From what I've read, tesla turbines are extremely lightweight and even more efficient (about 95% compared to 90% of turbopumps) when compared to turbopumps, and their bladeless design would also contribute to cost reduction. I would assume also that they would benefit from size reduction as the fluid being pump would have a higher viscosity where turbopumps start to experience inefficiencies. Have there been any proposals for a Tesla turbine-driven rocket engine, and if not, would it even be possible?
Do not have the maths skills to generate an example rocket engine turbine/pump system from first principles but the Wikipedia page indicates that efficient operation both as a pump and a turbine requires the plates to be close enough together to achieve laminar flow and that the fluid path make several circuits around the disks (to maintain laminar rather than turbulent flow) to achieve the claimed efficiency. So the fluids spend more time in the pump, and interact with more pump/turbine surface area than in a more conventional centrifugal system.
This would indicate that a system designed on the Tesla principles will by physically larger and heavier than a conventional turbine/pump system where achieving the same pressure/flow, which in a rocket engine quickly eats small working efficiency gains. .
The rocket equation means that a turbo pump system that burns 5% less fuel to function, but added weight is the same as the 5% saved fuel mass is a net loss, since the lower efficiency pump burnt the extra fuel mass and got it out of the rocket, while a higher efficiency/heavier system drags the extra pump mass all the way to burnout (has a worse mass ratio).
Also relevant is that in many systems the turbine exhaust is used to generate thrust, so goal is not extracting as close to 100% of the available potential energy as possible (as in a stationary steam generator) but instead at a low system weight efficiently extract a % of the energy from a suitably sized exhaust stream where the 'wasted' energy is usefully propelling the rocket.