A fully higher-order compact(HOC)finite difference scheme on the 9-point two-dimensional(2D)stencil is formulated for solving the steady-state laminar mixed convection flow in a lid-driven inclined square enclosure filled with water-Al2O3 nanofluid.Two cases are considered depending on the direction of temperature gradient imposed(Case I,top and bottom;Case II,left and right).The developed equations are given in terms of the stream function-vorticity formulation and are nondimensionalized and then solved numerically by a fourth-order accurate compact finite difference method.Unlike other compact solution procedure in literature for this physical configuration,the present method is fully compact and fully higher-order accurate.The fluid flow,heat transfer and heat transport characteristics were illustrated by streamlines,isotherms and averaged Nusselt number.Comparisons with previously published work are performed and found to be in excellent agreement.A parametric study is conducted and a set of graphical results is presented and discussed to elucidate that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by inclination of the enclosure at moderate and large Richardson numbers.
In this paper,an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated.Liouville transformation is applied to change the problem into an equivalent minimization problem.Finite element method is proposed and the convergence for the finite element solution is established.A monotonic decreasing algorithm is presented to solve the extremal eigenvalue problem.A global convergence for the algorithm in the continuous case is proved.A few numerical results are given to depict the efficiency of the method.
A compact alternating direction implicit(ADI) method has been developed for solving multi-dimensional heat equations by introducing the differential operators and the truncation error is O(τ 2 + h 4 ). It is shown by the discrete Fourier analysis that this new ADI scheme is unconditionally stable and the truncation error O(τ 3 + h 6 ) is gained with once Richardson's extrapolation. Some numerical examples are presented to demonstrate the efficiency and accuracy of the new scheme.
