Convective Instability of a Steady Flow in an Annulus Caused by Internal Heat Generation

Abstract

Linear stability of convective motion in a tall vertical annulus was analysed in the paper. The base flow was generated by a non-uniform distribution of heat sources in the radial direction. The base flow velocity and temperature were obtained analytically solving the system of Navier-Stokes equations under the Boussinesq approximation. The linear stability problem was solved for axi-symmetric and asymmetric perturbations by a collocation method based on the Chebyshev polynomials. Numerical results showed that there were three destabilising factors: (1) increase of the gap between the cylinders, (2) increase of the density of internal heat sources towards to the outer boundary of the annulus and (3) increase of the Prandtl number.

Linear Stability of a Steady Convective Flow between Permeable Cylinders

“…Wide gaps between the walls of the annulus are not considered in the present study (the values of η that are used in the paper are η = 0.6 and η = 0.7). As was shown in [1,20], asymmetric perturbations (depending on the angular coordinate ϕ) are the most unstable for very large gaps (small values of η), which is why only axisymmetric perturbations are considered in the present paper. In the future, we plan to investigate the role of the radius ratio on the stability boundary as well as to consider different Prandtl numbers by analyzing both axisymmetric and asymmetric perturbations.…”

“…where E is the activation energy, R is the universal gas constant, Q 0 is a constant and T is the absolute temperature. Models with internal heat generation given by (1) are used in practice in order to describe processes during biomass thermal conversion [20,21,23] with the objective to obtain cleaner and more efficient sources of energy. Instability is a desirable phenomenon in this case since it enhances fluid mixing, which, in turn, leads to more efficient energy conversion.…”

“…Linear stability of a steady convective flow in an annulus generated by non-homogeneous heat sources is studied in [20]. The effect of nonlinear internal heat sources on the stability boundary of a vertical flow in an annulus is investigated in [21] for both asymmetric and axisymmetric perturbations for wide range of radius ratios, Prandtl numbers and Frank-Kamenetskii parameters (proportional to the intensity of a chemical reaction).…”