incompressibility | math3402liamsnow

Hello again everybody, and welcome back to another addition in my blog! 😀 Before we start, I just need to clarify a few things:

1) No, I wasn’t on vacation, as delightful as that would have been for me! I’m afraid that the workload of uni caught up with me, and as such, I didn’t put as much time into my blogs as I should’ve done. Don’t worry though, you’ll hopefully be seeing a lot more of my blog posts in the next few days, so stay tuned!

2) As I mentioned in my last post, the rest of the content that is to be featured in my blog will be all about inviscid flow, as opposed to the viscous flow that we have been looking at previously.

With that out of the way, let us begin!

Flow Visualisation

It makes sense for us to be able to describe the properties of fluids as they flow, since we can see this natural phenomena around us all the time. The wind swaying the branches in the trees, and the raging torrents of a river, are all naturally occurring examples of fluid flows. The ability for us to mathematically deduce the properties of these types of flow, and how we can visualise them, can have a significant impact on our understanding of complex problems, such as modelling the weather.

Two such ways that we can represent the flows of fluids is through pathlines and streamlines (technically, there is another way through streaklines, but we won’t go into more depth with them here.)

Pathlines

We can think of pathlines as representing the trajectory of particles through the flow. For a 3-dimensional fluid flow, we can define the pathlines in the form of the ODE:

$\dfrac{d\vec{x}}{dt} = \vec{u} = (u,v,w)$

In this equation, the position vector $\vec{x}$ is defined as $(x(t),y(t),z(t))$ , while the time derivative of the position vector is simply the velocity vector $\vec{u}$ .

Pathlines are similarly defined in polar coordinates, but with the position vector defined as $(r(t),\theta(t),z(t))$ and the velocity vector as $(u_{r},u_{\theta},u_{z})$ .

Streamlines

Streamlines can be thought of as integral curves of the velocity field, which crucially means that a streamline is simply everywhere that is parallel to the velocity. From a physical perspective, a streamline is a snapshot of the flow of particles within the fluid at a fixed time, $t$ .

From some basic manipulation of the definition of the pathlines above, we can define the streamlines as the following:

$\dfrac{dx}{u} = \dfrac{dy}{v} = \dfrac{dz}{w}$

which can be rewritten as

$\dfrac{dy}{dx} = \dfrac{v}{u}, \dfrac{dz}{dx} = \dfrac{w}{u}, \dfrac{dz}{dy} = \dfrac{w}{v}$

Like the pathlines, streamlines can also be written in the form of polar coordinates. Once again, the time aspects of the equation are eliminated, to reveal the definitions that we want, i.e. (for 2-dimensions):

$\dfrac{1}{r}\dfrac{dr}{d\theta} = \dfrac{u_{r}}{u_{\theta}} \text{ or } r\dfrac{d\theta}{dr} = \dfrac{u_{\theta}}{u_{r}}$

Convected Derivative

In my introduction to viscous flow, I defined what it meant for a fluid to be incompressible, which relied upon the definition of a new form of derivative; the convected derivative. Using the definition of the convected derivative, we can now consider the acceleration of the fluid flow, since we can translate it as a convected time derivative of the velocity vector, like so:

$\dfrac{D\vec{u}}{Dt} = \dfrac{\partial\vec{u}}{\partial t} + \vec{u}.\nabla\vec{u}$

Alternatively, the convected acceleration can also be written as follows:

$\dfrac{D\vec{u}}{Dt} = \dfrac{\partial\vec{u}}{\partial t}$ $+ \dfrac{1}{2} \nabla (|\vec{u}|^2) - \vec{u} \text{ x curl(}\vec{u}\text{)}$

As another extension from viscous flow, it can be quite easily observed that the left hand side of the Navier-Stokes equations is simply the density of the fluid ( $\rho$ ), multiplied by our time derivative and convected acceleration. The same can also be said in polar coordinate form.

Example Sheet Questions

Below you will find my scanned answers to questions 1 and 2 of Example Sheet 0 and question 1 of Example Sheet 1.

Example Sheet 0 – Part 1

Example Sheet 0 – Part 2

Example Sheet 0 – Part 3

Example Sheet 1 – Part 1

Example Sheet 1 – Part 2

Reflection

The general principles highlighted during this particular week’s worth of lectures were fairly intuitive to understand, although I do need to make sure that I understand the physical interpretations of the flow visualisations, such as the pathlines and streamlines. I feel that the equations for these visualisations are straight-forward, so it should be a case of memorisation when it comes to revision.

In terms of the exercises, all of the processes were fairly repetitive, and as such, there weren’t many hidden surprises. In fact, the only aspect of the questions that I struggled with, was the interpretation of the first part of question 1, of Example Sheet 0. I was not confident with the graph, and I only convinced myself to put down what I did for it after experimentation with the equation in Maple.

Ideally, I would have liked to have covered more of the questions to further my understanding of the topic. However, with the current levels of workload, I will instead use these additional questions as part of my revision for the exam at the end of the year. I am glad though that I was able to significantly shorten my blog post (particularly after my last post) without leaving out the crucial details, which will be extremely useful when it comes to revision.

