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.
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.
a b s t r a c tThe combined effect of couple stresses and a uniform horizontal AC electric field on the stability of buoyancy-driven parallel shear flow of a vertical dielectric fluid between vertical surfaces maintained at constant but different temperatures is investigated. Applying linear stability theory, stability equations are derived and solved numerically using the Galerkin method with wave speed as the eigenvalue. The critical Grashof number G c , the critical wave number a c and the critical wave speed c c are computed for wide ranges of couple stress parameter c , AC electric Rayleigh number R ea 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 is found to be independent of AC electric Rayleigh number even in the presence of couple stresses but increases significantly with increasing c . Moreover, the effect of increasing R ea is to instill instability, while the couple stress parameter shows destabilizing effect in the stationary mode but it exhibits a dual behavior if the instability is via travelling-wave mode. The streamlines and isotherms presented demonstrate the development of complex dynamics at the critical state.
The hydrodynamic stability against small disturbances of plane Couette flow through an incompressible fluid-saturated fixed porous medium between two parallel rigid plates is investigated. The fluid flow occurs because of moving upper and lower plates with a constant speed in the opposite directions, and it is described by using the Brinkman-extended Darcy model with fluid viscosity different from the Brinkman viscosity. The resulting stability eigenvalue problem is solved numerically using the Chebyshev collocation method. The instigation of instability has been determined accurately by computing the critical Brinkman–Reynolds number as a function of the Darcy–Reynolds number. A comparative study between the plane porous-Poiseuille flow and the plane porous-Couette flow has been carried out, and the similarities and the differences are highlighted. For the Darcy and the inviscid fluid cases, the stability of fluid flow is analyzed analytically and found that the flow is always stable.
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