Navier-Stokes equations, boundary conditions and simple shear flow

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:

equation 1

Also since sigma(i)=sigma(ij)n(j) then the momentum conservation equation can be written as:

stokes equation

But if we simplify this further and set the surface integrals equal to volume integrals  and combining the terms, we get:

equation 2

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:

equation 3

These are the Navier-Stokes equations for an incompressible, constant viscosity fluid. This can be written in vector notation:

verctor 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.

Introdcution to viscous flow

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.


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:


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: Stress tensor picture.

Introduction: Bernoulli Effect

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



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: Picture of ping pong ball.


A fluid flow in my life

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.