As I also mentioned at the start of this post, I do apologise sincerely for not uploading this sooner, but the workload has been getting to me recently. If I have to take away anything from this term so far, it is that I have to make significant improvements to my overall time management, and not get too focused on a few activities! Oh, and that I really need to start on work as soon as I receive it, if it can be helped! 😛

Thank you to anybody who took the time to read this post! I hope you’ll be seeing more of this sooner rather than later! 🙂

References and Sources:

1) The vast majority of the concepts talked about in this blog post are derived from notes taken from Chapter 1 of the Inviscid Flow PowerPoints on flow visualisation, continuity and stream functions, and my own notes derived from the lectures.

Hello everybody, and welcome back to another instalment on my blog about fluid dynamics! 🙂

Clarifications (about change to planned upload, and differences in flow)

To those of you who are familiar with the plan of action for this module, you may note that there is an inconsistency with the subject of the title. Following on from my previous blog post, I should now be talking to you about the foundations of inviscid flow, but now I am writing to you about viscous flow.

The reason for this change is that, unfortunately, our first term lecturer David (who I also mentioned in my last blog post) has fallen ill, so our second term lecturer Jason has decided to take over for the time being. This means we will be looking at the foundations for the properties of viscous flow and why this is different to inviscid flow.

In fact, before I continue, I feel I should clarify perhaps the most obvious difference between a viscous and an inviscid flow. Also, in my first blog post, I mentioned viscosity, but I did not really give a definition of it. Thankfully, defining it now makes sense when comparing the two main types of flow.

The viscosity of a fluid is simply a means of measuring a fluid’s resistance to gradual deformation, which in the lecture notes given, we use the stress caused by shearing as a primary example (colloquially and in everyday life, we often refer to the viscosity of a fluid as its ‘thickness’).

Therefore, a viscous fluid is a fluid where viscosity has to be taken into consideration in order to understand it. An inviscid fluid, on the other hand, is often referred to as an ‘idealised fluid’, simply because we are making the assumption that the fluid has no viscosity. In reality however, there are no truly inviscid fluids since every fluid has some sort of viscosity, but there are still applications for when this assumption is useful.

Now that we are clear on our definitions for both viscous and inviscid flows, let us move onto some of the founding principles that make up our theory on viscous flow.

The Continuum Hypothesis

The Continuum Hypothesis is an assumption that allows us to regard the fluid as if it were continuous, no matter how small a portion of the fluid we are looking at. In layman’s terms, this means we can completely ignore the molecular, atomic (and sub-atomic) structure of the fluid in question.

In reality, we know that there are technically spaces (such as the spaces between individual fluid molecules, or on a more extreme level, the spaces between sub-atomic particles, such as electrons), and there could well be nothing there at any given point we are defining. This is problematic if we want to define physical properties, such as density, pressure, velocity and temperature. Therefore, by taking this assumption into account, we can now define the physical properties in terms of space, x, and time, t, since we are making sure that the fluid is continuously defined, regardless of where we are in the fluid.

Of course, by taking this assumption (and the others that follow) into question, we are only able to define an approximation, as opposed to an exact result, of a viscous flow system. This is a repeating occurrence within modern mathematics, and particularly in areas where we have to solve a system of equations.

This is because very few real world applications can be solved in a ‘simple’ manner (i.e. analytically) and therefore we need to use a combination of numerical methods and assumptions in order to produce reasonable solutions. However, these calculations should be fairly accurate to the exact solution, or what is observed naturally, since the assumptions and numerical methods are defined rigorously from appropriate theory.

Incompressibility

By assuming that our fluids are incompressible, we can ignore any and all velocity induced changes in the density, which we will write as $\rho(\vec{x},t)$ . In other words, the density of the fluid remains unchanged when any given volume in the fluid moves under any velocity.

In a similar manner to how, for the Continuum Hypothesis, our assumption does not obviously match up with that is observed in reality, the same occurs for incompressibility. This is because all fluids are somewhat compressible, however without our assumption, it would just make our solution unneccessarily more complicated without providing a significant benefit to our solution.

The incompressibility assumption is written in this form mathematically

$\dfrac{D\rho}{Dt} = 0$

where $\frac{D}{Dt}$ denotes the convected derivative, which represents the rate of change following the motion of a fluid particle.

This can be defined as

$\dfrac{D\rho}{Dt}\equiv\dfrac{\partial\rho}{\partial t} + \vec{u}.\nabla\rho = 0$

which can then be expanded in Cartesian coordinates as follows

$\dfrac{D\rho}{Dt}\equiv\dfrac{\partial\rho}{\partial t}$ $+ u\dfrac{\partial\rho}{\partial x}$ $+ v\dfrac{\partial\rho}{\partial y}$ $+ w\dfrac{\partial\rho}{\partial z} = 0$

where $\vec{x} = (x,y,z)$ is the position vector in space, $\vec{u} = (u,v,w)$ is the velocity vector and $\nabla$ or ‘nabla’ is the gradient operator derived from vector calculus.

