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| -rw-r--r-- | ph2hat_zf.tex | 478 |
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For +the avoidance of doubt, this paragraph does not form part of the +public licenses. + +Creative Commons may be contacted at creativecommons.org. + diff --git a/ph2hat_zf.pdf b/ph2hat_zf.pdf Binary files differnew file mode 100644 index 0000000..328183f --- /dev/null +++ b/ph2hat_zf.pdf diff --git a/ph2hat_zf.tex b/ph2hat_zf.tex new file mode 100644 index 0000000..55c9a33 --- /dev/null +++ b/ph2hat_zf.tex @@ -0,0 +1,478 @@ +\documentclass[a4paper, twocolumn]{article} + +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsthm} +% \usepackage{bm} + +\numberwithin{equation}{section} + +\usepackage{array} +\usepackage{parskip} +\usepackage{booktabs} + +\usepackage[margin=2cm]{geometry} + +\usepackage{siunitx} + +\usepackage{xcolor} +\usepackage[colorlinks = true, + linkcolor = red!50!black, + urlcolor = blue, + citecolor = black, + anchorcolor = blue]{hyperref} + +\usepackage{polyglossia} +\setmainlanguage{german} + + +\newtheoremstyle{hsr-def}% + {1em}% above + {1em}% below + {}% body + {0pt}% indent + {}% headfont + { }% headpunkt + { }% headspace + {\textsc{\thmname{#1} \thmnumber{#2}.} \textbf{\thmnote{#3}}}% headspec + +\newtheoremstyle{hsr-sub}% + {1em}% above + {1em}% below + {}% body + {0pt}% indent + {}% headfont + { }% headpunkt + { }% headspace + {\textsl{\thmname{#1} \thmnumber{#2}.} \textbf{\thmnote{#3}}}% headspec + +\newtheoremstyle{hsr-unnum}% + {1em}% above + {1em}% below + {}% body + {0pt}% indent + {}% headfont + { }% headpunkt + { }% headspace + {\textsc{\thmname{#1}} \textbf{\thmnote{#3}}}% headspec + +\theoremstyle{hsr-def} +\newtheorem{definition}{Definition}[section] + +\theoremstyle{hsr-sub} +\newtheorem{result}{Folgerung}[definition] +\newtheorem{example}{Beispiel}[definition] + +\theoremstyle{hsr-unnum} +\newtheorem{remark}{Bemerkung}[definition] + + +\newcommand{\dd}[1]{\ensuremath{\mathrm{d}#1}} +\newcommand{\di}[1]{\,\dd{#1}} +\newcommand{\deriv}[2]{\ensuremath{\frac{\dd{#1}}{\dd{#2}}}} +\newcommand{\pderiv}[2]{\ensuremath{\frac{\partial #1}{\partial #2}}} + +\renewcommand{\vec}[1]{\ensuremath{\mathbf{#1}}} +\newcommand{\uvec}[1]{\ensuremath{\vec{\hat{#1}}}} + +\newcommand{\unitsof}[1]{\ensuremath{\left[\,#1\,\right]}} + + + +\begin{document} +\section{Einf\"uhrung} + +\begin{definition}[Fluid] +Fl\"ussigkeiten und Gase werden under dem Oberbegriff \emph{Fluide} zusammengefasst. +\end{definition} + +% \begin{description} +% \item[Isotherm] +% \item[Isobar] +% \item[Isotrop] +% \end{description} + + +\begin{definition}[Druck und Schubspannung] +F\"ur einfache F\"alle der Cauchy Spannugstensor kann zu zwei Skalare \(p\) und \(\tau\) vereinfacht werden, sie werden Dr\"uck bzw. Schubspannung genannt. +\begin{align*} + pA + &= \vec{F}\cdot\uvec{n} + = F_\perp + & + \tau A + &= \vec{F}\cdot\uvec{T} + = F_\parallel + \stackrel{\text{statik}}{=} 0 + \\ + \unitsof{p} &= \si{\newton\per\square\metre} = \si{\pascal} +\end{align*} + +\begin{result}[Gesetzt von Pascal] +In ruhenden Fluiden \(\tau = 0\), somit ist die Kraft immer senkrecht. +\end{result} +\end{definition} + +\begin{definition}[Dichte] Ist die Masse pro Volumeneinheit. +\[ + \varrho = \frac{m}{V} \qquad \unitsof{\varrho} = \si{\kilo\gram\per\cubic\metre} +\] +\end{definition} + +\section{Hydrostatik} + +\begin{definition}[Schweredruck] +\begin{equation} \label{eqn:hydrostatic-pressure} + \dd{p} = \varrho \vec{g} \cdot \dd{\vec{y}} = - \varrho g \dd{y} +\end{equation} + +\begin{result}[Hydrostatischer Druck] +F\"ur Fl\"ussigkeiten, da die Dichte konstant ist folgt: +\[ + p = \varrho g h +\] +\end{result} + +\begin{result}[Schweredruck eines Gase] +Angenommen dass, die Dichte nur von Druck abh\"angt (barotrop) +\[ + \varrho(p) = \varrho_0 \frac{p}{p_0} +\] +Die L\"osung von \eqref{eqn:hydrostatic-pressure} ergibt die \emph{Barometrische H\"ohenformel} f\"ur eine isotherme Atmosp\"are. +\[ + p(h) = p_0 \exp\left(-\frac{\varrho_0}{p_0} gh\right) +\] +\end{result} +\end{definition} + +\begin{definition}[Gesetz von Boyle-Mariotte] +% \label{def:boyle-mariotte} +F\"ur ein ideales Gas gilt bei konstanter Temperatur +\[ + pV = (\text{konstant}) +\] + +\begin{result} +Die Dichte ist proportional zum Druck +\[ + \frac{V_2}{V_1} = \frac{p_1}{p_2} = \frac{\varrho_1}{\varrho_2} +\] +\end{result} +\end{definition} + + +\begin{definition}[Kompressibilit\"at] +Die Druckerh\"ohung \(\Delta p\) bewirkt in einem Fluid stets eine Volumenabname. +Die relative Volumen\"anderung ist proportional zur Druck\"anderung +\[ + \Delta V / V = - \kappa \Delta p +\] + +\begin{remark} + Eine ideale Fl\"ussigkeit ist reibungsfrei und inkompressibel. +\end{remark} + +\begin{remark} + In einer idealen Fl\"ussigkeit ist die Dichte konstant. +\end{remark} +\end{definition} + +\begin{definition}[Statische Auftriebskraft] Auch als Archimedische Prinzip bekannt. +\[ + F_A = G_f = \varrho_f V_k g + \qquad + \uvec{F}_a = - \uvec{g} +\] +Der auftrieb eines in ein Fluid eingetauchen K\"orper ist gleich dem Gewicht des von ihm verdr\"angten Fluids. +\end{definition} + +\subsection{Grenzfl\"acheneffekte} + +\begin{definition}[Oberfl\"achenspannung] +Zwischen zwei Atomen oder Molek\"ulen tritt die \emph{Van der Waals}-Kraft. +An der Oberfl\"ache der Fl\"ussigkeit ist der mittlere Abstand der Molek\"ule etwas gr\"osser als im Innern. Das bewirkt eine Parallel zur Oberfl\"ache gerichtete anzihende Kraft zwischen den Molek\"ulen. +\[ + \sigma = \frac{F}{\ell} + \qquad + \unitsof{\sigma} = \si{\newton\per\metre} +\] +\begin{remark} +Die Oberfl\"achenspannung kann auch als \emph{spezifische Oberf\"achenenergie} bezeichnet werden. +\[ + \sigma + = \frac{\Delta W}{\Delta A} + = \frac{F\Delta s}{\ell \Delta s} + = \frac{F}{\ell} +\] +Die \emph{Oberfl\"achenenergie} ist ein Ma{\ss} f\"ur die Energie, die zum Aufbrechen der chemischen Bindungen notwendig ist, wenn eine neue Oberfl\"ache einer Fl\"ussigkeit oder eines Festkörpers erzeugt wird. +\end{remark} + +\begin{result}[Grenzfl\"achenspannung] +Bei einer Vergr\"osserung der Grenzfl\"ache muss Arbeit geleistet werden, da die Grenzfl\"achenenergie vergr\"ossert wird. +Es gibt dann auch die Grenzfl\"achenspannungen +\( + \sigma_\text{sl}, + \sigma_\text{sg}, + \sigma_\text{lg} +\) (fl\"ussig = liquid, fest = solid, gas) die zwischen Festk\"orper und Fl\"ussigkeit wirken. \(\varphi\) ist dann der \emph{Kontaktwinkel}, und die Geometrie ergibt die Beziehung +\[ + \sigma_\text{sg} = \sigma_\text{sl} + \sigma_\text{lg} \cos \varphi +\] +\end{result} + +\begin{example}[Druck in Seifenblase] +\[ + p = \frac{2\sigma}{r} +\] +\end{example} +\end{definition} + +\begin{definition}[Kapillarit\"at] +Allgemein an die Grenze gilt: +\[ + F_\text{Oberfl\"ache} = F_{G,\text{Fl\"ussigkeit}} +\] +\begin{example}[In einem Rohr (Zylinder)] +\[ + 2\pi r\sigma = \varrho\pi r^2 hg \implies h = \frac{2\sigma}{\varrho g r} +\] +\end{example} +\end{definition} + +\section{Hydrodynamik} +\subsection{Einf\"uhrung} +\begin{definition}[Kontinuit\"atsgleichung] +\begin{equation} \label{eqn:continuity} + \pderiv{}{t}\int_V \varrho \di{V} + = \oint_{\partial V} \varrho \vec{v}\cdot\di{\vec{s}} +\end{equation} + +\begin{result}[Ideales Fluid] +Da die Dichte konstant ist (inkompressibel), man kann \eqref{eqn:continuity} durch \(\varrho\) teilen und folgt: +\[ + \dot{V} = \int_A \vec{v}\cdot\di{\vec{s}} = vA + \qquad + \unitsof{\dot{V}} = \si{\cubic\metre\per\second} +\] +\end{result} + +% \begin{remark}[Differentialform] +% \[ +% \nabla \cdot (\varrho \vec{v}) + \pderiv{\varrho}{t} = 0 +% \] +% \end{remark} +\end{definition} + +\begin{definition}[Bernoulli Gleichung] +Der Term \(\varrho v^2 / 2\) wird \emph{dynamische Druck} genannt. +\[ + p + \varrho g h + \frac{\varrho}{2} v^2 = (\text{Konstant}) +\] + +\begin{remark} +Bernoulli gilt f\"ur inkompressible Fluide, und gen\"ugt f\"ur Fl\"ussigkeite und Gase, sofern \(v \ll \) Schallgeschwidigkeit. +\end{remark} + +\begin{result} +\begin{gather*} + p_1 + \varrho g h_1 + \frac{\varrho}{2} v_1^2 + = + p_2 + \varrho g h_2 + \frac{\varrho}{2} v_1^2 + \\ + \text{oder}\quad -\Delta p = \varrho g \Delta h + \frac{\varrho}{2} \Delta \left( v^2 \right) +\end{gather*} +\end{result} + +\begin{result} +Wo die Geschwindigkeit am schnellsten ist, dort ist die Druck am tiefsten. +\end{result} + +\end{definition} + +\subsection{Reale Str\"omungen} + +\begin{definition}[Newtonsche Reibungsgesetz] +Die Proportionalit\"atskonstante \(\eta\) wird \emph{dynamische Viskosit\"at} oder \emph{Z\"ahingkeit} genannt. +\begin{gather*} + \tau = \eta \deriv{v}{z} + \stackrel{!}{=} \frac{F_\parallel}{A} \\ + \unitsof{\eta} + = \si{\kilo\gram\per\metre\second} + = \si{\newton\second\per\metre} + = \si{\pascal\second} +\end{gather*} +\end{definition} + +\begin{definition}[Formel von Stokes] +(Stokes'sche Reibung) Reibungskraft einer Kugel im \"Ol +\[ + F_R = 6\pi\eta Rv_0 +\] +\end{definition} + +\begin{definition}[Laminare Rohrstr\"omung] +Lauten die Gleichgewichtsbedingungen f\"ur die Kr\"afte innerhalb des Zylinders. +\begin{align*} + F_\text{Res,Druck} - F_\text{Reib} & = 0 \\ + \pi r^2 (p_1 - p_2) - 2\pi rl\tau &= 0 +\end{align*} + +\begin{result}[Geschwindigkeitsverteilung] +Innerhalb des Zylinders (\(r\) von \(0\) bis \(R\)) +\[ + v(r) = \frac{\Delta p}{4\eta\ell}\left(R^2-r^2\right) +\] +\end{result} + +\begin{result}[Gesetz von Hagen Poiseuille] +\[ + \dot{V} = \frac{\pi\Delta p R^4}{8\eta\ell} +\] +\end{result} + +\begin{remark} +Bei einer Zunahme des Rohrradius wird nicht nur die zur Verf\"ugung stehende Querschnittsfl\"ache gr\"osser, sondern zugleich w\"achst in der Rohrmitte auch die maximale Geschwindigkeit. +\end{remark} +\end{definition} + +\subsection{Turbulente Str\"omung} + +\begin{definition}[Reynolds