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This paper examines the effect of rotation on thermal instability under Hele-Shaw cell saturated by Casson nanofluid using both linear and nonlinear ways. The nanofluid model incorporates Brownian and thermophoresis diffusion. While conducting an analysis of nonlinear stability numerically using the truncated Fourier series method, analysis of linear stability is performed analytically using the normal mode methodology. The outcomes are all displayed graphically. The results demonstrate that the rotation has dual effect on Hele-Shaw parameter as well as Casson parameter, for higher value of rotation it has destabilizing effect and it stabilizing  the system for lower values of rotation. Lewis number and concentration Rayleigh number promote the onset of convective motion within the system. On the other hand, rotation stabilize the system. Understanding the behavior of heat and mass transportation, the concentration of nanoparticles and fluid phase, utilize the Nusselt number when Nusselt numbers are assessed as a function of time, it is found that the variation of the rotation, Hele-Shaw and Casson parameter has a major influence on the heat and mass transfer. Both steady and unsteady weakly nonlinear analyses are performed to understand the heat transport in the system. It is concluded that the Casson nanofluid parameter has both stabilizing and destabilizing impact depending upon the rate of rotation and therefore this work can be possibly utilized in both places, where heat removal and heat conservation are required. [ABSTRACT FROM AUTHOR]

This paper deals with a mathematical model of controlling heat transfer in nanofluids. The time-periodic vertical vibrations of the system are considered to effect an external control of heat transport along with internal heating effects. A weakly non-linear stability analysis is based on the five-mode Lorenz model using which the Nusselt number is obtained as a function of the thermal Rayleigh number, nano-particle concentration based Rayleigh number, Prandtl number, Lewis number, modified diffusivity ratio, amplitude and frequency of modulation. It is shown that modulation can be effectively used to control convection and thereby heat transport. Further, it is found that the effect of internal Rayleigh number is to enhance the heat and nano-particles transport.

The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (without) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the effect of rotation on the thermal instability in a horizontal layer of a Newtonian nanofluid. The nanofluid layer incorporates the effect of Brownian motion along with thermophoresis. The linear stability based on normal mode technique has been investigated.We observe that the value of Rayleigh number can be increased by a substantial amount on considering a bottom heavy suspension of nano particles. The effect of various parameters on Rayleigh number has been presented graphically.

The stability analysis of Soret driven double diffusive convection for electrically conducting couple stress liquid is investigated theoretically. The couple stress liquid is confined between two horizontal surfaces and a constant vertical magnetic field is applied across the surfaces. Linear stability analysis is used to investigate the effect of various parameters on the onset of convection. Effect of magnetic field on the onset of convection is presented by means of Chandrasekhar number. The problem is analyzed as a function of Chandrasekhar number (Q), positive and negative Soret parameter (S r) and couple stress parameter (C), mainly. The results show that the Q, both positive and negative Sr and C delay the onset of convection. The effect of other parameters is also discussed in paper and shown by graphs.