December 13, 2013 by liamsnow2013

Fluid Dynamics – Week 7: Euler’s Equations

Welcome everybody, to yet another blog post on fluid dynamics! 😀 This blog post will try and follow a slightly different format to my previous posts. Hopefully this one will be more succinct than past attempts, which should mean less sore eyes for you! 😛 Without further ado, let us begin.

Euler’s Equations, and their derivation

As you will see soon, Euler’s equations are very closely linked to the Navier-Stokes equations, which I mentioned in a previous blog post. However, we must still be aware that these equations are related to inviscid flow (as has most of the previous content I’ve covered), instead of the viscous flow that the Navier-Stokes equations cover.

Let me now describe the scenario of the derivation (but don’t worry, I’ll include a picture below from my lecture notes for clarity).

Visual overview of the derivation of Euler’s Equations, from the lecture notes.

Once again, we are examining but a small fraction of a moving fluid, which is composed of: a small volume $V$ , that has a small surface $\delta S$ (with corresponding normal vector $\vec{n})$ , all the while moving with a velocity $\vec{u}$ .

From this, we can deduce the other following properties:

(Convected) Acceleration = $\dfrac{D\vec{u}}{Dt}$
Mass = $\rho V$
Body force = $\vec{F}$
Pressure (due to force of moving fluid) = $-p\vec{n} \delta S$

By substituting these appropriately into an integral, we get:

$\dfrac{D\vec{u}}{Dt}\int_V \rho dV = \int_V \rho\vec{F} dV - \int_S p\vec{n} dS$

where the first integral equates to a total mass multiplied by the acceleration (a force), which is equivalent to the total body force minus the total surface force.

For convenience, we can now apply Gauss’ theorem to the integral over the surface, to convert it into an integral over the volume (like the other integrals), and thus produce:

$\dfrac{D\vec{u}}{Dt}\int_V \rho dV = \int_V (\rho\vec{F} - \nabla p) dV$

As stated above, we have already assumed that the volume is small, so if we now take $V \to 0$ , we can conclude that:

$\dfrac{D\vec{u}}{Dt} = \dfrac{-1}{\rho}\nabla p + \vec{F}$

The above is known as Euler’s equation (in its whole form), and is also obtained when we allow the viscosity to equal zero in the Navier-Stokes equations.

To be more explicit, here is the Cartesian version of Euler’s equation in summation form:

$\rho\left(\dfrac{\partial u_i}{\partial t} + u_j \dfrac{\partial u_i}{\partial x_j}\right) = \dfrac{\partial\rho}{\partial x_i} + \rho(F_B)_i$

A polar form is also shown similarly, but I will not state it here.

Example Sheet Questions

Below you will find my scanned answers to question 1 of Example Sheet 3.

Example Sheet 3 – Part 1

Reflection

In my honest opinion, I think that the derivation of Euler’s equations was not too bad (as I had mentioned, the process to deduce them is essentially a simplified Navier-Stokes method), although I still feel a bit apprehensive about certain details of it. In particular, I don’t think I quite understand the logic in deducing the force due to the pressure. Perhaps this is some anxiety I still hold from my experiences with similar properties in Vector Calculus last year? Either way, when it comes to revision, I will make sure that I fully understand the derivation.

Even though I’ve only attempted the first question of Example Sheet 3 here, I feel a bit more confident about using Euler’s equation now. The only problem I have with my uploaded solution is that, upon checking, it seems I have a factor of $\rho$ in front of the equation, whereas the uploaded solutions do not. If I have made a mistake, then perhaps I will clarify this with one of my peers, or through David directly.

On the whole, I am especially pleased with how this new ‘format’ went, as it seems to be much more succinct, and cuts to the chase. Of course, I still felt inclined to show a derivation, but that felt necessary in order to explain where the equations came from.

Thanks to anyone who has taken the time to read this blog post! See you soon! 🙂

References and Sources:

1) All of the concepts talked about in this blog post are derived from notes taken from Chapter 2 of the Inviscid Flow PowerPoints on Euler’s equation, and my own notes derived from the lectures.

October 31, 2013 by liamsnow2013

Fluid Dynamics – Week 3: Stress, Pressure and the Navier-Stokes equations

Hello everybody, and welcome back for the fourth instalment on my blog about fluid dynamics! 🙂 Following on from my previous blog post introducing viscous flow, we will continue our look at the properties of viscous fluids (yes, including that momentum conservation equation!), and we will finally derive the famous Navier-Stokes equations, which we will use extensively in the second term of the module for solving a variety of viscous flow systems.

