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: A nonlinear study of triple diffusive convection in a rotating couple stress liquid has been analysed. It is performed to study the effect of heat and mass transfer by deriving Ginzburg-Landau equation. Heat and mass transfer are quantified in terms of Nusselt number and Sherwood numbers, which are obtained as a function of thermal and solute Rayleigh numbers. The obtained Ginzburg-Landau equation is Bernoulli equation, and it has been elucidated numerically by using Mathematica. The effects of couple stress parameter, solute Rayleigh numbers, and Taylor number on the onset of convection and heat and mass transfer have been examined. It is found that the effects of couple stress parameter and Taylor number are to stabilize the system and to increase the heat and mass transfer.
Abstract: Method of multiple scales is used in the paper in order
to derive an amplitude evolution equation for the most unstable mode
from two-dimensional shallow water equations under the rigid-lid
assumption. It is assumed that shallow mixing layer is slightly curved
in the longitudinal direction and contains small particles. Dynamic
interaction between carrier fluid and particles is neglected. It is
shown that the evolution equation is the complex Ginzburg-Landau
equation. Explicit formulas for the computation of the coefficients of
the equation are obtained.
Abstract: Linear and weakly nonlinear analysis of shallow wake
flows is presented in the present paper. The evolution of the most
unstable linear mode is described by the complex Ginzburg-Landau
equation (CGLE). The coefficients of the CGLE are calculated
numerically from the solution of the corresponding linear stability
problem for a one-parametric family of shallow wake flows. It is
shown that the coefficients of the CGLE are not so sensitive to the
variation of the base flow profile.