Exact Solutions for Non-Isothermal Flows of Second Grade Fluid between Parallel Plates

In this paper, we obtain new exact solutions for the unidirectional non-isothermal flow of a second grade fluid in a plane channel with impermeable solid walls, taking into account the fluid energy dissipation (mechanical-to-thermal energy conversion) in the heat transfer equation. It is assumed that the flow is time-independent and driven by the pressure gradient. On the channel walls, various boundary conditions are stated. Namely, we consider the no-slip conditions, the threshold slip conditions, which include Navier's slip condition (free slip) as a limit case, as well as mixed boundary conditions, assuming that the upper and lower walls of the channel differ in their physical properties. The dependence of solutions on the boundary conditions is discussed in some detail. Moreover, we establish explicit relationships for the model parameters that guarantee the slip (or no-slip) regime on the boundaries.

Automated summary

In this paper, new exact solutions for the unidirectional non-isothermal flow of a second grade fluid in a plane channel with impermeable solid walls are obtained, considering the fluid energy dissipation in the heat transfer equation. Time-independent flow driven by pressure gradient is assumed. Various boundary conditions, including no-slip conditions, threshold slip conditions, and mixed boundary conditions, are considered. The dependence of solutions on the boundary conditions is discussed, and explicit relationships for the model parameters to guarantee slip or no-slip regime on the boundaries are established.