An extended Graetz problem is analyzed, with a semi-infinite axial domain and the Robin boundary condition on the heat-transfer wall. The heat-transfer problem examines a viscous fluid entering a cylindrical capillary from a reservoir. The capillary fluid is exchanging heat with the surrounding environment of prescribed temperature and thus the Robin boundary condition is employed on the wall. Since axial heat conduction is included in the analysis, a generalized Danckwerts boundary condition is shown to be most appropriate for the tube entrance. The energy equation is decomposed into a system of first-order partial differential equations, as in [4, 8, 9], to obtain a selfadjoint formalism. The Gram-Schmidt orthonormalization process is finally used to obtain what is technically an analytical solution, which is computationally simple and efficient. Other entrance boundary conditions are also discussed and analyzed.

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