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Biofluid Mechanics
Krishnan B. Chandran
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100% found this document useful (1 vote)
181 views50 pagesG1
Biofluid Mechanics
Krishnan B. Chandran
Uploaded by
Радић СтефанCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
/ 50
SECOND EDITION
THE HUMAN CIROULATION List of Symbols
English
A
Bo
c
G
c
149,570,2%
np
Re
R,
SA node
Cross-sectional area, amplitude (Chapter 6)
Main magnetic field strength (Chapter 10)
Compliance
Molar concentration of species
Wave speed
Diameter
Diffusion coefficient (Chapter 3); distensibility (Chapter 6)
Diffusing capacity
Elastic modulus (Young's Modulus)
Incremental elastic modulus
Pressure elastic modulus
Convective flux vectors (Chapter 11)
Viscous flux vectors (Chapter 11)
Shear modulus
Gravitational acceleration
Total head, total energy per unit volume, head loss
Hematocrit
Jacobian (Chapter 11)
Bessel function of first kind and vth order
Consistency index
Solubility coefficient
Spring constant
Bulk modulus
Initial length
Instantaneous length
Modulus (Chapter 6)
Dean number
Molar flux of species i
Hydrostatic pressure
Systolic pressure
Diastolic pressure
Axial load, force
Flow rate; vector containing flow variables (Chapter 11)
Radius
Cylindrical coordinates
Spherical coordinates
Reynolds number
Resistance—Eliminated the same four rows above
Sino atrial node
xvii LWZ
Zz
Zo
List of Symbols
‘Torque, truncation error (Chapter 11)
Velocity components
Volume
Initial volume
Cartesian coordinates
Impedance
Characteristic impedance
1D 2D, 3D) One-dimensional
Greek
mm aeee
oe
‘app
SEF EUS
Womersley parameter
Shear strain
Rate of shear strain, velocity gradient
Increment in length
Normal strain (engineering strain)
Phase angle (Chapter 6)
True strain
Normal rate of strain
Generalized coordinates (Chapter 11)
‘Tube wall displacements in the r, 0, z directions (Chapter 6)
Density
Viscosity coefficient
Apparent viscosity
Plasma viscosity
Poisson's ration (Solid); kinematic viscosity (\1/p) in fluids
Normal stress
Ultimate stress
Yield stress
Diagonal matrix (Chapter 11)
Shear stress
Angular velocity List of Abbreviations
AAA Abdominal aortic aneurysm
ALE Arbitrary Lagrangian-Eulerian (Chapter 11)
AV Atrioventricular valve (Chapter 3)
AVE Arteriovenous fistula
bpm Beats per minute
CDFM Color Doppler flow mapping
CFD Computational fluid dynamics
co Cardiac output
cT Computed tomography
cvP Central venous pressure
cw Continuous wave
cx Circumfiex coronary artery
DVT Deep vein thrombosis
EC Endothelial cell
ECG Electrocardiogram
EDV End diastolic volume
EF Ejection fraction
EMF Electromagnetic flow meter
ePTFE Expanded polytetrafluoroethylene (Teflon)
ESV End systolic volume
FSI Fluid-structure interaction
HDL High-density lipoprotein
HR Heart rate
1H Intimal hyperplasia
IMA Internal mammary artery
LAD Left anterior descending coronary artery
LDA Laser Doppler anemometry
LDL. Low-density lipoprotein
LDV Laser Doppler velocimetry
MAP Mean arterial pressure
MRI Magnetic resonance imaging
PET Polyethylene terephthalate (Dacron)
PIV Particle image velocimetry
PRF Pulse repetition frequency
PRU Peripheral resistance unit
PTA Percutaneous transluminal angioplasty
PTCA Percutaneous transluminal coronary angioplasty
PTFE Polytetrafiuoroethylene (Teflon)
RF Radio frequency
RMS Root mean square
RV Regurgitant volume
xix xx
SG
SMC
sv
Svc, IVC
SVG
SVHD
TEE
WSR
WSS
WSSG
ZCCIZCD
List of Abbreviations
Specific gravity
Smooth muscle cell
Stroke volume
Superior, inferior vena cava
Saphenous vein graft
Single ventricle heart defect
Trans-esophageal echocardiography
Wall shear rate
Wall shear stress
Wall shear stress gradient
Zero-crossing counter/zero-crossing detector 1 Fundamentals of Fluid
Mechanics
1.1. INTRODUCTION
Before considering the mechanics of biological fluids in the circulation, it is neces-
sary to first consider some key definitions and specific properties. Once established,
we will use these “pieces” to construct important laws and principles which are
the foundation of fluid mechanics. To begin, we will define what we mean by a
‘fluid. In general, a material can be characterized as a fluid if it deforms continu-
ously under the action of a shear stress produced by a force that acts parallel to the
Tine of motion. In other words, a fluid is a material that cannot resist the action of
a shear stress, Conversely, a fluid at rest cannot sustain a shearing stress. For our
applications, we will treat a fluid as a continuum (ie., it is a homogeneous mate-
rial) even though both liquids and gases are made up of individual molecules. On a
macroscopic scale, however, fluid properties such as density, viscosity, and so on are
reasonably considered to be continuous.
1.2 INTRINSIC FLUID PROPERTIES
‘The intrinsic properties of a fluid are considered to be its density, viscosity, compress-
ibility, and surface tension, We will discuss and define each of these individually.
1.2.1. Density
Density, commonly denoted by the symbol, p, is defined as the mass of a fluid per
unit volume and has units of [M/L?]. In the meter/kilogram/second (MKS) system,
this would be represented by (kg/m’) or by (g/cm!) in the centimeter/gram/second
(CGS) system where 1 g/cm? = 10? kg/m®, Values of density for several common
biofluids are
Paws = 999 kg/m? at 15°C («1 g/m’)
Pai = 1.22kg/m? at standard atmospheric temperature and pressure
Ponoiedioos = 1060kg/m? at 20°C (6% higher than water)
A related property of a fluid is its specific gravity, denoted as SG, which is defined
as its density divided by the density of water at 4°C (a reference value that is quite
repeatable). Thus, for whole blood at 20°C, SG = 1.06. The specific weight of a finid,
denoted as y (not to be confused with 7, the symbol for rate of shear), is the weight of
a fiuid per unit volume, or pg. For blood at 20°C, y = 1.04 x 10* N/m’. 4 Biofluid Mechanics: The Human Circulation
1.2.2 Viscosity
‘As we said earlier, a fluid is defined as a material that deforms under the action of a
shear force. The viscosity of a fiuid (or its “stickiness”), denoted by the symbol, 1,
js related to the rate of deformation that a fluid experiences when a shear stress is
applied to it. Just as with a solid, fluid shear stress is defined as shear force per unit
area applied tangentially to a surface and is denoted by «. To illustrate this, consider
two parallel plates each of cross-sectional area A (cm?) with fluid of viscosity }
between them as shown in Figure 1.1.
If a tangential force F, is applied to the top plate as shown, it will result in the
plate moving with a velocity U (cm/s) relative to the lower plate. The fluid adjacent
to the top plate will move with the same velocity as that of the plate since the fluid
is assumed to stick to the plate (known as the “no-slip” condition). Similarly, the
fluid adjacent to the bottom plate will be at rest since it sticks to a stationary surface.
