The Method of Parameter Differentiation Applied to Fluid Mechanics Problems

Abstract

This work applies the method of parameter differentiation (KPD) to nonlinear ordinary and partial differential equations of fluid mechanics.
It is shown that the differentiation parameter does not need to be a physical parameter of the problem, because it can be arbitrarily selected and placed in any nonlinear term of the differential equation, with the constraint that it takes the value one at the end of the integration procedure.
Emphasis is placed in two numerical aspects:
a) A nonlinear ordinary differential equation with boundary conditions can be transformed into a simpler problem, which consists of linear ordinary differential equations with initial conditions. The solution is then noniterative.
b) The solution of the steady stream function-vorticity scheme through finite differences with the overall iterative procedure (Gupta, 1980, p. 170) can be simplified because inner iterations are eliminated, and initialization functions are solutions of the problem at each previous outer parameter iteration.