Simple Methods For Classification And Construction Of Similarity Solutions Of Partial Differential Equations

Nonlinear Partial Differential Equations in Engineering

A first order approximation is derived in similarity coordinates for the velocity variation across a mixing zone between two streams of different velocities in an arbitrary pressure gradient. The velocity profiles obtained are functions of a single parameter, the ratio of the velocity of the slow stream to that of the fast stream. The first order velocity profiles are compared to those obtained from the complete solution to the zero pressure gradient case by Gortler and good agreement is found when the velocity ratio is greater than 0.5. The form of the eddy diffusivity appropriate to the first order theory is determined from analysis and experimental data. This form leads to an explicit transformation for the similarity coordinate requiring only a single empirical constant for all cases. This similarity coordinate is shown to reduce to Gortler's variable in the constant pressure case. Expressions for the shear and the mixing layer width for the general case are presented, and shown to check the data from several cases. A hot wire anemometer which was specially designed and constructed for taking the experimental data in a water channel is described. (Author).

The aim of this book is to provide a systematic and practical account of methods of integration of ordinary and partial differential equations based on invariance under continuous (Lie) groups of trans formations. The goal of these methods is the expression of a solution in terms of quadrature in the case of ordinary differential equations of first order and a reduction in order for higher order equations. For partial differential equations at least a reduction in the number of independent variables is sought and in favorable cases a reduction to ordinary differential equations with special solutions or quadrature. In the last century, approximately one hundred years ago, Sophus Lie tried to construct a general integration theory, in the above sense, for ordinary differential equations. Following Abel's approach for algebraic equations he studied the invariance of ordinary differential equations under transformations. In particular, Lie introduced the study of continuous groups of transformations of ordinary differential equations, based on the infinitesimal properties of the group. In a sense the theory was completely successful. It was shown how for a first-order differential equation the knowledge of a group leads immediately to quadrature, and for a higher order equation (or system) to a reduction in order. In another sense this theory is somewhat disappointing in that for a first-order differ ential equation essentially no systematic way can be given for finding the groups or showing that they do not exist for a first-order differential equation.