Chapter 4
FLOWING FLUIDS AND
PRESSURE VARIATION
Source:
http://www.geofys.uu.se/files/teacher/
Fluid Mechanics
Rotation and Vorticity
Rotation of a fluid element in a rotating tank of fluid
(solid body rotation).
Rotation of fluid element in flow between moving and
stationary parallel plates
Rotation: the average rotation of two initially mutually perpendicular faces
of a fluid element.
 The angle between the bisect line and the horizontal axis is the rotation, θ.
o You can think of the “plus signs” as small paddle wheels that
are free to rotate about their center.
o If the paddle wheel rotates, the flow is rotational at that point.
As
And similarly
The net rate of rotation
of the bisector about
z-axis is
The rotation rate we just found was that about the z-axis;
hence, we may call it
and similarly
Applicable to ideal flow theory
The rate-of-rotation vector is
Irrotational flow requires
(i.e., for all 3 components)
The property more frequently used is the vorticity
, which is a vector equal to twice the rate of rotation vector
Vortices
A vortex is the motion of many fluid particles around a common
center. The streamlines are concentric circles.
Choose coordinates such that z is perpendicular to flow.
In polar coordinates, the vorticity is (see p. 104 for details)
(V is function of r, only)
Solid body rotation (forced vortex):
or
Vortex with irrotational flow (free vortex):
A paddle wheel does not rotate
in a free vortex!
In a cyclonic storm:
Forced vortex (interior) and
free vortex (outside):
Good approximation to
naturally occurring
vortices such as
tornadoes.
Euler’s equation for
any vortex:
We can find the pressure variation in different vortices
(let’s assume constant height z):
In general:
1) Solid body rotation:
2) Free vortex (irrotational):
Application to forced vortex (solid body rotation):
with
Pressure as function of
z and r
p = 0 gives free surface