Spectral treatment of axially symmetric flow and heat transfer above an infinite rotating disk: Non-unique solutions with stability analysis

Zandbergen and Dijkstra (1977) reported dual solutions for the rotating flow above a solid rotating disk, leaving open questions about the nature of the second solution branch away from the disk and the stability of these branches. This work addresses these unanswered questions and enriches the current understanding of the problem, providing insights into the impacts of an externally applied uniform horizontal magnetic field and a shrinking disk on the flow and heat transfer characteristics of a viscous fluid, Cu/water nanofluid, and Cu−TiO2/water hybrid nanofluid. Employing the spectral quasilinearization method (SQLM), this study reports four solution branches in the case of a shrinking disk, with two branches corresponding to each primary solution branch of the solid disk configuration. Additionally, to assess the stability and instability of these multiple solution branches, different from the commonly adopted local approach, a Chebyshev collocation method-based global approach is proposed to address the generalized eigenvalue problem. The positive smallest eigenvalue solely for the first solution branch shows its stable nature, while the remaining three branches are unstable. This global method avoids the typical limitations found in the local approach and can be readily implemented to examine the mathematical stability of non-unique solutions in a wide range of nonlinear problems in science and engineering. Results further highlight the importance of the application of a horizontal magnetic field (α=±0°,±90°,45°) towards or opposing the fluid’s rotation in determining the critical value of rotation parameter βc; notably, |βc| is maximum when the magnetic field is applied at α=45° in a viscous liquid. Furthermore, compared to the Cu/water nanofluid, the |βc| value for the Cu−TiO2/water hybrid nanofluid consistently remains higher across all values of α.