Document Zbl 0698.76078

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Chicago, IL: Longman Scientific & Technical; New York etc.: Wiley. ix, 272 p. (1989). </div> <div class="abstract">This inviscid unsteady compressible flow is an interesting area in gas dynamics. It is governed by a system of quasi-linear hyperbolic differential equations. Since Riemann proposed and investigated the shock tube in 1860, problems in this area have attracted attention of many well-known applied mathematicians and physicists. In 1957, Lax proposed the concept of weak solution which laid the foundations for the rigorous analysis of the natural phenomenon of flow with discontinuities. It was recognized that the solution of the Riemann problem involves shock waves, centered refraction waves and contact discontinuities-elementary waves. The interaction of these waves exhibits essential features of these nonlinear waves and is important not only in establishing the existence theorem for a solution but also in providing the very detailed picture of solutions with rich geometric structure and interesting techniques of numerical computation. <br class="zbmathjax-paragraph">The present book by Zhang and Xiao on these subjects reports the collection of their research since 1963. Their original aim was to develop a method of analysis rigorous in the phase space. There are four chapters in this book. In chapter one, the authors considered the simplest model for hyperbolic equations. They intended to explain the ideas and motivations involved in later chapters through this simple model. Chapter two dealt with one-dimensional isothermal flow of an ideal gas, thus the energy equation can be conveniently eliminated and the mathematical analysis was simplified. Although this research may not be applied in gas dynamics, it can be applied to plastic-elasticity. Chapter three discussed the one-dimensional adiabatic flow. The authors tried to establish the results, previously obtained by others, in a rigorous manner meanwhile making them complete and accurate. Perhaps the most important part of the book is the chapter four where the authors considered the two-dimensional flow where not much has been known. They transferred some fundamental concepts in the one-dimensional problem to the two-dimensional case. They examined the interactions of one shock overtaking another shock in a steady flow, and also studied the shock diffraction around a wedge. <br class="zbmathjax-paragraph">It is gratifying to learn that (1) the criterion of transition between Mach reflection and regular reflection proposed by von Neumann in 1943 was refined and (2) the early work by Bleakney and Taub was unsuccessful due to mistakes in their calculations. This has been the point of concern when this review was working on Mach reflection of shocks related to free jet flows [AIAA J. 13 (1975), 762-766 (1976)].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A3236">W.L.Chow</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A194030">Cited in <strong>175</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A76N15" title="MSC2020">76N15</a> </td> <td class="space"> Gas dynamics (general theory) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76-02" title="MSC2020">76-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to fluid mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ainviscid+unsteady+compressible+flow">inviscid unsteady compressible flow</a>; <a href="/?q=ut%3Aquasi-linear+hyperbolic+differential+equations">quasi-linear hyperbolic differential equations</a>; <a href="/?q=ut%3Aweak+solution">weak solution</a>; <a href="/?q=ut%3ARiemann+problem">Riemann problem</a>; <a href="/?q=ut%3Ashock+waves">shock waves</a>; <a href="/?q=ut%3Acentered+refraction+waves">centered refraction waves</a>; <a href="/?q=ut%3Acontact+discontinuities-elementary+waves">contact discontinuities-elementary waves</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0332.76053">Zbl 0332.76053</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0698.76078" data-ciurl="/ci/00194030" data-biburl="/bibtex/00194030.bib" data-amsurl="/amsrefs/00194030.bib" data-xmlurl="/xml/00194030.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00194030.pdf" title="Zbl 0698.76078 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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