Scale analysis (mathematics)

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. Scale analysis ruled as follows:

Rule1- First step in scale analysis is to define the domain of extent in which we apply scale analysis. Any scale analysis of a flow region that is not uniquely defined is not valid.

Rule2- One equation constitutes an equivalence between the scales of two dominant terms appearing in the equation. For example,

In the above example, the left-hand side could be of equal order of magnitude as the right-hand side.

the order of magnitude of one term is greater than order of magnitude of the other term

then the order of magnitude of the sum is dictated by the dominant term

Rule4- In the sum of two terms, if two terms are same order of magnitude,

the order of magnitude of the product is equal to the product of the orders of magnitude of the two factors

Consider the steady laminar flow of a viscous fluid inside a circular tube. Let the fluid enter with a uniform velocity over the flow across section. As the fluid moves down the tube a boundary layer of low-velocity fluid forms and grows on the surface because the fluid immediately adjacent to the surface have zero velocity. A particular and simplifying feature of viscous flow inside cylindrical tubes is the fact that the boundary layer must meet itself at the tube centerline, and the velocity distribution then establishes a fixed pattern that is invariant. Hydrodynamic entrance length is that part of the tube in which the momentum boundary layer grows and the velocity distribution changes with length. The fixed velocity distribution in the fully developed region is called fully developed velocity profile. The steady-state continuity and conservation of momentum equations in two-dimensional are

this results in the well-known Hagen–Poiseuille solution for fully developed flow between parallel plates.

where y is measured away from the center of the channel. The velocity is to be parabolic and is proportional to the pressure per unit duct length in the direction of the flow.