Mass Conservation and the Continuity Equation

Mass conservation is a simple, and yet powerful, assumption: mass can neither be destroyed nor created. In the context of fluid dynamics, it means that our fluid can not appear out of nowhere, and it can not simply vanish.

In order to obtain what is known as the continuity equation for an incompressible fluid, we must first state that ‘the rate of change of mass inside an arbitrary, but fixed in space, volume = the net rate of inflow of mass into the arbitrary volume’.

If we were to state the above mathematically, and then apply Gauss’ Divergence Theorem to it (which is also derived from vector calculus), we can deduce the following continuity equation for a compressible fluid

$\dfrac{\partial\rho}{\partial t} + \text{div}(\rho\vec{u}) = 0$

where the divergence operator $\text{div}$ upon $\rho\vec{u}$ is defined as

$\text{div}(\rho\vec{u}) = \nabla . \rho\vec{u}$

By then applying the product rule for the divergence operator, this continuity equation can then be written in terms of the convected derivative

$\dfrac{D\rho}{Dt} + \rho\text{div}(\vec{u}) = 0$

Since we know for an incompressible fluid that the convected derivative must equal zero, and that the density of the fluid can never be equal to zero (since if a fluid had no density, then it would have no mass or volume), then we can further simplify the above to deduce the continuity equation for an incompressible fluid

$\text{div}(\vec{u}) = 0$

Just for completeness, the continuity equation can then be simply written in Cartesian coordinates as

$\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} + \dfrac{\partial w}{\partial z} = 0$

or in suffix notation, where the double suffix notation convention applies, as

$\dfrac{\partial u_i}{\partial x_i} = 0$

Momentum conservation

The concept of momentum conservation is fundamentally the same as mass conservation, but with momentum being the deciding factor to consider. In fact, the equation for momentum conservation is just an expression of Newton’s Second Law defined for a continuous medium (which is what we have assumed our fluids to be, due to the Continuum Hypothesis).

Therefore, we know that momentum, like mass, can not be created nor destroyed. In a more mathematical sense, we can state that ‘the rate of change of momentum inside an arbitrary volume = the net rate of inflow of momentum into the volume + the total force on the fluid inside the volume’. As before, an equation can be deduced from this statement, but I do warn you though, the equation is most certainly not a pretty one.

In order for us to make full use of the momentum conservation equation, we will first need to do some more physical reasoning in regards to the specification of stress. Secondly, we will then need to manipulate said equation, in order to get our ultimate goal of being able to derive what are called the Navier-Stokes equations, which fundamentally enables us to solve systems where a viscous flow is being considered.

Since this particular blog post is already quite long, I will deduce these elements when I start work on next week’s material, from the second chapter of the lecture notes.

Reflection

I feel that although I’ve grasped these founding principles well, and that they are intuitive in nature, I still feel that I struggle with the initialisation of the conservation assumptions in a mathematical sense, as outlined in the lecture notes. One assumption’s derivation that is confusing at first glance, is the derivation of the equation for momentum conservation, simply due to the number of terms and what it all means for the system as a whole. I will go over these notes again at a later point, to make sure that I fully understand the arguments being put forward.

Annoyingly, I can’t really comment on whether or not I understand the inviscid and viscous flows right now. I feel that this is a question that I may only be able to answer once I have gained enough experience in both particular fields, and that may not be until I am sufficiently into my second term of this module, or perhaps until near its completion.

In terms of my upload schedule, I’ve made significant progress in being able to upload this just the day after my previous post! 🙂 However, I’ve also taken a step backwards in that this blog post is significantly longer than my previous posts. As such, I am aware that I need to be more careful about the depth that I go into, for future reference. Maybe one day I’ll get the combination of brevity and punctuality right!

Hopefully this *not quite so* brief introduction to the concepts surrounding viscous flow was both useful and informative for you. Don’t worry, we’ve barely reached the tip of the iceberg here when it comes to the mathematics front! Just wait until you see that momentum conservation equation!

If you got this far, well, you’re a trooper. Thanks for reading, and I’ll see you in my next blog post! 😀

References and Sources:

1) Most of the concepts talked about in this blog post are derived from notes taken from Chapter 1 of the MATH3402 Fluid Dynamics – Viscous Flow lecture notes.

2) Another thank you goes to Darren Mooney again, this time for his helpful guide on writing $\LaTeX$ into WordPress, and for providing me with the necessary inspiration needed to write blog posts! You might find $\LaTeX$ useful too for your mathematical needs, so why not give it a try? Here’s the link: http://math3402darrenmooney.wordpress.com/2013/10/25/latex-in-wordpress-a-simple-guide/

math3402liamsnow

Nothing but a torrent of fluid dynamics.

Tag Archives: incompressibility

Fluid Dynamics – Week 4: Flow Visualisation, Streamlines and Pathlines, and the Convective Derivative

Fluid Dynamics – Week 2: Introduction to Viscous Flow