In order to develop mathematical models of fluid motion we first need to develop mathematical
equations which govern that motion.
To this end we will first analyze a Lagrangian systems or fluid parcel, for an easy formulation
of the governing equations based on physical laws.
Then we will move from the Lagrangian framework to the Eulerian framework in order to have
equations that can be more easily handled to obtain specific solutions.
This is done using the Reynolds transport theorem that provides a bridge between these
two frameworks.
So let's get started!
Consider a fluid parcel that is a fixed quantity of fluid moving through space.
We will analyze the parcel properties as well as the external effects like forces, work
and heat transfer acting on it.
Note that for the rest of the course we will assume that the direction vector normal to
the surface is always pointing outward the fluid parcel.
Let's start with the first equation of fluid motion: the conservation of mass.
The conservation of mass, or continuity equation, states that the mass is neither created or
destroyed in the fluid parcel.
This means that the mass variation in time is equal to zero.
A direct application of the mass conservation can be found in a common garden water hose.
The mass of water entering the hose is equal to the mass of water exiting the other end.
The second law of fluid motion is the conservation of momentum that is based on the Newton second
law.
This law states that the rate of change of momentum is equal to the sum of forces acting
on the fluid.
The time derivative represents the time rate of change of the parcel’s momentum and the
right hand side terms are the surface forces acting on the parcel’s surface, like pressure
and fiction, and the body forces acting on the parcel’s volume, like the gravity effects.
The conservation of momentum can help us analyze for example the thrust required by a rocket
to propel it outside the atmosphere into space.
Next is the conservation of energy that is based on the first law of thermodynamics.
This equation tells us that the change in total energy is equal to the net heat transfer
minus the net rate of work done by the fluid.
The heat transfer occurs due to thermal conduction between adjacent fluid, while the work can
be exchanged through viscous effects or mechanical mixing.
The fluid energy can be used to generate work that produces electricity.
For example, hydropower plants use the water energy to generate electricity.
Another example are wind farms, that convert the wind energy into electricity using wind
turbines.
Now, how do we connect the Lagrangian and Eulerian forms of the equations?
In order to move from one form to the other we will apply the Reynolds transport theorem
that will act as a bridge between these two frameworks.
In the Lagrangian point of view we're following a moving parcel, while in the Eulerian framework
we're just looking at a fixed control volume.
To help understand how we can relate these two frameworks we can see that the moving
parcel at a certain instant of time will suddenly occupy the same space of the control volume.
The Reynolds transport theorem relates the Lagrangian and the Eulerian frameworks as
follows.
The Lagrangian time derivative of F, that is an extensive property associated with the
fluid parcel, is equal to a combination of terms in the Eulerian framework.
The two terms represent the time variation of the f property in the control volume and
the flux of this property across the control volume boundaries.
The next lessons will show how to move from the governing equations in the Lagrangian
form into the governing equations in the Eulerian form.
We will be able to do this using first the Reynolds transport theorem, then apply the
Gauss divergence theorem to convert the surface integrals into volume integrals, for conciseness
we will refer to this as the “Divergence Theorem” for the rest of the course.
Once we have the entire equations described into volume integrals, we can remove their
dependency on the control volume size and shape.
Since the integral is satisfied for an arbitrary volume, hence the integrand must be zero and
finally get the governing equation in the Eulerian form.