Oh, and please don’t be misled by the title of this particular blog post. It most certainly isn’t going to be a gushing recount of my hardships at university, since I try to avoid drama whenever possible! 😉 With that out of the way, let the “fun” begin!

Definition of the Momentum Conservation Equation

As I mentioned previously, the deduced momentum conservation equation is vital in the derivation of the Navier-Stokes equations. As such, I feel that it is important that I introduce it now, especially since that I did not produce the result last time. Without further ado, here it is:

$\dfrac{d}{dt}\int_V \rho\vec{u} dV =$ $-\int_S \rho\vec{u}(\vec{u}.\vec{n}) dS$ $+\int_V \rho\vec{F}_B dV$ $+\int_S \vec{\sigma} dS$

I did warn you last time that it wasn’t pretty, so I apologise now if you have nightmarish visions of this engraved in your conscience. To make it somewhat less painful to read, let me try and explain what some of the individual terms mean, especially in regards to our expression of the principle in layman’s terms.

The first term (the component of the equation before the equals sign) relates to ‘the rate of change of momentum inside the volume’. Since momentum is defined as being the mass times the velocity, then thus the rate of change of momentum will have units matching those of mass times by acceleration (the same units as a force). From checking this first term, we can verify the units are correct.

As for the second term, the integrand represents ‘the net rate of inflow of momentum into the volume’. The term is also negative because we have to consider that the normal is acting outwards of the surface of the volume, when we are considering what is occurring inside the volume.

The final two components of the equation represent ‘the total force acting on the fluid inside the volume’, the first of which refers to $\vec{F}_B$ , which is the body force (per unit mass) acting on the fluid. For most applications we will look at, the body force is normally gravity. The latter term of the pair refers to stress (the force per unit area) acting upon the surface of the volume (represented by $\vec{\sigma}$ ), and it arises due to the motion of the surrounding fluid.

Thus that summarises the equation for momentum conservation. See, perhaps it’s not so bad after all? Well OK, maybe it is a bit. Regardless, the final term in this equation neatly leads on to the next aspect of the derivation of the Navier-Stokes equations, and that is by taking a closer look at what we mean by the stress.

Specifying stress and the stress tensor

As specified above, the stress given off by the fluid is the force that the fluid is exerting upon a particular surface. We can define the stress as a continous function with components of x, the position in space, t, the time, and n, which is the normal vector to the surface element in question, i.e.

$\vec{\sigma} = \vec{\sigma}(\vec{n},\vec{x},t)$

The stress is similarly defined for an inviscid fluid, where $\vec{\sigma}$ has a non-zero component of n only (in other words, $\vec{\sigma}$ and n are parallel), and therefore

$\vec{\sigma}(\vec{n},\vec{x},t) = -p(\vec{x},t)\vec{n}$

where the scalar quantity $p(\vec{x},t)$ is the mechanical pressure of the fluid. This pressure is considered to be a positive quantity when acting inwards on the surface of the fluid, hence why there is a minus sign present in the equation.

An important observation to take away from the definition of the stress of an inviscid fluid above is that, for a viscous fluid, $\vec{\sigma}$ will not in general, be defined in the same direction as the normal vector to the surface, n. This is simply because the force acting upon on the surface is unlikely to be perpendicular to that particular element.

Due to this observation that $\vec{\sigma}$ and n will generally be in different directions, we have to consider a concept known as the stress tensor to further clarify this point.

The stress tensor is initially defined as

$\sigma_{ij}(\vec{x},t) = \sigma_i(\vec{N}_j,\vec{x},t)$

where $\vec{N}_j$ is defined as being the unit vectors in the directions of the coordinate axes. An explanation of the above is that $\sigma_{ij}$ is the i-th component of the stress vector $\vec{\sigma}$ at x and t, on an element whose normal is in the j direction. As there are three axes, both i and j are summed from one to three respectively.

This initial derivation is found by considering properties of a tetrahedron (a triangular-based pyramid, if you will), but I will not cover it here. Instead, I will consider the main points of this derivation only, and how important factors that are deduced from it, will further our understanding of what we mean by stress.