‘Thus, a velocity gradient, or change in velocity per unit change in height, is produced
within the fluid as shown. The shearing force F, divided by the area, A, over which it
acts is defined as the shearing stress, t, having the units [ML“'T7]. The velocity gra-
dient, also referred to as the rate of shear, 7, is the ratio U/h where h is the distance
between the two parallel plates. Thus, the rate of shear has the dimension of s. In
general, the rate of shear is defined as Ou/dy where y is the distance perpendicular
to the direction of shear as shown in the figure. The viscous properties of all fluids
are defined by the relationship between the shear stress and the rate of shear over a
range of shear rates.
The relationship between viscous shear stress, 7, viscosity, 1, and shear rate,
duly, for flow in the x-direction is given in Equation 1.1 with the derivative taken in
the y-direction, perpendicular to the direction of flow:
au
My a)
‘The coefficient in this relationship is known as the dynamic viscosity [ML“"T-"] and
is usually expressed as either Pa-s (kg/m-s) in the MKS system or as Poise (g/cm-s)
in the CGS system, In many biological applications, it is convenient to define a cen-
tiPoise (cP) where 1 cP = 0.01 P due to the relatively low values of this property. The
constitutive relationship expressed in Equation 1.1 can be plotted in a shear stress
FIGURE 1.1 Fluid subjected to simple shearing stress. Fundamentals of Fluid Mechanics 5
Bingham plastic
Casson
fluid
Shear stress, +
‘Newtonian fluid
(dyn/cm?)
Power law
Yield} fluid
stress|
Rate of shear, ¥ (s“!)
FIGURE 1.2. Shear stress versus rate of shear plots for Newtonian and non-Newtonian fluids.
versus rate of shear plot as shown in Figure 1.2. (Note: Kinematic viscosity, » = /p,
is commonly used to condense the density and viscosity into a single variable, espe-
cially for liquids where the density is relatively constant.)
A fluid in which the viscosity is constant is known as a Newtonian fluid and the
relationship between viscous shear stress and shear rate is represented in the fig-
ure by a straight line passing through the origin with a slope equal to 1. In reality,
many fluids do not follow this ideal linear relationship. Those fluids in which the
shear stress is not directly proportional to the rate of shear are generally classified
as non-Newtonian fluids. In this case, the ratio of shear stress to the rate of shear at
any point of measurement is referred to as the apparent viscosity, Happ. The apparent
viscosity is not a constant but depends on the rate of shear at which it is measured.
‘There are several classes of non-Newtonian fluids whose constitutive relationships
are shown in Figure 1.2, For example, many fluids that exhibit a nonlinear relation-
ship between shear stress and rate of shear and pass through the origin are expressed
by the relationship
t= Kyi" (12)
where n # 1, Such fluids are classified as power law fluids. Another class of fluids
is known as Bingham plastics because they will initially resist deformation to an
applied shear stress until the shear stress exceeds a yield stress, t,. Beyond that point,
there will be a linear relationship between shear stress and rate of shear, The consti-
tutive relationship for a Bingham plastic is given by
t=
y+ Maly 3)
where
1, is the yield stress
Hy is the plastic viscosity 6 Biofluid Mechanics: The Human Circulation
Fluids that exhibit a yield stress and also a nonlinear relationship between shear
stress and rate of shear may be classified as Casson fluids. The specific empirical
relationship for such fluids that deviate from the ideal Bingham plastic behavior is
known as the Casson equation, or
t= Jt helt a4
As pointed out earlier, it is important in many biomedical applications to know
the theological characteristics of blood. In order to understand the relationship
between the shear stress and the rate of shear for blood, experimental measure-
ments are necessary. From those experiments, it has been determined that blood
behaves as a Newtonian fluid only in regions of relatively high shear rate (>100s”).
‘Thus, for flow in large arteries where the shear rate is well above 100s", a value
of 3.5cP is often used as an estimate for the (assumed) constant viscosity of
blood. In the microcirculation (i.e., small arteries and capillaries) and in veins
where the shear rate is very low, blood must be treated as a non-Newtonian fluid
(see Section 4.1.3).
‘The viscosity of a fluid is also strongly dependent upon its temperature. Generally,
the viscosity of liquids decreases with increasing temperature, while the viscosity of
gases increases with increasing temperature.
1.2.3 COMPRESSIBILITY
‘The compressibility of a fluid is quantified by the pressure change required to pro-
duce a certain increment in either the fluid’s volume or density. This property, known
as the bulk modulus, k, is defined as
Ap__ AP
aviv dplp
(1.5)
‘Thus, for an incompressible fluid, k = . For water, k = 2.15 x 10° Nim’, indicating
that it is practically incompressible. Since the transmission of sound through a fluid
such as air or water is simply the movement of pressure waves through the fluid, the
speed of sound, c, depends on the fluid’s bulk modulus and density as
kip (1.6)
Thus, for water at 15°C,
¢ = (2.15 107/999)" = 1470 m/s
‘The speed of sound in biological tissues and blood is used in various ultrasound
modalities to obtain anatomic images and also blood flow velocities (see Section 10.6). Fundamentals of Fluid Mechanics 7
FIGURE 1.3. Pressure-surface tension force balance for a hemispherical drop.
1.2.4 SuRFAcE TENSION
‘The surface layer between two different liquids behaves like a stretched membrane
due to intermolecular forces. An expression can be found for the surface tension in a
fiuid, 6, by considering a hemispherical fluid drop of radius, R, as shown as follows
(Figure 1.3).
‘Assuming static equilibrium between the pressure and surface forces and apply-
ing a force balance in the vertical direction,
2nRo = ApnR
and hence,
(7)
The effect of surface tension is particularly important in the pulmonary airways as it
is what maintains the openings in the alveoli (the smallest regions of the lung where
gas exchange occurs) and, thus, enables us to breath sufficient air in and out
1.3. HYDROSTATICS
While most fluids in the body exist in a state of continuous motion, there are impor-
tant effects on the fluid which are due to static forces. A fluid at rest in a gravitational
field, for example, is in hydrostatic equilibrium as shown in Figure 1.4, Under these
conditions, the weight of the fluid is exactly offset by the net pressure force support-
ing the fluid, Here, the pressure at the base of the fluid element is p while that at a
distance dz above the base is p plus the gradient of pressure in the z-direction, dp/dz,
times the incremental elevation, dz, where p is the gage pressure referenced to the
atmospheric pressure, p, (=1.01 x 105 N/m?). Thus, if we sum the pressure and gravi-
tational forces acting on the element in the vertical direction, we get
pda (p+ 2atc]an~pedade (8) 8 Biofluid Mechanics: The Human Circulation
+ Banana
FIGURE 1.4 Hydrostatic equilibrium of a fluid element,
which requires that the pressure gradient be
de
| 1
a8 a)
where g is the gravitational acceleration. The negative sign for the pressure gradient
indicates that p decreases as z increases, Integrating the previous equation between
the base and the top of the element, z, and z,, respectively, with corresponding pres-
sures, p, and p,, yields
”
fw -foeue G10)
or
P2~ Pi =~Pa(z2—%) ap
Thus,
Ap = pgh (1.12)
for any element of height h. An equivalent expression would be Ap = yh
Example 1.1
For the case of a column of mercury where py, = 1.35 x 10* kg/m, the pressure
increase over a height of 1mm would be
Ap = 135 x10%(kg/m’) x 9.81 (m/s?) x 107 (m)
= 133 Nim?
