NAZARIN B. NORDIN

What you will learn: Pascal’s law Incompressibility of fluids Pressure, force ratio Archimedes principle Density and relative density

Introduction to fluids A fluid is a substance that can flow and conform to the boundaries of any container in which we put them. e.g. water, air, glass.

A fluid is any substance that can flow such as a liquid or a gas. Fluids don’t have well defined shapes. A fluid takes on any shape to fit a container. The study of fluids can be divided into two categories : hydrostatics and hydrodynamics or fluid dynamics.

Basic properties of fluids Density (mass per unit volume) - Pressure (force per unit area) -

Basic properties of fluids Pressure (force per unit area) - Notice that from definition, pressure may depend on direction. However, this is not the case for static fluids. (why?).

Basic properties of fluids Pressure (force per unit area) - Unit of pressure: 1 pascal (Pa) = 1 Newton per square meter. 1 atm. = 1.01 x 10 5 Pa

Fluids at rest Pressure increases when we go “deeper” into water – why?

Fluids at rest Pressure of a fluid in static equilibrium depends on depth only

Example Which one of the four container + fluid has highest pressure at depth h? How about if (d) is move up (down) by distance h?

What we will cover: Pascal’s Law: For a confined fluid in a container, the change in pressure will be transmitted without loss to every point of the liquid and to the walls of the container Archimedes’ Principle: Any body that is completely or partially submerged in a fluid will experience an upthrust that is equal to the weight of the fluid displaced by the body

1. Pascal’s Principle A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of the container as a direct consequence of Newton’s Law.

3. Pressure and Pascal’s Law Pressure applied to a confined fluid increases the pressure throughout by the same amount. In picture, pistons are at same height: Ratio A 2 /A 1 is called ideal mechanical advantage

Example: Hydraulic level Applied force F i  change in pressure  p=F i /A i =F o /A o. Therefore output force is F o =F i A o /A i. Therefore F o > F i if A o > A i How about work done?

The Manometer An elevation change of  z in a fluid at rest corresponds to  P/  g. A device based on this is called a manometer. A manometer consists of a U-tube containing one or more fluids such as mercury, water, alcohol, or oil. Heavy fluids such as mercury are used if large pressure differences are anticipated.

Mutlifluid Manometer For multi-fluid systems Pressure change across a fluid column of height h is  P =  gh. Pressure increases downward, and decreases upward. Two points at the same elevation in a continuous fluid are at the same pressure. Pressure can be determined by adding and subtracting  gh terms.

Measuring Pressure Drops Manometers are well-- suited to measure pressure drops across valves, pipes, heat exchangers, etc. Relation for pressure drop P 1 -P 2 is obtained by starting at point 1 and adding or subtracting  gh terms until we reach point 2. If fluid in pipe is a gas,  2 >>  1 and P 1 -P 2 =  gh C

The Barometer Atmospheric pressure is measured by a device called a barometer; thus, atmospheric pressure is often referred to as the barometric pressure. P C can be taken to be zero since there is only Hg vapor above point C, and it is very low relative to P atm. Change in atmospheric pressure due to elevation has many effects: Cooking, nose bleeds, engine performance, aircraft performance.

4. Archimedes’ Principle When a body is fully or partially submerged in a fluid, a buoyant force F b from the surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight m f g of the fluid that has been displaced by the body.

Archimedes’ Principle Buoyant force – upward force in liquid because of increasing pressure in liquid as one goes down below the surface. (a) a hole in water. Notice that the hole is in static equilibrium if it is filled with water.

Archimedes’ Principle (a) a hole in water. Notice that the hole is in static equilibrium if it is filled with water. Therefore the upward force = m f g, m f = mass of displaced water.

Archimedes’ Principle (b) The hole in water is replaced by a solid with the same shape. Since nothing changes in water, therefore the upward force = m f g, m f = mass of displaced water = buoyant force

Archimedes’ Principle (c) The solid in water is replaced by a piece of wood with m w < m f.. In this case the wood float on the surface with F b =m w g.

Archimedes’ Principle Question: Imagine a large sphere of water floating in outer space. The sphere of water is formed under its own gravity. Is there any buoyant force if an object enters this sphere of fluid?

