4.2 Laws for Fluids at Rest
81
4.2.2 Pascal Applied
The original application of Pascal’s principle was to ‘hydraulics’, the study of the
mechanical aspects of liquids.
Consider a force F 1 pushing against a piston of area A 1 . In turn, suppose that
piston pushes on a contained fluid and that the fluid is connected through a tube to
act on larger piston of area A 2 , held by an opposing force F 2 . Pascal’s Principle
gives F 1 /A 1 = F 2 /A 2 . The resultant force is magnified by A 2 /A 1 . There is no such
principle as the conservation of force. But there is conservation of energy. If the
first piston displaces a distance x 1 and the second x 2 , then F 1 x 1 = F 2 x 2 , so x 2 =
(A 1 /A 2 )x 1 , i.e. the displacement has been reduced when the force is magnified.
Hydraulics is a useful replacement of levers to magnify forces, because fluid lines
can be flexible, small in diameter, and a long distance from the originating source.
A number of biological systems use hydraulics to apply forces. Starfish (sea
stars) use hydraulics to control their five radial feet and cups on those feet to attach
to a clam. Hydraulic pressure is generated by muscles that squeeze water-filled
ampoulae connected to canals extending through their feet and into their podia cups.
With valves, the pressure can be maintained with little muscle effort. Sea stars can
win the battle with clams, because the sea star does not have to expend much energy
to keep an opening force on the two sides of a clam, but the clam requires energy to
keep its shell muscles under tension. The clam eventually tires and succumbs.
A Venus fly trap uses hydraulics in the form of osmotic pressure to put its bimodal
leaves under flexure tension, and to trigger the closure of those leaves onto a fly in
only about 1/10th of a second. Amazingly, touching only one of the trigger hairs
does not cause the reaction. Touching one twice within a second, or touching one,
and then the second, will stimulate closure (Fig. 4.1).
The human arm from the skin surface down to the bone contains a large fraction
of water held by elastic and pliable membranes. If the pressure on the arm is
increased, that pressure increase is transmitted throughout the fluids of the arm and
into the membranes that hold fluids. Among these fluids is the blood in arteries.
A sphygmomanometer is a device which uses a cuff to determine blood pressure.
Whatever pressure is measured in the cuff air bag is the same as in the blood vessel.
The cuff is inflated until a major artery is constricted to close. As the pressure is
slowly released, there will be some pressure transmitted by the heart which pushes
blood through the constriction, causing a turbulent sound which can be detected
by listening through a stethoscope. During continued pressure release, the sound
stops. At this lower pressure, the heart-generated pressure is sufficient to move blood
through the artery and maintain its open condition. Then, far less noise is generated.
For a hypodermic needle, if the pressure is increased on the plunger, the pressure
of the fluid in the needle is correspondingly increased. However, viscous drag within
the needle results in a pressure near atmospheric when the fluid enters the body,
since this is the pressure on the surface of the skin, and Pascal would tell us this
pressure is communicated across all soft membranes.
81
4.2.2 Pascal Applied
The original application of Pascal’s principle was to ‘hydraulics’, the study of the
mechanical aspects of liquids.
Consider a force F 1 pushing against a piston of area A 1 . In turn, suppose that
piston pushes on a contained fluid and that the fluid is connected through a tube to
act on larger piston of area A 2 , held by an opposing force F 2 . Pascal’s Principle
gives F 1 /A 1 = F 2 /A 2 . The resultant force is magnified by A 2 /A 1 . There is no such
principle as the conservation of force. But there is conservation of energy. If the
first piston displaces a distance x 1 and the second x 2 , then F 1 x 1 = F 2 x 2 , so x 2 =
(A 1 /A 2 )x 1 , i.e. the displacement has been reduced when the force is magnified.
Hydraulics is a useful replacement of levers to magnify forces, because fluid lines
can be flexible, small in diameter, and a long distance from the originating source.
A number of biological systems use hydraulics to apply forces. Starfish (sea
stars) use hydraulics to control their five radial feet and cups on those feet to attach
to a clam. Hydraulic pressure is generated by muscles that squeeze water-filled
ampoulae connected to canals extending through their feet and into their podia cups.
With valves, the pressure can be maintained with little muscle effort. Sea stars can
win the battle with clams, because the sea star does not have to expend much energy
to keep an opening force on the two sides of a clam, but the clam requires energy to
keep its shell muscles under tension. The clam eventually tires and succumbs.
A Venus fly trap uses hydraulics in the form of osmotic pressure to put its bimodal
leaves under flexure tension, and to trigger the closure of those leaves onto a fly in
only about 1/10th of a second. Amazingly, touching only one of the trigger hairs
does not cause the reaction. Touching one twice within a second, or touching one,
and then the second, will stimulate closure (Fig. 4.1).
The human arm from the skin surface down to the bone contains a large fraction
of water held by elastic and pliable membranes. If the pressure on the arm is
increased, that pressure increase is transmitted throughout the fluids of the arm and
into the membranes that hold fluids. Among these fluids is the blood in arteries.
A sphygmomanometer is a device which uses a cuff to determine blood pressure.
Whatever pressure is measured in the cuff air bag is the same as in the blood vessel.
The cuff is inflated until a major artery is constricted to close. As the pressure is
slowly released, there will be some pressure transmitted by the heart which pushes
blood through the constriction, causing a turbulent sound which can be detected
by listening through a stethoscope. During continued pressure release, the sound
stops. At this lower pressure, the heart-generated pressure is sufficient to move blood
through the artery and maintain its open condition. Then, far less noise is generated.
For a hypodermic needle, if the pressure is increased on the plunger, the pressure
of the fluid in the needle is correspondingly increased. However, viscous drag within
the needle results in a pressure near atmospheric when the fluid enters the body,
since this is the pressure on the surface of the skin, and Pascal would tell us this
pressure is communicated across all soft membranes.