=133 Pa Fundamentals of Fluid Mechanics 9
1.4 MACROSCOPIC BALANCES OF MASS AND MOMENTUM
‘The ultimate goal of fluid mechanics is to identify relationships between variables so
that we can determine the value of one or more of these variables in terms of given
conditions. In order to do this, we will begin with basic balances of properties or
“conservation laws” which, in turn, involve multiple variables of interest. Initially,
we will do so on a macroscopic basis where we consider the dynamics associated
with a relatively large volume of fluid. Later, in Section 1.5, we will reevaluate these
laws relative to an infinitesimal, or microscopic volume. By doing so, we will be able
to determine relatively simple, gross parameters (ic., flow rate, reaction forces, etc.)
with the macroscopic approach but more complex, detailed parameters (i.c., local
velocities, pressures, etc.) with the microscopic approach. Classically, the three prop-
erties of a fluid that are considered in this analysis are: mass, momentum, and energy.
We will establish balances for each based upon the Reynolds transport theorem
This theorem relates the time rate of change of each of these properties in a system
relative to corresponding changes that occur within and across a control volume and
is given by
a)
ae J bpav + Jorvaa (1.13)
ev és
where
B denotes the extensive (i.e., absolute) property
b denotes the intrinsic property (j.e., the amount of that property per unit mass,
or, b = B/m)
CV denotes the control volume
CS denotes the control surfaces of that volume
1.4.1 Conservation oF Mass
‘The requirement, or “law,” of conservation of mass applies to all materials and is thus
the logical starting point for an introduction to fluid motion. To begin, we consider
a conceptual three dimensional (3D) space or, control volume, which is enclosed by
a surface across which fluid can move (Figure 1.5). This control volume is constant
in size and location over time and does not necessarily coincide with any physical
boundaries—that is, it is purely a tool for the analysis of physical systems. If we apply
Equation 1.13 for the property mass, then the time rate of change of B.,, would be
zero since mass of a system is constant by definition. Thus, the law of conservation of
‘mass (also known as the “continuity equation”) states that any change in mass within
this control volume must be equal to the mass of fluid that enters the volume (mass in)
minus the mass that exits (mass out). Therefore, for a given period of time, At,
((Rate of mass in) — (Rate of mass out) = Rate of change of mass within the tube]
The rate of mass carried across a surface is equal to the density times the flow rate,
pO, where Q is the volume flow rate, or pV,A, and V, is the component of velocity 10 Biofluid Mechanics: The Human Circulation
AL
FIGURE 1.5 Mass fiux balance for a stream tube.
normal to the cross section. Furthermore, the rate of change of mass within the tube
is given by
2 feav
ev
Thus, for the case shown in Figure 1.
Rate of mass in = Jovinaa
a
and
Rate of mass out = fvin dA
he
where V,,, and V,, are the velocities normal to the differential areas at cross sections
1 and 2, respectively.
Inserting these into the law of conservation of mass gives us
~[ ovina | pvinaa=2 frav (114)
a 4 v
For liquids in general and blood in particular, a very good assumption is that the fluid
is incompressible—that is, its density, p, is constant. This, together with the fixed
size of the control volume, causes the time rate of change of mass within the control
volume to be zero. Furthermore, if the flow is steady and there are only two surfaces
across which fluid flows (Surfaces 1 and 2), then the mean velocities (V) and cross-
sectional areas (A) can be related as shown in Equation 1.15:
AV, = 4,V, = Q (constant) (1s) Fundamentals of Fluid Mechanics "
If the velocity varies over the cross section as a function of radius, 7, then the mean
velocities must first be obtained by
1 eal
a [nom and Rag, fuera
Example 1.2
A patient is undergoing a cardiac catheterization procedure in which radiopaque
dye is injected into his heart through a 2-m long catheter to obtain x-ray images of
his left ventricle (Figure 1.6).
a. If the dye is injected from a syringe outside the body which is 2cm in
diameter, what must be the velocity of the plunger in order to deliver
8.5cm? in 152
b. What would be the average velocity of the dye as it exits the catheter tip if
the catheter has a diameter of 2mm (=6 Fr)?
Brachial artery
Alternative site
Guiding catheter
Introducer sheath
¢ A
Introducer sheath
in the groin or arm
FIGURE 1.6 Cardiac catheter being introduced into the left ventricle via an access site in
the iliac artery. (With permission from the Cleveland Clinic Foundation, Cleveland, OH.) 2 Biofluid Mechanics: The Human Circulation
Solution
a. Assuming that the dye has a constant density and that the injection is per-
formed steadily, we can simplify the conservation of mass to Equation 1.15,
Then, the volume flow rate, Q, can be calculated as
Q=85cem*/Is = 85cm /s
Since there is only one outlet at the catheter tip,
Q= Aounger * Vplunge
= [re (2cm)/4] x Vptunger
=3.14 XVpunger
‘Thus,
Vptunger = 8.5m? /s/3.1 4m?
b. If we further assume that the velocity exiting the catheter tip is uniform, that
is, constant over the cross section, then
Q= Aeatheter * Veatheter
= [(0.2 cm)?/4] x Veathorr
0.0314 X Vesinter
Thus,
Veatheter = 8.5 cm?/s/0.0314 cm?
= 27 1cm/s (=2.71m/s)
1.4.2 CONSERVATION OF MOMENTUM
The principle of conservation of momentum was initially formulated from Newton’s
second law of motion, which states that the sum of the forces (dF) acting on an
object is equal to its mass (m) times its acceleration (@), or
> Bema (1.16)
Rewriting @ as dV /dt and bringing m inside the differential (since it is constant for a
system) results in
SY F=n{ 2] a.)
dt dt
where the derivative is now the time rate of change of momentum. Fundamentals of Fluid Mechanics 13,
‘Therefore, an alternative way of stating Newton’s second law of motion is that for
a system:
[Sum of the external forces acting on the system|
= Time rate of change of linear momentum of the system|
me
R,-2 i
Dau=2 fia (18)
When the system and the control volume are coincident at an instant of time,
the forces acting on the system and the forces acting on the control volume are
identical, or
LED
Considering the change in linear momentum for such a system and coincident con-
trol volume (Equation 1.18 righthand side [RHS}), the Reynolds transport theorem
allows us to write
(1.19)
Time rate of change of linear momentum in a system
= Time rate of change of linear momentum in control volume
+ Net rate of change of linear momentum through control volume surfaces
or,
od Yoav = 5 J veavs | io? ida (1.20)
Therefore, for a control volume that is fixed (i.., with respect to an inertial reference)
and nondeforming,
Beant =
Sew cp hice) jidA an
The previous equation is called the linear momentum equation because we only
consider motion acting in an axial direction. The forces in this equation acting on
the control volume are both body forces and surface forces and can be expressed as,
Sins Joeav + fiaa (1.228)
ev bs
where
& is the body force per unit mass acting on the control volume contents
7 is the stress vector acting on the control volume surfaces 14 Biofluid Mechanics: The Human Circulation
When viscous effects are important, the surface area integral of the stress vector is
nonzero and must be determined empirically (i.e., from experimental data) and given
as friction factors ot drag coefficients (ic., constants relating drag forces to other vari-
ables). However, if viscous effects are negligible, then the stress vector is given by
ip (1.226)
i=
where
nis the outward directed unit vector normal to the surface
pis the pressure
Example 1.3
In Example 1.2, what is the peak force required to propel the radiopaque dye
(p = 1.3g/crm, 1 = 0.04P) into the heart if the mean systolic pressure, yy, is
120mmHg?
