Fluid Mechanics Study
Notes
Fluid Properties
DIMENSIONAL ANALYSIS
• Any quantity can be expressed in terms of the four basic dimensions as
follows:
o Length (L)
o Mass (M)
o Time (T)
o Force (F)
• For complex quantities in water engineering it is often helpful to
consider them in terms of their equivalent dimensional units (i.e.
F=MLT-2)
FLUID DEFINITIONS
• A fluid is any substance that deforms and flows when subjected to a
shear stress (applied pressure)
• The following quantities can be measured for each specific fluid:
o Mass: Amount of material (kg)
o Weight: Force exerted by fluid due to gravity (N)
o Density (ρ): Mass per unit of volume (kg/m3)
o Unit Weight (γ): Gravitational force exerted by fluid per unit of
volume (γ=gρ)
o Relative Density: Comparative density to water
o Specific Volume: Volume per unit of mass (1/ρ)
PRESSURE
• Pressure (Pa) is given as the normal force applied to a surface divided
by its area
• Absolute pressure is observed relative to absolute zero
• Gauge pressure is pressure recorded relative to atmospheric pressure
(101.3kPa at room temperature)
IDEAL GASES
• An ideal gas that has negligible volume and collision energy losses
follows the rule of PV=nRT
• Gases also follow the rule of P=ρRT where ‘P’ is absolute pressure
ELASTICITY AND COMPRESSIBILITY
• The Bulk Modulus of Elasticity (E) of a fluid is the ratio of pressure
change to the resulting fractional change in volume
• For gases ‘E’ is proportional to:
o E=p for isothermal processes
, o E=kp for adiabatic processes (the retention of heat energy
makes it more difficult to compress the gas)
• There are two formulas for sonic velocity:
o c=sqrt(E/ρ) for a liquid or solid
o c=sqrt(kRT) under adiabatic gas conditions
VISCOSITY
• Dynamic viscosity is the level of fluid flow in response to a shear
stress
• Highly viscous materials will flow extremely slowly even under high
shear stresses
• Shear stress = Viscosity*(Relative velocity)/(Flow Width)
• The viscosity of a fluid is highly temperature dependant
• Kinematic viscosity of a fluid is the ratio of its dynamic viscosity and
its density (v=V/ρ)
SURFACE TENSION
• The net inward intermolecular force between fluid molecules at the
surface leads to the creation of a membrane-like flat plane at that fluid
surface
• This tension arises due to an excess of pressure within the liquid
• For a curved surface, this pressure differential is calculated by equating
the equivalent forces:
o ‘pa2’ upwards
o ‘4σa2sin(θ/2)’ downwards
o Hence p=2σ/r
• Cohesion is the intermolecular forces between like particles
• Adhesion is the intermolecular forces between unlike particles such as
glass and water
• If adhesion exceeds the cohesion, the solid surface will be ‘wetted’
(small contact angle) and vice versa (large contact angle)
• Capillarity is the phenomenon of a fluid being carried up a narrow tube
due to adhesive forces and surface tension
• The capillary rise of a tube can be calculated be equating forces:
o ‘2πrσ’ upwards surface tension
o ‘πr2hγ’ downwards gravitational force
, o h=2σ/(γr)
VAPOUR PRESSURE
• Within a liquid, a certain amount of molecules will evaporate from a
free surface at every finite temperature and induce a vapour pressure
• Vapour pressure always increases with temperature
• Boiling of a liquid occurs when its vapour pressure is greater or equal
to the atmospheric temperature (e.g. water has a vapour pressure of
101.3kPa at 100 degrees Celsius, hence it boils)
• Cavitation occurs when a fluid flows at such a high velocity that its
pressure is low enough to equal the corresponding vapour pressure –
causing the liquid to rapidly evaporate
Hydrostatics
PRESSURE IN A STATIC FLUID
• Pressure is calculated as a limit of F/A as area becomes infinitesimally
small
• Weight force of a ‘parcel’ of fluid is given as: G=γ*δx*δy*δz
• The force exerted upon the fluid in each plane can be calculated by
multiplying the limit pressure minus its specific weight by the area
affected
• Pressure within a fluid increases with depth (decreases with height)
at a rate of γ*Pa/m
• Pressure within a static fluid does not vary with any movement along
the xz-plane in accordance with Pascal’s Law
MEASUREMENT OF PRESSURE
• A Piezometer column is a thin tube allowing for a liquid to rise without
overflowing which calculates gauge pressure from the height of the
piezometric head (p=γh)
• Manometers are narrow U-tubes that calculates pressure based off
the height difference from a predetermined datum (p=ρfgz+ρMgy)
Notes
Fluid Properties
DIMENSIONAL ANALYSIS
• Any quantity can be expressed in terms of the four basic dimensions as
follows:
o Length (L)
o Mass (M)
o Time (T)
o Force (F)
• For complex quantities in water engineering it is often helpful to
consider them in terms of their equivalent dimensional units (i.e.
