The extended Graetz problem for dipolar fluids

Thermally developing laminar flow of a dipolar fluid in a duct (pipe or channel) including axial conduction (Graetz problem extended) is investigated. The solutions are based on a self-adjoint formalism resulting from a decomposition of the convective diffusion equation for laminar flow into a pair of first-order partial differential equations. This approach, which is based on the solution method of Paputsakis et al. for a laminar pipe flow of a Newtonian fluid, is not plagued by any uncertainties arising from expansions in terms of eigenfunctions belonging to a non-self-adjoint operator. Then the eigenvalue problem is solved by means of the method of the weighted residual. Following this, the effect of the dipolar constant on the Nusselt number and temperature field are discussed in detail. Finally, it is shown that the Newtonian solution is a special case of the present result.