Assume you have created an indoor water fountain. You have connected pieces of pipe with different diameters into a path along which the water will flow. You also have inserted a pump into the circuit. A very simple circuit is shown in the figure below.
Running the pump for a while will accelerate the water and start it flowing. The pump creates a pressure gradient. If we look at a volume V of water in a straight section of pipe while the water is accelerating, then the pressure on side 1 of this volume is different than the pressure on side 2. This results in a net force on the volume of water in that section, and the volume of water accelerates.
Once the water is flowing at the chosen speed, the pump has to do much less work. If the pressure were the same on both sides of the volume V, then the net force would be zero, and the volume of water would continue to move with constant velocity. However, there will still be a small pressure gradient due to frictional forces. The pump now only has to do work against frictional forces. In a frictionless environment the pump would be no longer needed to keep the water flowing. Such a frictionless environment can actually be created. While most liquids freeze at near zero absolute temperature, liquid helium becomes a superfluid. It flows without friction.
For simplicity let us assume a frictionless environment and let us assume that the
water flows steadily through the circuit. The water in different sections of the circuit
has different gravitational potential energy per unit volume.
It also must have different kinetic energy per unit volume. In the narrower sections of
the pipe it must flow faster than in the wider sections, since the same amount of water
must flow across each cross sectional area in the same amount of time.
Is the pressure also different in different sections of the
pipe circuit?
Look at a particular volume of water. As it moves, the boundary 1 moves a distance l1 while boundary 2 moves a distance l2. Since water is incompressible, we have
Volume 1 = Volume 2 = V.
Area 1 ´ l1 = Area 2 ´
l2 = V.
Area 1 ´ dl1/dt = Area 2 ´
dl2/dt = dV/dt.
Area 1 ´ v1 = Area 2 ´
v2
This is the equation of continuity.
The potential energy of the water changes as it moves. While all the water moves, the change in potential energy is the same as that of a volume V, which has been moved from position 1 to position 2. The potential energy of the water in the rest of the pipe is the same as the potential energy of the water in this section of the pipe before the movement. We have
change in potential energy = (mass of water in V) ´
g ´ (change in height)
= density ´ V ´ g
´ (h2–h1) = rVg(h2–h1).
The kinetic energy of the water also changes. Again we only have to find the change in kinetic energy in the small volume V, as if the water at position 1 had been replaced by the water at position 2. The kinetic energy of the water in the rest of the pipe is the same as the kinetic energy of the water in this section of the pipe before the movement. We have
change in kinetic energy = ½mv22 – ½mv12 = ½rVv22 – ½rVv12.
If the force on the water at position 1 is different than the force on the water at
position 2, then work is done on the water as it moves. The amount of work done is
W = F1l1 – F2l2.
But force = pressure times area, so
W = P1A1l1 – P2A2l2
= P1V – P2V
.
The work must equal the change in energy. We therefore have
P1V - P2V = rVg(h2–h1)
+ ½rVv22 – ½rVv12,
or
P1V + rVgh1 + ½rVv12
= P2V + rVgh2 + ½rVv22.
Dividing by V we have
P1 + rgh1 +
½rv12
= P2 + rgh2 + ½rv22
or
P + rgh + ½rv2 =
constant.
This is Bernoulli’s equation.
What does Bernoulli's equation mean?
If a fluid or a gas which is not being compressed is flowing in a steady state, then the pressure depends on the speed of the fluid or the gas. The faster the fluid is flowing, the lower is the pressure at the same height. Phenomena which can be understood with the help of Bernoulli's equation include the Pitot tube, the Venturi effect, atomizers, hurricanes, flapping flags, etc.
 | The volume flow rate of water through a horizontal pipe is 2 m3/min.
