- Introduction
- Symbols
- Density
- Pressure
- Viscosity
- Surface Tension
- Compressibility
- Sonic Velocity (Mach No)

Fluids can be either liquids or gases. A liquid is hard to
compress and takes the shape of the vessel containing it. However it has a fixed volume
and has an upper level surface. Gas is easy to compress, and expands to fill
its container. There is thus no free surface. Liquids are generally assumed to
be incompressible fluids and gases compressible fluids. Liquids ar
only compressible when they are highly pressurised, and the compressibility of gases may be
disregarded whenever the change in pressure is very small.

Important characteristics of fluids from the viewpoint of fluid mechanics are
density, pressure, viscosity, surface tension, and compressibility. This section includes brief notes
on these important characteristics.

A = Area (m^{2})a = Speed of sound (m/s) g = acceleration due to gravity (m/s ^{2} )h = fluid head (m) K = Bulk modulus (MPa ) M = mach number u /a M = Molecular weight p = fluid pressure (N /m ^{2} )p _{abs} - absolute pressure (N /m^{2} )p _{gauge} - gauge pressure (N /m^{2} )p _{atm} - atmospheric pressure (N /m^{2} )p _{s}= surface pressure (N /m^{2} )Q = Volume flow rate (m ^{3} /s)R = Gas Constant (J/(kg.K) R _{o} = Universal Gas Constant (J/(kg.mol.K) ρ = fluid density (kg /m ^{2} )sg = Specific gravity RD = Relative density |
u = fluid velocity (m/s) v = fluid velocity (m/s) x = depth of centroid (m) β = Compressibility (1/MPa) γ = Surface Tension (N/m) θ =slope (radians) ρ = density fluid (kg/m ^{3})ρ _{w} = density water (kg/m^{3})ρ _{a} = density air (kg/m^{3})τ = shear stress (N /m ^{2})μ = viscosity (Pa.s) ν kinematic viscosity (m ^{2}s^{-1}) υ = Specific volume (m ^{3} / kg)γ= Ratio of Specific Heats |

The mass per unit volume of material is called the density, which is
generally expressed by the symbol ρ. The density of a gas changes according
to the pressure, but that of a liquid may be considered constant unless the relevant pressures changes are very high The units of density are kg/m^{3} (SI).

The ratio of the density of a material ρ to the density of water ρ_{w} ( at 4^{o} C ) ,
is called the relative density(RD).
This is often called the specific gravity (sg) a term which is sometimes confusing..

Note: The for liquids and solids the specific gravity is generally identified as the ratio of the materials density
relative to water both at the same conditions i.e 4^{o} and 1,013 bara ambient pressure.
For gases the specific gravity is the ratio of the materials density to the the density of air,
both at the same conditions as identified above. The term specific gravity is not as clear at term as the relative density.

For a liquid RD = ρ /ρ_{w}

For a gas RD = ρ /ρ_{a}

The apparent specific gravity sg_{a} is the ratio of the weight of a volume of a material to the weight of an equal volume of the reference fluid (water or air) again under the
same conditions. This is a practical value but essentially is the same as the true specific gravity.

The density of gases gases and vapours are greatly affected by the pressure . For so called perfect gases the density can be calculated from the formula .

R_{o} = the universal Gas constant = 8314 J/(kg.K) and M = Molecular weight.
Therefore R = 8314/M [ J/(kg.K) ]

The reciprocal of density, i.e. the volume per unit mass, is called the specific volume, which is generally expressed by the symbol υ

υ = 1/ρ

The dimensional formula for density = ML^{-3} and the
dimensional formula for specific volume = M^{-1}L^{3}

A fluid is always subject to pressure. Pressure is the force
per unit area at a point. The absence of pressure occurs in a complete
vacuum. A complete vacuum is really a theoretical concept.

The normal pressure experienced on the surface of the earth is called the atmospheric pressure
and, in general, pressures are measured relative to the local atmospheric pressure. These
measured pressures are called gauge pressures.
The absolute pressure is the pressure relative to that of a perfect vacuum .

The figure below shows the relationship between the gauge pressure
and the absolute pressure for two measurements : a pressure less than atmospheric (A)
and a pressure greater than atmospheric (B) are shown .

p_{abs} = p_{gauge} + p_{atm}

The SI unit of pressure is the Pascal (abrev.= Pa) (Newton /m^{2} ). The dimensional formula for pressure
is ML^{-1}T^{-2}.

In considering fluid pressures it has been found convenient in hydrostatics and in fluid dynamics
to use fluid head as a method of measuring pressure.
Considering the figure below. A quantity of fluid in an
open vessel is experiencing an atmospheric pressure on its surface. A tube is routed vertically
to a sealed container held at a pressure of absolute zero.

The liquid will be forced up the tube until
the gravity force resulting from the level of fluid in the tube balances the force due to the pressure at the bottom of the
head of fluid. Assuming the area of the tube is A _{t}, the density of the fluid = ρ,and the pressure at the top of the tube is zero.
The force at x-x

F_{xx} = 0 + hAρg.

