Scale analysis (mathematics)

Consider for example the momentum equation of the Navier–Stokes equations in the vertical coordinate direction of the atmosphere where R is Earth radius, Ω is frequency of rotation of the Earth, g is gravitational acceleration, φ is latitude, ρ is density of air and ν is kinematic viscosity of air (we can neglect turbulence in free atmosphere).

Other physical properties are approximately: Estimates of the different terms in equation (A1) can be made using their scales: Now we can introduce these scales and their values into equation (A1): We can see that all terms — except the first and second on the right-hand side — are negligibly small.

Thus we can simplify the vertical momentum equation to the hydrostatic equilibrium equation: Scale analysis is very useful and widely used tool for solving problems in the area of heat transfer and fluid mechanics, pressure-driven wall jet, separating flows behind backward-facing steps, jet diffusion flames, study of linear and non-linear dynamics.

Scale analysis is an effective shortcut for obtaining approximate solutions to equations often too complicated to solve exactly.

The object of scale analysis is to use the basic principles of convective heat transfer to produce order-of-magnitude estimates for the quantities of interest.

Scale analysis anticipates within a factor of order one when done properly, the expensive results produced by exact analyses.

Consider the steady laminar flow of a viscous fluid inside a circular tube.

Solving equation (7) subject to the boundary condition this results in the well-known Hagen–Poiseuille solution for fully developed flow between parallel plates.

The velocity is to be parabolic and is proportional to the pressure per unit duct length in the direction of the flow.