The best solution is probably one that already exists, like the jet bottle washers that attach to your laundry sink.
But if you want an engineering solution, I have seen a design for a home-made keg/carboy washer using a pond pump and plastic bucket/lid that could be adapted for your use:
1. Add an approximately 12-18" length of PVC pipe to the outflow of the pond pump. The PVC pipe should be thin enough to fit into the mouth of a bottle. This might require using a larger-size PVC pipe to fit to the pump, and then fitting the thinner PVC pipe into the larger one.
2. Partially fill the plastic bucket with with PBW, Oxy, or other cleaning solution.
3. Fit an upturned bottle over the pipe.
4. Hold the lid down on the bucket to keep the bottle from shooting up. You will need to play with the length of the thinner PVC pipe so that the bottom of the upturned bottle touches or almost touches the lid of the bucket, or design some other sort of way to hold the bottle in place (maybe a jig of some sort).
5. Turn on the pump.