2 different density liquids in deep space Let's say you are in deep space with basically zero gravity.  You have a ball of water floating in front of you.  If you carefully injected that ball with a syringe filled with vegetable oil, what would happen?  
Would the oil just sit in the middle, held by water's cohesiveness?  Or would it move to the outside to surround the water as it is less dense?  If it did move to the outside, would it even stay there?  
On Earth the oil would sit on top of the water, due to gravity?  In space, would it float away, or form a shell around the water, due to its own cohesiveness?
 A: The question whether two liquids mix or not is mainly to do with the intermolecular forces, not the densities. These forces lead to the bulk property known as chemical potential. 
Assuming the experiment is done in a pressurized cabin, then the ambient temperature and pressure may take ordinary values such as STP (room temperature, 1 atmosphere of pressure). 
At such ordinary temperature and pressure, what happens with oil and water is that the fluid consists of two parts, one of which is mostly (but not completely) water and the other is mostly (but not completely) oil. The entropy of system-plus-environment is maximised when the Gibbs function is minimised, with the result that the oil is not fully dissolved but rather the liquid consists of these two parts. The two parts will gather at separate locations so as to minimise any boundary areas where there is surface tension, and also in response to any ambient forces such as gravity.
In the absence of gravity, therefore, one expects a blob of one liquid and a blob of the other. There are three types of boundary: water-oil, water-air, oil-air, each with a different surface tension. I think the water-air boundary has the highest surface tension, which suggests the oil will surround the water so as to elliminate this boundary. Therefore if there is enough oil then I expect the equilibrium configuration is a sphere of water surrounded by a spherical shell of oil. At smaller amounts of oil I guess the oil will be spread out over the surface of the water, not quite covering it.
Perhaps the configuration with the least Gibbs potential is some other shape (depending on the proportions of oil and water), but the equilibrium configuration will certainly consist of two parts with different concentrations as I have said, unless the temperature is high. At high temperature the two fluids fully dissolve into one another and then you would have a single continuous substance, not two.
A: With no gravity and no impulse given to the blobs of fluid, my guess is that they would remain one inside the other and do nothing.  If present, gravity would play on the blobs of fluid by making the less dense fluid (the oil) rise relative to the denser fluid, due to buoyancy.  The oil would also "rise" if the spacecraft were rotating due to the greater inertia of the denser water.  If your oil blob was injected off centre, or the surrounding shell of water was thin such that the minimum distance between the oil-water and water-air interfaces is on the scale of the capillary length, something funky might happen.  My guess is that the water blob and oil blob might change places as the water (which has a higher surface tension than the oil) tries to minimise its surface energy by assuming the smallest size possible: a spherical blob of radius $r$ would have a smaller surface energy than a shell of inner and outer radii $r$ and $r+\delta r$, respectively.
A: The reason why oil and water do not mix on Earth doesn't really have to do with their densities.  Water is a polar molecule, which means it has distinct regions that are more positively or negatively charged.  Oil is non-polar, which means that it does not have differences in charge across the molecule.  As a general rule, “like dissolves like”; which means that a polar solvent will dissolve a polar solute, or a nonpolar solvent will dissolve a nonpolar solute.  Because water is polar but oil is not, they will not form a solution when mixed together.
So blame differences in polarity for the lack of mixing, but gravity does play a role in the way that oil stacks on top of water.  Water has a density of $1.0\  \mathrm {g/cm^3}$, where oil has a density of about $0.92 \  \mathrm {g/cm^3}$ depending on the oil.  As a result, the oil rests on top of the water on Earth.  However, even in the absence of gravity, water and oil would still be separated out into distinct liquids, but the way that these liquids arrange themselves would not be driven by gravity and density.
You suggest injecting the oil inside of the water.  If you do that extremely carefully and slowly you would probably be able to get the oil to form a  bubble inside the water.  However, if you inject too quickly, you would  disturb the water enough that it would become turbulent, and your oil and water would break apart into distinct “bubbles”, but would not layer or arrange themselves in any meaningful way.
A: Basically the oil would stay inside the water untill you over pressurized the injection of oil. Then It would become bubbles of water and bubbles of oil
A: https://www.nasa.gov/centers/marshall/news/background/facts/mfmg.html
This experiment for miscible fluids cites prior discovered information & the theories on immiscible fluids. Injected into the water it should first form a stream of droplets. There may be migration.
A: Less dense objects rise only if gravity is present. So the drops remain as such.    :) 
Any possible deviation would be due to gravitational forces on oil due to water and vise-versa. Since water has a strong cohesive force, much stronger than any possible gravitational pull (assuming an average sized drop), the drops should remain intact. 
A: This is something to do with surface tension let’s say you have a needle and then you put it on a glass of water it will float right. That’s because the water molecules are attracting molecules to them this is mainly due to cohesion. In zero gravity water and oil will mix not because of the densities. When drops are small gravity can’t overcome them so instead they mix.
