Apparently there are several companies in a great variety of fields (medical, biological, engineering, etc.) that need "consulting on mathematical modeling and PDEs" from applied mathematicians.

I'd like to gain some insight in this kind of industrial career prospectives.

- What kind of projects do these types of consultancy usually involve?

On J.D. Cook's website there's a brief sketch of some possibilities.

However I wonder:

Do they actually need only help with computational problems? Or do they sometime require a more theoretical insight into the qualitative behavior of solutions to certain PDEs?

Do they need help in finding a suitable PDE to describe a certain phenomenon?