Imperfectly conducting eigenflows in a vertical fluid layer

Abstract

The stability of buoyant flow in an infinite vertical fluid layer bounded by imperfectly conducting rigid walls, called imperfectly conducting eigenflows, is discussed. The third kind boundary conditions describing heat transfer to the external environment are applied to perturbations in temperature. The linear stability analysis is carried out numerically by employing the Chebyshev collocation method. Instability arises when the Grashof number G exceeds its critical value, which depends on the Prandtl number Pr and the Biot number Bi. It is found that the onset of instability changes dramatically depending on the magnitude of Prandtl and Biot numbers particularly when the instability is through the traveling-wave mode. The numerical results show that the Biot number plays a pivotal role in determining the transition Prandtl number PrT at which the instability switches over from one mode to another mode. The novel outcomes suggest the presence of a single PrT for Bi<2.1739 while three distinct values of PrT for Bi≥2.1739. The departure from the conventional single value typically observed at isothermal boundaries signifies the complexity of the instability mechanism.