Solution
First, we will consider a control volume, CV, defined by the boundary between
the dye and the syringe and catheter surfaces. Then, we can set up a force balance
according to Equation 1.21. Furthermore, if the syringe is moved steadily, the time
rate of change term becomes negligible, or
af,
= =0
a fvoav
&
Thus, Equation 1.21 reduces to
Die = [vernon
&
where the sum of the forces, Fey, consists of the pressure force produced by
the plunger, the pressure force of the blood resisting the motion and the shear, or
frictional, force along the dye/catheter interface. Thus,
> Fev = Funes ~ Feat — Fron
The desired unknown force, Finger acts at the interface of the dye and the syringe
plunger, while at the catheter tip, motion is resisted by the blood pressure force
Featetr = Pays * Acstotr
= (120mmHg-1333dyn/cm?/mmHg)[x(0.2.cm)?/4]
= (1.60- 10° dyn/cm?)(0.0314 cm?)
= 5023 dyn Fundamentals of Fluid Mechanics 15
The friction force depends upon the fluid viscosity, the fluid/wall shear rate, and.
the surface area over which it acts (see Section 1.5.3.2 for further details). In this,
example, this force is given as
Friction = BTUEV
= 8(3.14)(0.04 dyn: s/cr?)(200 cm)(2.7 1em/s)
= 545dyn
Again, if we assume uniform velocities at the syringe plunger and at the catheter
tip, the momentum integral over those control surfaces can be rewritten as
J%pi-hda=Svipv-Aya
&
= Vgtunger(PVatunger fi) Aptunge + Vester (P Veateter “F)Acanetr
Here, the term (pV-A) can be thought of as the axial momentum per unit
volume, which “carries” the intrinsic property of interest—in this case,
(see Equation 1.13).
momentum/unit mass =
Thus,
D0 - AVA = Vung (Vonge) sng + Vt (Vente) Act]
where (+) velocity is along the axis of flow.
If we substitute values into the simplified Equation 1.21, we obtain
Fotunger ~ 5023 ~ 545 = 1.3 g/cen? {[(2.7 lem/s) (-2.7 1cm/s) (3.14.crm*)]
+ [27 1cm/s)(27 1em/s) (0.0314 cm’ JI}
= 13 g/cm? (-23.1+ 2306}
= 2968dyn
Thus,
Folunger = 8536 dyn
The previous two balances of mass and momentum are called macroscopic or inte
gral balances because they consider the control volume as a large, discrete space and
are written in terms of bulk flow variables. In order to derive more general forms
of these equations which provide spatial detail throughout the flow field, we need
to take a microscopic or differential approach. Such an approach leads us to what
are commonly referred to in the fluid mechanics literature as the continuity and the
Navier-Stokes equations, for conservation of mass and momentum, respectively. 16 Biofluid Mechanics: The Human Circulation
1.5 MICROSCOPIC BALANCES OF MASS AND MOMENTUM.
1.5.1 CONSERVATION OF Mass
We begin by considering an infinitesimal control volume, CV, of dimension AxAyAz
(Figure 17a).
A fluid flowing through the control volume will have a velocity denoted by the
vector,
Veuityj+wk (1.23)
where each of these variables may be a function of time.
‘The conservation of mass principle states that for a system
‘The net rate of mass flux across the control surface
+ =
The rate of change of mass inside control volume
‘Treating each of the aforementioned terms individually, we get
1. The net rate of mass flux across the control surfaces (Figure 1.7b) in the
x-direction:
(pul pul...) AvAz
y-direction:
(pr|,—pr|,, Axdz
Iyeay
z-direction:
(pw|.—pr |, ..,)AxAy
vg Polete
Pullesax
Z
@ (b)
FIGURE 1.7 (a) Differential control volume in rectangular coordinates and (b) mass flux
across surfaces of control volume. Fundamentals of Fluid Mechanics 7
2. The rate of change of mass inside the control volume
ce yAxAyAz)
5; (Aray
Combining terms and rearranging yields
(Pu srs —Pul, AYAT+(PY| pray Pr, JAxAc+ (Pw| cose Pre], JAxAy
=- £ (pAxAyAz) (1.24)
Furthermore, each of the mass flux terms can be equivalently written as the gradient
of mass flux in that coordinate direction times the distance moved, or
(pu)
a
(eu... — Pal.)
and so on.
Since the volume within the control element is time invariant, we can divide each
term by AxAyAz, Then, in the limit as AxAyAz approaches zero, we obtain
2 ouy+2-(ovy+ 2 (pw) + 22 1.25)
ay PFS OHS OW) 5, 0 (1.25)
which is equivalent to
742?
VipV+=0 (1.26)
The previous equation is known as the continuity equation where the “del” vector
operator, V, is defined as
70 79 790
sot inc tk (1.27)
v=
ax Jay dz
A fluid such as blood is incompressible, and thus, p = constant, so that the continuity
equation becomes
pV-V=0P (1.28)
or
V-V=0 (1.29)
Note that this equation is valid for both steady and unsteady (including pulsatile, 3D)
flows of an incompressible fluid. 18 Biofluid Mechanics: The Human Circulation
If we return to Equation 1.25 and differentiate each numerator, we obtain
28 PP 422
ar ax ay a
=0 (1.30)
2 9( a or) e ay
Dt
ax * dy” dz ax dy
or,
(131)
where
Pua Py Day h
Dio“ ax* ay a
is called substantive derivative.
The substantive derivative represents the time derivative of a scalar or vector
quantity, which follows the motion of the fluid. For example, for a scalar quantity
such as temperature, 7, the substantive derivative would be expressed as
DT _ oT or or oT
a — — 1.32)
Dt at “ax ay az 4.32)
where
A7/0¢ is the local time rate of change of temperature
the terms u(87/0x) + v(OT/2y) + w(AT/82) represent the rate of change of tempera-
ture due to fluid motion (also known as convection)
Finally, for many fluid dynamic situations such as the consideration of liquids
or gases at low speeds, it is quite common to assume an incompressible fluid where
p= constant. In this case, Equation 1.31 can be further reduced to
V-¥=0
1.5.2 Conservation oF MoMENTUM
In deriving the differential form of this law, we once again consider the dynamics
associated with a fluid control volume ~AV = AxAyAz shown in Figure 1.7b. We now
apply Newton's second law of motion as written in Equation 1.16 in terms of the time
rate of change of momentum
Sum of the external forces acting on the control volume
= Net rate of efflux of linear momentum across control volume
+Time rate of change of linear momentum within the control volume| Fundamentals of Fluid Mechanics
19
In general, linear momentum per unit volume of fluid can be expressed as pV so that
by multiplying this term by the rate of volume change, we can obtain the time rate of
change of linear momentum, To determine the flux of a property across the control
volume surface, the appropriate expression for the rate of volume change is (V-7)dA
where fis the outward directed normal to a particular surface. For change of a prop-
erty within the control volume, the rate of volume change is simply given by dV/dt.
