• Login
    View Item 
    •   Home
    • Theses and Dissertations
    • Dissertations
    • View Item
    •   Home
    • Theses and Dissertations
    • Dissertations
    • View Item
    JavaScript is disabled for your browser. Some features of this site may not work without it.

    Browse

    All of KAUSTCommunitiesIssue DateSubmit DateThis CollectionIssue DateSubmit Date

    My Account

    Login

    Quick Links

    Open Access PolicyORCID LibguidePlumX LibguideSubmit an Item

    Statistics

    Display statistics

    Nonlinear Wave Motion in Viscoelasticity and Free Surface Flows

    • CSV
    • RefMan
    • EndNote
    • BibTex
    • RefWorks
    Thumbnail
    Name:
    Ussembayev Dissertation.pdf
    Size:
    1.005Mb
    Format:
    PDF
    Description:
    PhD Dissertation
    Embargo End Date:
    2021-12-03
    Download
    View more filesView fewer files
    Type
    Dissertation
    Authors
    Ussembayev, Nail cc
    Advisors
    Markowich, Peter A. cc
    Committee members
    Thoroddsen, Sigurdur T cc
    Tzavaras, Athanasios cc
    Bona, Jerry L.
    Program
    Applied Mathematics and Computational Science
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2020-07-24
    Embargo End Date
    2021-12-03
    Permanent link to this record
    http://hdl.handle.net/10754/664399
    
    Metadata
    Show full item record
    Access Restrictions
    At the time of archiving, the student author of this dissertation opted to temporarily restrict access to it. The full text of this dissertation will become available to the public after the expiration of the embargo on 2021-12-03.
    Abstract
    This dissertation revolves around various mathematical aspects of nonlinear wave motion in viscoelasticity and free surface flows. The introduction is devoted to the physical derivation of the stress-strain constitutive relations from the first principles of Newtonian mechanics and is accessible to a broad audience. This derivation is not necessary for the analysis carried out in the rest of the thesis, however, is very useful to connect the different-looking partial differential equations (PDEs) investigated in each subsequent chapter. In the second chapter we investigate a multi-dimensional scalar wave equation with memory for the motion of a viscoelastic material described by the most general linear constitutive law between the stress, strain and their rates of change. The model equation is rewritten as a system of first-order linear PDEs with relaxation and the well-posedness of the Cauchy problem is established. In the third chapter we consider the Euler equations describing the evolution of a perfect, incompressible, irrotational fluid with a free surface. We focus on the Hamiltonian description of surface waves and obtain a recursion relation which allows to expand the Hamiltonian in powers of wave steepness valid to arbitrary order and in any dimension. In the case of pure gravity waves in a two-dimensional flow there exists a symplectic coordinate transformation that eliminates all cubic terms and puts the Hamiltonian in a Birkhoff normal form up to order four due to the unexpected cancellation of the coefficients of all fourth order non-generic resonant terms. We explain how to obtain higher-order vanishing coefficients. Finally, using the properties of the expansion kernels we derive a set of nonlinear evolution equations for unidirectional gravity waves propagating on the surface of an ideal fluid of infinite depth and show that they admit an exact traveling wave solution expressed in terms of Lambert’s W-function. The only other known deep fluid surface waves are the Gerstner and Stokes waves, with the former being exact but rotational whereas the latter being approximate and irrotational. Our results yield a wave that is both exact and irrotational, however, unlike Gerstner and Stokes waves, it is complex-valued.
    Citation
    Ussembayev, N. (2020). NonlinearWave Motion in Viscoelasticity and Free Surface Flows. KAUST Research Repository. https://doi.org/10.25781/KAUST-25587
    DOI
    10.25781/KAUST-25587
    ae974a485f413a2113503eed53cd6c53
    10.25781/KAUST-25587
    Scopus Count
    Collections
    Applied Mathematics and Computational Science Program; Dissertations; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

    entitlement

     
    DSpace software copyright © 2002-2021  DuraSpace
    Quick Guide | Contact Us | Send Feedback
    Open Repository is a service hosted by 
    Atmire NV
     

    Export search results

    The export option will allow you to export the current search results of the entered query to a file. Different formats are available for download. To export the items, click on the button corresponding with the preferred download format.

    By default, clicking on the export buttons will result in a download of the allowed maximum amount of items. For anonymous users the allowed maximum amount is 50 search results.

    To select a subset of the search results, click "Selective Export" button and make a selection of the items you want to export. The amount of items that can be exported at once is similarly restricted as the full export.

    After making a selection, click one of the export format buttons. The amount of items that will be exported is indicated in the bubble next to export format.