$1.(30$ minutes$)$ Hot$-$wire anemometry is a measurement technique that is used for measuring the velocity of fluids. The working principle of this technique can be roughly explained as follows: an electrically$-$heated wire cools down under the effect of fluid flow $($typically air$).$ The energy balance between the heating and cooling of the wire can be considered as a function of fluid velocity because the dominant heat transfer mechanism is convection due to fluid flow. Hence, it can be stated that convective heat transfer coefficient is $\backslash ($ h$=$f$(\backslash$rho$,$ V$,$ D$,\backslash$mu$,$ k$)\backslash )$ where $\backslash (\backslash$rho $\backslash )$ is the fluid density, $\backslash ($ V $\backslash )$ is the fluid velocity, $\backslash ($ D $\backslash )$ is the diameter of the wire, $\backslash (\backslash$mu $\backslash )$ is the dynamic viscosity of fluid and $\backslash ($ k $\backslash )$ is the thermal conductivity of fluid. Typical SI units for $\backslash ($ h $\backslash )$ and $\backslash ($ k $\backslash )$ are $\backslash (\backslash$left$[$W $/$ m$^\{2\}\backslash$mathrm$\{$~K$\}\backslash$right$]\backslash )$ and $\backslash ([$W $/\backslash$mathrm$\{$mK$\}]\backslash ),$ respectively. Using the Buckingham$-$PI theorem, determine the non$-$dimensional parameters relevant to this problem.

Hint: In addition to three fundamental dimensions that are generally used in mechanics, there is an additional dimension you should take into account in this question.