Lecture 5 Hemodynamics Description of fluid flow
Hydrodynamics is the part of physics, which studies the motion of fluids. It is based on the laws of mechanics. Hemodynamics studies the motion of blood in the circulatory system. Some principles of hydrodynamics are stated below.
A line along which a fluid element moves be named a flowline. If every subsequent fluid element, which passes through a given point, takes the same path as the first fluid element, then the flowline is stable. A stable flowline is named a streamline. Streamlines may be straight or curved. The tangent drawn at any point off the streamlines represents the direction of velocity at that point.
The equation of continuity
Suppose incompressible fluid which fills completely a channel such as tube and flows along it. Then if some amount of fluid enters one end of the channel, an equal amount must leave the other end.
Fig. 1
Flow rate Q is used as the measure of this amount. It means a volume of fluid V, which moves through the cross section of a channel per second. If the fluid enters one end of a tube at the flow rate Q1, it must leave the other end at the flow rate Q2, which is equal to Q1. This principle is called the equation of continuity. Thus the equation of continuity can be written:
Q1 =Q2
const t S
l S t
Q=V = ⋅ = ⋅υ=
where V – volume, S – cross sectional area of tube,
υ t =
l – linear velocity of fluid flow .
The flow rate equals the velocity of liquid v multiplied by the cross-sectional area of the channel S:
1 2 2 1 2 2 1
1 υ
υ υ
υ = → =
S S S
S (5.2)
For a channel which cross section changes from S1 to S2 this yields another form of the equation of continuity
Sυ=const (5.3)
This the product of flow velocity and area of cross section is constant at every section of a tube. It may be concluded also that the area of cross section and the flow velocity are at inverse relations. Usually the flow velocity is not equal at the every point of cross section. But the equation of continuity still holds for such cases if it is written in terms of the average flow velocity.
Bernoulli’s Equation
For steady, irrotational flow, the speed, pressure, and elevation of an incompressible, nonviscous fluid are related by an equation discovered by Daniel Bernoulli (1700-1782).
const h
g
P+ + =
2 υ2
ρ ρ ,
where P –static pressure,
ρgh –hydrostatic pressure,
2
ρυ2 – dynamic pressure.
Bernoulli’s equations prove that total pressure remains constant along the tube of current during steady-state flow of perfect fluid.
Fluid viscosity
Viscosity is that property of fluids owing to which they oppose any motion of their neighbouring portions relative to one another. Viscosity is created by internal friction between the molecules. Such friction opposes the development of velocity differences within a fluid. The reciprocal to the viscosity is called fluidity. Various fluids differ greatly by the value of their viscosity. For instance, the viscosity of oil is greater than that of water. Viscosity is a major factor in determining the forces that must be overcome when fluids are transported in the tubes. Viscosity influences also the blood flow in the circulatory system.
In an ideal fluid there is no viscosity to hinder the fluid layers as they slide past one another. Within a pipe of uniform cross section, every layer of an ideal fluid
moves with the same velocity, even the layer next to the wall, as Fig. 2 (a) shows.
When viscosity is present, the fluid layers do not all have the same velocity, as part (b) of the drawing illustrates. The fluid closest to the wall does not move at all, while the fluid at the center of the pipe has the greatest velocity. The fluid layer next to the wall surface does not move, because it is held tightly by intermolecular forces.
Fig. 2. (a) In ideal (nonviscous) fluid flow, all fluid particles across the pipe have the same velocity. (b) In viscous flow, the speed of the fluid is zero at the surface of the
pipe and increases to a maximum along the center axis
Fig. 3
Fundamental law of viscous liquid was discovered by Newton (1687):
dy S
F =−ηdυ⋅ (Newton’s formula)
where η is the coefficient of viscosity and equal to the force of internal friction that acts on the unit area of the layer’s surface at the velocity gradient which is equal to one.
SI Unit of Viscosity: [ ]η =Pа⋅s
Common Unit of Viscosity: poise (P). 1 poise (P) = 0.1 Pa·s [ ]F =H – forces of internal friction;
c dx d =1
υ – velocity gradient;
[ ]S =м2 – area of tangent layers.
η normal = 0,004 – 0,005 Pа . s.
