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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