We assume that the answer is yes and that it would be of great importance in the theory of heat conduction and in experimental measurements of the thermal properties of different materials.
There's the Sherwood number (Sh=Kd/D) and the Schmidt number (Sc=μ/ρD). There is a table in the CRC Chem Physics Handbook that lists every dimensionless number. The table includes the parameters (like thermophysical properties) so that you can see every dimensionless number that contains a certain property. You can use this to cross-reference the dimensionless parameters. Reynolds Analogy (St=f/2) and the Chilton-Colburn Analogy (f/2=Sh/Re/Sc^(1/3)) are very useful. For example, you won't find a correlation for the convective heat transfer coefficient (Nusselt number) for the entrance region to a tube sheet or through a pipe fitting, but you can find the pressure drop, which means you can calculate the friction factor, from which you can calculate the Stanton number, and ultimately calculate the Nusselt number. I have mass transfer examples that you can get free online.
There are several animations of mass transfer solutions at this web page. The first is simple domains. At the bottom is a link to more complex domains. All the software is free. http://dudleybenton.altervista.org/projects/MassTransfer/MassTransfer1.html
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