Natural convection within a non-uniformly heated cavity partly filled with a shear-thinning nanofluid and partly with air

•The liquid in the lower phase stimulates the particle movement in the upper phase.•Heat transfer is the best when the lower phase is largely filled with nanofluid.•The shear-thinning nanofluid's flow index effect outweighs that of its consistency.•Stress intensity factors at the crack tip in gradient theory.•Instabilities appear for critical Rayleigh number and solid volume fraction.•Useful correlations to predict heat transfer rate within the enclosure are presented.