Thermal management of unstable convective heat transfer with temperature dependent viscosity

This paper presents a comprehensive analysis of the effect of non - linear motion on the mixed convection viscous fluid flow with viscosity inversely proportional to linear function of temperature in the presence of oscillating thermal waves and vorticity. The governing partial differential equations for continuity, momentum, and energy are derived and transform into dimensionless form with the help of suitable scaling variables and obtained parameters like Prandtl number, viscous diffusion parameter, buoyancy force parameter and viscosity variation parameter. To examine flow and thermal responses, the equations are classified into steady and unsteady forms. For the unsteady case, a primitive variable transformation is applied to decompose the equations into real and imaginary parts. Numerical solutions are obtained using a finite difference scheme combined with Gaussian elimination, and results are illustrated through graphs and tables. The findings demonstrate that, in steady state, increasing the viscosity variation parameter ε reduces temperature distribution while enhancing fluid velocity, with skin friction increasing by over 42% and heat transfer decreasing by nearly 168% as the ε rises. For the unsteady state, at ε=1.5, there is a notable increase in the amplitudes of transient shear stress τs, transient heat transfer τt and then followed by a phase angle shift at higher values. These insights are particularly relevant to engineering applications such as heat exchanger optimization, cooling of electronic devices, thermal insulation design, and other thermal management strategies under unstable convective regimes.