The convective instability of the boundary-layer flow over a rotating cone in and out of a uniform magnetic field

Abstract

We present convective instability analyses of the boundary-layer flows originated over cones (with half-angle ψ≤90∘) rotating in an otherwise still conducting fluid. A uniform magnetic field (with magnetic strength parameter, m≥0) is acting normally on the surface of each cone. In the non-magnetic case of cones, comparison of present results with existing experimental and theoretical studies lead *to propose that onset of instabilities may be attributed to crossflow (type I) modes and streamline curvature (type II) modes for broad cones (with ψ≥40∘) and more slender cones (with ψ≤15∘), respectively. For slender cones with half-angle ψ in the vicinity of 25° the presented results validate the hypothesis of centrifugal (type III) modes ( discovered by Garrett et al. (2014)). In the magnetic case, increasing magnetic strength m∈(0,11] considerably delays the onset of instabilities (due to both type I and type II modes) over each rotating cone with fixed ψ. The stabilising influence of magnetism is in agreement with the meagrely available theoretical studies for magnetic rotating disk case ψ=90∘. In addition, our results lead to suggest that streamline curvature mode over each cone with fixed half-angle ψ become sensitive with increasing m. A minimum m0∈[0,11] exists for each fixed half-angle ψ0≤90∘ such that whenever m≥m0 the onset of instabilities (in the sense of minimum critical Reynolds numbers) over every cone with half-angle (ψ≤ψ0) are commenced by type II mode instead of type I mode whilst the aforementioned conjecture of stabilising effect of magnetism is not violated in the considered range of parameters. Under non-stationary vortices assumption with magnetism, we show that crossflow modes travelling at around 75% of the cone surface are likely to be selected in applications where smooth and frictionless surfaces are used. This finding is in complete agreement with the non-magnetic case of rotating cone in existing studies.