From defining the stress tensor as we have, we now need to consider two directions to help us define the overall stress imposed by the fluid:

1) The direction in which the stress acts, and

2) The direction of the normal of the surface element on which the stress is acting.

From here, when taking geometrical considerations into account, we were able to determine that having knowledge of the stress tensor was sufficient enough to write down the stress vector on a surface of arbitrary orientation (the full derivation is in the notes for those interested.)

Also of note is that the stress tensor is often displayed as

$\sigma_{ij} =$ $\begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33}\end{pmatrix}$

although it should be made clear that the stress tensor is not a matrix!

Using the same derivation, we can equally determine the stress tensor for an inviscid fluid to be

$\sigma_{ij} = -p\delta_{ij}$

where $\delta_{ij}$ , as you all know by now, is the Kronecker delta, where

$\delta_{ij} = \begin{cases} 0 & \text{if i} \neq \text{j} \\ 1 & \text{if i = j}\end{cases}$

For the sake of completeness, we can then express the stress tensor of the inviscid fluid like so

$\sigma_{ij} =$ $\begin{pmatrix} -p & 0 & 0 \\ 0 & -p & 0 \\ 0 & 0 & -p\end{pmatrix}$

Just quickly, it should also be noted that the conservation of angular momentum does come into play, as it shows that the stress tensor is symmetric, i.e.

$\sigma_{ij} = \sigma_{ji}$

however, we are mostly just interested in this as a result. As such, no rigorous derivation was given for this result.

The Constitutive Equation for an Incompressible Newtonian Fluid

The individual components of the stress tensor depend on what is known as the rate of deformation of the fluid. In fact, what distinguishes one fluid from another is the precise form of said dependence. As this particular subtitle states, we are looking at Newtonian fluids, which satisfy what is known as Newton’s Law of Viscosity. For an incompressible fluid, this law is represented mathematically as

$\sigma_{ij} = -p\delta_{ij} + \mu\left(\dfrac{\partial u_i}{\partial x_j} + \dfrac{\partial u_j}{\partial x_i}\right)$

where $\mu$ is known as the dynamic viscosity and is constant, provided that the temperature and pressure are fixed, for any given Newtonian fluid. You should also clearly see that the inviscid form for the stress tensor is found if you allow for $\mu = 0$ .

No attempt was given within the notes to justify the above form for the stress tensor, but a simple analysis was used to hint towards the argument from which the above equation is derived.

To round off this particular subsection of the blog, I will include details of two important properties of any fluid, that is the aforementioned dynamic viscosity, and the kinematic viscosity.

A change in the pressure will do little to vary dynamic viscosity, but changing the temperature can have quite a dramatic effect on its value. As I mentioned way back in my first post, viscosity (I should have specified dynamic viscosity at the time) has S.I. units of kilogram per metre-seconds ( $\text{kg m}^{-1}\text{s}^{-1}$ ), which is equivalent to Pascal-seconds (Pa.s). This translates, as you can see, to dimensions of $\dfrac{\text{mass}}{\text{length} \times \text{time}}$ .

The kinematic viscosity is related to the dynamic viscosity by this equation

$\nu = \dfrac{\mu}{\rho}$

Since the density, $\rho$ , is defined as $\dfrac{\text{mass}}{\text{volume}}$ , then it follows simply that it must have dimensions of $\dfrac{\text{mass}}{\text{length}^{3}}$ . From this, it is pretty easy to deduce that the kinematic viscosity, $\nu$ , has dimensions of $\dfrac{\text{length}^{2}}{\text{time}}$ . In S.I. units, this is equivalent to metres-squared per second ( $\text{m}^{2}s^{-1}$ ).

The Navier-Stokes Equations

At last, we’ve finally got here! It took us long enough, didn’t it? We’ve now defined just about everything we need in order to deduce the equations we need to finally solve those pesky viscous flow systems! 😀

Annoyingly though, the derivation is still quite long, so I’ll try and summarise the main points below:

1) Do you remember that momentum conservation equation? Yep, the one at the start of this blog. Sorry to bring that up again, but we need it! Thankfully we’ve already defined that, so I won’t post it again here, but just keep it in mind.

2) Since the volume of our fluid is fixed (i.e. it does not vary with time), and only the density and velocity vector do change with respect to time, then we can switch the order of the first term of the momentum conservation equation, so that the derivative with respect to time is brought inside the integrand. We also know that the stress vector can be represented as a product of the stress tensor and the normal vector to the surface element, so the final term is replaced with this product.

