Question

Drag force F on a high speed air craft depends on the velocity of flight V, the
characteristic geometrical dimension of the air craft L, the density ρ, viscosity
μ and isentropic bulk modulus of elasticity Es of ambient air. Using
Buckingham’s π theorem, find out the independent dimensionless quantities
which describe the phenomenon of drag on the aircraft.

Answer 1

There would be dimensionless constant of proportionality if the area is not a plane surface normal to the direction of travel.

Given:
Air craft L
The density ρ
Viscosityμ
Bulk modulus of elasticity Es
To find: Have to describe phenomenon of drag on the aircraft.
Solution:
The product of density by area is:

Mass/volume • area = mass/length.

In MKS units this is kg/mA

By equating the equalities we can get our equation through,

$F = kg m/s^2 = A • kg/m,$

The unknown A units are

$A = kg m/s^2 / kg/m \\= m^2/s^2$

So we see that the force dependence is on the square of velocity.

In this there is another dimensionless constant of proportionality if the area is not a plane surface normal to the direction of travel. This is the coefficient of drag, and its importance is what gives rise to streamlined shapes.

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