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Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe

The nonsteady Navier–Stokes equations are considered in a thin infinite pipe with the small diameter ϵ in the case of the Reynolds number of order ϵ. The time-dependent flow rate is a given function. The complete asymptotic expansion is constructed and justified. The error estimate of order O(ϵ J ) for the difference of the exact solution and the J-th asymptotic approximation is proved for any real J.