Primary and secondary instabilities in Rayleigh-Bénard convection of water-copper nanoliquid

Abstract

A study of primary and secondary instabilities in Rayleigh-Bénard convection of water-copper nanoliquid is made using a generalized two-phase model. Boussinesq approximation and small scale convective motion are assumed to be valid. The Brownian motion effect is assumed to be negligibly small and a weak thermophoretic effect is included in the investigation. The parameter regimes for the existence of pitchfork, Takens-Bogdanov and Hopf bifurcations are reported. Small-amplitude modulation is considered to derive the Newell–Whitehead–Segel equation and using its phase-winding solution the condition for the occurrence of Eckhaus and zigzag secondary instabilities are obtained. The influence of copper nanoparticles on the secondary instability region is reported. The travelling wave solutions for the Newell–Whitehead–Segel equation are also presented. Oscillatory convection is not, in general, preferable in the problem yet in those cases where it can exist the necessary condition for the occurrence of Benjamin-Feir instability is discussed. The present investigation sheds light on useful parameters’ ranges wherein a desired instability can be made to manifest depending upon the need of an engineering application. The Rayleigh-Bénard convection can also be used as a rheometric tool for the measurement of viscosity and thermal diffusivity of the nanoliquid in a dynamic situation.