The fluids whose viscosity may be defined by Newton’s equation (F ~ddxυ) are called Newtonian fluids. They are homogeneous fluids, such as water, spirits, solution of electrolytes etc. There do exist, however, more complicated heterogeneous fluids for which the Newtonian description is inadequate. The viscosity of these fluids depends at high rate on the velocity of flow. They are called non-Newtonian fluids. This category of fluids includes suspensions, emulsions, foams and even the solutions of macromolecules such as proteins.
Relative viscosity is equal to the ratio of the coefficient of viscosity of the given fluid to the coefficient of viscosity of distilled water at one and the same temperature:
w rel ηη η = .
Kinematic viscosity is ratio of the coefficient of viscosity to density:
ρ
ν=η (ρ– density of the liquid). [ ]
s m2 ν = . Laminar and turbulent flow
If each fluid layer slips over the other, different layers do not get mixed then the flow called laminar. In the laminar flow every particle of fluid follows the path of its preceding particle. The velocity of flow at any point of fluid remains stable. The streamlines do not intersect each other.
The other kind of flow is named turbulent. The turbulent flow is unstable. The flow of fluid is curled and all layers merge in one stream. The flowlines become zigzag. The velocity of a particle crossing particular point of fluid is not constant and varies with time. The turbulent flow needs more energy than the laminar one because additionally energy is used in producing currents through the fluid.
English physicist Reynolds investigated the conditions under which a flow becomes laminar or turbulent. The transition from laminar flow into turbulent one depends on the value of dimensionless quantity called the Reynolds number (Re).
Reynolds number for a liquid flowing in a cylindrical tube is defined by the equation:
η υ ρ D Rе =
here υ is an average velocity of flow;
D – diameter of a tube; ρ – density of fluid; η – viscosity.
The critical value of Reynolds number for cylindrical tubes at which laminar flow turns into turbulent is 2000 - 2400. Blood flow in the circulatory system is laminar with the exception of aorta. In aorta it may become turbulent during a physical work which greatly increases velocity of blood. Blood flow may be turbulent also in arteries the cross-section area of which is diminished by some pathological process.
If Re <Recr, then flow is laminar. If Re >Recr, then flow is turbulent.
The change of blood viscosity value (for example, in patients with anemia) may be diagnosed owing to turbulent noises. It may be explained by the fact that at anemia viscosity coefficient decreases by a factor of 2-3 and even more.
Correspondingly, Reynolds number increases since Re ~1/η/ As a result Reynolds
number becomes greater than its critical value and the transition from laminar blood flow to turbulent one tarts place.
Medical application of transition between laminar and turbulent flow of blood is connected with measuring of blood pressure by Korotkov method. In accordance with this method systolic (upper) pressure is measured at the moment when blood begins to squeeze through the hole in artery compressed by the cuff. Exactly at this moment noises appear resulting from turbulent flow of blood. Diastolic (lower) pressure is fixed at the moment when these noises disappear as a result of release of cuff and transition of flow from laminar to turbulent.
Pressure (P) - is the force exerted by blood on blood vessels per unit area:
S
P=F , [P] = Pа. .
Volume velocity (Q) is called a value numerically equal to the volume of fluid passing per unit time through this section:
[ ] s Q m t
Q=V, = 3 .
Linear velocity (υ) is the path traversed by the particles of blood per unit time:
υ=tl ; [ ]υ = ms .
Formula of dependence the linear and volume velocity:
υ
⋅
=S
Q ,
where S – cross-sectional area of fluid flow.
Formula of Poiseuille
Let us find linear and volume velocity of flow for steady-state stream of viscous fluid through a vessel with radius R, length l, pressure differential at its ends P1 – P2:
( )
l P P Q r
η π
8
2 1
4 −
= (formula of Hagen-Poiseuille) η is dynamic coefficient of viscosity.
Quantity 4
8 r X l
π
= η is called hydraulic resistance:
Q=∆ΡX , (formula of Poiseuille)
X Q⋅
=
∆Ρ .
Distribution of mean pressure Mean arterial pressure determined by the formula:
3
d s d mean
Р P Р
P = + − ,
wherе Рs – systolic pressure, Рd – diastolic pressure.
