Dimensional analysis

From Chemepedia

The Navier-Stokes Equation

The Navier-Stokes Equation can be written for a Newtonian fluid in the following compact form:

            (1)

Under Cartesian Coordinates

Using Cartesian coordinates to identify the velocity vector field as containing the x-component of in order to non-dimensionalize the component vector. This same methodology can be utilized for solving any coordinate system chosen for the Navier-Stokes, albeit with differing nondimensionalized variables; this includes the other component vectors of the cartesian coordinate system, the cylindrical coordinate system as well as spherical coordinate system. For the sake of simplicity, only the x-component of the cartesian coordinate system will be discussed.

The following dimensionless variables will be used to substitute values in the equation above on the right hand side. The capital length and velocity variables denote a specific characteristic length and and average velocity, with capital variable of time denoting a reference time:

Velocity (x, y, z): , ,

Length (x, y, z): , ,

Pressure:

Time:

Substituting the variables above and simplifying the resulting equation, the following is obtained:

The following dimensionless variables will be used to substitute values in the equation above on the left hand side.

Velocity differentials (x, y, z): ; ;

Pressure differential (x):

Gravity (x):

Substituting the variables above and simplifying the resulting equation, the following is obtained:

Dividing by a common factor of :

Taking the now nondimensionalized Navier-Stokes equation and substituting known dimensionless numbers, the final equation will be obtained:

Where the Stanton (St), Reynolds (Re) and Froude (Fr) numbers replace variable values to finalize the resultant equation.