### 01 The Nature of Fluids

(Water Resources I)

Dave Morgan

Prepared using Lyx, and the Beamer class in LA_{TEX2ε,}

### Recommended Text

A recommended text to accompany these notes is Applied Fluid Mechanics by Mott:

Read sections:

1.3, 1.4, 1.6 (omit US units), 1.7, 1.8, 1.9, 1.11 Study Example Problems: 1.5 - 1.9

### Elementary Properties of Fluids

Fluids can be either liquid or gas

A liquid tends to ow and conform to the shape of its container

Liquids are not readily compressible (for the purpose of this course, we consider them to be imcompressible)

A gas tends to expand to ll the closed container it is in (or to disperse if not contained).

Gases are readily compressible

### Primary Units

SI units are used. Four primary units will be used extensively in this course:

Quantity SI unit Dimension Length metre, m L

Mass kilogram, kg M Time second, s T Temperature Kelvin, K θ

### Derived Units

Quantity SI Unit Dimensions
velocity m/s LT−_{1}
acceleration m/s2 _{LT}−_{2}
force N, newton MLT−_{2}
kg·m/s2
energy J, joule
(work) N · m ML2_{T}−2
kg m2_{/}_{s}2
power N · m/s ML2_{T}−_{3}
J/s

pressure Pa, pascal ML−_{1}_{T}−_{2}

(stress) N/m2

### Derived Units:

Quantity SI Unit Dimensions
Volume ow rate, Q m3_{/s} _{L}3_{T}−1

L/s

Weight ow rate, W N/s ML−1_{S}−2

kg/m/s2

Mass ow rate, M kg/s MT−_{1}

Specic weight, γ N/m3 _{ML}−_{2}_{T}−_{2}

### Pressure

Pressure is given by:

p = F A

It is the force per unit area on a surface, where
1 N/m2_{=}_{1 Pa (pascal)}

Blaise Pascal (1623 - 1662), after whom the Pascal programming language was named, determined the following principles:

1 Pressure acts uniformly in all

directions on a small volume of a uid at rest

2 _{In a uid conned by solid}

boundaries, pressure acts

### Pressure

Pressure is given by:

p = F A

It is the force per unit area on a surface, where
1 N/m2_{=}_{1 Pa (pascal)}

Blaise Pascal (1623 - 1662), after whom the Pascal programming language was named, determined the following principles:

1 Pressure acts uniformly in all

directions on a small volume of a uid at rest

2 _{In a uid conned by solid}

boundaries, pressure acts

### Pascal's Laws

Pressure acts uniformly in all directions on a small volume of a uid at rest. The forces must balance out

(i.e. Σ Fx= ΣF_{y}=0); otherwise the
volume of uid will not be in equilibrium
and cannot remain at rest.

Also, the volume must be suciently small that we do not have to consider the mass, and therefore the weight, of the volume of uid. If the weight is not negligible, the upward pressure on the bottom of the volume will have to be greater than the downward pressure on the top of the volume so that

### Pascal's Laws

In a uid conned by solid boundaries, pressure acts

perpendicularly to the boundaries. Why is this true?

(Consider a small volume of uid at rest against one of the boundaries? If this volume remains at rest, what are the forces that act upon it?)

### Density

Density is mass per unit volume: ρ = m

V

The density of water between 0◦_{C and 15}◦_{C is close to}

1000 kg/m3_{.}

It has a maximum density at 4◦_{C.}

Above 15◦_{C, the density drops steadily to a density of 958 kg/m}_{3}

at 100◦_{C.}

(There is a table of values for the properties of water at the back of Applied Fluid Mechanics by Mott, or from numerous other sources)

### Density

Density is mass per unit volume: ρ = m

V

The density of water between 0◦_{C and 15}◦_{C is close to}

1000 kg/m3_{.}

It has a maximum density at 4◦_{C.}

Above 15◦_{C, the density drops steadily to a density of 958 kg/m}_{3}

at 100◦_{C.}

(There is a table of values for the properties of water at the back of Applied Fluid Mechanics by Mott, or from numerous other sources)

### Specic Weight

Specic weight is weight per unit volume: γ = w

V

Water has a specic weight of 9.81 kN/m3 _{between 0}◦_{C and 15}◦_{C.}

Since w = mg, it follows that: γ = w

V = mg

### Specic Weight

Specic weight is weight per unit volume: γ = w

V

Water has a specic weight of 9.81 kN/m3 _{between 0}◦_{C and 15}◦_{C.}

Since w = mg, it follows that: γ = w

V = mg

### Specic Weight

Specic weight is weight per unit volume: γ = w

V

Water has a specic weight of 9.81 kN/m3 _{between 0}◦_{C and 15}◦_{C.}

Since w = mg, it follows that: γ = w

V = mg

### Specic Gravity

Specic gravity is the ratio of the density (or specic weight) of a
substance to the density (or specic weight) of water at 4◦_{C.}

Then, the specic gravity of a substance s is given by sg = γs

γ_{w@4}◦_{C} =

ρ_{s}
ρ_{w@4}◦_{C}

The density of gasoline at 25◦_{C is 680 kg/m}3 _{and the density of}

water at 4◦_{C is 1000 kg/m}_{3} _{. Therefore, the specic gravity of}

gasoline at 25◦_{C is sg= 680/1000 = 0.68.}

The specic weight of mercury at 25◦_{C is 132.8 kN/m}3 _{and the}

specic weight of water at 4◦_{C is 9.81 kN/m}_{3} _{so the specic}

### Specic Gravity

Specic gravity is the ratio of the density (or specic weight) of a
substance to the density (or specic weight) of water at 4◦_{C.}

