liquid-statephase-transitionsurface-tensiontemperaturethermodynamics
Critical temperature is the highest temperature at which a gas can be turned into liquid using pressure.
Again critical temperature is the temperature at which the surface tension of a liquid reaches zero.
Is it that thermodynamics and fluid dynamics have different concepts for critical temperature or there is any relation between them?
Your description of critical temperature isn't quite right.
If you increase the temperature of a liquid beyond the critical point, the atoms are moving so quickly that persistent structure fails to form and so you have something that behaves a lot like a very dense gas.
Similarly, if you increase the pressure of a gas beyond the critical point, it becomes very dense so that it's like a liquid but without persistent structure.
In other words, it's not so much that the liquid phase does not form, but rather the liquid and gas phases become indistinguishable (rather intuitively) and you end up with what's called a supercritical fluid.
Here's the phase diagram of CO$_2$ for clarity:
If the temperature is below critical temp and the pressure is also below its critical value then there is a possibility that we see an equilibrium between the liquid and gaseous states.
When one of them is above their critical value it is impossible to change the state of the substance from gas to liquid by merely changing the other parameter. For instance if $T > T_c$, then no matter how much you try to increase the pressure, the gas will not liquefy. Similarly if $P > P_c$, you can't have a gas-liquid equilibrium in any temperature. It's clear from the given picture.
Best Answer
Find a "closed" container, and pull a vacuum on it in order to remove substantially all of the air from the container. Then take a liquid (assume water for this discussion) and fill the container to the half-way mark. Obviously, the vapor space above the liquid contains water vapor, and the amount of water vapor depends on the temperature. Now, slowly heat the container. As the temperature increases, the water expands and the liquid density decreases. In addition, the vapor pressure of the water increases, so the vapor density increases. As the temperature continues to increase, the liquid density continues to go down while the vapor density continues to go up. At the critical temperature, the liquid and vapor densities become equal, and only one phase exists at and above the critical temperature. Naturally, this means that there is no longer a surface tension because there is no longer a surface.
Regarding turning gas into liquid, the compression work done on a gas will cause it to heat up. This means that in addition to compressing a gas, you also need to cool it off after compressing it, in order to turn some of it into liquid. Naturally, above the critical temperature, there will be no liquid regardless of the pressure, and below the critical temperature, if the pressure is higher than the vapor pressure of the liquid, there will be no gas.