Heat transfer of power-law fluids with variable thermal conductivities on a horizontal rough surface

Power-law fluids have recently received increasing attention because of their applications in different industrial fields. In previous works, the energy and momentum equations for power-law fluids were considered the same as those for Newtonian fluids. However, as the heat transfer of fluids results from thermomolecular motions, the heat-transfer behavior of non-Newtonian power-law fluids should be different from that of Newtonian fluids. The flow of fluids on a smooth plate is a classical problem. In most situations, the plates are rough. In particular, in industrial fields, many plates are deliberately designed to be rough to enhance heat transfer. Herein, according to the Taylor expansion and boundary-layer theory, the boundary-layer equations for the Ostwald–de Waele power-law fluids with a variable thermal conductivity along a horizontal wavy surface are reduced to partial differential equations. An energy equation with a variable thermal conductivity is constructed, where the heat-conduction coefficient is assumed to be a power-law function dependent on the temperature gradient. Through the introduction of a series of transformations, including nondimensional and coordinate transformations, the original wavy-surface problem is transformed into a system of partial differential equations describing the flow problem with boundary conditions on a flat plate, which is solved numerically using the Keller-box method. The effects of some parameters, such as the amplitude–wavelength ratio $ \alpha $ , power-law index $ n $ , and generalized Prandtl number $ {N_{{\rm{zh}}}} $ , on the local friction coefficient and heat-transfer coefficient are discussed. Numerical results show that the velocity of power-law fluids on the surface and pressure gradient varies periodically along the wavy plate. Furthermore, the cycles of the velocity and pressure gradients are the same as the one of the wavy-shape plate. The results show that the local Nusselt number and the friction coefficient vary periodically in a wavelike manner and increase gradually with the amplitude–wavelength ratio, although a sudden change exists near the zero point. With the increasing amplitude, the friction coefficient oscillates more considerably. With the increasing power-law index, the local Nusselt number decreases. For a special case in which the plate is flat, the local Nusselt number and friction coefficient are in a stable state for a short distance along the plate, although initial oscillations appear near the zero point. Owing to the effects of different parameters on the periodicity, the peak and trough of the local Nusselt number and friction coefficient are not consistent, despite occurring in the same cycle.