On The Stability of The Non-Newtonian Boundary-layer Flow Over a Rotating Sphere

The majority of this work is concerned with the local-stability of incompressible non-Newtonian boundary-layer flows over rotating spheres. Convective stability given by the power-law fluids is considered for fluids that adhere to a non-Newtonian governing viscosity relationship. The velocity distribution of the base flow is described by ordinary differential equations with the power-law flow index as the only parameter. The laminar boundary layer flow is studied by extending the perturbation method suggested by Howarth and used by Banks. The laminar mean velocity profiles are obtained by solving the resulting ordinary differential equations assuming that the flow is axisymmetric and time independent. Having the base flow solutions, we investigate the convective instabilities associated with flows in the limit of large Reynolds number inside the threedimensional boundary-layer. The local rotating sphere analyses are conducted at various latitudes from the axis of rotation (θ). A linear stability analysis is conducted using the Chebyshev collocation method. Extensive computation results are given for the flow index 0.6 ≤ n ≤ 1.0. Akin to previous Newtonian studies at sphere latitudes θ = 10◦−70◦ in ten degrees increments, it is found that there exists two primary modes of instability; the upper-branch Type I modes (cross-flow) and the lower-branch Type II modes (streamlinecurvature). The results of the convective instabilities are presented in terms of neutral curves and growth rates. The predictions of Reynolds number, vortex angle and vortex speed at the onset of convective instability when the power-law index n = 1 are consistent with previous Newtonian studies. Our predictions and neutral curves for convective instability of the power-law boundary layer on rotating spheres approach those of the rotating disk as we approach the pole. Roughly speaking, our findings reveal that shear-thinning power-law fluids over rotating spheres have a universal stabilising effect.