F=MLT-2)
FLUID DEFINITIONS
• A fluid is any substance that deforms and flows when subjected to a
shear stress (applied pressure)
• The following quantities can be measured for each specific fluid:
o Mass: Amount of material (kg)
o Weight: Force exerted by fluid due to gravity (N)
o Density (ρ): Mass per unit of volume (kg/m3)
o Unit Weight (γ): Gravitational force exerted by fluid per unit of
volume (γ=gρ)
o Relative Density: Comparative density to water
o Specific Volume: Volume per unit of mass (1/ρ)
PRESSURE
• Pressure (Pa) is given as the normal force applied to a surface divided
by its area
• Absolute pressure is observed relative to absolute zero
• Gauge pressure is pressure recorded relative to atmospheric pressure
(101.3kPa at room temperature)
IDEAL GASES
• An ideal gas that has negligible volume and collision energy losses
follows the rule of PV=nRT
• Gases also follow the rule of P=ρRT where ‘P’ is absolute pressure
ELASTICITY AND COMPRESSIBILITY
• The Bulk Modulus of Elasticity (E) of a fluid is the ratio of pressure
change to the resulting fractional change in volume
• For gases ‘E’ is proportional to:
o E=p for isothermal processes
, o E=kp for adiabatic processes (the retention of heat energy
makes it more difficult to compress the gas)
• There are two formulas for sonic velocity:
o c=sqrt(E/ρ) for a liquid or solid
o c=sqrt(kRT) under adiabatic gas conditions
VISCOSITY
• Dynamic viscosity is the level of fluid flow in response to a shear
stress
• Highly viscous materials will flow extremely slowly even under high
shear stresses
• Shear stress = Viscosity*(Relative velocity)/(Flow Width)
• The viscosity of a fluid is highly temperature dependant
• Kinematic viscosity of a fluid is the ratio of its dynamic viscosity and
its density (v=V/ρ)
SURFACE TENSION
• The net inward intermolecular force between fluid molecules at the
surface leads to the creation of a membrane-like flat plane at that fluid
surface
• This tension arises due to an excess of pressure within the liquid
• For a curved surface, this pressure differential is calculated by equating
the equivalent forces:
o ‘pa2’ upwards
o ‘4σa2sin(θ/2)’ downwards
o Hence p=2σ/r
• Cohesion is the intermolecular forces between like particles
• Adhesion is the intermolecular forces between unlike particles such as
glass and water
• If adhesion exceeds the cohesion, the solid surface will be ‘wetted’
(small contact angle) and vice versa (large contact angle)
• Capillarity is the phenomenon of a fluid being carried up a narrow tube
due to adhesive forces and surface tension
• The capillary rise of a tube can be calculated be equating forces:
o ‘2πrσ’ upwards surface tension
o ‘πr2hγ’ downwards gravitational force
, o h=2σ/(γr)
VAPOUR PRESSURE
• Within a liquid, a certain amount of molecules will evaporate from a
free surface at every finite temperature and induce a vapour pressure
• Vapour pressure always increases with temperature
• Boiling of a liquid occurs when its vapour pressure is greater or equal
to the atmospheric temperature (e.g. water has a vapour pressure of
101.3kPa at 100 degrees Celsius, hence it boils)
• Cavitation occurs when a fluid flows at such a high velocity that its
pressure is low enough to equal the corresponding vapour pressure –
causing the liquid to rapidly evaporate
Hydrostatics
PRESSURE IN A STATIC FLUID
• Pressure is calculated as a limit of F/A as area becomes infinitesimally
small
• Weight force of a ‘parcel’ of fluid is given as: G=γ*δx*δy*δz
• The force exerted upon the fluid in each plane can be calculated by
multiplying the limit pressure minus its specific weight by the area
affected
• Pressure within a fluid increases with depth (decreases with height)
at a rate of γ*Pa/m
• Pressure within a static fluid does not vary with any movement along
the xz-plane in accordance with Pascal’s Law
MEASUREMENT OF PRESSURE
• A Piezometer column is a thin tube allowing for a liquid to rise without
overflowing which calculates gauge pressure from the height of the
piezometric head (p=γh)
• Manometers are narrow U-tubes that calculates pressure based off
the height difference from a predetermined datum (p=ρfgz+ρMgy)
