For a two-dimensional velocity field in the form

the streamlines are:

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:

### Conservation of Momentum

For steady flow, resolved into components:

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

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, :

Major losses, :

Minor losses, :