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Commits (4)
 \renewcommand{\lastedityear}{2019} \renewcommand{\lasteditmonth}{05} \renewcommand{\lasteditday}{01} \renewcommand{\lasteditday}{10} \renewcommand{\numberofthischapter}{4} \renewcommand{\titleofthischapter}{\namechapterfour} ... ... @@ -25,7 +25,7 @@ %%%% \subsection{Magnitude of the pressure force} \youtubethumb{ke4Rf9FrHnw}{when pressure-induced forces in static fluids matter: 24 hours of heavy tonnage transit through the \textit{Miraflores} locks in Panama. Can you quantify the force applying on a single lock door?}{Y:XyliboxFrance (\styl)} What is the force which which a fluid pushes against a wall? What is the force with which a fluid pushes against a wall? When the pressure $p$ exerted is uniform and the wall is flat, the resulting force $F$ is easily calculated: \begin{IEEEeqnarray}{rCcCl} ... ... @@ -111,7 +111,7 @@ \subsection{The direction of pressure} An important concept is that in continuum mechanics, the flat surface is imaginary. More precisely, a fluid is able to exert pressure on not only on solid surfaces, but also upon and within itself. In this context, we need to rework eq.~\ref{eq_first_def_pressure_two} so that now pressure is defined as perpendicular force per area on an \emph{infinitesimally small} surface of fluid: An important concept is that in continuum mechanics, the flat surface is imaginary. More precisely, a fluid is able to exert pressure not only on solid surfaces, but also upon and within itself. In this context, we need to rework eq.~\ref{eq_first_def_pressure_two} so that now pressure is defined as perpendicular force per area on an \emph{infinitesimally small} surface of fluid: \begin{IEEEeqnarray}{rCl} p &\equiv& \lim_{A \to 0} \frac{F_\perp}{A} \label{eq_def_pressure} \end{IEEEeqnarray} ... ...
 \renewcommand{\lastedityear}{2019} \renewcommand{\lasteditmonth}{03} \renewcommand{\lasteditday}{31} \renewcommand{\lasteditmonth}{05} \renewcommand{\lasteditday}{10} \renewcommand{\numberofthischapter}{5} \renewcommand{\titleofthischapter}{\namechapterfive} ... ... @@ -35,20 +35,21 @@ When the shear $\tau$ exerted by the fluid is not uniform (for example, because more friction is occurring on some parts of the surface than on others), the situation is more complex: the force must be obtained by integration. The surface is split in infinitesimal portions of area $\diff S$, and the corresponding forces are summed up as: \begin{IEEEeqnarray}{rCcCl} F_\text{{shear, direction } i} & = & \int_S \diff F_{\text{shear, direction } i} & = & \int_S \tau_{\text{direction } i} \diff S \label{eq_shear_force_scalar} F_{\text{shear, direction } i} & = & \int_S \diff F_{\text{shear, direction } i} & = & \int_S \tau_{\text{direction } i} \diff S \label{eq_shear_force_scalar} \end{IEEEeqnarray} \begin{equationterms} \item for a flat surface, \item where the $S$-integral denotes an integration over the entire surface. \end{equationterms} What is required to calculate the scalar $F$ in eq.~\ref{eq_shear_force_scalar} is an expression of~$\tau$ as a function of~$S$. In a simple laminar flow, this expression will often be relatively easy to find, as we see later on. Typically, in two dimensions $x$ and $y$ we re-write $\diff S$ as $\diff S = \diff x \diff y$ and we may then proceed with the calculation starting from What is required to calculate the scalar $F$ in eq.~\ref{eq_shear_force_scalar} is an expression of~$\tau$ as a function of~$S$. In a simple laminar flow, this expression will often be relatively easy to find, as we see later on. Typically, in two dimensions $x$ and $y$ we re-write $\tdiff S$ as $\tdiff S = \diff x \diff y$ and we may then proceed with the calculation starting from \begin{mdframed} \begin{IEEEeqnarray}{rCcCl} F_\text{{shear, direction } i} & = & \iint \tau_{\text{direction } i (x, y)} \diff x \diff y \label{eq_shear_force_twod_integration} \end{IEEEeqnarray} \end{mdframed} %%%% \subsection{Direction and position of the shear force} The above equations work only for a flat surface, and in a chosen direction $i$. When we consider a two- or three-dimensional object immersed in a fluid with non-uniform shear, the integration must be carried out with vectors. We will not attempt this in this course, but the expression is worth writing out in order to understand how computational fluid dynamics (\cfd) software will proceed with the calculation. ... ... @@ -144,13 +145,13 @@ \end{IEEEeqnarray} Given that $S_3 = S_6 = \diff x \diff y$, that $S_2 = S_5 = \diff x \diff z$ and that $S_1 = S_4 = \diff z \diff y$, this is re-written as: \begin{IEEEeqnarray}{rCl} \vec F_{\text{shear}\ x} &=& \diff x \diff y (\vec \tau_{zx\ 3} - \vec \tau_{zx\ 6})\nonumber\\ && + \diff x \diff z (\vec \tau_{yx\ 2} - \vec \tau_{yx\ 5})\nonumber\\ && + \diff z \diff y (\vec \tau_{xx\ 1} - \vec \tau_{xx\ 4})\label{eq_fshear_xdir} \vec F_{\text{shear}\ x} &=& \diff x \diff y \ (\vec \tau_{zx\ 3} - \vec \tau_{zx\ 6})\nonumber\\ && + \diff x \diff z \ (\vec \tau_{yx\ 2} - \vec \tau_{yx\ 5})\nonumber\\ && + \diff z \diff y \ (\vec \tau_{xx\ 1} - \vec \tau_{xx\ 4})\label{eq_fshear_xdir} \end{IEEEeqnarray} In the same way we did with pressure in \chapterfourshort (\S\ref{ch_pressure_and_depth} p.\pageref{ch_pressure_and_depth}), we express each pair of values as derivative with respect to space multiplied by an infinitesimal distance: \begin{IEEEeqnarray}{rCl} \vec F_{\text{shear}\ x} &=& \diff x \diff y \left(\diff z \partialderivative{\vec \tau_{zx}}{z}\right) + \diff x \diff z \left(\diff y \partialderivative{\vec \tau_{yx}}{y}\right) + \diff z \diff y \left(\diff x \partialderivative{\vec \tau_{xx}}{x}\right)\nonumber\\ \vec F_{\text{shear}\ x} &=& \diff x \diff y \left(\tdiff z \partialderivative{\vec \tau_{zx}}{z}\right) + \diff x \diff z \left(\tdiff y \partialderivative{\vec \tau_{yx}}{y}\right) + \diff z \diff y \left(\tdiff x \partialderivative{\vec \tau_{xx}}{x}\right)\nonumber\\ &=& \diff \vol \left(\partialderivative{\vec \tau_{zx}}{z} + \partialderivative{\vec \tau_{yx}}{y} + \partialderivative{\vec \tau_{xx}}{x}\right)\label{eq_shear_force_x_tmp} \end{IEEEeqnarray} ... ...
