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  • 1.
    Kozlov, Vladimir
    et al.
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Lokharu, Evgeniy
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    N-Modal Steady Water Waves with Vorticity2018Ingår i: Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, E-ISSN 1422-6952, Vol. 20, nr 2, s. 853-867Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation.

  • 2.
    Kozlov, Vladimir
    et al.
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Kuznetsov, N.
    Russian Academic Science, Russia.
    Lokharu, Evgeniy
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    On the Benjamin-Lighthill conjecture for water waves with vorticity2017Ingår i: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 825, s. 961-1001Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin-Lighthill conjecture for flows with values of Bernoullis constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

  • 3.
    Lokharu, Evgeniy
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Small-amplitude steady water waves with vorticity2017Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    The problem of describing two-dimensional traveling water waves is considered. The water region is of finite depth and the interface between the region and the air is given by the graph of a function. We assume the flow to be incompressible and neglect the effects of surface tension. However we assume the flow to be rotational so that the vorticity distribution is a given function depending on the values of the stream function of the flow. The presence of vorticity increases the complexity of the problem and also leads to a wider class of solutions.

    First we study unidirectional waves with vorticity and verify the Benjamin-Lighthill conjecture for flows whose Bernoulli constant is close to the critical one. For this purpose it is shown that every wave, whose slope is bounded by a fixed constant, is either a Stokes or a solitary wave. It is proved that the whole set of these waves is uniquely parametrised (up to translation) by the flow force which varies between its values for the supercritical and subcritical shear flows of constant depth. We also study large-amplitude unidirectional waves for which we prove bounds for the free-surface profile and for Bernoulli’s constant.

    Second, we consider small-amplitude waves over flows with counter currents. Such flows admit layers, where the fluid flows in different directions. In this case we prove that the initial nonlinear free-boundary problem can be reduced to a finite-dimensional Hamiltonian system with a stable equilibrium point corresponding to a uniform stream. As an application of this result, we prove the existence of non-symmetric wave profiles. Furthermore, using a different method, we prove the existence of periodic waves with an arbitrary number of crests per period.

    Delarbeten
    1. On bounds and non-existence in the problem of steady waves with vorticity
    Öppna denna publikation i ny flik eller fönster >>On bounds and non-existence in the problem of steady waves with vorticity
    2015 (Engelska)Ingår i: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 765, nr R1Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    For the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoullis constant. (ii) If only one parallel shear flow exists for a given value of Bernoullis constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

    Ort, förlag, år, upplaga, sidor
    Cambridge University Press (CUP): STM Journals, 2015
    Nyckelord
    surface gravity waves; waves/free-surface flows
    Nationell ämneskategori
    Matematik
    Identifikatorer
    urn:nbn:se:liu:diva-114417 (URN)10.1017/jfm.2014.747 (DOI)000348130700001 ()
    Anmärkning

    Funding Agencies|Swedish Research Council; G.S. Magnusons Foundation of the Royal Swedish Academy of Sciences; Linkoping University

    Tillgänglig från: 2015-03-02 Skapad: 2015-02-20 Senast uppdaterad: 2017-12-04
  • 4.
    Kozlov, Vladimir
    et al.
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Kuznetsov, N.
    Russian Academic Science, Russia.
    Lokharu, Evgeniy
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    On bounds and non-existence in the problem of steady waves with vorticity2015Ingår i: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 765, nr R1Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoullis constant. (ii) If only one parallel shear flow exists for a given value of Bernoullis constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

  • 5.
    Kozlov, Vladimir
    et al.
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Kuznetsov, N.
    Russian Academy of Sciences, St Petersburg, Russia.
    Lokharu, Evgeniy
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Steady water waves with vorticity: an analysis of the dispersion equation2014Ingår i: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 751Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Two-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm-Liouville problem considered by Constantin and Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481-527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133-175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm-Liouville problem mentioned in (i) has an eigenvalue is obtained.

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