The thermostat restricts flow of the coolant, so that is why it works without. However, you have no heat, so you need the thermostat. Can you tell me if the car overheats all the time, only at highway speeds, or only when sitting? As far as the radiator, you may want to flush it out. And the heater core will not help with an overheating problem.
Monday, January 14th, 2013 AT 1:48 AM