### 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-**Law**Poiseuille**

### 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, :