- inviscid
- g is only body ‘force’
- no surface tension
- displacements, velocities small – can ignore products
- water at free surface stays there
- starts from rest

Bernoulli’s Equation

top surface – far from distance

Incompressible flow

- Divergence zero

Irrotational flow

- curl zero
- Implies existence of velocity potential

Ideal flow: incompressible and irrotational

- Laplace’s equation

Stream function

Velocity Potential

– Cauchy-Riemann equations

Velocity potential example:

Velocity potential:

Streamfunction

Velocity Potential

Source – Streamfunction

Velocity potential:

We know that the flow velocity must be tangential to the surface of the cylinder, since no fluid can penetrate through the solid surface of the cylinder. As this is the only surface condition for an inviscid flow we can assume that for any inviscid flow, streamlines can be thought of as solid surfaces and vice versa.

Let’s look at an example:

RHS = top surface (1), LHS=tap (2)

This is Bernoulli’s potential flow equation.

Bernoulli’s Principle is not always valid. For example, if we consider the Rankine Vortex Tornado Model, there is low pressure at the origin, where the speed is also lowest. This is because the speed is constant along the streamlines for this flow.

Bernoulli’s Principle states, that for an inviscid flow, if the speed of the fluid is increased, then the pressure or potential energy of the fluid must decrease.

I found that this clip helped explain Bernoulli’s Principle better, and has 6 more examples to look at:

Dimensional Analysis is used to identify the physical dimensions of variables in an equation. It determines the units and analyses the relationship between the quantities.

We will be looking mainly at the fundamental units Mass (M in kilogrammes), Length (L in meters) and Time (T in seconds).

Dimensional Homogeneity is where both sides of the equation must have the same dimensions.

A relationship between m physical variables can be expressed as a relationship between m-n non-dimensional groups of variables (called p groups), where n is the number of fundamental dimensions required to express the variables.

Each p group is a function of n governing or repeating variables* *plus one of the remaining variables.

Sources:

*MATH3402 Viscous Flow lecture notes*- http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section5/dimensional_analysis.htm
- Definitions from http://www.physics.uoguelph.ca/tutorials/dimanaly/
- Definitions from http://en.wikipedia.org/wiki/Dimensional_analysis