The pressure at x-x=

P_{xx} = 0 + hAρg. / A = hρg.

For a fluid with a known fixed density the height h can be conveniently used to identify the pressure.
For water the atmospheric pressure is about 10,5m. In practice water vaporises into the vacuum at the top of
the tube reducing the vacuum this reduces the column height by about 180mm.

Mercury is used for measuring pressure and the height of a column of mercury which
can be supported by atmospheric pressure is about 0,760m. Mecurey has a low vapour pressure and the vacuum is only reduced
by about 0,16 Pa, (very small compared to atmospheric pressure of 10^{5} Pa ).
It is clear that gauge pressures and vacuum pressures are easily obtained using this method. The barometer
identifies pressure readings in mm Hg.

Additional ref notes.. Viscosity

Tables of fluid viscosities Fluid Viscosities

Perfect fluids cannot in theory transmit shear stresses.
All real fluids resist shear flow. The viscosity property of the fluid defines
the degree of resistance to flow it possess. This is illustrated using the figure below. A cylinder is
located on a shaft and the space between is filled with a fluid. The cylinder is rotated
at an angular velocity ω. The velocity distribution in the fluid as shown.
The torque required to rotate the cylinder is an indication of the viscosity of the fluid.

Consider an element of fluid STQR which is subject to a shear stress τ

In a short period of time dt the fluid element distorts to S'T'QR. The fluid will experience a strain φ in time dt.

μ ( dφ /dt ) = μ τ

Note: The rate of shear strain is also measured as the deflection dx divided by the distance dy i.e dx/dy occuring over a time intervel dt. It is is effectively the velocity gradient dv /dy .. (dv = dx/dt)

= μ (dv/dy) = τ .... therefore ..... μ = τ / (dv/dy)

If the element where an elastic solid it would distort a fixed amount proportional to the shear stress and the proportionality
constant is called the Modulus of Rigidity (G). The fluid element distorts
at a rate based on the viscosity of the fluid.

The SI unit for viscosity is the Pa.s (Pascal Second).
This is simply derived from the units pressure /( velocity/ length) = Pa / (m /s / m )= Pa.s. The dimensional formula
= ML^{-1}T ^{-1}.
The centipoise , a cgs unit, is commonly used because water has a viscosity
of 1,0020 cP (at 20 C;). 1 cP = 10^{-2} Poise. 1 Poise = 1g.cm.s^{-1} = 0,1 Pa.s........ Therefore 1 Pa.s = 1000cP

__Kinematic viscosity __

The viscosity μ and the density ρ are both properties
of a fluid. The ratio μ/ρ is called the kinematic viscosity and is
also a property. Kinematic viscosity .ν can be completely
defined in terms of length and time and has a dimensional equation L^{2}T^{-1}.
The SI units for kinematic viscosity is the (m^{2}s^{-1}). The cgs physical
unit for kinematic viscosity is the stokes (abbreviated S or St). It is sometimes
expressed in terms of centistokes (cS or cSt).

1 centistokes stokes = 10^{-2} stokes. ... 1 stokes = 10^{-4} ms^{-1}.

__Newtonion /Non-Fluids__

Solids which distort an amount which is proportional to the stress are called elastic solids.
Fluids which deform at a **rate** which is proportional to the tangential stress
are called Newtonion fluids.
Fluid mechanics generally relates to Newtonion fluids.
Fluid with high viscosities are called thick or heavy fluids and include tar, treacle and grease. Fluids of low viscosity are called thin
fluids and include water, paraffin and petrol. Gases have very low values of viscosity.
Non Newtonion fluids are studied under the heading of rheology.

Typical Non-Newtonion fluids include.

Pseudo plastic fluids e.g. solutions including gelatine, clay, milk and blood
often have reduced viscosity when the rate of shear is increased.

Some fluids experience increased viscosity when the rate of shear is increased. This group includes concentrated
solutions of sugar, and aqueous suspensions of starch.

Some materials, including metals, deform continuously with little increase in stress when stessed
above their yield point. These behave as plastically above
the yield point.

The surface of a liquid is the interface between the liquid volume
and the fluid above the liquid. Generally the liquid is water and the fluid
above the liquid is air. The molecules within the liquid attract each other and
at the interface there are more attractive forces towards the bulk of the liquid than
there are towards the adjacent gas molecules. The molecular forces tend
to pull into the fluid bulk. The surface of a liquid is apt to shrink,
and its free surface is in such a state that each section pulls another as if an
elastic film is being stretched. The surface behaves like a flexible membrane.
This property is evident when overfilling a cup with water. The level of
water in the cup will be higher than the cup edge before it overflows.

If a double line is drawn on the surface of a liquid there is a force normal to
the lines holding the lines together.

The tensile strength per unit
length of assumed section on the free surface is called the surface tension (symbon γ).

