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In this work, the problem of the unsteady natural convection in an Al 2 O 3 -water filled nanofluids influenced by an uniform magnetic field is analysed numerically. Governing equations are given in terms of the streamlines, the vorticity and the temperature of the fluid. Dual Reciprocity Boundary Element Method (DRBEM) is performed as a solution technique. Both the fundamental solutions of Laplace equation ((1/2π) ln(x)) and modified Helmholtz equation (1/2π)K 0 (x)) are used in the derivation of the method. The numerical analysis are done for physical parameters Rayleigh number (10 3 ≤ Ra ≤ 10 6 ), Hartmann number (0 ≤ Ha ≤ 75) and particle volume fraction (0 ≤ φ ≤ 0.2). From the results, it is observed that the behaviours of all the variables are influenced by the changing values of parameters. The solution procedure of the all variables needs considerably small number of iterations and large time increments with suitable values of relaxation parameters which occur in the argument of Bessel function K 0 (x).

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The behavior of unsteady mixed convection flow of Cu−water based nanofluids is investigated numerically inside a square lid-driven partially heated flow below. Dual Reciprocity Boundary Element Method is used to solve stream function-vorticity form of the governing equations of the problem. The need of time integration scheme is eliminated by transforming the vorticity transport and energy equations to modified Helmholtz equations. This procedure also diminishes the stability problems. The numerical results are given for several values of Reynolds number (Re), Rayleigh number (Ra), heat source location (D), heat source length (B) and solid volume fraction (φ). The steadystate results are in good agreement with the results available in the literature.