As with the derivation of the continuity equation, the control volume considered is
constant and we can express the aforementioned equality as
} J a. J J ae
lim. = lim |}*—
av50 AxAyAg vod) AxAyAz— * ar AxAyAz
if we take the limit of AV = AxAyAz as it approaches zero.
Each of these terms can be evaluated separately as follows:
1. Sum of the external forces
AM Rayne
2. Net rate of momentum efflux across the control volume
(1.33)
a,.ia a
RPO; O45. (0)
and reduce this limit to
‘AxAyAz May ec
v0,
eee ee
3. Time rate of change of momentum within the control volume
pn [IRB So oF ve
(1.34) 20 Biofluid Mechan
: The Human Circulation
Substituting for the limits and combining terms gives
dB oof ov ov Ell ass)
‘We have now expressed the change of momentum in terms of its component veloci-
ties. Let us now look at the external forces in more detail. The external forces con
sist of the sum of the body forces, F's, and the surface forces, Fs. The body forces
are typically due to the presence of gravitational, electromagnetic, and electrostatic
fields. If the only body force is gravity, then
peAxAyAz (1.36)
‘The surface forces acting on the control volume are those due to the normal, o, and
the shear, ¢, stresses. These stresses can be assumed to vary continuously from their
nominal value at the center of the control volume in each of the coordinate direc-
tions. Figure 1.8 depicts the normal and shear stresses acting on the control volume
in the x-direction alone. Similar figures can be constructed for the normal and shear
stresses acting in the y- and z-directions. Thus, the net surface force acting in the
2-direction is given by
Fa= (= Janay vc+{ 2 Vana pac (8) arava (1.37)
a ay. a:
‘The total force acting on the control volume in the x-direction then becomes.
Fy = Fox + Foe (1.38)
2
FIGURE 1.8 Normal and shear stresses along the x-coordinate in the control volume. Fundamentals of Fluid Mechanics 2
which, in the limit, can be expressed as
(1.39a)
(1.39b)
(1.39¢)
Substituting the results of the previous expressions (1.39a through 1.39¢) back into
the expression for Newton’s second law (Equation 1.17) yields
(40a)
Ay (24, Ov
ad (z wate tws } (1.40b)
11 yA ay) (1.40c)
In this form, we can see that the RHS of the previous equations actually represents
density (mass/volume) x acceleration, or force/volume, where the acceleration terms
can be separated into local acceleration (@u/2t, etc.) and convective acceleration
(udu/ax, etc.) components. The total acceleration can be expressed in terms of the
substantive derivative as
Du _Ou, du, du, du
Dre ar oe tay oe (41)
Equations 1.40a through 1.40c represent the complete form of the differential
conservation of momentum balances. These equations cannot be solved, how-
ever, because there are more unknowns (i.¢., dependent variables) than equations.
Thus, it is necessary to derive additional information in order to provide those
equations. In practice, we will consider the special, although not uncommon, 22 Biofluid Mechanics: The Human Circulation
case of incompressible, Newtonian fluids. Here, the normal and shear stresses
can be expressed as
ou ov
On =—pt ote *) (1.42a)
oy =—p+ 2” (1.42b)
oy
ow ow
“pees + =) (1.42)
By substituting these relationships into Equations 1.40a through 1.40c, we obtain the
equations of motion in scalar form along the three coordinate axes as
Ou, eu, eu du, ou, du du
Pao ou $e Zs 2) e of Sted “yt 2) (1.43a)
av)_ (ay, av, av, ay
+n Sor )o( Seu de nv wo) (1.43b)
ap, (aw, aw, Fw)_ (daw, dw. dw. dw
eT [eee Ee eee eae 1.43
Pas Pou( Ss a ar) Plan ax vay ae) O°
‘The equivalent vector form of the previous equations is
a 3 _ DV
pB~Vp+HV'V =p 44)
which can be alternatively written as
vig 1 ay}
Spt VW =-" Vp re +e"V) (1.45)
P
by expanding the material derivative for acceleration and dividing by p.
The previous equations (in either scalar or vector form) are commonly called the
Navier-Stokes equations for Newtonian incompressible fluids in honor of the math-
ematicians who originally derived these relationships.
The continuity and Navier-Stokes equations can also be derived in other coordi-
nate systems and, thus, are given in Cartesian, cylindrical, and spherical coordinates
in Table Ll. Fundamentals of Fluid Mechanics 23
TABLE 1.1
Continuity and Momentum Equations in Cartesian, Cylindrical
(Polar), and Spherical Coordinate Systems
a. Car
coordinate system
and
aw, aw, aw
p) eu
a att
b. Cylindrical coordinate system
re
(2
and
(continued) 24 Biofluid Mechanics: The Human Circulation
TABLE 1.1 (continued)
Continuity and Momentum Equations in Cartesian, Cylindrical
(Polar), and Spherical Coordinate Systems
c. Spherical coordinate system
Lay
2°
my 2%
or r
av. , Ye ave, Ve ave
or r 0 rsin@ ag
cto av, 1 8, 2 ay
20 Fin? Og? 7? 3
Wav , Yo aM
Ye Me
r 90" rsind ap
aM 22% __ Ve
a
2Y,__2eot® aM
a0 sind dp
and
av
rsind d9
BY, | cot 0 BY,
Poe Fo
2cor0 ava,
7 sind 39)
1.5.3 MATHEMATICAL SOLUTIONS
As mentioned earlier, in order to solve a set of equations, we must have at least
as many constraints as we have dependent variables. Examination of Equations
1.43a through 1.43c shows that there are four dependent variables—pressure (p) and
the three velocity components (u, v, and ») defined in terms of four independent
variables—time (0) and the three position coordinates (x, y, and 2) but only three
equations. (Note: Similar results would be found with the cylindrical and spheri-
cal coordinate systems as well.) However, by including the continuity equation, we Fundamentals of Fluid Mechanics 25
obtain a fourth constraint which will allow us to uniquely define each dependent
variable. Mathematically, Equations 1.31 (continuity) and 1.44 (Navier-Stokes)
are first and second order partial differential equations, respectively. Furthermore,
Equations 1.43a through 1.43c is nonlinear because of the presence of product terms
such as udu/x, vdu/dy, and so on. Unfortunately, no exact analytical solution has
been defined for equations of this form. Therefore, in practice, two approaches have
been taken. One is to first simplify the equations until they have a mathematical form
for which there is a solution while the other is to solve them numerically.
1.5.3.1 Couette Flow
One example of obtaining a solution to the continuity and Navier-Stokes equations
by simplifying the number of spatial and temporal variables used is that of flow
between two parallel plates as shown in Figure 1.9. We will assume that the upper
plate is moving at a steady velocity U while the lower plate is fixed. Furthermore,
there is no variation into or out of the plane (z-axis), so we can consider this as a
two-dimensional (2D) problem with flow along only one axis. These two assump-
tions, then, allow us to eliminate all terms involving 0/8¢ due to steady flow and two
of the velocity components (v and w) due to uniaxial flow. Since the flow is in the
x-direction only, Equation 1.31 reduces to
au
5.70
or,
u=f(y)+C
where C is the integration constant.
