to the Venturi flow regime when $y_{1}=y_{2}$ (Fig. 7.8), so

$\begin{aligned} P_{1}+\frac{1}{2}\rho u_{1}^{2}=P_{2}+\frac{1}{2}\rho u_{2}^{2}. \end{aligned}$

(7.15)

Because the continuity of flow in such a Venturi tube is $A_{1}u_{1}=A_{2}u_{2}$

$\begin{aligned} u_{2}=\frac{A_{1}}{A_{2}}\mathrm { }u_{1}. \end{aligned}$

(7.16)

Therefore we find

$\begin{aligned} P_{1}+\frac{1}{2}\rho u_{1}^{2}=P_{2}+\frac{1}{2}\rho \left( \frac{A_{1}}{ A_{2}}\mathrm { }u_{1}\right) ^{2} \end{aligned}$

(7.17)

and

$\begin{aligned} P_{2}-P_{1}=\frac{1}{2}\rho u_{1}^{2}\left( 1-\left( \frac{A_{1}}{A_{2}} \right) ^{2}\right) . \end{aligned}$

(7.18)

With $A_{2}<A_{1}$ , we see that $$$u_{2}>u_{1}$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_7_Chapter_IEq90.gif”></SPAN> and <SPAN id=IEq91 class=InlineEquation><IMG alt=$ . This shows that the flow becomes faster and the pressure becomes lower in clogged blood vessels.

Fig. 7.8

Flow in a tube when the tube cross-sectional area changes. This is a Venturi tube, for which pressure and flow speed are related by Bernoulli’s equation in the limit of constant height

7.2.5 Interactions Among the Flow Parameters

Pressure P, volume V, and flow rate Q are all interrelated in flow through vessels, be it blood flow in the circulatory system or air flow in breathing [22, 27, 28]. Resistance $R_{\mathrm {flow}}$ is the pressure difference or drop $\Delta P$ along a vessel, such as from $P_{1}= P(x=0)$ to $P_{2}= P(x=L)$ for a vessel of length L, needed to cause a given flow rate Q

$\begin{aligned} R_{\mathrm {flow}}=\frac{\Delta P}{Q}. \end{aligned}$

(7.19)

Compliance $C_{\mathrm {flow}}$ (or volume compliance) is the local change in volume ${\Delta V}$ caused by a local change in pressure ${\Delta P}$ in a vessel

$\begin{aligned} C_{{\mathrm {flow}}}=\frac{{\Delta V}}{{\Delta P}}. \end{aligned}$

(7.20)

Sometimes , the inertance $L_{\mathrm {flow}}$ is also defined. It is the change in pressure along a vessel caused by a flow rate that changes with time

$\begin{aligned} L_{\mathrm {flow}}=\frac{\Delta P}{\partial Q/\partial t}. \end{aligned}$

(7.21)

We will now consider the physics behind resistance, compliance, and inertance separately. In the next two chapters we will usually be able to analyze flow in vessels assuming that either resistance or compliance dominates, but for some vessels both resistance and compliance need to considered (and we will do this in the next chapter). Inertance is not important when the flow is constant in time. We will usually examine the overall flow properties of specific lengths of vessels, using models with these above “lumped” parameters. Occasionally we will analyze cases in which the parameters are distributed per unit length, such as flow resistance per unit length for volumetric flow along an artery in Chap. 8, using “distributed” or “transmission-line” models, as described in Appendix D. Such models are sometimes needed to examine time-dependent flow along a vessel, such as caused by the heart beating. Appendix D gives the differential form of these distributed equations when the effects of resistance, compliance, and inertance must all be considered and describes an analog between fluid flow and electrical circuits.

7.2.6 Resistance in Flow: Viscous Flow and Poiseuille’s Law

Bernoulli’s equation would predict that the pressure does not change during flow if the tube cross-section and height do not change. This is true for an ideal, nonviscous fluid. Viscosity is the friction during flow. It is always present and causes the pressure to drop during flow.

The coefficient of (dynamic or absolute) viscosity $\eta$ is formally defined in (7.22), which gives the tangential or shear force F required to move a fluid layer of area A at a constant speed v, in the x direction, when that layer is a distance y from a stationary plate (Fig. 7.9) [8, 10]

$\begin{aligned} F=\eta \frac{A}{y}v. \end{aligned}$

(7.22)

This equation is also written as

$\begin{aligned} \tau =\eta \frac{\mathrm{d}v}{\mathrm{d}y}, \end{aligned}$

(7.23)

where $\tau =F/A$ is the shear stress, as in (4.5) and Figs. 4.10 and 4.11, and $\mathrm{d}v/\mathrm{d}y$ is called the shear rate. (Check that the units of the shear rate are those that a rate should have, 1/s.) Fluids that are characterized by (7.22) and (7.23) are called “Newtonian fluids” and are said to undergo “Newtonian flow.”

The SI units of $\eta$ are (N/m $^{2})$ s, which is equal to kg/m-s and Pa-s; this is called a Poiseuille (PI), but this unit is not often used. More commonly used is the poise (P) which is $10\times$ smaller. It is a natural unit in the CGS units system with 1 poise $$=$$

1 g/cm-s

0.1 (N/m $^{2})$ s $$=$$

0.1 kg/m-s $$=$$

0.1 Pa-s. Also common is the centipoise (cP), with 1 cP $$=$$

0.01 poise $$=$$

0.001 Pa-s, because the viscosity of water at 20 $^{\circ }$ C is almost equal to 1 cP (and is actually 1.002 cP). We will usually use the units of Pa-s. Also, note that this viscosity coefficient is often called $\eta$ by physicists (and is used as such here), whereas it is often called $\mu$ by biomedical engineers. It is also related to, but different from the viscosity damping constant for the dashpot c in (4.48).

