I have a hot wire anemometer calibrated for air. I want to measure velocity of Argon gas. To get an approximate value, can I somehow use any mathematical relation such as ratio of heat conductivity of gases, etc.
You can basically get 2 equations (for 2 gases) with the resistance of the wire as the common term for both the gases and deduce velocities for a different gas. I cannot surely say that it will work, but it can give you a direction for sure. If you validate it let us know here!!
That's something I've tried before, by treating all the non-gas related terms (characteristic length, etc.) in the equations as constants and the gas-related terms as variables. Sadly, in the end the value of the non-gas related terms was still important, as they functioned as weights for the gas-related variables. Deriving a relation between both gases' velocity from the equations (without knowing the value of all variables) was not possible.
Therefore, we've calibrated the anemometer experimentally. The relation between measured air velocity and argon velocity seems to be linear (and temperature dependent). At T=24.15, Vargon = Vair_measured/0.4456. At T=29.45, Vargon = Vair_measured/0.473. Assuming a linear relation between temperature and the ratio Vargon/Vair enables you to calculate the argon flow velocity at a given temperature. Note that this calibration was conducted for a PCE-423 hot-wire anemometer, so it might not be valid in your application.