Applications Of Pade' Approximation Theory In Fluid Dynamics

Although Pade' presented his fundamental paper at the end of the last century, the studies on Pade' approximants only became significant in the second part of this century. Pade' procedure is related to the theory of continued fractions, and some convergence theorems can be expressed only in terms of continued fractions, but Pade' approximants have some advantages of practical applicability with respect to the continued-fraction theory. Moreover, as Chisholm notes, a given power series determines a set of approximants which are usually unique, whereas there are many ways of writing an associated continued fraction. The principal advantage of Pade' approximants with respect to the generating Taylor series is that they provide an extension beyond the interval of covergence of the series. Pade' approximants can be applied in many parts of fluid-dynamics, both in steady and in nonsteady flows, in incompressible and compressible regimes. This book is divided into four parts. The first one deals with the properties of the Pade' approximants that are useful for the applications and illustrates, with the aid of diagrams and tables, effectiveness of this technique in the field of applied mathematics. The second part recalls the basic equations of fluid-dynamics (Navier-Stokes, Euler, and Prandtl) and gives a quick derivation of them from the general balance equation. The third shows eight examples of application of the Pade' approximants to steady flows, also taking into account the influence of the coupling of heat conduction in the body along which a fluid flows with the conduction and convection in the fluid itself. The fourth part considers two examples of application of the Pade' approximants to unsteady flows.