Chapter 4

FLOWING FLUIDS AND PRESSURE VARIATIONFluid MechanicsSource:

http://www.geofys.uu.se/files/teacher/

Rotation and VorticityRotation of a fluid element in a rotating tank of fluid(solid body rotation).

Rotation of fluid element in flow between moving andstationary parallel platesYou can think of the plus signs as small paddle wheels that are free to rotate about their center.

If the paddle wheel rotates, the flow is rotational at that point.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, .

AsAnd 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 itand similarlyThe rate-of-rotation vector isIrrotational flow requires (i.e., for all 3 components)Applicable to ideal flow theory

The property more frequently used is the vorticity, which is a vector equal to twice the rate of rotation vector

VorticesA 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) andfree vortex (outside):

Good approximation to naturally occurring vortices such astornadoes.Eulers equation for any vortex:

We can find the pressure variation in different vortices(lets assume constant height z):

In general:

Solid body rotation:

Free vortex (irrotational):

Application to forced vortex (solid body rotation): with Pressure as function ofz and rp = 0 gives free surface