Nonlinear dynamics of modulated flow between a porous injector and an impermeable substrate

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Abstract
The stable, time‐periodic flow between a porous injector disk and an impermeable substrate disk is explored for the case where fluid is injected into the gap region with a spatially uniform time‐periodic velocity V(τ)=V0(1+α cos(στ)), where V0 is the mean injection velocity, α is the flow modulation amplitude, and σ is the flow modulation frequency. Fourier series expansions in time (τ) are combined with regular perturbation expansions of the Fourier coefficients in powers of α to describe the linear and nonlinear frequency dispersion in the system when 0<α<1. Generalized analytical expressions are obtained for the quasisteady response, and the method of matched asymptotic expansions is used to analyze the high frequency linear response of the system. Finite difference methods are used to calculate the frequency dispersion of the system for a wide range of modulation frequencies (σ) and Reynolds numbers. Oscillating harmonics are shown to interact destructively (via nonlinear inertial terms), resulting in the nullification of certain Fourier modes in the flow field. The Reynolds number and frequency dependence of harmonic nullification events are explored and their implications for creating multilayered alloys are briefly discussed.

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