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# Fluid mechanics-have you ever drunk mis mash cocktail?

Have you ever drunk mis mash cocktail? This is a cocktail of unmixed orange juice and wine...Firstly, pour chilled orange to half pint straight glass. Then carefully add red wine, with a spoon, so it settles at the top of orange. Cocktail is prepared, drink it with a straw. This is the recipe for mis mash cocktail, that is known because of the fact that the two liquids stay unmixed.

Orange will stay on down half while wine will fill upper half of the glass. What's the trick? The reason why the two liquids stay unmixed lays in difference in their density. Orange is here more dense liquid, because of sugar, so it stay at the lower half of glass.

This article addresses the physics underlining fluid mechanics. This branch of physics explains phenomena like the mentioned fact that only 11% of an iceberg is visible, communication vessels or hydraulic press. The essential notion of the whole fluid mechanics is pressure and firstly we are going to explain this important physical quantity.

## The notion of pressure

When there is a great amount of snow, it is hard to walk in the show. With ordinary shoe we will drop in the snow. That is why in such a situation people use snowshoes. Although the same force act on the snow in both of cases, with snowshoes one wouldn't drop. How come? Surely, in case of snowshoe our weight is distributed on bigger area and that makes the difference.

Definition of pressure. Physical quantity pressure is defined as the ratio of a force and an area where this force act. Usually we denote pressure with the letter P while force by F and area could be represented by A. Then it holds

p = F / A .

Unit of measure for pressure is Pascal, in honor to French physicist and mathematician Blaise Pascal (1623-1662). The unit Pascal is denoted by 'Pa'; Pa = N / m2.

The opposite example of snowshoes is a needle, which punctures the skin. If we push the skin with some blunt object, we can't harm it. However, if we weight on the skin with a needle, very small force is enough to puncture the skin. So, in case of snowshoes the area is enlarged in order to reduce pressure whereas in case of needle very small area enables huge pressure.

It is worth mentioning that 'bar' is also a unit of measure for pressure, but not SI one. 1 bar = 100000 Pa.

A cocktail. Mis mash is a cocktail with unmixed orange and rad wine. Contrary to this picture, orange is down and wine is up. (source: corbisimage.com)

## Do it yourself - physical experiment in your home!

With a little skill one can prepare a mis mash cocktail. Ingredients are fanta and wine.

1. Pour fanta up to the half of a glass.

2. Carefully add red wine, with a spoon, so it stay at the top of juice.

3. Serve it with a straw.

Water in a container in weightless condition. Then the Pascal's law can be clearly observed. (source: flysky)

## Hydraulic pressure and Pascal's law

Imagine a ball shape container of water. Let it be in a space without the gravity. On the ball container there are several holes of different radius. Let one of the hole be shaped as cylinder and piston. What would happened with water in space without the gravity? - whether the water will drain? The answer is no, there is no any movement of water in this situation, because no one force act of it. If we now press the piston towards the water, what would happen? Surely, water will start to flow through the holes, forming straight streams. This means that the initial force is transmitted throughout the whole liquid. It can be shown that there is a rule on this transmission: the force differentiate from point to point of the liquid but the created pressure is the same in every point of the liquid.

This rule is clearly expressed by the Pascal's law: An external force is transmitted throughout the fluid in all directions and it causes the equal pressure in all points of the fluid. Applied to the mentioned experiment Pascal's law gives

F/A = F1/A1 = F2/A2 = … = p

The pressure created this way is called hydraulic pressure. We were discussing quite unusual situation, in order to emphasize the fact that would be hidden otherwise. In everyday life hydraulic pressure usually comes with pressure cause be the gravity.

Hydrostatic pressure depend on the depth, density of the liquid and gravitational acceleration. (source: flysky)

## Hydrostatic pressure

The related figure shows a pot filled out by some liquid. Obviously, there is pressure on openings, since liquid is poured. Furthermore, the deeper level the bigger pressure. This pressure is caused by the gravity and additionally by atmospheric pressure. More precisely, pressure at certain depth is caused by the weight of the liquid above.

The weight of the liquid above depend on its mass i.e. it is m *g . Having in mind defining relation for density, the pressure p at the level h is

p = pa+ ρ V g / A ,

where V is volume and ρ density of the liquid above. Knowing that the height of the liquid column is h, if follows V= h * A. This leads to the relation for hydrostatic pressure at he depth h is

p = pa + ρ * g * h .

