Quartic B-spline and two-step hybrid method applied to boundary value problem AIP Conf. Proc. 1522, 744 (2013); 10.1063/1.4801200Modified predictor-corrector multistep method using arithmetic mean for solving ordinary differential equations AIP Conf.A three-stage explicit two-step Runge-Kutta-Nyström method for solving second-order ordinary differential equations AIP Conf.In this paper, a numerical solution of two dimensional nonlinear coupled viscous Burger equation is discussed with appropriate initial and boundary conditions using the modified cubic B-spline differential quadrature method. In this method, the weighting coefficients are computed using the modified cubic B-spline as a basis function in the differential quadrature method. Thus, the coupled Burger equation is reduced into a system of ordinary differential equations. An optimal five stage and fourth-order strong stability preserving Runge-Kutta scheme is applied for solving the resulting system of ordinary differential equations. The accuracy of the scheme is illustrated by taking two numerical examples. Computed results are compared with the exact solutions and other results available in literature. Obtained numerical result shows that the described method is efficient and reliable scheme for solving two dimensional coupled viscous Burger equation. C 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.

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a b s t r a c tThe stability of buoyancy-driven parallel shear flow of a couple stress fluid confined between vertical plates is investigated by performing a classical linear stability analysis. The plates are maintained at constant but different temperatures. A modified Orr-Sommerfeld equation is derived and solved numerically using the Galerkin method with wave speed as the eigenvalue. The critical Grashof number G c ; critical wave number a c and critical wave speed c c are computed for wide ranges of couple stress parameter K c and the Prandtl number Pr. Based on these parameters, the stability characteristics of the system are discussed in detail. The value of Prandtl number, at which the transition from stationary to travelling-wave mode takes place, increases with increasing K c . The couple stress parameter shows destabilising effect on the convective flow against stationary mode, while it exhibits a dual behaviour if the instability is via travelling-wave mode. The streamlines and isotherms presented demonstrate the development of complex dynamics at the critical state.

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