This week we learnt about Water Waves.
Some assumptions are made when looking into these waves. We assume that the flow is incompressible and irrotational, and therefore:
We also assume that g is the only force implied, and that there is no surface tension. Also, it is an inviscid flow, the velocities are small, the water at a free surface stays there and it starts from rest.
We use Bernoulli‘s equation:
Let’s start with a quote.
“When a fluid is both frictionless and irrotational, pleasant things happen” – F.M White
The Potential Flow or velocity potential exists as the velocity field as the gradient of a scalar function. If a flow is irrotational, then its curl will equate to zero, and if a flow is incompressible, then its divergence of velocity will equal zero. We use a flow that consists of both of these to
When thinking about potential flow, I found this video extremely useful and informative.
I liked this topic as it is a very practical vision of how fluid work.
References
Euler’s equations are a set of equations similar to the Navier-Stokes set, but looking at conservation and non conservation results.
This week was consolidation week, and we have to talk about dimensional analysis.
As with any equation, we have variables, and in this topic, we use mass, length, time, and temperature. For those of you that would argue for the fifth and sixth variables, electric current and luminosity intensity, they are not applicable in fluid dynamics, so therefore we will eliminate them.
Now we will work through an example
To continue smoothly from last week, we are now looking at the Stream Function. This is a function used to map the streamlines of a fluid, which we’ve previously discussed.
This will be a constant, as it is tangential to the velocity vector at each point.
For incompressible flows, we use velocity and density.
To check incompressibility, 
Examples from my notes include
My understanding of this needs to be improved. Although going through my notes has helped, I will need to revise more thoroughly.
References
We’ll now start to look properly into the world of inviscid flows, and in particulary here, how we map them out.
We can map flows by plotting streamlines, and pathlines.
Streamlines are points which show the instantaneous direction of the velocity at any given time, and pathlines map out the trajectory of a particle in the fluid.
Here is question 1a on the first exercise sheet about streamlines and pathlines.
I am not 100% on this yet, so I will definitely be doing more of the exercises as revision and to further my understanding.
References
The Phenomena of Fluid Motions, by Robert Brodkey. Published in 2004.
This week we were talking about Dimensional Analysis. It forms relationships between factors involved in a physical situation. In fluid dynamics, we use
Length=L (Metre in SI units)
Mass=M(Kilogramme in SI units)
Time=T(Second in SI units)
Temperature= Θ(Kelvin in SI units)
Electric current=A(Amp in SI units)
Luminous intensity=I(Candela in SI units)
Here is a table given in our notes:
We’ll also look at Buckingham’s Theorems.
Firstly, a relationship between m 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.
Secondly, each p group is a function of n governing or repeating variables plus one of the remaining variables.
References
http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section5/dimensional_analysis.htm
The Navier-Stokes equations help determine its motion.
There are different equations for a compressible and in compressible fluid and they are as follows.
We learnt to derive these using the continuity equation.
There are no exercises for this as we will be going back to inviscid flows next week.
Currently I think I understand this, however I will definitely need to recap before the second term.
References
So, this week we had a change about. We had a lecture on viscous flow and a taster into our term two syllabus.
Viscous flow takes into consideration the viscosity of a fluid. Viscosity essentially being the thickness of the fluid, for example syrup has a higher viscosity than milk. The viscosity of most liquids will change according to temperature, so honey has a lower viscosity when hot, but a higher one when cold.
We were introduced to various assumptions, first of which is that the fluid is incompressible. This implies that the density of the fluid in a fixed volume will stay the same regardless of movement.
One of the main assumptions, was the Continuum Hypothesis. This says that at any point in our fluid, we can calculate the density, pressure and temperature. This essentially dismisses the fact that fluids are made up of molecules, so if a pin was placed in a fluid, there would be a chance that we haven’t actually got a part of the fluid, however the continuum hypothesis allows us to calculate the characteristics as an extremely close estimate.
Momentum conservation is another trait of a viscous fluid. Its is quite simply the fact that we cannot create momentum, and we cannot eliminate it. Essentially,
rate of change of momentum inside volume (V) = net rate of inflow of momentum into V + total force on fluid inside V
We also consider the conservation of mass. It is said that the the mass of one element before an action is equivalent to the mass after, therefore, (rate of change of mass inside V = net rate of inflow of mass into V).
This is only a small introduction to viscous flows, and although the assumptions seem relatively straight forward, I will definitely have to recap before term 2.
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