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Summary. The interaction of purely periodic mean flow with a peristaltic induced flow is investigated within the framework of a two-dimensional analogue. The mathematical model considers a viscous incompressible fluid under the effect of a transverse magnetic field through a porous medium between infinite parallel walls on which a sinusoidal traveling wave is imposed. A perturbation solution to the complete set of Navier-Stokes equations is found for the case in which the frequency of the traveling wave and that of the imposed pressure gradient are equal. The ratio of the traveling wave amplitude to channel width is assumed to be small. For this case a first-order steady flow is found to exist, as contrasted to a second-order effect in the absence of the imposed periodic pressure gradient. The effect of the magnetic parameter, permeability parameter and the various parameters included in the problem are discussed numerically.

The impetus for the current investigation relates to the implementation of an efficient novel technique to obtain a time-delayed vibration control analytical solution. The current technique follows simple and easy-to-apply criteria. This technique is based on converting the nonlinear time-delayed Van der Pol (VDP)-Duffing oscillator to an equivalent linear one. Details of the conversion to an equivalent linear ordinary differential equation are mentioned. The convergence between the numerical outcomes and the analytic solution is achieved and gives a satisfying accuracy of the equivalent result.

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