Zahl] +Ist ein dimensionslose Koeffizient aus der \emph{Navier-Stokes} Gleichung, der das Verh\"altnis zwischen kinetischer Energie des Fluides und dessen innerer Reibung (proportional zur Viskosit\"at) beschreibt. +\[ + \mathcal{R} = \frac{E_k}{E_r} = \frac{\varrho}{\eta} v^*\ell^* + %\frac{\Delta p}{\tau} +\] +\(v^*, \ell^*\) sind eine charakteristische L\"ange bzw. Geschwindigkeit. Sie sind dimensionslose Variablen f\"ur geometrische und physikalische Gr\"ossen. + +\begin{result}[Rohrstr\"omung] +Wird bei der Str\"omung durch ein Rohr mit kreisf\"ormigem Querschnitt der Durchmesser \(d\) als charakteristische Abmessung gew\"ahlt, sot ist die Reynolds-Zahl +\[ + \mathcal{R} = \frac{\varrho v d}{\eta} +\] +\end{result} + +\begin{result}[Kritische \(\mathcal{R}\)] + +\end{result} +\end{definition} + +\begin{definition}[Druckwiederstand] +\end{definition} + + +Druckdifferenz +\[ + \Delta p \propto \varrho \Delta v \cdot \bar{v} +\] + +Schubspannung +\[ + \tau \propto \eta\frac{\Delta V}{\ell} +\] + + +Prandtl'sche Genzschicht +\[ + D = \sqrt{\frac{\eta\ell}{\varrho v}} +\] + +Druckabfall laminar +\[ + \Delta p = 32\eta\ell \frac{v}{d^2} +\] + +\(\lambda\) nach Blasius, \emph{turulent} +\[ + \lambda_\text{t} = \frac{0.316}{\sqrt[4]{\mathcal{R}}} +\] + +\(\lambda\) nach Hagen-Poiseuille, \emph{laminar} +\[ + \lambda_\text{l} = \frac{64}{\mathcal{R}} +\] + +Turbulente Rohrst\"omung, je nach turbulent oder laminares \(\lambda\) +\[ + \Delta p = \lambda \frac{\varrho\ell}{2d} v^2 +\] + +\section*{Kapitel 6} +Druckwiederstand +\[ + F_D = c_W \frac{\varrho}{2}v^2 A_\perp +\] + +Auftriebskraft nach Kutta-Jukowski +\[ + F_A = \varrho v \ell \Gamma +\] + +Zirkulation +\[ + \Gamma = \oint \vec{v} \di{\vec{l}} +\] + +Induzierter Widerstand +\[ + F_W = \dot{c}_W \frac{\varrho}{2} v^2 A_\parallel +\] + +Dynamischer Auftrieb +\[ + F_A = c_A \frac{\varrho}{2} v^2 A_\perp +\] + +Gleitwinkel +\[ + \tan(\varphi) = \frac{F_W}{F_A} = \frac{c_W}{c_A} = \frac{v_V}{v_H} +\] + +\begin{thebibliography}{2} +\bibitem{hsr} + \textsc{Hochschule f\"ur Technik Rapperswil (HSR)}. + \textit{\texttt{Ph2HAT} Vorlesungen und die dazugeh\"orige Unterlagen,} + Sourlier David, + Fr\"uhlingssemester 2020, + Rapperswil. + +\bibitem{bucher-ruh} + \textsc{Arthur Ruh, Benno Bucher}. + \textit{Physik 1: Mechanik, Fluide, W\"armelehre}. + Vol I, HSR, 2014, Rapperswil. + +\bibitem{feynman1} + \textsc{Richard Feynman}. + \textit{Mainly Mechanics, radiation, and heat}. + \textit{The Feynman Lectures on Physics}, + Leighton, Sands, + New Millenium Edition, + Vol I, + Basic Books, + California Institute of Technology (Caltech). + +\bibitem{feynman2} + \textsc{Richard Feynman}. + \textit{Mainly electromagnetism and matter}. + \textit{The Feynman Lectures on Physics}, + Leighton, Sands, + New Millenium Edition, + Vol II, + Basic Books, + California Institute of Technology (Caltech). + +\end{thebibliography} + + +\section*{License} +{ \tt +Ph2HAT-ZF (c) by Naoki Pross +\\\\ +Ph2HAT-ZF is licensed under a Creative Commons Attribution-ShareAlike 4.0 Unported License. +\\\\ +You should have received a copy of the license along with this work. If not, see +\\\\ +{\small\url{http://creativecommons.org/licenses/by-sa/4.0/}} +} + +\end{document} |