Examples of Archimedes Principle

The Golden Crown of Hiero II, King of Syracuse Archimedes, B.C. Hiero, B.C. Hiero learned of a rumor where the goldsmith replaced some of the gold in his crown with silver. Hiero asked Archimedes to determine whether the crown was pure gold. Archimedes had to develop a nondestructive testing method

The Golden Crown of Hiero II, King of Syracuse The weight of the crown and nugget are the same in air: W c =  c V c = W n =  n V n. If the crown is pure gold,  c =  n which means that the volumes must be the same, V c =V n. In water, the buoyancy force is B=  H2O V. If the scale becomes unbalanced, this implies that the V c ≠ V n, which in turn means that the  c ≠  n Goldsmith was shown to be a fraud!

3. PRESSURE Pressure is the quantity that is related to the force acting on the walls of the balloon and is defined as the normal force per unit area. If F is the force perpendicular to the surface area A, the pressure P is therefore The pressure at a point in a fluid depends on the depth. Greater depths result in greater pressures

Fluids in Motion An ideal fluid is one that (i) flows smoothly, (ii) is non-viscuous, (iii) is incompressible, (iv) is irrotational. The path of steady flow can be visualized using streamlines. Under steady-state flow conditions, for a given time interval, the volume of liquid flowing into the tube must equal the volume of liquid flowing out of a tube.This is known as the Continuity Equation.

4. Bernoulli’s Equation i.Bernoulli’s Equation relates the elevation y, speed v and pressure P of a fluid at any point in a tube. ii.According to Bernoulli’s Equation: for an inviscid flow of a non-conducting fluid, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in iii.However, Bernoulli’s Equation is not applicable to viscous fluids

Bernoulli’s Equation Bernoulli’s Equation is a consequence of conservation of energy in steady flow.

Bernoulli’s Equation Bernoulli’s Equation is a consequence of conservation of energy in steady flow.

Bernoulli’s Equation Adding together, we obtain (Bernoulli’s Equation)

Water pressure in a home (Bernoulli’s Principle II) Consider this example:

Example What is the speed v of the water emerging from the hole? Show that v 2 =2gh (same as free fall)

The continuity equation for flowing liquids The continuity equation – conservation of mass in a incompressible liquid flow. v = velocity of fluid flowing through area A in the tube

Example What is the volume flow rate of water if A o =1.2cm 2, A=0.35cm 2 and h=45mm.

5. DENSITY AND RELATIVE DENSITY The density is an important factor that determines the behaviour of a fluid. The density of a fluid is defined as the mass m per unit volume V: The SI unit for the density is

Densities of common substances—Table 8.1 Table 8.1

A piece of glass is broken into two pieces of different size. How do their densities compare? A.  1 >  3 >  2. B.  1 =  3 =  2. C.  1 <  3 <  2. QuickCheck 8.1 Slide 15-23

A piece of glass is broken into two pieces of different size. How do their densities compare? A.  1 >  3 >  2. B.  1 =  3 =  2. C.  1 <  3 <  2. QuickCheck 8.1 Slide Density characterizes the substance itself, not particular pieces of the substance.

Liquids in Hydrostatic Equilibrium  No!  A connected liquid in hydrostatic equilibrium rises to the same height in all open regions of the container. Slide 15-40

What can you say about the pressures at points 1 and 2? A. p 1 > p 2. B. p 1 = p 2. C. p 1 < p 2. QuickCheck 8.2 Slide 15-41

What can you say about the pressures at points 1 and 2? A. p 1 > p 2. B. p 1 = p 2. C. p 1 < p 2. QuickCheck 8.2 Slide Hydrostatic pressure is the same at all points on a horizontal line through a connected fluid.

Liquids in Hydrostatic Equilibrium  No!  The pressure is the same at all points on a horizontal line through a connected liquid in hydrostatic equilibrium. Slide 15-43

An iceberg floats in a shallow sea. What can you say about the pressures at points 1 and 2? A. p 1 > p 2. B. p 1 = p 2. C. p 1 < p 2. Slide QuickCheck 8.3

An iceberg floats in a shallow sea. What can you say about the pressures at points 1 and 2? A. p 1 > p 2. B. p 1 = p 2. C. p 1 < p 2. Slide QuickCheck 8.3 Hydrostatic pressure is the same at all points on a horizontal line through a connected fluid.