Liquid seperation in space We know if we take immiscible liquids of different densities then they form different layers on earth. 
But when we take them to outer space, they form tiny droplets which are energetically most favourable for them. This is because there is no gravity, so no preferred direction for the liquids to settle.
But if we take miscible liquids of different densities, then at least on earth they form different levels (salt water, fresh water). Although there is a mixing between the two layers, that is a fairly slow process.

My question is what would happen if we take a container containing
  unmixed (in two seperate layers) miscible liquids of different densities and switch gravity off.  I think there would be mixing in this case too. But would it be
  faster or slower?

I think if the liquids are already mixed then there won't be any considerable change in their configuration. 
It would be better if the answers do not have a very high mathematical rigour. 
 A: The slow mixing you are referring to, is called diffusion. On the microscopic level, the molecules move in random direction, this is called Brownian motion. Without gravity, this microscopic motion is still present (unless you go to the absolute zero temperature).
With gravity (suppose you have the heavier layer on the bottom), this is an unmixing force, which slows down the mixing process (or increases separation). So without gravity, this diffusion is the only process playing a roll, and will thus be faster. (If you put the heavier phase on top with gravity, you will get all kinds of interesting instability phenomena).
As an useful analogy you could consider temperature diffusion in solid materials. This is obviously not affected by gravity. Putting two layers of different temperature together, will eventually 'mix' the temperature, such that it is uniform. 
A: Assuming the fluids completely fill the closed container and the temperature is not near zero, nothing noticeable would happen. Each molecule in the fluids experience three basic types of forces. One is simply the force due to the net electrostatic potential between each molecule. For example Van der Waals. Another is simple collisions. The third is gravitational interactions (earth and other molecules). Gravity is by far the smallest of these forces. The reason is the mass is so small and the velocities are so high. 
