I am starting this post by giving a brief description of the dynamic viscosity and kinematic velocity. I will then go on to talking about the Navier-Stokes equations and their derivation.
The dynamic viscosity, μ, is a property of the fluid:
-
It varies little with pressure, but can vary significantly with temperature
- It has dimensions mass/(length x time)
- In S.I. units, μ kg/ms
- These units correspond to pascal-seconds abbreviated to Pa.s
The kinematic velocity, v, is another property of the fluid:
- It is defined as v = μ/ρ
- It has dimensions (length^2)/time
- In S.I. units, v has the units (m^2)/s
The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. Navier-Stokes equations are very useful when modelling airflow around a plan wing, weather, ocean currents and many more. They are also used when designing such things as cars, and aircraft’s.
Brief derivation of the Navier-Stokes equations
Writing the momentum conservation equation using suffix notation gives:
This equation applies for each coordinate direction, i=1,2,3 and for each of these there is a summtion for j=1,2,3.
Since V is fixed,
Also, we know that,
Using this information, We can write the momentum conservation equation as,
Using the Divergence theorem, we can obtain the momentum conservation principle for a general fluid. This is given below,
Inserting the expression for σij and simplifying further gives the Navier-Stokes equations for an incompressible Newtonian fluid.
In vector notation, they may be written as:
Solving Navier-Stokes equations – Boundary Conditions
To solve the Navier-Stokes and continuity equations, we require boundary conditions. A good example of these conditions is the no-slip boundary condition, which requires that at a solid surface the fluid velocity and surface velocity are identical.
Examples of no-slip boundary conditions
- At a fixed surface, the no-slip boundary condition requires u=0
- At a surface which is moving with constant velocity U in the x-direction, then the no-slip boundary condition requires u=(U,0,0).
Initial conditions are also required when the problem has a time dependent flow.
What to work on:
This week I am going to try to work on memorizing some of the definitions given in lectures. I am also going to try and find some exercises online for the given material and work through them. This should help me when it comes to any exercises we are given in class or even in the exams.
Acting upon what to work on:
From last weeks blog I said I would read ahead on lecture notes and also look back at previous notes before the lectures. I did this and It helped an awful lot when I got to the lecture. All of the material seemed very familiar which was a great bonus as if I hadn’t I can imagine I would of struggled to keep up.
Links to resources
- A look at the Navier-Stokes equations and Newtonian fluids –http://www.efunda.com/formulae/fluids/navier_stokes.cfm
- Derivation of the Navier-Stokes equation – http://www.cfd-online.com/Wiki/Navier-Stokes_equations
Fluid Dynamics 