of fluids, viscosity
a stationary state fluid if an obstacle is not present moves in
parallel layers. Each fluid can flow stationary if certain conditions
are fulfilled: the speed small enough and obstacles so that do not
cause too big change in speed. If those conditions are not fulfilled,
the flow of fluid is much more complex and is called turbulent flow.
properties of stationary state will simply observe by the device
Hele-Shaw. That device consists of two containers - for water and
ink - in which small holes are pierced. Water or ink leak through
holes into a very narrow space between the two glass plates. If
water (or ink) stream slowly, the stream is stationary, and in the
area between the two glass plates there are parallel dark and light stripes coming as a
result of the stream of water and ink: those layers do not mix,
but slide parallel. The parallel dark and light stripes show so
called streamers, in fact imagined lines which show the direction
of the streaming of the fluid. Streamers can be defined more exactly,
by explaining that those are the line whose tangent in every point
is parallel to the direction of speed of motion of liquid.
What is important here is the stationary motion and it's
temporal continuousness. The particles of fluid change its speed
and direction, what can be seen in the example of narrowing the
flowing area. But, streaming in one given point of the fluid is
always done in the same direction, with the same speed. Each particle
of the fluid that passes from the particular point of fluid passes
with the same speed and in the same direction. That property is
the basis of the fact that stationary stream can be presented mathematically
through the function that does not depend upon time, but only on
the position of that particular particle. That spatial function
is called the potential of the streaming field of fluid and presents
the solution of one differential equation analog to those we meet
in electrostatics. Because of this property of stationary streaming
we often call it potential streaming. It's connection to electrostatics
will be explained deeper when studying it.
real fluid can in given conditions stream stationary. At higher
speeds forces inside molecules in fluid are more expresses. Those
forces are seen as friction among the layers of the fluid. When
dealing with real fluids we ignore those forces: an ideal fluid
is the fluid where the streaming of layers is done with no friction.
The streaming of the ideal fluid is stationary as friction is an
important condition for forming of whirls.
We will now derive the equation for streaming of fluid or Bernoulli's
equation, which will relate pressure, speed and elevation in each
point of a fluid.
The Bernoulli's equation says that when an ideal fluid streaming takes
place the sum of hydrostatic and hydrodynamic pressure is constant.
The static pressure in the liquid
that is streaming is less that in the liquid that is in the
standstill state. As the speed of streaming is bigger the static
pressure is smaller.
The inner friction or viscosity can be described as force used to make
one layer friction on other layer of liquid. The molecules of each
layer work on the molecules of the neighbouring layer with forces
that are on such distances magnetic. Those forces will try to prevent
and slow down the moving of layers and influence the inner friction.
Viscosity is also meet with gases and liquids, but with liquids
those forces are much stronger. The experimental inner friction
we see as resistance to move through water or oil.
The coefficient of viscosity
measuring of the inner friction or viscosity will be measured by
the quantity called coefficient of viscosity. We will reach that
quantity in the following way. Drown the waxed parallel glass plates
into liquid. One plate is moving and the other is in standstill
position. In that way only the layers closer to the plate move.
The shift of layers is bigger as layers are closer to the glass
plate. We must use certain work for that moving as layers resist
to the moving by molecular forces. That is friction.
smaller speeds, the streaming of fluids is stationary. In that case
the speed of fluid motion in each point is constant and does not
change through time. When the speed grows and overpasses a certain
value, called critical, the nature of streaming becomes more complex.
There arouses the local irregular circulation, called whirls.
The speed of fluid in the given point changes through time, and
the profile of streaming is flat, the fluid does not move in laminas.
This kind of streaming is called the turbulent streaming. The critical speed
is expressed by the derivation : page 176.
The border layer.
The forming of whirls under turbulent streaming can be explained
by acting of border layer. The friction among fluids and membrane
of the tube, in other words fluid and obstacle is the biggest in
the narrow part of fluid closest to the obstacle. In that layer
the motion of the fluid is the slowest. So there is a gradient of
speed distribution; from the border layer to the inside of the fluid
the speed is bigger.
bodies lighter then water float on it's surface. At the same time
the bodies lighter then air rise up. The force that lifts them up
is called buoyancy, and it is equal to the weight of the displaced
fluid. That force is also called static thrust, and comes from the
fact that hydrostatic pressure in fluid grows as depth grows. But,
the plane flying is not kept in air because of static thrust: the
plane that is not moving falls
down. So this kind of thrust will be called dynamic thrust. According
to the Bernoulli's equation the body that relatively moves is under
the influence of hydrodynamic pressure. If the motion of the fluid
is symmetric to the position of the body, the pressure from both
sides is annulled. But, if the streaming of fluid is such that the
speed of streaming is higher above the body, than the one below
the body, then according to Bernoulli there must be a resultant
force upward present, responsible for the dynamic thrust.