Then, the specic gravity of a substance s is given by sg = γs

γ_{w@4}◦_{C} =

ρ_{s}
ρ_{w@4}◦_{C}

The density of gasoline at 25◦_{C is 680 kg/m}3 _{and the density of}

water at 4◦_{C is 1000 kg/m}_{3} _{. Therefore, the specic gravity of}

gasoline at 25◦_{C is sg= 680/1000 = 0.68.}

The specic weight of mercury at 25◦_{C is 132.8 kN/m}3 _{and the}

specic weight of water at 4◦_{C is 9.81 kN/m}_{3} _{so the specic}

### Specic Gravity

Specic gravity is the ratio of the density (or specic weight) of a
substance to the density (or specic weight) of water at 4◦_{C.}

Then, the specic gravity of a substance s is given by sg = γs

γ_{w@4}◦_{C} =

ρ_{s}
ρ_{w@4}◦_{C}

The density of gasoline at 25◦_{C is 680 kg/m}3 _{and the density of}

water at 4◦_{C is 1000 kg/m}_{3} _{. Therefore, the specic gravity of}

gasoline at 25◦_{C is sg= 680/1000 = 0.68.}

The specic weight of mercury at 25◦_{C is 132.8 kN/m}3 _{and the}

specic weight of water at 4◦_{C is 9.81 kN/m}_{3} _{so the specic}

### Nature of Fluids

Example

Calculate the pressure produced in the oil in a closed cylinder by a piston with diameter 7.5 cm exerting a force of 11175 N

7.5 cm
11175 N _{Solution}
p = F_{A}
= 11175 N
π (0.075)2/4 m2
= 2529500 Pa
= 2.53 MPa

### Nature of Fluids

Example

Calculate the pressure produced in the oil in a closed cylinder by a piston with diameter 7.5 cm exerting a force of 11175 N

7.5 cm
11175 N
Solution
p = F_{A}
= 11175 N
π (0.075)2/4 m2
= 2529500 Pa
= 2.53 MPa

### Nature of Fluids

Example

Calculate the pressure produced in the oil in a closed cylinder by a piston with diameter 7.5 cm exerting a force of 11175 N

7.5 cm
11175 N _{Solution}
p = F_{A}
= 11175 N
π (0.075)2/4 m2
= 2529500 Pa
= 2.53 MPa

### Nature of Fluids

Example

Calculate the weight of 1 m3 _{of kerosene if it has a mass of 823 kg}

Solution

W = mg

= 823 kg × 9.81 m/s2 = 8070 N

Note: In general, use 5 signicant gures for interim calculations and 3 signicant gures for displayed solutions.

### Nature of Fluids

Example

Calculate the weight of 1 m3 _{of kerosene if it has a mass of 823 kg}

Solution

W = mg

= 823 kg × 9.81 m/s2 = 8070 N

Note: In general, use 5 signicant gures for interim calculations and 3 signicant gures for displayed solutions.

### Nature of Fluids

Example

Calculate the density and the specic weight of benzene if its
specic gravity is 0.876.
Solution
0.876 = ρb
ρ_{water@4}◦_{C}
ρ_{b} = 0.876 × 1000 kg/m3
= 876 kg/m3
0.876 = γb
γ_{water@4}◦_{C}
γ_{b} = 0.876 × 9.81 kN/m3
= 8.59 kN/m3

### Nature of Fluids

Example

Calculate the density and the specic weight of benzene if its
specic gravity is 0.876.
Solution
0.876 = ρb
ρ_{water@4}◦_{C}
ρ_{b} = 0.876 × 1000 kg/m3
= 876 kg/m3
0.876 = γb
γ_{water@4}◦_{C}
γ_{b} = 0.876 × 9.81 kN/m3
= 8.59 kN/m3

### Nature of Fluids

Example

A cylindrical tank with diameter 12.0 m contains water at 20◦_{C to}

a depth of 4.0 m. If the water is heated to 65◦_{C, what is the depth}

of the water? (Assume that the tank dimensions remain constant and that there are no losses due to evaporation.)

Solution
Volume at 20◦_{C:}
V20=πd
2_{h}_{20}
4 =
π (12.0 m)2_{(4.0 m)}
4 =452.39 m3
Mass of water in the tank:

m = ρV20=998 kg/m3×452.39 m3=451490 kg

### Nature of Fluids

Example

A cylindrical tank with diameter 12.0 m contains water at 20◦_{C to}

a depth of 4.0 m. If the water is heated to 65◦_{C, what is the depth}

of the water? (Assume that the tank dimensions remain constant and that there are no losses due to evaporation.)

Solution
Volume at 20◦_{C:}
V20= πd
2_{h}_{20}
4 =
π (12.0 m)2_{(4.0 m)}
4 =452.39 m3
Mass of water in the tank:

m = ρV20=998 kg/m3×452.39 m3=451490 kg

### Nature of Fluids

Solution (continued) V20= πd 2_{h}

_{20}4 = π (12.0 m)2

_{(4.0 m)}4 =452.39 m3 m = ρ20V20=998 kg/m3×452.39 m3=451490 kg Volume at 65◦

_{C:}V65= m ρ

_{65}= 451490 kg 981 kg/m3 =460.23 m3 Depth at 65◦

_{C:}h65=4V65 πd2 = 4 × 460.23 m3 π (12.0)2 =4.0639 m The depth at 65◦

_{C is 4.06 m}