Furthermore, Equation 1.43a reduces to
op _ ( %u
2 = 4( 24
7
a
FIGURE 1.9 Flow between two infinite, flat plates. 26 Biofluid Mechanics: The Human Circulation
where it is now possible to solve for u(y) as an explicit function of p. Integrating
twice, we obtain
and
1 op|
5-2 |¥ +Gy+e,
2 ax \y Wy 2
where C, and C; are constants of integration which can be evaluated by applying
specific boundary conditions for the problem. In this case, for example, we know that
the near-wall velocity is the same as that of the walls due to the “no-slip” criteria.
‘Thus, u(y = 0) = 0 and u(y = A) = U, Furthermore, there is no pressure gradient in
the x-direction. Applying these boundary and driving force conditions, C, = 0 and
C, = U/h, yielding a velocity solution or “profile” of
_(¥
uly) = ( h
as shown.
An alternate configuration is that the flow is driven by a pressure gradient in the
x-direction between two stationary plates. In that case, dp/dx # 0 but u(y = 0) =
u(y = h) = 0. Applying these constraints, we find that C, = 0 but now, C, = [(1/2)
Ap/ax]h. The resulting velocity profile becomes
1 P|
=|— aT
u(y) [ze 2 lo y)
This is the classical Couette relationship for steady flow between stationary
boundaries.
1.5.3.2. Hagen-Poiseuille Flow
If we now apply the Navier-Stokes equations in cylindrical coordinates (Table 1.1)
to the case of steady flow in a straight, circular, horizontal tube (Figure 1.10), the
momentum balance in the z-(axial) direction reduces to
(1.46)
since the time rate of change (i.e., 0/81), secondary velocity (ie., V, and Vg), and
circumferential velocity gradient (i.e., 0V,/9®) terms are zero. As a consequence, the
conservation of mass balance results in aV./0z also being zero. Fundamentals of Fluid Mechanics 27
(4 Bde) amir rbd
FIGURE 1.10 Force balances for steady flow through a straight, horizontal, circular tube.
Rearranging terms then yields
a _ fb (,a% ua
az Lr ar” or Ca
Since pressure is only a function of length and axial velocity is only a function of
radius, this equation can equivalently be written in terms of ordinary derivatives, or
a _ [ial ay.
i yflal Mv 1.48
dz of} ag dr )I Cy
We can further observe that, for the two terms of the equation (ie., left-hand side
[LHS] and RHS) to be equal for all values of the independent variables r and z (each
of which is only present in one of the terms), each term must be constant. Equation
1.48 can then be integrated twice to yield
(oa
“dr az r
-2{2)rsamree (1.49)
The constant terms, c, and c,, can be evaluated by applying known values of axial
velocity at specific boundary locations. For example, V, = 0 at r = R due to the
“no-slip” condition at the tube wall. The value of V,, however, is not known at the
tube center, r = 0, although we can assume that it is a maximum at that point due to
overall symmetry of the tube. Thus, the appropriate boundary condition here is that
dVJar = 0, which requires that c, = 0 and which also constrains all velocities to be
finite
Evaluating c, and substituting it into Equation 1.49 results in
le (aps tate
—|- -R -
2( AG ] (1.50) 28 Biofluid Mechanics: The Human Circulation
If we replace the differential pressure gradient term by the pressure gradient along
the entire tube, Ap/L, then the velocity variation or “profile” in a tube is given by
woffa
or,
ry
V(r) = mf}-(3) asl)
where
V (nis) is the velocity of the fluid at a distance r (m) from the center of the tube
Vmax (mn/S) is the maximum (centerline) velocity
R (m) is the radius of the tube
Ap (Pa) is the pressure drop along a length Z (m) of the tube
By integrating this velocity profile over the tube’s cross section and dividing by the
area, we can obtain the average velocity
(1.52)
Since flow rate in the tube, Q (im°Js), is equal to the average velocity, V,,.. times the
cross-sectional area, we can write
= Vola?) = Yak)
(1.53)
Apnd*
O= Togu. (1.54)
Solving for the pressure difference, we obtain
dp = 12g HO (1.55)
md’
which is commonly known as the Hagen—Poi:
uille equation. Fundamentals of Fluid Mechanics 29
Finally, the shear rate along the inner tube wall can be obtained by differentiating
Equation 1.51 and evaluating it at r = R, or
where the negative sign indicates that the shear stress acts in the direction opposite
to the flow.
1.5.3.3 Numerical Solutions
‘This approach is more complex but provides the ability to solve problems without mak-
ing unrealistic simplifying assumptions. The basic technique is to first subdivide the
flow into many small regions, or elements, over which the governing equations are
applied. Rather than use the differential form of the equations, however, they are rewrit-
ten in algebraic form in terms of changes that occur in variables due to incremental
changes in position and time. Solutions are then obtained locally at specific locations,
or nodes, on the finite elements across a mesh of elements (Figure 1.11). This set of
1 3
FIGURE 1.11 CFD mesh for a reconstructed human coronary artery with a stenosis seg-
‘ment. It can be observed that a finer mesh is employed in the area of stenosis in order to obtain
more accurate results in this region of interest. 30 Biofluid Mechanics: The Human Circulation
solutions is then updated at subsequent time intervals over the entire mesh until some
acceptable level of accuracy, or tolerance, is achieved based upon convergence between
successive values of certain output variables. Obviously, this can be a very detailed
and time-consuming process depending upon the complexity of the geometry being
analyzed and the initial and boundary conditions imposed, Furthermore, additional
features are sometimes included in the equations, for example, to allow for simulation
of non-Newtonian fluids and turbulent flow conditions. While it is always important to
validate such results, these computational fluid dynamic (CED) software programs are
increasingly being used to solve challenging biomedical flow problems unapproachable
by any other means. A thorough discussion of this topic is given in Chapter 11.
1.6 BERNOULLI EQUATION
One particularly useful tool known as the Bernoulli equation can be obtained by
simplification of the Navier-Stokes equations (Equations 1.43a through 1.43c)
Specifically, flow is assumed to be inviscid (i.e., the viscous forces are negligible),
which then allows the Navier-Stokes equations to be simplified to what is known as
Euler’s equation of motion (Equation 1.46):
av
eee
V-V)V =-—Vp+3 i
3 TUM) pre (1.56)
As can be seen, the shear stress terms have been set equal to zero since there is no
friction in the system. To further simplify the equation of motion, we will focus on
flow along a streamline (Figure 1.12). A streamline is an imaginary line in the flow
field drawn so that it is always tangential to the velocity vectors. The sum of the
forces acting on a fluid particle along the streamline is given by
x
where sin @ = dz/ds,
tt-(p+ as ak —pelsindy dads (1.57)
s
FIGURE 1.12 Force balance for a fluid particle along a streamline. Fundamentals of Fluid Mechanics 31
Substituting this and simplifying Equation 1.57 results in
ve (- ® as
ds
Furthermore, fluid acceleration along the streamline curve consists of temporal and
spatial components given by
pg acs (1.58)
a
dt
=i (1.59)
or as
where V has been substituted for 43/41.
