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:   ### Conservation of Momentum  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:

• 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:

• 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, : Major losses, : Minor losses, : Pump head, :   