Commit bf81afb0 authored by Olivier's avatar Olivier

Complete appendix section on field operators (div, grad, lap., curl)

parent 518eca1b
......@@ -134,7 +134,7 @@
\partialderivative{p}{x} \vec i + \partialderivative{p}{y} \vec j + \partialderivative{p}{z} \vec k & = & \rho \vec g \label{eq_threedgradientp}
\end{IEEEeqnarray}
We can simplify the writing of this last equation by using the mathematical operator \vocab{gradient}, defined as so:
We can simplify the writing of this last equation by using the mathematical operator \vocab{gradient} (see also Appendix~\ref{appendix_field_operators} p.\pageref{appendix_field_operators}), defined as so:
\begin{IEEEeqnarray}{rCl}
\gradient{} \equiv \vec i \partialderivative{}{x} + \vec j \partialderivative{}{y} + \vec k \partialderivative{}{z} \label{eq_def_gradient}
\end{IEEEeqnarray}
......
......@@ -107,7 +107,7 @@
&=& \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}
If we make use of the operator \vocab{divergent} written $\divergent{}$:
If we make use of the operator \vocab{divergent} (see also Appendix~\ref{appendix_field_operators} p.\pageref{appendix_field_operators}), written~$\divergent{}$~:
\begin{IEEEeqnarray}{rCcCl}
\divergent{} &\equiv& \partialderivative{}{x} \vec i \cdot \ + \ \partialderivative{}{y} \vec j \cdot \ + \ \partialderivative{}{z} \vec k \cdot \label{eq_def_divergent}\\
\divergent{\vec A} &\equiv& \partialderivative{A_x}{x} \ + \ \partialderivative{A_y}{y} \ + \ \partialderivative{A_z}{z}\\
......
......@@ -306,7 +306,7 @@
&=& \mu \left(\secondpartialderivative{u}{x} + \secondpartialderivative{u}{y} + \secondpartialderivative{u}{z}\right) \vec i\label{eq_tmp_shear_der}
\end{IEEEeqnarray}
Using the \vocab{Laplacian} operator to represent the spatial variation of the spatial variation of an object:
Using the \vocab{Laplacian} operator (see also Appendix~\ref{appendix_field_operators} p.\pageref{appendix_field_operators}) to represent the spatial variation of the spatial variation of an object:
\begin{IEEEeqnarray}{rClll}
\laplacian{} &\equiv& \divergent{\gradient{}}\label{eq_def_laplacian}\\
\laplacian{A} &\equiv& \divergent{\gradient{A}}\\
......
......@@ -80,7 +80,7 @@
With potential flow, these two conditions are addressed as follows:
\begin{enumerate}
\item We restrict ourselves to \vocab{irrotational} flows, those in which the curl of velocity (see Appendix~\ref{appendix_vector_operators} p.\ref{appendix_vector_operators}) is always null:
\item We restrict ourselves to \vocab{irrotational} flows, those in which the curl of velocity (see Appendix~\ref{appendix_field_operators} p.\pageref{appendix_field_operators}) is always null:
\begin{IEEEeqnarray}{rCl}
\curl{\vec V} &=& \vec 0
\end{IEEEeqnarray}
......
......@@ -9,6 +9,9 @@
\clearpage
\renewcommand{\theequation}{A/\arabic{equation}}
\setcounter{equation}{0}
\section{Notation}
......@@ -24,42 +27,148 @@
\item[straight subscripts]%
Points in space or in time (temperature~$T_\A$ at point~A).\\
Subscripts “cst” indicate a constant property, “in” indicates “incoming” and “out” is “outgoing”.\\
Subscribt “av.” indicates “average”.
Subscript “av.” indicates “average”.
\item[lowercase symbols]%
Specific values: property per unit mass. For example, $ b \equiv B/m $.
\item[operators] Differential $\diff$, partial differential $\partial$, finite differential $\delta$, total (alt.: subtantial) derivative $\text{D}/\text{D}t$ (def. eq.~\ref{eq_totaltimederivative} p.\pageref{eq_totaltimederivative}), exponential $\exp x \equiv e^x $, natural logarithm $\ln x \equiv \log_e x$.
\item[vectors] Vectors are always written with an arrow. Velocity is $\vec V \equiv (u, v, w)$, alternatively written $u_i \equiv (u, v, w)$. The norm of a vector $\vec A$ (positive or negative) is $|\vec A|$, its length (always positive) is $||\vec A||$.
\item[vector calculus]%
Dot product $\vec A \cdot \vec B$; cross product $\vec A \wedge \vec B$; gradient $\gradient{A}$ (def. eq.~\ref{eq_def_gradient} p.~\pageref{eq_def_gradient}); divergent $\divergent{\vec A}$ (def. eq.~\ref{eq_def_divergent} p.\pageref{eq_def_divergent}); Laplacian $\laplacian{\vec A}$ (def. eq.~\ref{eq_def_laplacian} p.\pageref{eq_def_laplacian}); curl $\curl{\vec A}$ (def. eq.~\ref{eq_def_curl} p.\pageref{eq_def_curl}).