Determine the speed of flow at a point where the diameter of the pipe is (a) 10 cm, (b) 5 cm.
| ||
| A Venturi tube may be used as a fluid flow meter. If the difference
in pressure P1 - P2 = 21 kPa, find the fluid flow rate in m3/s
given that the radius of the outlet tube is 1cm, the radius of the inlet
tube is 2cm, and the fluid is gasoline (r =
700kg/m3).
|
To accelerate the water in a circuit and to overcome frictional forces while maintaining a steady flow, you use a pump. Pumps create pressure gradients. To accelerate a column of water, a pump either increases the pressure on one side of the column, or decrease the pressure on the other side of the column.
| Consider a simple U-shaped pipe filled with water. The water level is the same in each
leg and the pressure on each surface is the atmospheric pressure. How can you make the
water level rise in the right leg and sink in the left leg?
|
In a vertical, water-filled pipe gravity creates a pressure gradient in the water. The water below has to support the weight of the water above. The pressure exerted by 10 m or 33 feet of water overhead is equal to the atmospheric pressure at sea level. From Bernoulli's equation we have P1+rgh1 = P2+rgh2, where P1 is the pressure at height 1 and P2 is the pressure at height 2. If h1 is zero and the pressure on top of the pipe is 1atm, then P1 = 1atm+rgh2, i.e. the pressure at the bottom of the pipe increases with the height of the pipe. One way of maintaining pressure in plumbing is to have tall columns of water connected to the pipes. Many municipalities use a water tower build at a relative high site within their service region to maintain pressure in the water mains.
Suppose a U-shaped piece of pipe is completely submerged in water, filled with water, and then turned upside down under water. As you slowly pull the top of U-shaped piece of pipe out of the water, the water does not run out of the pipe. Why?
Air cannot enter the pipe. As the water starts running out of the pipe, a near vacuum is created in the topmost region of the inverted U. The pressure here drops to near zero. The atmospheric pressure on the surface of the water in the bucket pushes the water into the U-shaped pipe.
If a U-shaped hose or pipe connects a liquid-filled container at a higher altitude to a container at a lower altitude over a barrier, the liquid can be siphoned into the container at lower altitude. Atmospheric pressure helps to push the liquid over the barrier. In the diagram below P1 > P2, and the fluid is siphoned from the left to the right bucket.
Fluid in a pipe experiences frictional forces. There is friction with the walls of the pipe, and there is friction within the fluid itself, converting some of its kinetic energy into heat. The frictional forces that try to prevent different layers of fluid from sliding past each other are called viscous forces. Viscosity is a measure of a fluids resistance to relative motion within the fluid. We can measure the viscosity of a fluid by measuring the viscous drag between two plates.
If you measure the force to keep the upper plate moving with constant velocity v0, you find it is proportional to the area of the plate, and to v0/d, where d is the distance between the plate.
F/A = hv0/d
The proportional constant h is called the viscosity. The units of h in SI units are Pa-s.
Under all circumstances where it has been experimentally checked, the velocity of a real fluid goes to zero at the surface of a solid object. A thin layer of fluid next to the walls of the pipe does not move at all. The speed of the fluid increases with distance from the walls of the pipe. If the viscosity of the fluid is low or the pipe has a large diameter, a large central region will flow with uniform velocity. For a high viscosity fluid the transition takes place over a large distance and in a small diameter pipe the velocity may vary throughout the pipe.
If the water is flowing smoothly through the pipe, it is in laminar flow. The velocity at a given point does not change in magnitude and direction. The water is flowing in a steady state. A small volume of fluid follows a streamline, and different streamlines do not cross. For laminar flow of fluids (and gases under certain conditions) Bernoulli's equation tells us that in the regions where the speed is higher the pressure is lower. If the streamlines are squeezed together in a region, the pressure is lower in that region. (In gases Bernoulli's equation can be applied to laminar flow if the flow speed is much smaller than the speed of sound in the gas. In air we can apply it if the flow speed is less than 300 km/h.)
If a fluid in laminar flow flows around an obstacle, it exerts a viscous drag on the obstacle. Frictional forces accelerate the fluid backward (against the direction of flow) and the obstacle forward (in the direction of flow).
While flying in an open cockpit airplane, you feel the air rushing past you. A person on the ground observes you moving through fairly stationary air. An object moving through a stationary fluid or gas is equivalent to a stationary object submerged in a fluid or gas flowing with the same speed in the opposite direction in another reference frame. The picture above can be viewed as a fluid flowing past a stationary sphere in laminar flow in one reference frame, or a sphere moving through the fluid in another reference frame.
Not all flow is laminar. In turbulent flow, water swirls erratically. The velocity at a given point can change in magnitude and direction. The onset of turbulent flow depends on the fluids speed, its viscosity, its density, and the size of the obstacle it encounters. A single number, called the Reynolds number, can be used to predict the onset of turbulent flow. For the flow past a smooth cylinder of diameter D we have
The Reynolds number has no units, the units on the right hand side of the equation all cancel out. It increases with the flow speed and decreases with the viscosity. Turbulence appears when the Reynolds number is about 2300.
