For now, we are moving back to inviscid flows. I will remind myself of what a fluid actually is.

What is a fluid?

A fluid is a substance that continues to deform in the presence of sheer stress.

We can model a fluid with a continumm model, where the fluid properties are defined by:

• velocity u(x,t) [vector]
• temperature T(x,t) [scalar]
• density p(x,t) [scalar]
• viscosity mu(x,t) [scalar]
• and many more..

We can also visualise fluids. One way we can do this is by using pathlines and streamlines.

Pathlines are defined as trajectories of marked fluid parcels. These can be written mathematically: This can also be adapted for polar coordinates: Streamlines are defined as a curve which is always tangential to the fluid velocity. These can be written mathematically: We can now use these equations to calculate fluid flows and we can show the graphically.

We have also seen that pathlines and streamlines are identical if they do not depend on time.

Please find below examples on pathlines and streamlines.  This week I need to work on a few more of these examples and try to find some useful links which would help me understand more about how to display the pathlines and steamlines on a graph.

To start of with, I was really struggling to understand pathlines and streamlines, but since we have been working through the example sheets it feels like I’m starting to understand it more and have a go at some of the harder examples.

This week we are still looking at viscous flow and more on the Navier-Stokes equations.

We also will touch on Newtonian viscous fluids. These satisfy Newton’s Law of Viscosity. For an incompressible fluid, this law takes the form of: Note that the term is called the dynamic viscosity and is constant, at a fixed temperature and pressure, for a given Newtonian fluid. Also if this is equal to 0 then the inviscid form of sigma(ij) is immediately recovered. This has dimensions: mass/(length*time).

If we write the momentum conservation equation using suffix notation, we obtain: But since V is fixed, we can write: Also since sigma(i)=sigma(ij)n(j) then the momentum conservation equation can be written as: But if we simplify this further and set the surface integrals equal to volume integrals  and combining the terms, we get: This equation expresses the momentum conservation principle for a general fluid.

If we insert the expression we have for sigma(ij) (at the start of this blog) gives the Navier-Stokes equations for an incompressible fluid, which can be written as: These are the Navier-Stokes equations for an incompressible, constant viscosity fluid. This can be written in vector notation: To find solutions of the Navier-Stokes and continuity equations, we require boundary conditions (and initial conditions if the problem is unsteady).

We also touched on simple shear flow which can be solved using the Navier-Stokes equations with boundary conditions, since we are not continuing on this next week, I will come back and explain this in more detail later in the year.

Unfortunately there has been a change in the structure of this module and we are now going to jump and look at viscous flow.

Viscousoity is a measure of how a fluid’s resistance to flow.

Incompressibility

We have learnt that all fluids will be assumed to be incompressible. This assumption is not too restrictive for liquids and is valid for gases if the flow velocities are very small (i.e. if the Mach number is small). Mathematically this is written as:

Dp/Dt=0

D/Dt denotes the convected derivative and this is the reate of change dollowing the motion of a particle.

Mass conservation

The mass conservation equation states that a fluid cannot simply appear and disappear. This equation is:

rate of change of mass inside V = net rate of inflow of mass into V (where V is the volume of the fluid)

If we use Gauss’ Divergence Therorem, we can manipulate the equation and get:

Continuity equation for an incompressible fluid: div(u)=0

The mass conservation equation makes sense since something cannot disappear and reappear like magic!

Momentum conservation

The momentum conservation equation expresses Newton’s Second Law for a continuous medium. We have seen that the principle of conservation of momentum may be expressed as:

rate of change of momentum inside V = net rate of inflow of momentum into V + total force on fluid inside V (where V is the volume of the fluid)

This can be written mathematically: To make use of this, we must:

(i) specify sigma in terms of fluid velocity, u, and pressure p.

(ii) manipulate the above equation into a more ‘solvable’ partial differential equation.

The equations we get from following i and ii are called the Navier-Stokes equations.

Specification of stress

This is the force on an element of surface due to the motion of the surrounding motion. This all depends on the location, the time, and the orientation. We have also been introduced to the stress tensor: Reflecting on the week of the course, it was very ‘wordy’ and lots of equations in. I plan to make sure I know where each of the equations are formed from since I didn’t quite understand how they were formed in the class. Hopefully it all clicks into place!

Sources:

http://principles.ou.edu/eq_seismo/stress_tensor.gif Stress tensor picture.

In our first lecture we had a short introduction to Inviscid Flow and outlined the topics we would be covering for the year

. Within this introduction we looked at a few different fluid flows. One of which, is where you have a straw and a ping pong ball (above). The process is hovering the ball above the straw then blowing through the straw. You would think looking at this, the ball would fly up a little and right but then fall on the floor, but this wasn’t the case. I was quite shocked to find that the ball had ‘bobbed’ up and down but stayed above the straw. We tried to replicate this ourselves, but unfortunately were unable to do so, since the process is fairly tricky! Although I did wonder why this happened. I did some research and found that this was due to Bernouilli’s principle. When we blow through the straw it creates high pressure below the ball therefore there is a slower airflow, but this also is the opposite on top, where as there is lower pressure, but faster airflow, therefore the ball floats. This effectively means where there is high pressure, it creates a slow airflow. But where there’s low pressure there is also faster air flow!

High pressure = Slow airflow; Low pressure = Fast airflow

Reflection:

Upon reflection on the first week, I now understand why a ping pong ball can float with a straw! On a more important note, it’s made me think more about other types of fluids effect us and other objects.

What to work on:

I need to start looking at other types of fluids and how they can be used.

How can we visual these types of flows.

Sources:

http://i.stack.imgur.com/SyXfE.jpg Picture of ping pong ball. One fluid of my life is the drink ‘Lucozade sport’. This drink has helped in every sport I have played giving me the energy I needed to compete at a high level. It doesn’t just supply the energy needed but also tastes great too!

The brand ‘Lucozade’ was first founded in 1927 by William Owen, who was attempting to make a drink to supply energy for the ill and sick people. In 1983 they had decided to try and take the brand’s association with illness and ever since have been improving the formula to supply people with energy.

‘Lucozade’ has very similar properties to water.

The speed of my fluid (at 70F) is roughly 1m/s-2.5m/s.

The viscosity of my fluid (at 70F) is 1 centipoise.

The density of my fluid would be slightly more then in pure water, roughly around 1,000-1,100 kg/m³.