1Parallel viscous flow
IB Fluid Dynamics
1.1 Stress
We first look at stresses. These are the forces that exist inside the fluid. If we
have a boundary, we can classify the stress according to the direction of the force
— whether it is normal to or parallel to the boundary. Note that the boundary
can either be an actual physical boundary, or an imaginary surface we cook up
in order to compute things.
Suppose we have a fluid with pressure
p
acting on a surface with unit normal
n, pointing into the fluid. This causes
n
τ
p
solid
fluid
pressure p
Definition (Normal stress). The normal stress is
τ
p
= −pn.
The normal stress is present everywhere, as long as we have a fluid (with
pressure). However, pressure by itself does not do anything, since pressure acts
in all directions, and the net effect cancels out. However, if we have a pressure
gradient, then gives an actual force and drives fluid flow. For example, suppose
we have a pipe, with the pump on the left:
high pressure low pressure p
atm
∇p
force −∇p
Then this gives a body force that drives the water from left to right.
We can also have stress in the horizontal direction. Suppose we have two
infinite plates with fluid in the middle. We keep the bottom plane at rest, and
move the top plate with velocity U .
U
h
By definition, the stress is the force per unit area. In this case, it is the horizontal
force we need to exert to keep the top plate moving.
Definition
(Tangential stress)
.
The tangential stress
τ
s
is the force (per unit
area) required to move the top plate at speed U.
This is also the force we need to exert on the bottom plate to keep it still.
Or the horizontal force at any point in fluid in order to maintain the velocity
gradient.
By definition of a Newtonian fluid, this stress is proportional to the velocity
gradient, i.e.
Law. For a Newtonian fluid, we have
τ
s
∝
U
h
.
Definition
(Dynamic viscosity)
.
The dynamic viscosity
µ
of the fluid is the
constant of proportionality in
τ
s
= µ
U
h
.
We can try to figure out the dimensions of these quantities:
[τ
s
] = ML
−1
T
−2
U
h
= T
−1
[µ] = ML
−1
T
−1
.
In SI units, µ has unit kg m
−1
s
−1
.
We have not yet said what the fluid in the middle does. It turns out this
is simple: at the bottom, the fluid is constant, and at the top, the fluid moves
with velocity U. In between, the speed varies linearly.
U
h
u
We will derive this formally later.
For a general flow, let
u
T
(
x
) be the velocity of the fluid at position
x
. Then
the velocity gradient is
∂µ
T
(x)
∂n
.
Hence the tangential stress is given by
τ
s
= µ
∂u
T
(x)
∂n
,
and is in the direction of the tangential component of velocity. Again, the normal
vector n points into the fluid.