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  Config.pm:302> INFO - This is Biber 2.9  Config.pm:305> INFO - Logfile is 'fluidmech.blg'  biber:313> INFO - === Wed May 1, 2019, 14:16:29  Biber.pm:371> INFO - Reading 'fluidmech.bcf'  Biber.pm:857> INFO - Found 19 citekeys in bib section 0  Biber.pm:3981> INFO - Processing section 0  Biber.pm:4154> INFO - Looking for bibtex format file '/home/olivier/.fluidmechbib.bib' for section 0  bibtex.pm:1468> INFO - LaTeX decoding ...  bibtex.pm:1294> INFO - Found BibTeX data source '/home/olivier/.fluidmechbib.bib'  UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable'  UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized'  Biber.pm:3809> INFO - Sorting list 'ynt/global//global/global' of type 'entry' with template 'ynt' and locale 'en-US'  Biber.pm:3815> INFO - No sort tailoring available for locale 'en-US'  bbl.pm:617> INFO - Writing 'fluidmech.bbl' with encoding 'UTF-8'  bbl.pm:720> INFO - Output to fluidmech.bbl  biber:313> INFO - === Fri May 10, 2019, 10:07:34  Biber.pm:371> INFO - Reading 'fluidmech.bcf'  Biber.pm:857> INFO - Found 19 citekeys in bib section 0  Biber.pm:3981> INFO - Processing section 0  Biber.pm:4154> INFO - Looking for bibtex format file '/home/olivier/.fluidmechbib.bib' for section 0  bibtex.pm:1468> INFO - LaTeX decoding ...  bibtex.pm:1294> INFO - Found BibTeX data source '/home/olivier/.fluidmechbib.bib'  UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable'  UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized'  Biber.pm:3809> INFO - Sorting list 'ynt/global//global/global' of type 'entry' with template 'ynt' and locale 'en-US'  Biber.pm:3815> INFO - No sort tailoring available for locale 'en-US'  bbl.pm:617> INFO - Writing 'fluidmech.bbl' with encoding 'UTF-8'  bbl.pm:720> INFO - Output to fluidmech.bbl
 This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2018.10.12) 1 MAY 2019 14:19 This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2018.10.12) 10 MAY 2019 10:10 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. ... ... @@ -2992,16 +2992,9 @@ Package biblatex Warning: 'babel/polyglossia' detected but 'csquotes' missing. (./fluidmech.aux (./0/titlepage.aux) (./0/introduction.aux) (./1/chap1.aux) (./1/exercises1.aux) (./2/chap2.aux) (./2/exercises2.aux) (./3/chap3.aux) (./3/exercises3.aux) (./4/chap4.aux) (./4/exercises4.aux) (./5/chap5.aux) (./5/exercises5.aux) (./6/chap6.aux LaTeX Warning: Label eq_continuity_der' multiply defined. LaTeX Warning: Label eq_continuity_der@cref' multiply defined. ) (./6/exercises6.aux) (./7/chap7.aux) (./7/exercises7.aux) (./8/chap8_placeholder.aux) (./9/chap9.aux) (./9/exercises9.aux) (./10/chap10.aux) (./10/exercises10.aux) (./11/chap11.aux) (./5/exercises5.aux) (./6/chap6.aux) (./6/exercises6.aux) (./7/chap7.aux) (./7/exercises7.aux) (./8/chap8_placeholder.aux) (./9/chap9.aux) (./9/exercises9.aux) (./10/chap10.aux) (./10/exercises10.aux) (./11/chap11.aux) (./11/exercises11.aux) (./a/appendix.aux)) \openout1 = fluidmech.aux'. ... ... @@ -3609,7 +3602,7 @@ LaTeX Font Warning: Size substitutions with differences LaTeX Font Warning: Some font shapes were not available, defaults substituted. LaTeX Warning: There were multiply-defined labels. LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. Package logreq Info: Writing requests to 'fluidmech.run.xml'. \openout1 = fluidmech.run.xml'. ... ... @@ -3618,7 +3611,7 @@ Package logreq Info: Writing requests to 'fluidmech.run.xml'. Here is how much of TeX's memory you used: 62600 strings out of 494214 1148350 string characters out of 6166503 1748200 words of memory out of 5000000 1748161 words of memory out of 5000000 64429 multiletter control sequences out of 15000+600000 762686 words of font info for 348 fonts, out of 8000000 for 9000 500 hyphenation exceptions out of 8191 ... ...
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