Steady Streamlines
For a two-dimensional velocity field in the form
the streamlines are:
Unsteady Streamlines & Pathlines
For a two-dimensional unsteady field in the form
the parametrised pathlines are:
Forces in Fluids
Shear stress, :
Pressure force, :
Viscous Force, :
Kinematic viscosity, :
Fluid Statics
Hydrostatic Equation:
Archimedes’ Principle:
The magnitude of upthrust is equal to the weight of water displaced
Resultant Pressure Force, :
Point of application, :
Manometry

Mass Flow Rate

When velocity is perpenduclar:
Reynolds Transport Theorem
- The first term is the rate of change of N in the system
- The second term is the rate of change of N in the control volume and system
- The third term is the net flow rate of N out of the control volume
Where is the property being conserved and
.
is the property of
per unit volume.
Conservation of Mass
Algebraic formulation:
Steady, uniform, incompressible flow:
Steady Flow:
Conservation of Momentum
For steady flow:
For steady flow, resolved into components:
For steady, uniform, incompressible flow:
When mass is conserved for one inlet and one outlet:
The Bernoulli Equation
The four conditions for the Bernoulli Equation:
- Steady flow
- Inviscid flow
- Incompressible flow
- The two points are on the same streamline or have the same Bernoulli constant,
Stagnation point flow:

Laminar Flow between Horizontal Plates
Where
- This is known as the Pouiselliu Law
Laminar Flow in a Circular Pipe
Where
- This is known as the Hagen-Poiseuille Law
Conservation of Energy for Steady Flow
For a uniform velocity profile:
First law of thermodynamics:
Reynolds substitution:
The Pipe Flow Energy Equation (PFEE)
is the lost energy
, the pump work
is the lost head
is the pump head
The pipe flow energy equation only applies to flow that is:
- Steady
- Adiabatic
- Incompressible
- Uniform velocity field
- Between a single inlet and outlet
Turbulent Flow in Circular Pipes
Reynolds Number, :
Mean velocity, :
Friction factor, :
Relative roughness, :
Lost Head
Lost head, :
Major losses, :
Minor losses, :
Pump Head
Pump head, :