Chapter 4 FLOWING FLUIDS AND PRESSURE VARIATION Fluid Mechanics Source:

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 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. 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.

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 The rate-of-rotation vector is Irrotational 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

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