A non-iterative transformation method for boundary-layer with power-law viscosity for non-Newtonian fluids

In this paper, we have defined and applied a non-iterative transformation method to an extended Blasius problem describing a 2D laminar boundary-layer with power-law viscosity for non-Newtonian fluids. Let us notice that by using our method we are able to solve the boundary value problem defined on a semi-infinite interval, for each chosen value of the parameter involved, by solving a related initial value problem once and then rescaling the obtained numerical solution. This is, of course, much more convenient than using an iterative method because it reduces greatly the computational cost of the solution. Furthermore, by using Richardson’s extrapolation we define a posteriori error estimator and show how to deal with the accuracy question. For a particular value of the parameter involved, our problem reduces to the celebrated Blasius problem and in this particular case, our method reduces to the Töpfer non-iterative algorithm. In this case, we are able to compare favourably the obtained numerical result for the so-called missing initial condition with those available in the literature. Moreover, we have listed the computed values of the missing initial condition for a large range of the parameter involved, and for illustrative purposes, we have plotted, for two values of the related parameter, the numerical solution computed rescaling the computed solution. Finally, we have indicated the limitations of the proposed method as it seems not be suitable, for values of n>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>1$$\end{document}, to compute the values of the independent variable where the second derivative of the solution becomes zero or goes to infinity.