This paper presents new equations for simulating three-dimensional convective heat transfer using the complete viscous dissipation function in the energy equation to account for the viscous heating and incorporating the Boussinesq approximation for the thermal buoyancy term in the Navier-Stokes equations. In addition, new implementation of fourth order Stiffly Stable schemes were achieved and tested for time integration. In order to provide dual-level mesh refinement, for flexible spatial resolutions, a modal spectral element method was implemented in solving these equations in three dimensions. Simulation results demonstrate that these new equations and implementations are accurate enough for investigating convective heat transfer with viscous heating subject to complex thermal and flow boundary conditions in three dimensional irregular domains using the high order finite element method.

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