Is it time to say good bye to Bernoulli's
principle while speaking about a lift of wing?
For horizontal fluid flow, an increase in the velocity
of flow will result in a decrease in the static pressure. The equation
describing this effect is known as Bernoulli's law.
Bernoulli's law describes the behavior of a fluid under
varying conditions of flow and height. It states
where P is the static pressure (in Newton's per square
meter), rho is the fluid density (in kg per cubic meter),
v is the velocity of fluid flow (in meters per second) and
h is the height above a reference surface. The second term
in this equation is known as the dynamic pressure. The effect described
by this law is called the Bernoulli effect, and sometimes is known
as Bernoulli's equation.
For gases effects of gravitation are neglected:
P + 1/2·rho·v2
= Constant;
Derivation of Bernoulli's
equation from Newton's Second Law and
the assumption, that total Kinetic energy of n ideal gas molecules
is sum of energy of each molecule, and for isolated system
it remains constant
Mathematically it can be written as the relation:
K = 1/2·
mm·v i2 = Constant; (1)
where mm is single molecule mass and vi is speed of molecule. K is
total kinetic energy of ideal gas.
is sum over index i.
i = n molecules, where n is a concentration of molecules.
K = 1/2·
mm·vi2 = 1/2·mm·
vi2 = 1/2·mm·n·
((vxi + v)2 + vyi2
+ vzi2); (2)
Here is assumed that gas volume is moving along axis x with
transitional speed v. Then
vi2 = (vxi + v)2
+ vyi2 + vzi2; (3)
All speeds are independent. Symmetry gives that there are equal
number of vxi in positive and negative directions. So
v·vxi = 0;
K = 1/2·mm·
(vxi2 + vyi2 +
vzi2 + v2) = 1/2·mm·
(vxi2 + vyi2 +
vzi2) + 1/2·mm·
v2; (4)
For every speed component for each x,y,z direction
vxi2 = n·<vx2>
,
vyi2 = n·<vy2>,
vzi2 = n·<vz2>
(5)
Now we obtain equation
K = 1/2·mm·n·< vxyz2>
+ 1/2·mm·n·< v2>, (6)
where < vxyz2>
= < vx2>
+ <vy2> + < vz2>; (7)
First member gives static pressure P = 1/2·mm·n·<
vxyz2>, and second member gives dynamic
pressure, as density: rho = mm·n;
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To derive Bernoulli's relation P= 1/2·rho·v2,
we first apply Newton's Second Law to the fluid in a segment
of the pipe. During a particular time interval, the fluid
travels the length of the segment.
F = m·a;
where F is force (newtons), m is mass of fluid (kilograms),
a is acceleration (meters / second2)
The fluid mass, which hits a wall with area A during time interval dt is:
dm = rho·dV;
Change in volume is
dV=A·dx;
where A - area (meter2),
dF = a·dm = a·rho·dV = a·rho·A·dx;
By definition P = F/A, where
P is static pressure (newtons per meter2),
F = P·A; dF=A·dP;
A·dP = a·rho·A·dx;
dP = a·rho·dx = rho·dx·dv/dt = rho·v·dv;
where dv is fluid velocity difference (meters per second)
dt - time interval (seconds)
Taking into account that density rho = mm·n,
where mm is mass of molecule, n is concentration
of molecules
and after integration we obtain final Bernoulli's expressions
P= 1/2·rho·v2; for dynamic pressure
P= 1/2·mm·n·<v2>; for static
pressure
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From (6)
K = P + 1/2·rho·v2 , and taking into account (1)
the final Bernoulli's equation is obtained:
P + 1/2·rho·v2 = Constant; (8)
Comparing (1) and (8) we find that, assumption of system isolation,
made in (1) is still valid. So (1) and (8) physics is the same. The
Bernoulli equation is derived for isolated ideal fluid systems, such
as gas in tubes.
Lets see mechanism, which shows how static
pressure energy (internal chaotic molecules motion kinetic energy)
is transformed into translation kinetic energy of the gas volume
Molecules in the cube have velocities of random directions.
Absolute average speed of molecules is about the same for X,Y,Z directions.
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(INSIDE BOX) |
Total speed of gas volume (don't confuse it with speed
of molecules) inside the cube is zero, until the cylinder hole is
closed. Total energy of gas volume V in Bernoulli equation is represented
by static pressure potential energy pV. See,
what happens when cylinder input is opened. The
only molecules, which are moving about along the cylinder axis Z can
get out of the cube. The natural selection rule makes Z axis
special - motion of molecules in the cylinder is mainly orientated along
cylinder axis Z.
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(INSIDE CYLINDER) |
Domination of the speed direction also means that gas static
pressure into cylinder walls is reduced. Gas stream moves to output and
speed of gas inside cylinder is nonzero. Pressure energy of the gas is
transformed into stream kinetic energy of the same gas. Thus pressure potential
energy "feeds" Kinetic one as described below
Total energy of gas remains constant. Bernoulli principle works well for isolated systems.
What happens when gas gets to cylinder output? Reduced
static pressure at x, y directions of the gas is quickly compensated
by motion of external air molecules into the gas stream. It looks
like the gas stream sucks air from atmosphere. (For an example,
blow air with your lips and you see that air stream becomes cold
when increasing the stream speed). Static pressure in a free air stream
becomes the same as in surrounding atmosphere. Due to interaction
with air molecules the gas stream cannot be isolated, so the total
energy of the gas stream is no more constant. Bernoulli
principle cannot be applied for open air applications.
