Fully developed flow of a higher-gradient nanofluid in a vertical channel: Mixed and natural convection

In the present work, we study the steady Poiseuille flow and heat transfer of a viscous fluid containing nano- sized particles in a vertical channel. The two walls of the infinitely long channel are kept at different constant temperatures. Particles and fluid may have different densities, and account is taken of the thermal expansivity of the fluid by invoking the Boussinesq approximation. The momentum equation describing the fluid differs from the Navier-Stokes equations by containing a bi-Laplacian term of the velocity, as proposed by Fried and Gurtin. The higher-order terms in the momentum equation require additional boundary conditions (strong, weak, general adherence). Several velocity profiles are presented also for real nanofluid suspensions. The found velocities are compared with the velocity of nanofluids relative to the Buongiorno model.