Fundamentals of Fluid Dynamics

Continuum Assumption : We assume that a fluid is a continuous medium; down to the smallest dimensions of interest to us. For example: Air is a fluid and we assume its continuous for fluid dynamic purposes though it is made of discrete molecules. Some important numbers when dealing with air, M.W. 28.8 g/mol, density, 1.2 kg/m^3, Avogadro number : 6.023e23. With these numbers we get a particle density of 2.5e7 molecules per micrometer.

Streamline : A streamline is the tangent to the local flow direction at every point in the fluid.

If the tangent is defined as

Then, the condition of tangency requires for U = u i + v j + w k

Which gives

dy/dx = v/u ; dz/dy = w/v ; dz/dx = w/u.

Basic fluid motion can be described as some combination of

1. Translation, U


This is the motion of the center of mass of the fluid packet

2. Dilatation, Divergence of U, volume change
3. Rotation, or vorticity ,
4. Shear strain,

Circulation :

The negative sign is included such that positive circulation on a body corresponds to positive lift, and the integral is evaluated counter-clockwise.

Important Points :

a) The circulation around a closed contour with rotation and/or shear will be non-zero. It is, however, always possible to have a combination of rotation and/or shear that gives a zero circulations. See the examples below.
 
 

where is the undisturbed freestream density, is freestream speed , is the ratio of specific heats and is the freestream Mach number.

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