Liquid | Surface Fluid | Surface Tension N/m |

Water | Air | 0,0728 |

Mercury | Air | 0,476 |

Mercury | Water | 0,373 |

Paraffin | Air | 0,027 |

Water | Paraffin | 0,027 |

Methyl alcohol | Air | 0,048> |

For large volumes of liquid the forces due to gravity and inertia are
large compared to the surface tension forces. Therefore the surface tension is not considered in most hydrostatic and hydrodynamic
calculations.

For small volumes and areas of fluid the surface tension
becomes important and results in spherical water droplets and the capillary effect.

The volume of a fluid changes from V to V + δV as a result of the applied pressure changing from p to p + δp. The compressibility (β) is basically (δV / V ) /δp i.e. the ratio of the proportional change of volume to change of pressure . This is the reciprocal of the bulk modulus K. The bulk modulus K is similar to the spring factor , that is K .(δV/V) = δp

The volume of the fluid clearly decreases if the pressure increases and is proportionate assumed
that the fluid does not change state during the process (it remains a liquid, solid or gas.

For water of normal temperature/pressure K = 2,06 x 10^{9} Pa,
and for air K = 1.4 x l0^{5} Pa assuming adiabatic change. In
the case of water, 1/K = 4.85 x l0^{-10}Pa^{-1}.Water compresses by about
0.005% when the pressure is increased by 1 atm (10^{5} Pa).

The product of density ( ρ) and volume is the mass i.e. ρ V = m = constant. ,
and therefore volume V = m /ρ. The bulk modulus can be expressed in terms of density explained below

For gases the bulk modulus is very much dependent on the conditions : if the compression takes
place at constant temperature the bulk modulus is called the isothermal bulk modulus and if the
compression takes place with no transfer of heat across the system boundary the bulk modulus is colled
the isentropic bulk modulus. The ratio of isentropic/isothermal bulk modulii is γ which is
the ratio of specific heats.

The propagation speeds of traveling waves are characteristic of the media in which they travel and are generally not dependent upon the other wave characteristics such as frequency, period, and amplitude. The speed of sound in air and other gases, liquids, and solids is predictable from their density and elastic properties(bulk modulus). In a fluid medium the wave speed takes the general form

Consider a fluid in which a sound wave is being transmitted at a velocity c. The fluid velocity is u. To simplify the assessment this has been resolves such that the wave is stationary and the fluid has a velocity u-c. See figure below.

Taking a small area normal to the wave front ΔA continuity requires that

.....(equation a)

For the volume enclosed by ΔA

the force to the right = (p + δ p)ΔA -pΔA

= The rate of increase of momentum towards the right

= ρ(u-c)ΔA (- δ u)

Therefore ...δp = ρ(c - u)δ u.....(equation B)

Elimination of δ u from (A) and (B) above

For a weak pressure wave with δp and δρ --> zero

This equation states that a sound wave which is a weak pressure surge of
value √(∂ p /∂ ρ )
move through a fluid at a velocity of

( c-u ) =a (the speed of sound ) relative to the
fluid ahead of it moving with a velocity u. The assumption is that the friction is
low and the resulting temperature difference across the wave is small. The movement
of the wave is considered to be isentropic. (not heat transfer and no friction).

Now the bulk modulus is defined (see above) in terms of density by K = ρ ( ∂ p /∂ ρ )and therefore

Considering gases subject to isentropic processes. The law pv^{γ} = constant (k) applies. Therefore

The bulk modulus as defined above K = - v.(dp/dv) and therefore

K = γp and therefore for a perfect isentropic gas

The mach number M is the ratio of the velocity of gaseous flow in relation to the sonic velocity

Fluids velocities less than the speed of sound are called sub-sonic (M < 1) and fluid velocities greater than the speed of sound are called supersonic (M >1 )

Solid | Velocity of Sound(a) bar /bulk |
Liquid | Velocity of Sound(a) |
Gas | Velocity of Sound(a) |
||

(m/s) | (m/s) | (m/s) | |||||

Aluminium | 5100 / 6300 | Water-Fresh | 1430 | Air | 331 | ||

Copper | 3700 / 5000 | Water-Sea | 1510 | Oxygen | 315 | ||

Iron | 3850 | Alcohol | 1440 | Hydrogen | 1263 | ||

Steel | 5050-6100 | Mercury | 1460 | Carbon Monoxide | 336 | ||

Lead | 1200 | Carbond Dioxide | 258 | ||||

Glass | 5100 / 5600 | ||||||

Rubber | 30 | ||||||

Wood | 04-5000 |

- Thermophysical Properties of Fluid Systems.. Quality fluid property information from NIST
- Wikipedia _Fluid dynamics.. Lots of relevant information
- Fluid Physical Properties and Constants.. A set of links to detailed tables and calculators -seems up to date
- Engineers Edge Fluid Characteristics.. A table show density, viscosity , and vapour pressure
- Roymech -Liquid properties.. My table showing a lot of info on liquids
- Roymech -Gas properties.. My table showing a lot of info on gases

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