Abstract: In the paper we make linear and non-linear stability
analyses of Rayleigh-Bénard convection of a Newtonian nanoliquid
in a rotating medium (called as Rayleigh-Bénard-Taylor convection).
Rigid-rigid isothermal boundaries are considered for investigation.
Khanafer-Vafai-Lightstone single phase model is used for studying
instabilities in nanoliquids. Various thermophysical properties of
nanoliquid are obtained using phenomenological laws and mixture
theory. The eigen boundary value problem is solved for the Rayleigh
number using an analytical method by considering trigonometric
eigen functions. We observe that the critical nanoliquid Rayleigh
number is less than that of the base liquid. Thus the onset of
convection is advanced due to the addition of nanoparticles. So,
increase in volume fraction leads to advanced onset and thereby
increase in heat transport. The amplitudes of convective modes
required for estimating the heat transport are determined analytically.
The tri-modal standard Lorenz model is derived for the steady state
assuming small scale convective motions. The effect of rotation on
the onset of convection and on heat transport is investigated and
depicted graphically. It is observed that the onset of convection is
delayed due to rotation and hence leads to decrease in heat transport.
Hence, rotation has a stabilizing effect on the system. This is due to
the fact that the energy of the system is used to create the component
V. We observe that the amount of heat transport is less in the case
of rigid-rigid isothermal boundaries compared to free-free isothermal
boundaries.
Abstract: A local nonlinear stability analysis using a eight-mode
expansion is performed in arriving at the coupled amplitude equations
for Rayleigh-Bénard-Brinkman convection (RBBC) in the presence
of LTNE effects. Streamlines and isotherms are obtained in the
two-dimensional unsteady finite-amplitude convection regime. The
parameters’ influence on heat transport is found to be more
pronounced at small time than at long times. Results of the
Rayleigh-Bénard convection is obtained as a particular case of
the present study. Additional modes are shown not to significantly
influence the heat transport thus leading us to infer that five minimal
modes are sufficient to make a study of RBBC. The present problem
that uses rolls as a pattern of manifestation of instability is a needed
first step in the direction of making a very general non-local study of
two-dimensional unsteady convection. The results may be useful in
determining the preferred range of parameters’ values while making
rheometric measurements in fluids to ascertain fluid properties such
as viscosity. The results of LTE are obtained as a limiting case of
the results of LTNE obtained in the paper.
Abstract: Rayleigh-B´enard convection of a nanoliquid in shallow, square and tall enclosures is studied using the Khanafer-Vafai-Lightstone single-phase model. The thermophysical properties of water, copper, copper-oxide, alumina, silver and titania at 3000 K under stagnant conditions that are collected from literature are used in calculating thermophysical properties of water-based nanoliquids. Phenomenological laws and mixture theory are used for calculating thermophysical properties. Free-free, rigid-rigid and rigid-free boundary conditions are considered in the study. Intractable Lorenz model for each boundary combination is derived and then reduced to the tractable Ginzburg-Landau model. The amplitude thus obtained is used to quantify the heat transport in terms of Nusselt number. Addition of nanoparticles is shown not to alter the influence of the nature of boundaries on the onset of convection as well as on heat transport. Amongst the three enclosures considered, it is found that tall and shallow enclosures transport maximum and minimum energy respectively. Enhancement of heat transport due to nanoparticles in the three enclosures is found to be in the range 3% - 11%. Comparison of results in the case of rigid-rigid boundaries is made with those of an earlier work and good agreement is found. The study has limitations in the sense that thermophysical properties are calculated by using various quantities modelled for static condition.
Abstract: The effect of internal heat generation is applied to the Rayleigh-Benard convection in a horizontal micropolar fluid layer. The bounding surfaces of the liquids are considered to be rigid-free, rigid-rigid and free-free with the combination of isothermal on the spin-vanishing boundaries. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. It is shown that the critical Rayleigh number decreases as the value of internal heat generation increase and hence destabilize the system.