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A representation of the solution of the Gromeko problem for viscous and elastico-viscous fluids

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Abstract

The solution of the Gromeko problem [1] on unsteady flow of a viscous fluid in a long circular pipe is among the few exact solutions of the Navier-Stokes equations. Its effective solution is obtained only when the longitudinal pressure gradient is given as an arbitrary time function. However, in practice we encounter cases when the flow rate is a known time function. This sort of problem arises, in particular, in rheological experiments using viscometers with a given flow rate. In this case the determination of the pressure gradient from the given flow rate leads in the general case to a very unwieldy expression. Below we present an effective solution of this problem for viscous and elasticoviscous media using the method of solving the inlet flow problem for a steady flow of a viscous fluid in a semi-infinite pipe. It is shown that for the case of a viscous fluid these two problems are actually equivalent.

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References

  1. I. S. Gromeko, “Theory of fluid motion in narrow cylindrical pipes”, Kazansk. universitetsk. tipografia, 1882.

  2. A. G. Fredrickson, “On stress-relaxing solids; I. Flow and normal stress effects”, Chem. Engng. Sci., vol. 17, p. 155, 1962.

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  3. A. I. Leonov, “Some problems of unsteady flow of an incompressible viscoelastic (Maxwellian) fluid”, All-Union Congress on Theoretical and Applied Mechanics, Moscow, 27 Jan.–3 Feb. 1960 [in Russian], report summaries.

  4. A. I. Leonov, “Steady flow of a viscous fluid in plane and circular pipes of finite length”, Izv. AN SSSR, Mekhanika i mashinostroenie, no. 6, 1962.

  5. Saul Vela, J. W. Kalb, and A. G. Fredrickson, “On stress-relaxing solids; Part III. Simple harmonic deformation”, Amer. J. Chem. Engng., vol. 11, no. 2, 1965.

  6. A. I. Leonov, “Unsteady motion of an incompressible Maxwellian fluid in the gap between infinite parallel plates”, Izv. AN SSSR, Mekhanika i mashinostroenie, no. 3, 1962.

  7. S. M. Targ, Fundamental Problems of Laminar Flow Theory [in Russian], Gostekhizdat, 1951.

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Authors and Affiliations

  1. Moscow

    A. I. Leonov

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  1. A. I. Leonov
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Leonov, A.I. A representation of the solution of the Gromeko problem for viscous and elastico-viscous fluids. Fluid Dyn 2, 35–38 (1967). https://doi.org/10.1007/BF01024798

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  • DOI: https://doi.org/10.1007/BF01024798

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