When I work outside in near zero degree temps and mittens or gloves are too cumbersome to work with my fingers get so cold they ache. My toes may get cold and ache as well depending on how long I'm outside and if moister wicking socks are worn. My question is not on prevention. Common sense helps most of the time. I've found that applying dry heat can bring your extremities back to normal quite effectively; hair dryer or wood stove etc. But why if you put your toes or fingers under even Luke warm water to below "hot to the touch" water can bring on excruciating pain. Similar to hitting your fingers/thumb with a hammer. Just wondering why?
When your hands are cold then your nerves will adapt to the cold, so lukewarm water will often feel boiling hot if your hands are freezing (the old hand in hot water, hand in cold water then both hands in warm water trick shows this.)
As to why water hurts but a dry heat doesn't, I suspect this is predominantly because water is a much more effective conductor of heat than air, so water at the same temperature will warm your hands much more suddenly than warm air, not giving your brain time enough to adapt to the change. It's similar to the reason why it feels much colder putting your hand in ice cold water rather than surrounding it with air at the same temperature.
A secondary, though I suspect lesser reason is that you can easily move your hands closer or further to a hot air stream, so there's much more of a continuous temperature gradient where you can choose to put your hands (and again this helps with the gradual change.) With water, it's pretty much an instant change.