What's the most efficient strategy to shake a bottle to create a global flow? Consider a bottle of orange juice with pieces of the fruit gathered at the bottom of the bottle. You need to shake the bottle to evenly distribute the pieces in the bottle.
The first part of the process seems straight forward, just giving the bottle a few sharp shakes to create turbulence and destroy the conglomerate of fruity pieces.
However, what's the most energy-efficient strategy to create a global flow to distribute the stuff? 
Since at the start of the process, the interesting parts are gathered at the bottom of the bottle, a strong axial component would be neccesary, for energy reasons a laminar flow would be interesting (assuming the dispersal has allready happened). 
Edit to add: Efficient means the least energy input from me. If your mixing method involves throwing the bottle off a cliff, the energy for cyrrying it up there counts.
Where I live, juice comes in round bottles or in boxy containers.
 A: I have two methods for you. For simplicity I'm calling the stuff at the top the liquid, and the stuff at the bottom the pulp.

  
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*Pick the bottle up, and turn it by 180 degrees clockwise then anti-clockwise in the vertical plane repeatedly(About 4-5 times) and slowly. With the first turn the pulp at the bottom will gain a downward velocity, and when you turn it the other way once again, the liquid will gain a downward velocity, in effect mixing the two(Liquid and pulp).
  

Note: The motion should be slow enough to allow gravity to act on the pulp. And the container should be partially empty.
I think this is more efficient than shaking the bottle vigorously, because first of all you don't require to accelerate the bottle to a very high speed and then decelerate it repeatedly, like you do in shaking it vigorously. But this might not be the most energy-efficient, and you might not get a uniform mixture at the end.

  
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*If it is not important to keep the juice in the bottle after mixing it, then you can create some sort of tube mechanism in which you pour the juice and it comes out uniformly mixed.
  You can take a container with a hole at the bottom which can be shut. Pour the juice into the container with the hole shut, and let the pulp settle. Connect the hole with a tube containing something which slows the pulp down but can easily allow the liquid to pass through.
  Then when you open the hole, the pulp will go through the tube, being slowed down by the tube, and the liquid, being able to easily pass through, will collide and effectively mix with the pulp. You can then collect this in a glass.
  For the tube mechanism, you can use multiple thin sieves, like those you use to separate mud and gravel.
  

This method is very energy efficient as it just requires you to pour the liquid into the mechanism. Gravity will do the rest. But there are technical problems such as maintenance and cleaning of the tube. It might happen that the pulp gets stuck altogether inside the tube, and you only get the liquid out. So you'll have to experiment with sieve sizes.
Note: The two methods I described do not include laminar, but turbulent flow. Personally I think laminar flow(Even nearly laminar) will take a lot of time and precision.
A: Simple two step process:


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*Lay it on its side

*Roll it
Obviously this would work better with a cylindrical bottle, as opposed to the more common square bottle.
A: Ok..
I think the most eficient way cannot deviate much from 'shaking the bottle' as, as you said, you need an axial component and anything else seems to much energy consuming.
The most energy-efficent way in my opinion would be a vertical circular motion, where you dont use energy to stop the motion, but let it stop itself.
Otherwise i would go for turning it around like udiboy said.
