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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Homentrop</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p><b>Homentrop</b> ist ein Begriff aus der <a href="Str%C3%B6mungslehre" class="mw-redirect" title="Strömungslehre">Strömungslehre</a> und bezeichnet eine <a href="Isentrope" class="mw-redirect" title="Isentrope">isentrope</a> Strömung:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {D} s}{\mathrm {D} t}}=0,}">
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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} s}{\mathrm {D} t}}=0,}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d06fae8ef4fa03d4c36a2693455b0482dededf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.61ex; height:5.176ex;" alt="{\displaystyle {\frac {\mathrm {D} s}{\mathrm {D} t}}=0,}" loading="lazy"></span></dd></dl>
<p>in der die spezifische <a href="Entropie_(Thermodynamik)" class="mw-redirect" title="Entropie (Thermodynamik)">Entropie</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">
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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span>, d. h. die Entropie pro Masseteilchen, <a href="Homogenit%C3%A4t" title="Homogenität">homogen</a> verteilt ist:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla s=0}">
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<annotation encoding="application/x-tex">{\displaystyle \nabla s=0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/428040c5898afa822c3f5c10fc7eed7a7f5280d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.287ex; height:2.176ex;" alt="{\displaystyle \nabla s=0}" loading="lazy"></span></dd></dl>
<p>mit dem <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla .}">
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<mi mathvariant="normal">∇<!-- ∇ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \nabla .}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46f06d4d98a887e6430b4b67aab1a7c3f191b9a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="{\displaystyle \nabla .}" loading="lazy"></span>
</p><p>Anders ausgedrückt: die Entropie ist gleich verteilt, sowohl über der Zeit als auch im Raum. <i>Homentrop</i> beinhaltet somit auch die Vereinfachungen <a href="Reibungsfrei" class="mw-redirect" title="Reibungsfrei">reibungsfrei</a> und <i>keine</i> <a href="W%C3%A4rmeleitung" title="Wärmeleitung">Wärmeleitung</a>.
</p><p>Eine weitere Bedingung für Homentropie ist:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} p=a^{2}\cdot \mathrm {d} \rho }">
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<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} p=a^{2}\cdot \mathrm {d} \rho }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bb14cdf40bdb9b98ebc6e0f80d613c1197d2a5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.018ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} p=a^{2}\cdot \mathrm {d} \rho }" loading="lazy"></span></dd></dl>
<p>mit
</p>
<ul><li>dem <a href="Druck_(Physik)" title="Druck (Physik)">Druck</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> und</li>
<li>der <a href="Dichte" title="Dichte">Dichte</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho .}">
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<annotation encoding="application/x-tex">{\displaystyle \rho .}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ae3f23f76f614ab4dc47bfc296699c2be740666.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.849ex; height:2.176ex;" alt="{\displaystyle \rho .}" loading="lazy"></span></li></ul>
<p>Die <a href="Schallgeschwindigkeit" title="Schallgeschwindigkeit">Schallgeschwindigkeit</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> ist auf diese Weise definiert:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow a^{2}=\left({\frac {\partial p}{\partial \rho }}\right)_{s}}">
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow a^{2}=\left({\frac {\partial p}{\partial \rho }}\right)_{s}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3798990faafaf79041c5fb822fbc47fd699eb755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.132ex; height:6.176ex;" alt="{\displaystyle \Leftrightarrow a^{2}=\left({\frac {\partial p}{\partial \rho }}\right)_{s}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Bernoullische_Gleichung">Bernoullische Gleichung</h2></div>
<p>Für eine homentrope und <a href="Inkompressibles_Fluid" class="mw-redirect" title="Inkompressibles Fluid">inkompressible</a> Strömung kann über die <a href="Bernoullische_Gleichung" class="mw-redirect" title="Bernoullische Gleichung">Bernoullische Gleichung</a> der Zusammenhang zwischen Druck und Geschwindigkeit zwischen zwei Punkten berechnet werden:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \Phi }{\partial t}}+{\frac {1}{2}}\;\nabla \Phi \;\nabla \Phi +{\frac {p}{\rho }}+\psi =C(t)}">
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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Phi }{\partial t}}+{\frac {1}{2}}\;\nabla \Phi \;\nabla \Phi +{\frac {p}{\rho }}+\psi =C(t)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5375f2bab1c1691c75477c70f5f6a21edc7abf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.935ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \Phi }{\partial t}}+{\frac {1}{2}}\;\nabla \Phi \;\nabla \Phi +{\frac {p}{\rho }}+\psi =C(t)}" loading="lazy"></span></dd></dl></div><!--htdig_noindex--><div><div class="zim-footer">
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