Effect of cubic temperature gradient on onset of thermal convection in a micropolar fluid

Thermal convection with shear flow has received widespread attention due to its importance in geophysical flows as well as several technological applications, such as heat exchangers and chemical vapour deposition. The thesis deals with two types of thermal convection in a fluid layer that is Rayleigh Benard convection (driven by buoyancy) and Marangoni convection (driven by surface tension). The Rayleigh-Benard and Marangoni stability problem for a fluid bound by bottom and top wall which are heated and cooled, respectively are studied numerically. The fluid layer with various boundary conditions at the different lower and upper boundary are investigated theoretically based on linear stability theory. The various boundary conditions are assumed for lower and upper boundaries to be free isothermal and free isothermal (FIFI), free isothermal and free adiabatic (FIFA), free isothermal and rigid adiabatic (FIRA), rigid isothermal and free isothermal (RIFI), rigid isothermal and free adiabatic (RIFA), and rigid isothermal and rigid adiabatic (RIRA). The effect of cubic temperature gradient, internal heat generation, feedback control, and electric field on the onset of Rayleigh-Bénard and Marangoni convection in an Eringen’s micropolar fluid has been examined. Three types of non-uniform basic temperature gradients which are linear, cubic 1 and cubic 2 are considered. The single-term Galerkin method is applied to obtain the eigenvalue for FIFI, FIFA, FIRA, RIFI, RIFA, and RIRA boundary combination. Closed form analytical solutions, of the full governing equations, are derived and the governing parameters of the problem are the thermal critical Rayleigh number, the critical Marangoni number, couple stress, coupling, and micropolar heat conduction, on the onset of convection has been analysed. It is found that cubic 1 is the most stabilizing temperature gradient and linear temperature gradient is the most destabilizing temperature gradient.

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