CHAPTER 8

ADDITIONAL SUBJECTS INFUNDMENTALS OF FLOW

Dr . Ercan Kahya

Vorticity for the z axis:

When vorticity is zero, irrotational flow

Case (a): rotational flow

Case (b): irrotational flow Although liquid makes a rotary movement, its microelements

always face the same direction without performing rotation.

In natural vortices such as hurricanes, tornados, eddying water currents has abasic structure with a forced vortex at the center and free vortex at the periphery.

Figure shows howthe wood chips float

Given closed curve s, the integrated vs’ along this same curve is called circulation Γ . Taking counterclockwise rotation positive,

Circulation dΓ of elemantary rectangle ABCD (area dA)

& the circulation is equal to the product of vorticity by area. Integration gives

Stokes theorem: surface integral of vorticity equal to the circulation

If there is no vorticity inside a closed curve, then circulation around it is zero.

Flow of viscous fluid

Mass flow rate at inlet & outlet sections in x- & y-directions

Fluid mass stored in the fluid element per unit time (x-dir.) :

Fluid mass stored in the fluid element per unit time (y-dir.) :

Mass change in unit time (right hand side) :

or

Continuity EquationUnsteady flow & compressible fluid& for real and ideal fluid

For steady flow and incompressible fluid:

For steady flow and incompressible fluid for axially symetric flow using cylindrical coordinates:

For axially symetric flow using cylindrical coordinates:

The strict solutions obtained for N-S equations to date are only for some special flows. Two such examples are:

Flow of a viscous fluid between two parallel plates

A flow in a long circular pipe is a parallel flow of axial symmetry. In this case, it is convenient to use the Navier-Stokes equation using cylindrical coordinates

Under the same conditions as in the previous section, N-S equation simplifies to