Fluid Dynamics

, Volume 1, Issue 4, pp 81–86 | Cite as

Wave resistance of a viscous liquid

  • R. A. Gruntfest
  • A. K. Nikitin
Article
  • 20 Downloads

Abstract

In this paper we examine the resistance encountered by a system of normal stresses during its rectilinear motion along the surface of a viscous liquid of infinite depth. The problem is solved in the linear formulation, i.e., it is assumed that amplitudes of the waves which arise are small and the waves are shallow. The solution for the two-and three-dimensional problems is obtained in the general case in closed form. In the two-dimensional case a detailed study is made of the case when a constant pressure p0, moving with the constant velocity U, is given on a segment of length 2l. In the three-dimen-sional problem the case is studied when the normal stress is concentrated on a segment of a straight line of length 2l, which can replace a ship moving along a straight course with the constant velocity U. The integrals obtained in both cases are studied using the stationary phase method, the application of which for the three-dimensional integrals with respect to a volume with boundaries is justified in §1 of the paper. As a result we obtain equations for the wave resistance in the two- (§2) and three-dimensional (§3) cases.

Keywords

Stationary Phase Normal Stress Closed Form Linear Formulation Constant Pressure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. J. Stoker, Water Waves [Russian translation], Izd-vo inostr. lit-ry, 1959.Google Scholar
  2. 2.
    M. I. Kontorovich and Yu. K. Murav'ev, “Derivation of the laws of reflection of geometric optics on the basis of the asymptotic treatment of the diffraction problem”, Zh. tekhn. fiz.,22, no. 3, 1951.Google Scholar
  3. 3.
    M. V. Fedoryuk, “The stationary phase method for multi-dimensional integrals”, Zh. vychislit. matem. i matem. fiz.,2, no. 1, 1962.Google Scholar
  4. 4.
    V. I. Smirnov, Course in Higher Mathematics [in Russian], Fizmatgiz, 1960.Google Scholar
  5. 5.
    A. K. Nikitin and R. A. Gruntfest, “On the two-dimensional problem of waves on the surface of a viscous liquid of infinite depth”, collection: Numerical Methods of Solution of Differential and Integral Equations and Quadrature Formulas [in Russian], Izd-vo Nauka, 1964.Google Scholar
  6. 6.
    A. K. Nikitin and S. A. Podrezov, “On the three-dimensional problem of waves on the surface of a viscous liquid of infinite depth”, PMM.28, no. 4, 1964.Google Scholar
  7. 7.
    A. I. Lur'e, Operational Calculus [in Russian], Moscow-Leningrad, Gostekhizdat, 1960.Google Scholar
  8. 8.
    N. de Bruijn, Asymptotic Methods in Analysis [Russian translation], Izd-vo inostr. lit-ry, 1961.Google Scholar
  9. 9.
    I. S. Gradshtein and M. I. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Moscow, Fizmatgiz. 1963.Google Scholar

Copyright information

© The Faraday Press, Inc. 1969

Authors and Affiliations

  • R. A. Gruntfest
    • 1
  • A. K. Nikitin
    • 1
  1. 1.Rostov-on-Don

Personalised recommendations