3) Now using Gauss’ Divergence Theorem, and some clever manipulation, we can change all of our surface integrals into integrals over the volume. Now every single integral is being integrated over the volume, we can now combine all these integrands into one large integral, and move them all onto the left hand side of the equation. Thus the right hand side of the equation now equals zero. Since we are considering our volume to be arbitrary, then it must mean that our integrand must be identically equal to zero.

4) Now by re-arranging our equation so that the stress tensor and the body force terms are on the right hand side, we have now successfully expressed the momentum conservation principle for a general fluid! Congratulations! Well OK, we aren’t quite done yet.

5) By inserting the definition of the stress tensor in terms of Newton’s Law of Viscosity, we will eventually arrive at the Navier-Stokes equations for an incompressible Newtonian fluid. However, there is some algebraic manipulation involved for this. Just for convenience though, I’ll state the results below.

Here are the Navier-Stokes equations for an incompressible, constant viscosity fluid using the summation notation (this form of the equation is particularly useful for re-defining it in terms of Cartesian co-ordinates.)

$\rho\left(\dfrac{\partial u_i}{\partial t} + u_j\dfrac{\partial u_i}{\partial x_j}\right) =$ $-\dfrac{\partial\rho}{\partial x_i} + \mu\dfrac{\partial^2 u_i}{\partial (x_j)^2} + \rho(F_B)_i$

The above can be expressed in a more condensed form in terms of vectors

$\rho\dfrac{D\vec{u}}{Dt} = -\nabla p + \mu\nabla^2\vec{u} + \rho\vec{F}_B$

That’s it! Hallelujah! 🙂 With the above defined equations, and the continuity equation that I defined in my last post, we can now solve systems of viscous fluids (subject to appropriate boundary and initial conditions of course).

Just a final note though on boundary and initial conditions. It is strongly advised to consider applying the no-slip boundary condition, which requires that at a solid surface, the fluid velocity should be equal to the surface velocity. If the flow is unsteady (i.e. the flow is time-dependent), then initial conditions are also required.

Reflection

Although it took a long time to get there, it was fairly clear why we were doing what we were doing; to get to the Navier-Stokes equations, which we will use extensively throughout the second term of the module. However, there are a lot of components to consider for its derivation, with lots of subtleties, especially in relation to the stress vector and tensor, and the momentum conservation equation. I feel more comfortable with the aspects of the derivation that were covered in last week’s lectures.

Everyone knows that the best way to practice mathematics is by doing it, so I’ll have to wait until next term when we cover this material again, before I can say with any degree of certainty as to whether I understand it or not. Although it should be stated that, while we need to have a grasp of the derivation, the vast majority of the second term will be to do with applying the appropriate Navier-Stokes equations and continuity equation to multiple scenarios.

While from a revision point of view, I’m glad that I’ve covered a significant amount of material in this post, I feel that I’ve ultimately sacrificed brevity for additional detail and clarification. Just for future reference to any readers, I won’t be writing to this level of depth in my future blog posts. The time investment was far more significant that was expected of me, and simply put, a blog post of such length can get a little tiring for most people. At least I’m uploading on a more regular schedule now, so progress has been made there! 🙂

Now that we are moving back onto the inviscid flow part of the module, maybe, just maybe, my blog posts might be slightly shorter? Well, we can always hope! 😉 Thanks for reading all of this! If you didn’t…well…I hope it was good enough to send you into a peaceful night’s sleep. See you next time!

References and Sources:

1) The vast majority of the concepts talked about in this blog post are derived from notes taken from Chapter 1 and 2 of the MATH3402 Fluid Dynamics – Viscous Flow lecture notes.

2) I have found and used this particularly useful book on $\LaTeX$ , in order to help me write appropriate matrices and piecewise functions as part of this blog post. Perhaps you’ll find something useful too? Here’s the link: http://en.wikibooks.org/wiki/LaTeX/Mathematics

October 28, 2013 by liamsnow2013

Fluid Dynamics – Week 2: Introduction to Viscous Flow

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: viscous flow

Fluid Dynamics – Week 7: Euler’s Equations

Fluid Dynamics – Week 3: Stress, Pressure and the Navier-Stokes equations

Fluid Dynamics – Week 2: Introduction to Viscous Flow