Taking the mass of the particle as pdA ds and inputting Equations 1.58 and 1.59
into Newton’s second law of motion gives us
-7 ds pede}aa (1.60)
Is
Since an equivalent mathematical expression for the term VaV/ds is 0/0s(V7/2), we
will now substitute this into Equation 1.59 and integrate between any two arbitrary
points 1 and 2 to obtain
2g 2 _y2
of Bacay” *) str py tpetes—a)=0 (6b)
He ld 2
Equation 1.61 is known as the unsteady form of the Bernoulli equation where p,, V,, and
2; are the pressure, velocity, and height at the upstream location, and p,, Vs, and z, are
the pressure, velocity, and height at the downstream location. The integral term accounts
for the flow acceleration or deceleration between the two locations and for steady flow
conditions (where 0/dt = 0), it becomes zero. Furthermore, since the points | and 2 are
arbitrary, we can write a general expression of the steady flow Bernoulli equation as
pept-+pec=H (1.62)
where p, V, and z are evaluated at any point along the streamline. Here, H1 is a con-
stant and is referred to as the “total head” or total energy per unit volume of fluid.
This terminology is used because even though we derived this equation based on
conservation of momentum principles, the terms in Equation 1.62 all have the units
of F/L? which can equivalently be written as FL/L®, ot energy/volume.
In the absence of frictional losses, then, the Bernoulli equation states that the total
mechanical energy per unit volume at any point remains constant. The mechanical
energy of a system consists of a pressure energy component (due to static pressure),
a kinetic energy component (due to motion), and a potential energy component (due
to height) assuming there are no thermal energy changes. Thus, between any two
points, the form of the mechanical energy may change from kinetic to potential and
vice versa, for example, but the total mechanical energy does not change. 32
Biofluid Mechanics: The Human Circulatio
Example 1.4
A syringe is filled with water and held vertically as shown in Figure 1.13. A pres
sure of 100 mmblg is applied to the fluid by pushing on the plunger. What is the
velocity of the fluid leaving the syringe tip and how high will the stream go?
Using Bernoulli equation (1.62), we can solve for the total head, H, by assum-
ing that the elevation in the barrel is zero and that its velocity is negligible (this
is reasonable since the cross-sectional areas of the barrel and tip are in a ratio of
100:1). Thus,
H = p, = 100 mmHg x 1,330 dyn/cm? mmHg,
= 133,000 dyn/cm?
The water velocity leaving the syringe tip, Vp, can be calculated knowing that the
local (gauge) pressure, p,, is zero since it is the same as atmospheric. Thus,
o+ tev +pgh, = 133,000dyn/cm?
and
FIGURE 1.13 A jet of flow from a pressurized syringe. (Image purchased from 123rf.com Fundamentals of Fluid Mechanics 33
Finally, the elevation of the stream, hy, can be calculated by knowing that it occurs
when the local velocity, Vs, becomes zero, Thus,
0+0+pgh; = 133,000 dynécm*
and
hy = 137 cm (=137 m)
(Note: By conservation of mass, the velocity in the syringe barrel, V,, would actu-
ally be V,/100 = 5.14cm/s and lead to an error of <1%,)
The Bernoulli equation also states that the pressure drop between two points
located along a streamline at similar heights is only a function of the velocities at
these points. Thus, for z, = z,, Bernoulli equation simplifies to
1
Pi- P= 7 PVE -W) (1.63)
Finally, when the downstream velocity is much higher than the upstream velocity
(ie., V; > V,), pV" can be neglected, and Equation 1,63 can then be further simpli-
fied to
1
Pim P= 5PVE (1.64)
In the specific case of blood flow where we can assume a density of 1.06 g/cm’,
Equation 1.64 becomes
Pi~Pr (1.65)
where
p;is in mmHg
Vy is in mis
the constant term, 4, has units of (mmHg/(m/s))
Equation 1.65 is referred to as the simplified Bernoulli equation, and has found wide
application clinically in determining pressure drops across stenoses, or vascular
obstructions, using velocity measurements made from noninvasive instruments such
as Doppler ultrasound and magnetic resonance phase velocity mapping (see Sections
10.6 and 10.8).
It is important to keep in mind that the Bernoulli equation is not valid in cases
where viscous forces are significant, such as in long constrictions with a small lumen
diameter, or in flow separation regions. In addition, it does not account for turbulent
energy losses in cases of severe stenosis since they are time varying. Such losses
should be taken into account in any calculation because they may substantially
reduce the energy content of the fluid. 34 Biofluid Mechanics: The Human Circulatic
Example 1.5,
A patient has a stenotic aortic valve producing a pressure drop between the left
ventricle and the aorta. The mean velocity in the left ventricle proximal (upstream)
to the valve is 1m/s while the mean velocity in the aorta distal (downstream) of
the valve is 4mm/s. Applying Equation 1.65, the pressure drop across the valve is
~ Pr = Ave
= 44s?’
= 64mmHg
By comparison, if we include the proximal velocity in the calculation, then the
pressure drop reduces to
~pr=4(v? -v?)
(4s)? — (1 m/s}?)
= 60mmHg
This example demonstrates an important clinical use of the Bernoulli equation, sin
the degree of pressure drop is a good indicator of the severity of the stenosis, a
thus, of the need for treatment. Unfortunately, it is a somewhat risky and comp
cated procedure to obtain pressure data directly since it requires inserting a cat
eter manometer (pressure meter) into the aorta, A much simpler and noninvasi
technique (see Section 10.6) is capable of measuring the flow velocity in the val
thus enabling an estimate of the pressure drop by means of the simplified Bernou
‘equation. Several cautions should be observed in using this approach, however, due
the basic assumptions made in deriving the original Bernoulli equation and tho
made in further simplifying it which may either over- or underestimate its true valt
1.7 DIMENSIONAL ANALYSIS
If we return to the general form of the conservation of momentum or the Navie
Stokes equations, we can write
pee pP- VP =—Vp+ pe + RV’
1.6
Por :
where
p8V/2r is the local acceleration
pV-VV is the convective acceleration
-Vp is the pressure force per unit volume
8 is the body force per unit yolume
uV°P are the viscous forces per unit volume Fundamentals of Fluid Mechanics 35
If we represent
& (1.67)
where
Q =~ (Gxt BF Bz) (2.68)
then
‘Vp-pV9=—V(p+p9)=—-Vp (1.69)
or
ow spy VV =-Vp+pv°V (1.70)
Based on the principles of dimensional analysis, the dependent variables, V and
p, are functions of the independent variables, x,y,z, and f, as well as the constants,
pH, and 3, along with any other parameters that appear in the boundary conditions.
