Equations for Ideal Fluids
Contents
Equations for Ideal Fluids¶
We will derive the equations of fluid dynamics somewhat more systematically later in the course. Our goal here is to write down some basic equations and make them plausible. Recommended texts: [Chorin et al., 1990, Falkovich, 2018, Salmon, 1998].
1. Variables¶
A. Background.¶
We are going to imagine fluids in spac which is
some domain of
B. Lagrangian description¶
In the continuum hypothesis we can consider a fluid as a collecion of
infiniesimal parcels. Each parcel will have a trajectory
We then assign each fluid particle a density
C. Eulerian description¶
In practice we often cannot or do not follow fluid parcels; rather we take
measurements at points in space and time as determined by our apparatus:
oceanographic moorings, measurements taken from ships or airplanes, and so on.
The description of fluid quantities
Trajectories and streamlines. In this language, we have two
interesting sets of integral curves which
are discussed often. The first is the trajectory
has a unique solution which describes a line in
The second is the streamline. This is a representation of the velocity field
at fixed time. Here we introduce a parameter
These are the integral curves for te velocity at a given time. If we placed a large collection of little grains which moved with the fluid, a time lapse photo of their motion over an infinitesimal time would show streamlines.
In general, if
2. Equations¶
where
A. Continuity equation¶
Consider a fluid inside a fixed volume
Because the volume is fixed,
On the other hand, the total rate of mass flux into the volume is the
density times the velocity flux through the boundary
surface
where we have used the divergence theorem to rewrite the surface integral.
These definitions of
The focus of this course (not exclusive!) will be on incompressible fluids. We can define these as fluids for which the density of a parcel does not change along a trajectory. In this case,
Note that here we have defined he convective derivative
Combining this with the continuity equation gives us
Note that Acheson defines an ideal fluid as having constant density. This plus the continuity equation implies that the velocity is dovergence-free. In other references incompressibility, and not constant density, is part of the crierion for an ideal fluid. We should note that there are important examples of incompressible fluids which have variable density.
B. Euler’s equation¶
The next step is to write dynamical equations for the motion of the fluid. We start with Newton’s laws for a parcel. If the momentum is
then combining mass conservation with the chain rule, we find:
This is set equal to the force on the parcel.
To complete the equations, we need an expression for the force on a fluid parcel. The force will get contributions from sources external to the fluid, such as gravity, and from the force due to neighboring parcels.
If we take the external force per unit mass on the particle to
be
An ideal fluid, in addition to being incompressible, is one in which the force exerted on a fluid parcel by neighboring parcels takes a specific form. In particular, there is no shear stress exerted by one parcel on another: the force is always perpendicular to the surface of the parcel, and takes the form
where
where
C. Boundary conditions¶
To solve the above equations we will have to impose boundary conditions. Since the equations of motion are first order in space, we can guess that we only need a single boundary condition. This is not meant to be completely obvious!
A physical boundary condition is
where