Solids & Fluids

FLUIDS (return to CONTENTS)

FLUIDS AT REST (return to start of page)

Definitions

• Pressure equals force per unit area

The SI unit of pressure is N/m2 (= Nm-2), which is called the pascal (Pa). i.e. 1 Pa = 1 Nm-2.
From the definition, we could also use, for example, Ncm-2.
Also, pounds per square inch is a commonly used unit for vehicle tyre pressure.
• Density equals mass per unit volume

The SI unit of density is kg/m3 (= kgm-3), but from the definition we could also use, for example, gmcm-3.

Pressure in gases

Observation 1

In the following, a thin metal can has a little water put in it. With the lid off, the water is bought to the boil, and boiled for a short time.

The heat is turned off, and the lid quickly secured. After a while the can collapses.

This is because:

• the steam from the water displaces the air from the can, but as long as the can is open, the inside and outside pressures are the same
• when the heat is removed and the lid is put on the can, the steam condenses back to water, leaving a vacuum inside the can
• the outside air pressure then crushes the can
This experiment demonstrates the reality of the air pressure, which we are not normally aware of.

Observation 2

A tube, sealed at one end, about 1m long, is filled with mercury (‘Hg’) and its open end is placed under mercury in a dish. It is then raised to the vertical. A gap appears in the tube – and this must be a vacuum since no air has been able to enter.

It is the vertical height that matters – no gap appears until it is more than about 76cm.

The only thing that can be holding the column of mercury up is the air pressure acting on the mercury in the dish.

This arrangement can be used as a simple barometer (a device for measuring air pressure) since, as the air pressure varies a little daily, the height of the Hg column also varies.

• 76cm Hg is called ‘standard’ or ‘normal’ atmospheric pressure or 'one atmosphere’ (1 atm) of pressure
Note: The density of mercury is 13.6 g/cm3, while that of water is 1g/cm3, so to do the above experiment with water, would require a tube over 10m long.

Pressure in liquids

Observation 1

The holes are at the same level and water shoots out equal distances from each.

This indicates that:

• the pressure acts equally in all directions at a given depth
Observation 2
This indicates that:
• the pressure increases with depth
Equation for pressure

It can be shown that the pressure below the surface of a fluid is given by:

Example

U-tube manometer

The height h represents the ‘excess’ pressure of the gas – the amount by which it is greater than atmospheric pressure.
The density of water is 1000 kgm-3. Suppose that h = 21.5cm:

Liquids in a U-tube

If two immiscible liquids (i.e. which do not mix, such as oil and water) are put in a u-tube, they will settle to different levels, unless they have the same density:

The pressure at the same horizontal level is the same, so:
Archimedes’ principle
• The upthrust on an object equals the weight of fluid displaced by the object

A block of steel cannot float, because steel is denser than water, and so the block does not displace enough water to equal its own weight. However, when steel is formed into the shape of a boat, it can displace enough water to enable it to float.

A cork needs only displace a small amount of water to balance its own weight, so it sits high in the water.

Hot air balloons also make use of Archimedes’ principle.

Consider:

The forces on the opposite vertical sides of the object balance each other.

But the force up is greater than the force down, since pressure increases with depth, and the difference between them equals the upthrust.

FLUIDS IN MOTION (return to start of page)

Steady flow

Consider a liquid flowing slowly through a pipe:

Fluid molecules (liquid and gas molecules) tend to stick to solid surfaces. In the above set-up, the molecules immediately next to the pipe are at rest. But towards the centre they get faster. This is indicated by the arrows, representing velocity, getting longer towards the middle.

This also occurs in rivers. The water flows faster in the middle than near the banks, and faster at the top than near the bottom.

Since layers of molecules immediately next to each other are moving at different speeds, they must be sliding over each other. This relative motion is opposed by forces between the molecules, i.e. there is fluid friction. The situation is similar to dragging a block of wood across a bench - there is friction between the wood and the bench producing a resistive force.

The viscosity of a fluid influences:

• how readily the fluid flows, whether it is flowing along a pipe or poured out of a beaker - imagine pouring water or treacle - they have different viscosities, and pour at different rates
• how readily an objects falls through the fluid - consider a ball bearing falling through air, water or treacle - its acceleration would be different in each case.
Usually the viscosity of a fluid decreases as the temperature rises - because the average separation of the molecules is greater, so it is easier for molecules to move past each other.

Poiseuille’s formula

A section of pipe carrying a moving fluid:

Poiseuille showed that for steady flow, if volume V passed a fixed point in time t, then the volume flow rate (V/t) is given by:

Expression for mass flow rate:

Example

V = the rate of flow of a liquid along a pipe
V/ = the rate of flow when the length is doubled, the radius is halved, the pressure difference between the ends is doubled, the viscosity is halved. What is V/ in terms of V?

Streamline and turbulent flow

The following indicates how the paths of water flowing along a pipe can be made visible. The coloured liquid is released slowly through a small hole, and it follows a streamline in the water.

There are two types of fluid flow:

• If water (or other fluid) flows slowly, the flow is called steady flow or streamline flow. The above represents streamline flow in which the streamlines are straight and parallel
• If the flow rate gets too fast, then turbulent flow occurs. The water becomes churned up and the streamlines are no longer straight and parallel, and eddies are formed, as represented below

The resistance of a fluid to flow increases a lot when the flow becomes turbulent. This applies not just to a fluid flowing through a pipe, but when an object moves through a fluid. This is an important consideration in the design of cars etc.

The equation of continuity

Consider a fluid undergoing streamline flow though a pipe whose cross-sectional area decreases.

Setting these masses equal to each other:

Thus, the equation of continuity says that the mass per second passing all points in a pipe is the same.

Now, a liquid (unlike a gas) is almost incompressible, i.e. it has a constant density.
Thus, for a liquid, the above equations reduce to:

Unit of Av = m2 m s-1 = m3 s-1º volume per second (or, volume flow rate)

So, for a liquid, the equation of continuity says that the volume per second passing all points in a pipe is the same.

This implies that the smaller the area A, the bigger the velocity v (imagine ‘nipping’ the end of a hose pipe - the area gets smaller and the water shoots out quicker)

Example

What is the velocity of the emerging spray?

The Bernoulli effect

• The Bernoulli effect is that the pressure of a fluid decreases if its speed increases
a) A piece of paper can be folded as indicated below:
The top moves down if you blow into the tunnel. This is because the moving air inside has a lower pressure than the still air outside, and so the extra pressure on the top outside pushes the top down.

b)

c) The Venturi meter

The above type of arrangement can be inserted into a pipe carrying a liquid, and the flow rate can be determined from the height difference, h.

d) Aerofoil

The top part of the aerofoil is made longer than the bottom part:

As the aerofoil moves through the air, the air has to move further over the top than the bottom, and so has to move faster to ‘keep up’, and so has less pressure. The pressure on the bottom is therefore greater than that on the top, hence a net upward force, called ‘lift’, occurs.

An aeroplane flying horizontally at a constant speed has four basic forces acting on it, which are balanced in pairs:

• lift = weight  and  thrust = drag (air resistance)
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A Level Physics - Copyright © A C Haynes 1999 & 2004