Graetz problem

Remarks

The problem is solved by separation of variables. With $\eta = r/R$ and $\theta = (T - T_w)/(T_0 - T_w)$, the solution is $$\theta = \sum_{n \ge 0} c_n \, \varphi_n(\eta) \exp\!\Big(-\lambda_n^2 \frac{x}{R \, \mathrm{Re} \Pr}\Big),$$ where $\mathrm{Re} = 2RU/\nu$, $\Pr = \nu/\alpha$, and $\varphi_n$, $\lambda_n$ are the eigenfunctions and eigenvalues of the Sturm–Liouville problem $(\eta \varphi')' + \lambda^2 \eta (1 - \eta^2) \varphi = 0$, $\varphi'(0) = 0$, $\varphi(1) = 0$. The coefficients $c_n$ come from the orthogonality of the $\varphi_n$ with the weight $\eta(1 - \eta^2)$. Far downstream only the first term survives, and the Nusselt number tends to about $3.66$. Near the entrance the series converges slowly, and the Lévêque approximation is used instead. The dimensionless group $\mathrm{Gz} = \mathrm{Re} \Pr D / x$, with $D = 2R$, is called the Graetz number. Wilhelm Nusselt later treated the same problem independently.