Adoption of the hyperbolic Cattaneo–Christov heat-flow model in place of the more usual parabolic Fourier law is shown to raise the possibility of oscillatory convection in the classic Bénard problem of a Boussinesq fluid heated from below. By comparing the critical Rayleigh numbers for stationary and oscillatory convection, Rc and RS respectively, oscillatory convection is found to represent the preferred form of instability whenever the Cattaneo number C exceeds a threshold value CT≥8/27π2≈0.03. In the case of free boundaries, analytical approaches permit direct treatment of the role played by the Prandtl number P 1 , which—in contrast to the classical stationary scenario—can impact on oscillatory modes significantly owing to the non-zero frequency of convection. Numerical investigation indicates that the behaviour found analytically for free boundaries applies in a qualitatively similar fashion for fixed boundaries, while the threshold Cattaneo number CT is computed as a function of P 1 ∈[10 −2 ,10 +2 ] for both boundary regimes.
|Pages (from-to)||1 - 18|
|Number of pages||18|
|Journal||Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences|
|Early online date||18 Feb 2015|
|Publication status||Published - Mar 2015|
Bissell, J. J. (2015). On oscillatory convection with the Cattaneo-Christov hyperbolic heat-flow model. Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 471(2175), 1 - 18. [2014.0845]. https://doi.org/10.1098/rspa.2014.0845