Thus, the hydrostatic pressure at certain depth depend on the density of liquid and gravitational acceleration.

Hydrostatic pressure is defined as the pressure in a fluid caused by the force of gravity. It appear in both liquids and gases. However, it is more stronger in liquids because the higher density. We feel hydrostatic pressure when dive. The pressure on our body is much stronger than in our atmosphere. While we might aren't aware of this when ordinary dive, a special equipment is needed for the depth. Even with equipment human being can dive few dozen of meters. Recommended technical diving limit is 100 meters while world record is set up at few hundreds meters.

Communicating vessels. (source: flysky)
Substance
Density (kg/m3)
water
1000
sea water
1030
mercury
13600
air
1.29
oxygen
1.43
Densities of some common fluids

## Communicating vessels

The law of communication vessels is quite intuitive: in communicating vessels the levels of liquid are on the same height, regardless on the shape of vessels (assuming there is no capillary action). This rule follows from the fact that the hydrostatic pressure is equal in all points of the given depth.

However, if vessels are fulfilled by two different liquids, then situation is different. In such a system, the height of column of one liquid denote by h1 and the column of another liquid denote by h2. The system is in equilibrium which means that the two hydrostatic pressures are equal: p1=p2. Thus, γ1 h1 g = γ2 h2 g which leads to γ1 h1= γ2 h2.

## How does a hydraulic press work?

A hydraulic press is a machine based on Pascal's law. With relatively small force acting on one side it produce a big force on another side. A hydraulic press consists of two cylinders of different radius (see figure). Smaller force should act on smaller area in order to produce huge force on bigger cylinder. Namely, according to he Pascal's law, the pressure is equal throughout the liquid which result with big force on the bigger side. However, the distance that pistons traveled is bigger on smaller side.

If the force F1 act down on piston of area A1 than the force F2 act up on piston with area A2 and it holds

F1 / A1 = F2 / A2 .

An example of usage of hydraulic presses is forgery, where it is used to shape metal. The video above shows how a metal ring can be completely flattened with a small press.

Hydraulic press, based on the fact that pressure is transmitted throughout the fluid equally in all directions (Pascal's law). (source: flysky)

## Case studies

Hydraulic press

The smaller piston of an hydraulic press has an area of 25 cm2 and the bigger one has an area of 200 cm2. What the force should be applied to the smaller piston in order to raise cargo of 1 ton?

Solution: Firstly we are going to calculate the weight of the cargo, which is the force acting on the bigger side of the press. Thus, F = m*g = 1000 kg * 9.81 m/s2 = 9810 N. Now, applying the Pascal's law it holds F1 / A1 = F2 / A2. Furthermore, F1= F2 / S2 * S1 gives result of 1226.25 N.

Water and mercury in a vessel

Amount of water and mercury of equal weights are poured in an cylindrical vessel. Since mercury is more dense than water, is will stay on the lower part in vessel. Total height of both liquids is h = 29.2 cm. Density of water is 1000 kg / m3 and density of mercury is 13600 kg/m3. What is the pressure at the bottom of vessel?

Solution: Both liquids contribute to the pressure at the bottom, p = p (Hg) + p (H2O). Since the system is in a balance, we have

Fg(Hg) = Fg(H2O),

which gives

mHg * g = mH2O * g

VHg * γHg * g = mH2O * γH2O* g

S * h1 * ρHg= S * h2 * ρH2O

h1 *(ρHg +ρH2O) = h * ρHg

h1 = 0.272 m

Finally, we have result for the pressure at the bottom

p = (13600 * 9.81* 0.02 + 1000 * 9.81 * 0.272) Pa

p = 5336.64 Pa

## Summary

• Pressure p is the ratio of a force F and an area A where this force act, p = F / A.
• Pascal's law: An external force is transmitted throughout the fluid and it causes the equal pressure in every point of the fluid, p1 = p2=... = p.
• As a consequence of Pascal's law, the levels of liquid in communicating vessels are equal.
• A well known application of Pascal's law is hydraulic press. An action with small force on a smaller side results with a huge force on a bigger side of the press. It holds F1 / A1 = F2 / A2.
• Hydrostatic pressure p at certain depth h depend on the density of liquid γ and gravitational acceleration g, p = γ g h.
• The dencity of water is 1000 kg /m3.

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