Graetz-Brinkman problem in laminar core-annular flow of two immiscible liquids

Thermally-developing flow with the inclusion of the viscous dissipation is known as the Graetz-Brinkman problem. Here, the problem of thermally-developing, hydrodynamically-developed laminar core-annular flow in a circular duct with a prescribed inlet temperature distribution and viscous dissipation is considered. Both fluids are assumed to be Newtonian, and the effects of interfacial waves, flow eccentricity, and axial heat conduction are neglected. An external convection boundary condition is considered, which recovers the constant wall temperature boundary condition as the Biot number approaches infinity. The problem is solved using a combined analytical and numerical solution, with a general inlet temperature distribution expanded by the method of quasi-orthogonal functions. Results are presented for the case of an oil-water flow, with oil occupying the core region of the flow. The effects of the different boundary conditions and the inlet temperature distribution of the fluid are discussed and compared to the well-known results for the single-fluid Graetz-Brinkman problem.