From the intellectual ferment around Neal Russell Amundson, his colleagues and his students in 1959-1974 grew a vision of computer-enabled functional analysis — a name that echoes W. W. Sawyer's 1978 lucid little book on “Numerical Functional Analysis” — serving the fluid mechanical theory then to be worked out for the viscous free-surface flows at the heart of important industrial processes. Those are the ubiquitous processes of depositing liquid layers to be solidified into permanent coatings on their substrates, or into strippable sheets to be used as films or membranes. The functional analysis is that of solving by Galerkin's powerful method, or relatives of it, the Navier-Stokes equations, or variants, in terms of the marvelous sets of nearly orthogonal local basis functions developed in the middle of the Twentieth Century by engineers and soon embraced by some mathematicians: finite element basis functions. Evaluating their coefficients rests on matrix mathematics espoused by The Chief, and doing so efficiently rests on modern computational mathematics that evolved with computers. Part of the vision was the promise of superseding Nineteenth Century coordinate systems, restricted domain shapes, inefficient domain-spanning basis functions, heroics to deal with free surfaces — including those in shape design, and retreat to approximation by finite difference equations.

The vision has evolved into a coherent assemblage of methods, and a standard operating procedure for using them. Not only are these comprehensive and notably successful; they may be to the field of coating processes as were Amundson et al.'s to reaction and separation processes. That is the subject of this presentation.