Furthermore, examination of the previous equation shows that each term has units of
force per unit volume, or F/L?. Multiplying numerator and denominator by L results
in an equivalent expression (F - L)(L?-L), or energy/(volume- 1). Now, if we divide
each term by a constant having those same units, we would obtain a dimensionless
equation, For the moment, let us designate a characteristic (.e., reference or repre-
sentative) length and a characteristic velocity as [., and V,, respectively. Now, the
ratio of pV2/L, also has the dimension of (F-Z)/(Z>-L), so multiplying every term by
L,/pV2 would result in
aviv.) v)_ Hippy)
BOV,ME) "| (Fhevfz -)- wf; sir} (az je {F) an
where individual terms have been grouped into dimensionless variables defined as
dimensionless velocity
dimensionless vector gradient operators
dimensionless pressure
dimensionless time
dimensionless parameter called the Reynolds number 36 Biofluid Mechanics: The Human Circulatio
Writing the equation in terms of these dimensionless variables, we get the dimer
sionless form of the Navier-Stokes equation:
ov
< =, 41
sot: VY =-Vp+— Ve es
ar Pw Re :
For incompressible flows, the dimensionless continuity equation becomes
Vev=0 a7.
Examination of Equations 1.72 and 1.73 shows that, for a given value of Re, there
only one solution to these equations for each driving function, Vp. One consequenc
of this is that by matching values of Re for various systems that are similar but «
different scales, it is possible to achieve dynamic similarity and obtain predictiy
information about prototypes from scaled model measurements.
Example 1.6
It is desired to study the fluid dynamics in a coronary artery (diameter = 3 mm)
using a laser Doppler velocimetry (see Section 10.7). However, the resolution of
the LDV system is 300,m, resulting in a maximum of 10 velocity points across
the artery diameter and low accuracy for obtaining wall shear stresses from the
fitted velocity profile. In order to improve the resolution by x10, an in vitro model
is constructed with a diameter of 3.cm. What flow rate should be used to ensure
dynamic similarity between the model and the prototype? (Assume the working
fluid has the same kinematic viscosity as blood: v = 0.035 cm’/s.)
Since D, = 3mm, D,, = 30mm, dynamic similarity can only be achieved if
Rem = Rep
or
For Uy) =
Vn = 2ev, = 0.1V,
Therefore, flow in the model must be
2n=v4( 22)
o.v,[ MP" | 100,
1.8 FLUID MECHANICS IN A STRAIGHT TUBE
Much of the blood flow in the human circulation occurs within tubular struc
tures—arteries, capillaries, veins, and so on. It is for that reason that the study 0
fluid mechanics in a straight tube is of particular interest in biofluid mechanics Fundamentals of Fluid Mechanics 37
While the human vasculature is not strictly a series of straight tubes of constant
diameter, results from this analysis do provide good estimates or starting points for
further evaluation. Before we look at this topic in detail, however, we need to make
several definitions of terms that are relevant to common fiow conditions,
Blood flow in the circulatory system is invariably unsteady and, in most regions
(eg, the systemic arteries and the microcirculation), it is pulsatile, The term
“unsteady” is very general and refers to any type of flow, which is simply not constant.
If the flow has a periodic behavior and a net directional motion over the cycle (ie.,
the average flow is >0), then it is called pulsatile. On the other hand, if the flow has a
periodic behavior but oscillates back and forth without a net forward or reverse output
(ie. the average flow = 0), itis called oscillatory flow. Despite the fact that blood flow
is unsteady, itis helpful to first describe the principles of fluid flow under more simpli-
fied conditions, before moving to complex physiological situations. The simplest case
to consider, therefore, is that of steady flow of a Newtonian fluid through a straight,
rigid, circular tube aligned in a horizontal position (see Section 1.5.3.2).
1.8.1 Frow Stasitity AND RELATED CHARACTERISTICS.
‘The nature of flow of a Newtonian fluid ina straight, rigid, circular tube is controlled
by the inertial (accelerating) and the viscous (decelerating) forces applied to the fluid
elements. When viscous forces dominate, the flow is called laminar, and is character-
ized by a smooth motion of the fluid. Laminar flow can be thought of as if the fluid is,
divided into a number of layers flowing parallel to each other without any disturbances
or mixing between the layers. On the other hand, when inertial forces strongly domi-
nate, the flow is called turbulent. Here, the fluid exhibits a disturbed, random motion
in all directions, which is superimposed on its repeatable, main motion.
1.8.1.1. Steady Laminar Flow
The key characteristic of laminar flow is that it is very orderly and very energy
efficient, whereas turbulent flow is chaotic and accompanied by high energy losses.
Therefore, turbulent flow is undesirable in the blood circulation because of the exces-
sive workload it would put on the heart and also because of potential damage to
blood cells. A useful index used to determine whether the flow in a tube is laminar
or turbulent is the ratio of its inertial properties to its viscous properties. This ratio
is classically known as the Reynolds number (Re), which is dimensionless since both
terms have the same units [F-T/L?]. It is defined as
_ Inertial forces _ pV
Re (1.74)
Viscous forces
where
p (kg/m?) is the density of the fluid
V (m/s) is the average velocity of the fluid over the cross section of the tube
d(m) is the tube diameter
1 (kg/m-s) is the dynamic viscosity of the fluid 38 Biofluid Mechanics: The Human Circulation
Although inertial forces obviously begin to dominate for Re > 1, it has been deter-
mined experimentally that in a smooth-surfaced tube, flow is laminar for all con-
ditions where Re < 2100. Within this range, the viscous or “retarding” forces are
sufficient to suppress any tendency for the flow to gain so much inertia as to become
“chaotic.” Due to our assumption in Section 1.5.3.2 that the flow was steady, Equation
1.55 will only be valid for laminar flow conditions—that is, Re < 2100, Furthermore,
if the tube is long enough to have stabilized any entrance effects that are present (see
Section 1.8.1.3), then the velocity profile takes on a parabolic shape and the flow is
called fully developed laminar flow.
1.8.1.2 Turbulent Flow
For Re > 2100, the flow starts to become turbulent in parts of the fluid and par-
ticle motions begin to vary with time. Such flow is called transitional flow, and
the range of Re for which this type of flow exists varies considerably, depend-
ing upon the application (i.e., system configuration and stability) and the rough-
ness of the tube surface. Usually, flows in a smooth-surfaced tube with 2100 <
Re < 4000 are characterized as transitional. Fully developed turbulent flow is
observed starting at Re = 10‘ for a rough-surfaced tube and starting at Re = 10°
for a smooth-surfaced tube. The velocity profile of fully developed turbulent
flow is flatter than that for laminar flow, although it is not considered uniform or
“plug” flow (Figure 1.14).
10
tos
FIGURE 1.14 Plots of laminar (Poiseuille) and turbulent velocity profiles for n = 6, 8, and 10. Fundamentals of Fluid Mechanics 39
In order to quantitatively describe turbulent flow, we think of the instantaneous
velocity, V(0, at a given radial location as the sum of a time-averaged mean velocity,
V(m/s), and a randomly fluctuating velocity component, V’(m/s), or
V(t) =V4V" (1.75)
Expressing turbulent flow in this way allows us to separate out one component
of a given variable which can be quantified and expressed as an analytic func-
tion while also allowing us to modify each variable by superimposing a random,
disorderly motion on top of it. In the case of velocity, the mean velocity profile
(e., the quantifiable component) of a fully developed turbulent flow at Re < 10°
is described by
V = Vox (1 =
where
V (m/s) is the time-averaged velocity at a distance r from the center of the tube
Troy (m/s) is the maximum (centerline) time-averaged velocity
(1.76)
The exponent, L/n (where 6