Pulsating laminar flow in a duct with EPR

The laminar pulsatile flow in a plane duct can be treated as the simplest unsteady problem in basic fluid mechanics with applications, for instance, to engineering vibrators [199, 507], In biological and medical fluid mechanics, one meets the pulsating blood flow in arteries and the air flow in lungs. The generalization of such flows was suggested in literature [598, 601], and is of interest for this chapter. 

Let the pressure gradient in a duct between two parallel plates periodically fluctuate, 

The flow under consideration is mathematically more simple in a plane duct than in an axisymmetric round tube. However, no readily solution for smooth walls, i.e. in the case where the EPR is absent, is given in the literature. The formulations and solutions of the problem of a pulsating viscous flow to be taken for a comparison have been presented only for round tubes. The known solutions employ mainly the Laplace transformation and small-parameter methods. The present investigation follows the standard technique of mathematical physics of finding the periodic solutions described particularly by Schlichting [566] with references to some original sources. Their solution for smooth walls will serve for a comparison. 

In the general statement of the problem (3.1) and (3.2), a pulsating pressure-driven flow in a plane channel with easily penetrable layers symmetrically placed near both walls can be described by the following equation  

It is easy to see that this fluctuating addition is to satisfy to the following two problems 

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