1Parallel viscous flow

IB Fluid Dynamics



1.0 Preliminaries
This section is called preliminaries, not definitions, because the “definitions” we
give will be a bit fuzzy. We will (hopefully) get better definitions later on.
We start with an absolutely clear an unambiguous definition of the subject
we are going to study.
Definition (Fluid). A fluid is a material that flows.
Example.
Air, water and oil are fluids. These are known as simple or Newtonian
fluids, because they are simple.
Paint, toothpaste and shampoo are complex or non-Newtonian fluids, because
they are complicated.
Sand, rice and foams are granular flows. These have some fluid-like properties,
but are fundamentally made of small granular solids.
In this course, we will restrict our attention to Newtonian fluids. Practically
speaking, these are fluids for which our equations work. The assumption we will
make when deriving our equations will be the following:
Definition
(Newtonian fluids and viscosity)
.
A Newtonian fluid is a fluid
with a linear relationship between stress and rate of strain. The constant of
proportionality is viscosity.
This, and other concepts we define here, will become more clear when we
start to write down our equations.
Definition (Stress). Stress is force per unit area.
For example, pressure is a stress.
Definition
(Strain)
.
Strain is the extension per unit length. The rate of strain
is
d
dt
(strain) is concerned with gradients of velocity.
As mentioned in the introduction, for fluid dynamics, we tend to study
velocities instead of displacements. So we will mostly work with rate of strain
instead of strain itself.
These quantities are in fact tensor fields, but we will not treat them as
such in this course. We will just consider “simplified” cases. For the full-blown
treatment with tensor fields, refer to the IID Fluid Dynamics.
In this course, we are going make a lot of simplifying assumptions, since
the world is too complicated. For example, most of the time, we will make
the inviscid approximation, where we set the viscosity to 0. We will also often
consider motions in one direction only.