A numerical solution of high accuracy is obtained for the large Reynolds number, thin film flow over a horizontal flat plate, cylinders and spheres resulting from a vertical jet of liquid falling on the surface. A coordinate transformation is used which simultaneously maps the film thickness onto the unit interval and removes the singularity at the leading edge. The resulting equations are parabolic and these are solved using the Keller box method which is modified to accommodate the outer, free boundary. Using an extrapolation technique, results of sixth order accuracy are obtained with an estimated accuracy of six significant figures. For the flat plate, solutions for both 2-dimensional and axisymmetric jets are obtained and, for cylinders and spheres, different Froude numbers and volume flows are considered. The method can be used to solve any parabolic problem with a free boundary. © 1989.
Hunt, R. (1989). The numerical solution of parabolic free boundary problems arising from thin film flows. Journal of Computational Physics, 84(2), 377–402. https://doi.org/10.1016/0021-9991(89)90239-8