Fig. 7.9

Viscous fluid flow, with a linear gradient of fluid speed with position between a fixed and moving plate. This is shown for Newtonian flow

Because of this drag, there must be a pressure difference (gradient) to maintain fluid flow in a tube. The relation between this pressure drop and the volumetric flow rate Q is given by Poiseuille’s Law (or Hagen-Poiseuille’s Law)

$\begin{aligned} Q=\frac{\pi R^{4}}{8\eta L}(P_{1}-P_{2}), \end{aligned}$

(7.24)

where R is the radius of the tube and L is its length (Fig. 7.10). This relation can be viewed as the flow rate for a given pressure drop. Alternatively, it can be viewed as the pressure drop when there is a flow Q in the tube

$\begin{aligned} P_{1}-P_{2}=\Delta P=\frac{8\eta L}{\pi R^{4}}\;Q. \end{aligned}$

(7.25)

We will use this expression in Chap. 8 to determine the pressure drops in blood vessels during circulation. We will derive it soon as an advanced topic.

Fig. 7.10

Calculation of Poiseuille’s Law for a tube in (a), using the cylindrical shell in (b), and balancing forces between the hydrostatic flow pressure force and the differential shear stress on the shell in (c)

Equation (7.25) is formally analogous to Ohm’s Law for resistors, $V=IR_{\mathrm {elect}}$ (or in a manner more parallel to this equation, $V=R_{\mathrm {elect}}I$ ), where V is the voltage or potential difference across the resistor and is the driving term (which is analogous to $\Delta P$ ), $R_{\mathrm {elect}}$ is the electrical resistance (analogous to the resistance of flow $8\eta L/\pi R^{4}$ here, which we will call $R_{\mathrm {flow}}$ ), and I is the electrical current, which is the flow resulting from the driving term (analogous to the volumetric flow Q here).

Consider a tube with cross-sectional area A. The net force on the fluid in it is $(\Delta P)A$ . If this force moves the fluid a distance L, the work done on it is $FL=(\Delta P)AL$ . If this volume AL is moved in a given time, the work needed to do this in this given time—the power—is

$\begin{aligned} P_{\mathrm {power,\;flow}}=(\Delta P)Q, \end{aligned}$

(7.26)

or $P_{\mathrm {power,\;flow}}=Q^{2}R_{\mathrm {flow}}=(\Delta P)^{2}/R_{\mathrm {flow}}$ . These expressions are analogous to those for the power dissipated by an electrical resistor: $P_{\mathrm {power,\;elect}}=VI=I^{2}R_{\mathrm {elect} }=V^{2}/R_{\mathrm {elect}}$ .

The coefficient of viscosity for water is $1.78 \times 10^{-3}$ Pa-s at 0 $^{\circ }$ C and it decreases with temperature, dropping to $1.00 \times 10^{-3}$ Pa-s at 20 $^{\circ }$ C and $0.65 \times 10^{-3}$ Pa-s at 40 $^{\circ }$ C. At 37 $^{\circ }$ C, $\eta$ is $1.5 \times 10^{-3}$ Pa-s for blood plasma and $4.0 \times 10^{-3}$ Pa-s for whole blood, which are both higher than that for water at the same temperature. (Blood is really thicker than water.) The coefficients of viscosity of common human body fluids and other materials are listed in Table 7.3. As is clear from the table, the viscosity of liquids decreases with increasing temperature T, because the kinetic energy of molecules increases with T and this can overcome intermolecular forces that slow down motion between the dense, adjacent layers. In contrast, viscosity increases with temperature for gases, as T(in K) $^{1/2}$ , because diffusion between adjacent layers increases with T.

Table 7.3

Coefficient of viscosity $\eta$ of common materials, in Pa-s (1 poise = 0.1 Pa-s)

Material	T ( $^{\circ }$ C)	$\eta$
Water	0	1.78 $\times$ 10 $^{-3}$
	20	1.00 $\times$ 10 $^{-3}$
	37	0.69 $\times$ 10 $^{-3}$
	50	0.55 $\times$ 10 $^{-3}$
	100	0.28 $\times$ 10 $^{-3}$
Blood plasma	37	1.5 $\times$ 10 $^{-3}$
Whole blood $^{{\mathrm{a}}}$	37	$\sim$ 4.0 $\times$ 10 $^{-3}$
Low shear rate, Hct = 45%		$\sim$ 100 $\times$ 10 $^{-3}$
Low shear rate, Hct = 90%		$\sim$ 1,000 $\times$ 10 $^{-3}$
High shear rate, Hct = 45%		$\sim$ 10 $\times$ 10 $^{-3}$
Low shear rate, Hct = 90%		$\sim$ 100 $\times$ 10 $^{-3}$
Cerebrospinal fluid	20	1.02 $\times$ 10 $^{-3}$
Interstitial fluid	37	1.0–1.1 $\times$ 10 $^{-3}$
Human tears	37	0.73–0.97 $\times$ 10 $^{-3}$
Synovial fluid $^{{\mathrm{b}}}$	20	Share this: Click to share on Twitter (Opens in new window) Click to share on Facebook (Opens in new window) Like this: Like Loading... Related Related posts: Statics of the Body Muscles Cardiovascular System Metabolism: Energy, Heat, Work, and Power of the Body Stay updated, free articles. Join our Telegram channel Tags: Physics of the Human Body Jun 11, 2017 \| Posted by admin in GENERAL & FAMILY MEDICINE \| Comments Off on Fluid Pressure, Fluid Flow in the Body, and Motion in Fluids Full access? Get Clinical Tree Get Clinical Tree app for offline access Get Clinical Tree app for offline access