Combustion in liquid state Normally, we think of combustion as occurring only in the gas phase. I think I understand the main reason for this - in the gas phase it's very hard for a bimolecular reaction to produce a single product, which makes it more likely for radical chain reactions to propagate rather than being terminated - but I'm curious about whether it's possible for combustion to occur in the liquid state at all.

For example: suppose I mixed liquid $\ce{CH4}$ with liquid $\ce{O2}$ under high pressure and reasonably low temperature. Would they react immediately? If not, could I 'ignite' the mixture in the same way as I could in the gas phase? (Note that the combustion of $\ce{CH4}$ into $\ce{H2O}$ and $\ce{CO2}$ doesn't change the mole number, so the volume change should be relatively small.) Would the reactions that occur be similar to the reactions that occur during combustion of $\ce{CH4}$ in the gas phase, or would they be completely different?

If this reaction wouldn't resemble combustion, is there some other example of a self-propagating oxidation reaction in liquid or aqueous phase with similar properties to combustion?
Your question is addressed by Explosive Systems Containing Liquid Oxygen: Liquid Oxygen-Liquid Methane Mixtures. J. Chem. Eng. Data, 1959, vol. 4, pp. 127–131.

If liquid methane and liquid oxgygen at 90K are mixed, they are miscible in all proportions.  Mixtures over a wide range of proportions detonated with use of a blasting cap. 100 grams of mixture punched a 10cm hole in a 3/16 inch steel plate. 

For comparision, the TNT control only dented the metal plate!

Further tests on the stoichimetric liquid methane/oxgygen mixture found that it was detonated by a .22 caliber copper bullet at 40 foot range, flame and spark.

So nothing happens if the liquids are simply mixed, but a powerful explosion occurs if a stimulus is applied.  

Liquid oxygen and liquid methane have also been used in rocket engines.  See Liquid Methane/Liquid Oxygen Injectors for Potential Future Mars Ascent Engines for example.