An Eulerian-Lagrangean integral transform technique for solving advective-diffusive problems

In this paper, we present a Eulerian-Lagrangean analytical method for solving advective-diffusive problems. The basic idea is to transform the advective part of the equation into a total derivative, similar to what is done in the Method of Characteristics, where the resulting equation is valid along the characteristic lines. The spatial part of the equation is then converted to an eigenvalue expansion, using the Integral Transform Technique, and solved analytically in time. The correction of the results to obtain the potential field in other regions outside the characteristic lines is done implicitly in the eigenvalue space, resulting in a fully analytical expression. The solution can be converted back from the eigenvalue space to the physical space at any time the results are needed. Results for the thermally developing problem are presented, where the steady-state and transient solutions are compared with other analytical and numerical techniques, showing a good agreement.