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For the second part, using the definition of heat transfer coefficient $h$ and thermal conductivity: $$\dot{q} = h(T_{wall} - T_{mean}) \\ \dot{q} = k\left( -\frac{\partial T}{\partial y} \right)_{y=0}$$ we will have the relation $$\frac{h}{k}= \frac{\left( -\frac{\partial T}{\partial y} \right)_{y=0}}{T_{wall} - T_{mean}}=\mathrm{constant}$$ So if using the equation of $Pr = c_p \mu /k$, then when Pr increases, k decreases. But when k decreases, $\left( -\frac{\partial T}{\partial y} \right)_{y=0}$ increases. Is this what it supposed to be?
@ChesterMiller Hi, can you please guide me through the reasoning? For the first part, which we assuming temperature-independent viscosity, do we also have the same velocity profile for all of the Pr values, and we have the same momentum diffusivity for different Pr?
@Emil You did point me to the right direction! Thank you!!!It's the stagnation point of free convection boundary layer flow on a horizontal cylinder. url:core.ac.uk/download/pdf/159188580.pdf
It turns out I misunderstood the statement. At first I thought "number of states with the same set of $n_k$ " means "how many k will have the same number of $n_k$", now I realize it actually means the combinatorics of the states...