Under conditions of turbulent flow Bernoulli's equation is not applicable. It was derived by equating the work done by pressure forces to the change in the potential energy and the ordered kinetic energy of the fluid. Under conditions of turbulent flow the fluid gains disordered kinetic energy. More work is done on the fluid and a greater pressure difference is needed to move the fluid at the same rate.
Only recently have scientist been able to gain a better understanding of the patterns observed in turbulent flow under different circumstances. The study of chaos is providing new insights into many related phenomena exhibiting turbulence, such as global weather patterns, the atmosphere of Jupiter, etc.
When a fluid or gas in laminar flow streams past an object, it exerts a viscous drag on the object. When the objects shape changes the direction of flow, the object is acted on by lift as well as drag forces.
In the above picture, the airplane wing redirects the air flow. The air flowing over the airplane wing moves with greater speed than the air flowing underneath. The pressure above the wing is lower that the pressure below the wing. This results in a net upward force on the wing.
Airflow past ordinary size objects is only laminar at low speed. As the airspeed increases, a boundary layer forms. This boundary layer fills the region behind the object with a turbulent wake.
The figure above shows the airflow past a cylinder as the airspeed and therefore the Reynolds number increase. In pictures 1-3 the Reynolds number is below 2000, in picture 4 it is approximately 10000, and in picture 5 it is above 100000. The first two pictures show laminar flow at low speed. The air directly before and behind the cylinder comes to a stop. The pressure is highest here, but the net force on the cylinder due to pressure differences is approximately zero. There is no pressure drag. In the fourth picture a turbulent wake has formed. The air behind the cylinder no longer slows down and the pressure no longer rises behind the cylinder. Due to the high pressure in front of the cylinder it now experiences a large pressure drag. This happens for Reynolds numbers of approximately 2000 to 100000. The pressure drag is much larger than the viscous drag. It can decrease the forward component of velocity of an object moving through a fluid or gas very rapidly. A thrown object can seem to stop in midair and drop straight to the ground. You can easily observe this by throwing an air-filled balloon. As the airspeed increases and the Reynolds number becomes larger than 100000, the turbulent region works itself forward. We have what is called a turbulent boundary layer. The flow lines now separate from the cylinder and follow the turbulent boundary layer, as shown in the fifth picture. We have something similar to laminar flow around an object of a different shape. The pressure behind the object rises again and the pressure drag is drastically reduced.
The critical Reynolds number at which a turbulent boundary layer forms depends on the condition of the surface. The rougher the surface, the lower is the critical Reynolds number. The surface of many balls used in a variety of sports is intentionally roughed up. Golf balls have dimples and grooves, tennis balls have hair, etc. This decreases the critical Reynolds number, so that a turbulent boundary layer forms even at moderately high speeds. In this way pressure drag can be largely eliminated and only the viscous drag acts on the ball.
The shape of an object can redirect airflow, thus producing lift. Spinning symmetrical objects can also produce lift. Even for laminar flow, a thin layer of air next to the object does not move with respect to the object. A thin layer of air next to a spinning ball spins with the ball. As the distance from the ball increases, the speed of the air changes, so that the airflow around the ball exhibits the patterns shown in the figure below.
If the ball is spinning clockwise as seen in the diagram, the air flows faster over the top of the ball. The pressure on the top is lower than the pressure on the bottom of the ball, and there is a net lift force on the ball towards the top of the diagram. This force is called the Magnus Force. Note: The direction of the lift force depends on the direction of the ball's spin. The lift force does not have to point upwards.
Turbulent regions can also form. The wake behind the ball can be deflected because the spinning surface pulls air with it. This again can result in a net lift force. This force is called the wake deflection force.
For an airplane wing, the lift force depends on the shape of the wing and on the angle of attack. Shapes that produce more lift also produce more drag.
When the airspeed falls below a critical speed, (i.e. the Reynolds number falls below approximately 100000), a turbulent wake develops. The wing looses all lift, and pressure drag increases dramatically. The airplane stalls, and without intervention by the pilot falls nearly straight to the ground.
Link:
|
Bernoulli
Ball (This is an experiment you can try at home.) |
Links to other Web material:
| Fluid Dynamics and Bernoulli's equation |
| Aerodynamics of Animals |
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