Rare exact solution to a model of two-phase flow consisting of a nanofluid adjacent to a clear fluid

We consider a theoretical model of a two-phase flow consisting of a nanofluid adjacent to a clear fluid. A method of obtaining the exact solution to the two-fluid vertical channel flow and convective heat transfer model is presented. The results constitute a rare case in which completely exact solutions are possible for a nonlinear flow problem involving nanofluids, as the governing equations are often highly nonlinear. We show rigorously that the nanofluid can modify the fluid velocity at the interface of the two fluids, and can be used to reduce shear at both the surface of the clear fluid and the interface of the two fluids. Upon exploring the existence of the exact solutions, we discover that in some situations there exist two mathematical solution branches, one of which is the physically relevant solution. We then discuss the behavior of the velocity and thermal profiles with the important parameters dictated by the nanoparticles.