If apply resistor of 2K over a battery of 6v, will I double the time that the battery can give a 3v flow of current?
3 V is not a current, but I think you want to know if the 6 V battery will last twice as long as the 3 V battery.
Batteries are usually rated in mAh. Then the 6 V battery will have a current of 6 V / 2 kΩ = 3 mA, while the 3 V battery will have a 1.5 mA current. Then if both batteries have the same mAh rating the 3 V battery will last twice as long.
This may go against intuition, as the energy of the 6 V battery is twice that of the 3 V battery: at 1000 mAh that's 6 Wh versus 3 Wh. But the power in the resistor is proportional to voltage squared, so for the 6 V that's four times the power at 3 V: 18 mW versus 4.5 mW. So the energy is double, but the power is times four, therefore x 2 / 4 = / 2.