Current of a liquid on a horizontal pipe. Puazal’s law.
Let's
consider
current
of a viscous, moistening wall of a liquid on a horizontal
pipe
of constant section therefore, it is important for understanding of
process of movement of blood on vessels. At rather low speed
current
thus has laminar character: the layer of molecules,
adjoining
to
a wall of a pipe, sticks to it and remains motionless.
Following
layers
of molecules under action of force
of
pressure and at counteraction of force
of
internal friction between layers it is displaced rather a
previous
layer
and moves
in
relation to walls of a pipe with some small speed. Each
subsequent
layer of molecules, mixing up
concerning
the
previous layer, moves
in
relation to
a
wall of a pipe with
gradually
increasing
speed
which
reaches
the
greatest value in the center of a pipe.
The mathematical analysis of this phenomenon shows, that distribution of speeds on section of a round pipe has parabolic character
The maximal speed in the center of a pipe
.
In these
formulas:
— pressure in the beginning and in the end of a site of a pipe in
length
- factor of viscosity of a liquid, R
-
radius of a pipe and r
-
radius of a considered layer of liquids.
Gradient of speed, on absolute size equal
,
gradually decreases and addresses
in
a zero on an axis of a pipe.
L
aminar
current
is established in pipes with smooth walls, without sharp changes of
the area of section or bends of a pipe, and as
at
absence of plural branching. At infringement of these
conditions
and especially at high speeds current passes in turbulent: speeds of
particles of a liquid thus vary randomly, formed
local
turbulences - there is a mixing of
particles
of a liquid.
For turbulent current local changes of pressure in the liquids causing oscillatory movement of particles, accompanied the sound phenomena (noise, murmur, etc.) owing to which turbulent current is easily found out are characteristic. Turbulent current is connected with additional expenses of energy at movement of a liquid, therefore in blood system it can lead to additional loading on heart. Noise arising at turbulent current of blood, can be used for diagnostics of diseases. At defeat of valves of heart there are the so-called intimate noise caused by turbulent current of blood. Turbulent current of blood in an aorta can be caused, first of all, by turbulence of a blood-groove in an input in it: whirlwinds of a stream exist, when blood is pushed out from ventricle in an aorta. At places of a branching of vessels, and also at increase of speed of a blood-groove current can become turbulent in arteries. Turbulent current can arise in a vessel in the field of its local narrowing, for example, at formation of a blood thrombus.
Speed
(critical
speed) of transition of laminar current in turbulent it is possible
to define, using the criterion certain from experience -
Reynolds's
Re
number
(dimensionless
size) which connects it with density
,
viscosity of a liquid
and diameter
D
of
a pipe:
.
If
Re ≤ 1000, then the current is laminar,
if Re > 1000, that
current
is turbulent.
Current
of blood on vessels depends upon its viscosity. Current of blood can
be laminar or turbulent. Laminar current of blood creates smaller
loading on heart as at it work of heart is directly proportional to
volumetric speed of a blood-groove, at turbulent movement of blood
between these parameters almost square-law dependence is established
(А~
Q1,8)
.
At current of a real liquid on a horizontal pipe potential energy of its particles is spent for work on overcoming internal friction and consequently static pressure along a pipe gradually falls. To provide current of a liquid, it is necessary to create the pressure a little bit exceeding pressure drops along all pipe in the beginning of a pipe. In a pipe of constant section pressure falls in regular intervals. If to connect levels of a liquid in the manometrical tubes established along a pipe, the direct line will turn out. (fig). If it to continue up to the basic vessel A it will divide height h a column of a liquid in a vessel on two parts h1 and h2.
The
height
h1
corresponds
to initial static pressure
,
height
h2
-
to dynamic pressure
causing speed
of
current of a liquid. The
tangent
of
a corner of an inclination of this straight line
characterizes
a gradient of pressure
(pressure
drop on unit
of
length of pipe):
.
Puazal
has established, that average speed
of
laminar
current of a liquid on to rather narrow horizontal round pipe of
constant section
it is directly proportional to a difference of pressure
at
an input and an output from
a
pipe, to a square of radius
R
of
a pipe and inversely proportional to length l
pipes and to factor of viscosity
of liquid (law
Puazal
);
Average
speed
of
current of a liquid defines the quantity of liquid
Q
proceeding through cross-section section S of a pipe in unit of time:
Hence,
This formula also name usually formula Gagen - Puazal. It can give and such kind:
,
where
Value ω is called hydraulic resistance.