Stability Analysis Of Rayleigh-B(?)nard Convection With Viscoelastic Fluid Under Temperature Modulation

The stability of thermal convection has been concerned by many researchers because of its widespread application in daily living,work and industry.The temperature gradient across the fluid layer causes the change in the density of the fluid micelles,thus buoyancy force is generated to drive the convection,which is the principle of Rayleigh-Benard convection.And this paper focuses on the stability of thermal convection when the temperature gradient is not uniform across the fluid layer.The stabilities of thermal convection in viscoelastic fluid saturated porous media and electro-thermal convection in a pure dielectric fluid layer with an electric field are analyzed respectively under temperature modulation.In the first part,Darcy-Brinkman convection model is considered due to the low-speed permeability of fluid in the porous layer.To keep the temperature gradient nonuniform across the layer,transient heating or cooling is applied at the boundary The problem of solving the velocity and temperature fields of the flow are converted into ordinary differential equations of time-varying coefficients by Galerkin method,from which the coefficients are obtained to represent the pattern and stability of the flow.The difference of stability made by temperature modulation is studied by using Rayleigh number,a dimensionless quantity,as the control parameter to describe the conditions for the onset of convection and the instability of steady convection.It turns out that there are two different modes for static solution losing its stability determined by the viscoelastic parameters,one of those is the direct convert to steady convection solution,and the other is a transition with oscillation solution.Under the influence of temperature modulation,although the onset of convection changes little,the stability of the convective solution decreases,which means the critical Rayleigh number for chaos' occurrence getting smaller.The system state at different Rayleigh Numbers is presented by the phase plane projection reflecting the correlation between time-varying coefficients.Thus the changes caused by temperature modulation are also shown visually,such as the finite-amplitude oscillation and the appearance of chaotic attractors.For the second study,the stability of pure viscoelastic dielectric fluid convection in external electric field under temperature modulation is analyzed by using perturbation method to expand the velocity and Rayleigh number into power series for the amplitude of temperature modulation.By that thermal Rayleigh number can be divided into the Rayleigh number in unmodulated system and the corrected Rayleigh number caused by temperature modulation.Here three types of temperature modulation are considered,including the in-phase modulation,out-phase modulation and unilateral modulation which only applied on the lower boundary.The corrected Rayleigh number caused by three types of temperature modulation with any frequency is obtained to represent the influence of temperature modulation on the stability of convection,and the characteristics of different types of temperature modulation are analyzed respectively.It is found that the in-phase modulation always weakens the stability of the system,while the other two types of temperature modulation can enhance the stability of the system at low frequency and weaken the stability at high frequency.The instable effects of three high-frequency temperature modulations on the stability of system all decrease with the increasing frequency.By controlling the electric Rayleigh number which characterizes the driving effect of the external electric field on the convection,it turns out that the electric Rayleigh number has a decreasing effect on the stability of the unmodulated system,and in the other hand makes the stability less influenced by temperature modulation.Furthermore,the effects of stress relaxation time and strain retardation time on the convective stability under temperature modulation are also analyzed with a constant electric Rayleigh number.The result shows that the stress relaxation time enhances the damage effect of temperature modulation on the stability of the flow system,thus weakens the stability of the modulated system,while the strain retardation time has opposite effect.