A fully three-dimensional time-dependent Navier-Stokes model with forced convection is developed to numerically investigate the heat and flow patterns of the two-roll mill system with two inner rotating cylinders. Such direct numerical simulations are usually limited by the difficulties from huge computational cost and complex boundary treatment. For a fast numerical process, we can use the operator-splitting scheme with the BTD term to advance the solution in temporal evolution. To implement the calculation over a Cartesian grid, the hybrid Cartesian/immersed-boundary finite-element method is employed for spatial discretization. In the authors' previous study [D.L. Young, C.L. Chiu, C.M. Fan, A hybrid Cartesian/immersed-boundary finite-element method for simulating heat and flow patterns in a two-roll mill, Numer. Heat Transfer B 51 (3) (2007) 251-274]. we have developed a simplified 2D numerical model to analyze the heat and flow patterns on the cross section of two-roll-mill flow under the assumption of infinite length in the third (vertical) direction. However, the 2D solutions could not completely represent the realistic physical phenomena unless a 3D algorithm is developed. In this study we then paid the particular attention to develop a 3D model to investigate the vertical heat and flow behaviors, including 3D features of the vortex structure, periodic oscillation and chaotic instabilities. It is found that the proposed 3D model is able to cover the 2D features if the assumptions of 2D conditions are fulfilled. (C) 2009 Published by Elsevier Ltd.