· PDF fileReynolds used the device ... There is no transportation of fluid particles from one layer to another ... liquid is called an incompressible fluid,

Pressure

When fluid is at rest, at any point of a supporting surface , the force exerted by a fluid is normal to the surface.

This normal force exerted by a fluid at a point per unit area of the surface is called Pressure Intensity or Unit Pressure or Specific Pressure or Hydrostatic Pressure.

In this case, p is the pressure and P is the pressure force.

The unit of pressure is the Pascal (Pa), but it is also expressed in bars or meters of water column (mH2O).

Pressure of fluid at rest

In general, in a fluid at rest the pressure varies according to the depth.

When the base point is set at Zo below the upper surface of liquid as shown in figure,

and Po is the pressure acting on that surface, then P = Po when Z = Zo, so

Thus it is found that the pressure inside a liquid increases in proportion to the depth.

Example:01

What is the water pressure on the sea bottom at a depth of 6500m? The specific gravity of sea

water is assumed to be 1.03.

Example :02

What depth of oil of specific gravity 0.9 will produce a pressure intensity of 9 Kg/cm2?

Example :03

Convert a pressure head of 40 m 0f oil to corresponding head of water if the specific gravity of oil is 0.8?

Absolute pressure and gauge pressure There are two methods used to express the pressure: one is based on the perfect vacuum and the other on the atmospheric pressure. The former is called the absolute pressure and the latter is called the gauge pressure. Then,

For example, in the case of measuring the pressure of liquid flowing inside a pipe, the pressure p

can be obtained by measuring the height of liquid H coming upwards into a manometer made to

stand upright as shown in figure.

Differential Manometer or U-tube manometer

When the pressure P is large, this is inconvenient because H is too high.

So a U-tube manometer, as shown in following figure, containing a high-density liquid such as

mercury is used.

In this case, when the density is p’,

In the case of measuring the pressure difference between two pipes in both of which a

liquid of density p flows, a differential manometer as shown in figure is used.

Fig. Differential manometer

In the case of Fig. a, where the differential pressure of the liquid is small, measurements are

made by filling the upper section of the meter with a liquid whose density is less than that of

the liquid to be measured, or with a gas. Thus

Figure (b) shows the case when the differential pressure is large. This time, a liquid column of a

larger density than the measuring fluid is used.

Example:

Obtain the pressure p at point A in Figs (a), (b) and (c).

Example:

Obtain the pressure difference P1 - P2 in Figs (a) and (b).

Buoyancy

1. Fluid pressure acts all over the wetted surface of a body floating in a fluid, and the resultant

pressure acts in a vertical upward direction. This force is called buoyancy.

2. The buoyancy of air is small compared with the gravitational force of the immersed body, so

it is normally ignored.

Cube in liquid

•Suppose that a cube is located in a liquid of density p as shown in figure.

•The pressure acting on the cube due to the liquid in the horizontal direction is balanced right

and left.

•For the vertical direction, where the atmospheric pressure is Po,

The force Fl acting on the upper surface A is expressed by the following equation:

The force F2 acting on the lower surface is:

When the volume of the body in the liquid is V, the resultant force F from the pressure acting

on the whole surface of the body is

From this equation, the body in the liquid experiences a buoyancy equal to the

weight of the liquid displaced by the body. This result is known as Archimedes’

principle.

The centre of gravity of the displaced liquid is called ‘centre of buoyancy’ and is the

point of action of the buoyancy force.

Example:01

An ice berg of specific weight 900 Kg/m3 floats in sea water of specific weight 1025 Kg/m3. Find

the ratio of the volume of the iceberg above the sea water level to its total volume?

Example :02

A cubical body of side 0.25 m and specific gravity 2.5 is immersed in water. Find the least force

required to lift the body?

Archimedes's Principle

1. An object is subject to an upward force when it is immersed in liquid. The force is equal to

the weight of the liquid displaced.

2. The apparent weight of a block of aluminum (1) immersed in water is reduced by an amount

equal to the weight of water displaced.

3. If a block of wood (2) is completely immersed in water, the upward force is greater than the

weight of the wood. (Wood is less dense than water, so the weight of the block of wood is

less than that of the same volume of water.)

4. So the block rises and partly emerges to displace less water until the upward force exactly

equals the weight of the block.

Fig. Stability of a ship

Figure shows a ship of weight W floating in the water with an inclination of small angle θ.

The location of the centroid G does not change with the inclination of the ship.

•But since the centre of buoyancy C moves to the new point C’, a couple of forces Ws = Fs is

produced and this couple restores the ship’s position to stability.

•The intersecting point M on the vertical line passing through the centre of buoyancy C’ (action

line of the buoyancy F) and the centre line of the ship is called the metacentre, and GM is called

the metacentric height.

• If M is located higher than G, the restoring force acts to stabilize the ship, but if M is located

lower than G, the couple of forces acts to increase the roll of the ship and so make the ship

unstable.

Fundamentals of flow

1. A flow whose flow state expressed by velocity, pressure, density, etc., at any position, does

not change with time, is called a steady flow.

2. On the other hand, a flow whose flow state does change with time is called an unsteady

flow.

Smoke from a chimney

On a calm day with no wind, smoke ascending from a chimney looks like a single line as shown

in figure (a).

However, when the wind is strong, the smoke is disturbed and swirls as shown in figure (b) or

diffuses into the peripheral air.

One man who systematically studied such states of flow was Osborne Reynolds.

Reynolds used the device shown in figure.

Colored liquid was led to the entrance of a glass tube.

Figure: Reynolds' experiment

As the valve was gradually opened by the handle, the colored liquid flowed, as shown in figure,

like a piece of thread without mixing with peripheral water.

When the flow velocity of water in the tube reached a certain value, as shown in figure that the

line of colored liquid suddenly became turbulent on mingling with the peripheral water.

The former flow is called the laminar flow, the latter flow the turbulent flow, and the flow

velocity at the time when the laminar flow had turned to turbulent flow the critical velocity.

Laminar Flow

1. Laminar flow is a type of flow in which the fluid particles move in layers.

2. There is no transportation of fluid particles from one layer to another.

3. The fluid particles in any layer move along well defined paths or stream lines.

Turbulent Flow

1. Turbulent flow is the most common type of flow that occurs in nature.

2. There is a general mixing up of the fluid particles in motion.

3. There is continuous collision between fluid particles involving transference of momentum

between them

Whenever water is allowed to flow at a low velocity by opening the tap a little, the

water flows out smoothly with its surface in the laminar state.

But as the tap is gradually opened to let the water velocity increase, the flow becomes

turbulent and opaque with a rough surface.

Compressible and Incompressible Flow

1. In general, liquid is called an incompressible fluid, and gas a compressible fluid.

Nevertheless, even in the case of a liquid it becomes necessary to take compressibility into

account whenever the liquid is highly pressurized, such as oil in a hydraulic machine.

2. Similarly, even in the case of a gas, the compressibility may be disregarded whenever the

change in pressure is small.

Rotational and Irrotational flows

1. As fluids moves the fluid particles may be subjected to a rotatory displacements. Suppose a

particle which is moving along a stream line rotates about its own axis also then the particle

is said to have a rotational motion.

2. If the particles as it moves along the stream lines does not rotate about its own axis the

particle is said to have irrotational motion.

Irrotational flow Rotational flow

Calculate Reynolds' Number and decide what type of flow is this?

Example: Lubricating Oil at a velocity of 1 m/s (average) flows through a pipe of 100 mm ID. Determine whether the flow is laminar or turbulent.