Study On Rayleigh-Bénard Convection In The Absence And Presence Of Time-Periodic Body Force

Energy is rated as the most important issue facing mankind in the next 50 years.Of all the forms of energy we use,more than 70 percent come directly or indirectly from heat.Therefore,strengthening the heat transfer process has become an increasingly difficult task in view of the rapidly increasing energy requirements worldwide.In this thesis,we study linear and weakly nonlinear stability analyses of Rayleigh-Bénard convection using stress-free and isothermal boundary condition.The equations which govern the aforementioned system are system of partial differential equations which have Rayleigh number as eigenvalue.The eigenvalue problem is converted into the Lorenz and the Ginzburg-Landau models using minimal mode truncated Fourier-series representation and the method of multiscales.The solution of the Ginzburg-Landau model is used to quantify heat transport.When there is not time-periodic body force,the study of regular and chaotic Rayleigh-Bénard convective motions is made,we consider methanol and water as working medium.For linear stability analysis,the stationary mode of convection is shown to be the preferred one at the onset of convection in the case of both the liquids.The Rayleigh numbers at which the onset of regular and chaotic convective motions occur are reported for both methanol and water.For weakly nonlinear stability analysis,using a higher-order truncated Fourier series representation we arrive at the energy-conserving penta-modal Lorenz model and then the tri-modal Lorenz model is obtained as a limiting case of it.To keep the study analytical the Ginzburg-Landau model is derived from the penta-modal Lorenz model.It is shown that the tri-and the penta-modal Lorenz models predict exactly the same results leading to the conclusion that the tri-modal Lorenz model is a good enough truncated model for a weakly nonlinear study of convection.The behavior of the dynamical system is studied using the maximum Lyapunov exponent,the bifurcation diagram and phase-space plot.The Hopf bifurcation Rayleigh number is obtained analytically.It is shown that the thresholds for onset of regular and chaotic motions are smaller in the case of methanol compared to water.Another very important finding is to show the existence of a developing region for chaos before becoming fully-developed.We use the Nusselt number to quantify heat transport and we find that methanol can be a better coolant compare to water.As an extension,the effect of time-periodic vertical modulation with trigonometric sine,triangular and square wave-forms on Rayleigh Bénard convection in water,water-Al₂O₃-Cu and water-Al₂O₃ nanoliquids are studied by using a single-phase model.Trigonometric sine,triangular and square wave-forms of vertical modulation are considered in the study.Among these three types of modulations the comparison on onset of convection and the heat transport is made.Using perturbation method,linear stability analysis is performed for all the three wave-form.The study from linear stability analysis reveals that the critical Rayleigh number obtained in the case of triangular wave-form is less compared to the value obtained in the case of trigonometric sine and square wave-forms.The study shows that compared to trigonometric sine and square wave-forms,the triangular wave-form facilitates enhancement in onset of convection.For weakly nonlinear stability analysis,a generalized Lorenz model which is having influence of hybrid nanoliquids and modulation is derived using the Fourier series representation.The Lorenz model is transformed to a Ginzburg-Landau model using the method of multiscales and the solution of which helps to study heat transport.It is found that such an enhancement in heat transport increases with amplitude and decreases with frequency of modulation.Thus,the modulation is treated here as a regulating mechanism of heat transport.Further,we observe that water-Al₂O₃-Cu has maximum heat transport than water-Al₂O₃ and water,which lead to the conclusion that hybrid nanoliquids facilitates heat transport significantly than that of mono nanoliquids.