\item[units] Units are typed in roman (normal) font and colored gray (\SI{1}{\kilogram}). In sentences units are fully-spelled and conjugated (one hundred \si{watts}). The \si{liter} is noted \si{\liter} to increase readablility ($\SI{1}{\liter} \equiv \SI{e-3}{\metre\cubed}$). Units in equations are those from \textit{système international} (\textsc{si}) unless otherwise indicated.
\item[numbers] The decimal separator is a comma, the decimal exponent is preceded by a dot, integers are written in groups of three ($\SI{1,234e3} ~=~ \num{1234}$). Numbers are rounded up as late as possible and never in series. Leading and trailing zeroes are never indicated.
Dot product $\vec A \cdot \vec B$; cross product $\vec A \wedge \vec B$; gradient $\gradient{A}$ (def. eq.~\ref{eq_def_gradient} p.\pageref{eq_def_gradient}); divergent $\divergent{\vec A}$ (def. eq.~\ref{eq_def_divergent} p.\pageref{eq_def_divergent}); Laplacian $\laplacian{\vec A}$ (def. eq.~\ref{eq_def_laplacian} p.\pageref{eq_def_laplacian}); curl $\curl{\vec A}$ (def. eq.~\ref{eq_def_curl} p.\pageref{eq_def_curl}).
\item[units] Units are typed in roman (normal) font and colored gray (\SI{1}{\kilogram}). In sentences units are fully-spelled and conjugated (one hundred \si{watts}). The \si{liter} is noted \si{\liter} to increase readability ($\SI{1}{\liter} \equiv \SI{e-3}{\metre\cubed}$). Units in equations are those from \textit{système international} (\textsc{si}) unless otherwise indicated.
\item[numbers] The decimal separator is a comma, the decimal exponent is preceded by a dot, integers are written in groups of three ($\SI{1,234e3} ~=~ \num{1234}$). Numbers are rounded up as late as possible and never in series. Leading and trailing zeros are never indicated.
\end{description}
\clearpage
\section{Vector operators}
\label{appendix_vector_operators}
The mathematical operator \vocab{curl} (sometimes named \vocab{rotational}) is written~$\curl{}$ and defined as:
\begin{IEEEeqnarray}{rCCCl}
\curl{} &\equiv& \begin{vmatrix}
\vec i &\vec j &\vec k \\
\partialderivative{}{x} &\partialderivative{}{y} &\partialderivative{}{z} \\
& & \end{vmatrix}\\
\curl{\vec A} &\equiv& \begin{vmatrix}
\vec i &\vec j &\vec k \\
\partialderivative{}{x} &\partialderivative{}{y} &\partialderivative{}{z} \\
A_x &A_y &A_z \end{vmatrix} &=& \left(\partialderivative{A_z}{y} - \partialderivative{A_y}{z}\right) \vec i \ \ + \ \ \left(-\partialderivative{A_z}{x} + \partialderivative{A_x}{z}\right) \vec j \ \ + \ \ \left(\partialderivative{A_y}{x} - \partialderivative{A_x}{y}\right) \vec k \nonumber\\\label{eq_def_curl}
\end{IEEEeqnarray}
With this definition, we the curl of the velocity field is:
\begin{IEEEeqnarray}{rCCCl}
\curl{\vec V} &=& \begin{vmatrix}
\vec i &\vec j &\vec k \\
\partialderivative{}{x} &\partialderivative{}{y} &\partialderivative{}{z} \\
u &v &w \end{vmatrix} &=& \left(\partialderivative{w}{y} - \partialderivative{v}{z}\right) \vec i \ \ + \ \ \left(-\partialderivative{w}{x} + \partialderivative{u}{z}\right) \vec j \ \ + \ \ \left(\partialderivative{v}{x} - \partialderivative{u}{y}\right) \vec k \nonumber\\
\end{IEEEeqnarray}
\section{Field operators}
\label{appendix_field_operators}
Four operators which apply on vector or scalar fields are important in fluid mechanics: gradient, divergent, Laplacian and curl.
\subsection{Gradient}
The mathematical operator \vocab{gradient} (first introduced as eq.~\ref{eq_def_gradient} p.~\pageref{eq_def_gradient}) is written~$\gradient{}$~. It applies on a scalar field and produces a vector field. It is defined~as:
\begin{IEEEeqnarray}{rCcCl}
\gradient{} &\equiv& \vec i \partialderivative{}{x} + \vec j \partialderivative{}{y} + \vec k \partialderivative{}{z}\\
\gradient{A} &\equiv& \partialderivative{A}{x} \vec i + \partialderivative{A}{y} \vec j + \partialderivative{A}{z} \vec k %
&=& \left(\begin{array}{c}%
\partialderivative{A}{x}\\
\partialderivative{A}{y}\\
\partialderivative{A}{z}\\
\end{array}\right)
\end{IEEEeqnarray}
For example, the gradient of a pressure field is the vector field $\gradient{p}$:
\begin{IEEEeqnarray}{rCcCl}
\gradient{p} &\equiv& \partialderivative{p}{x} \vec i + \partialderivative{p}{y} \vec j + \partialderivative{p}{z} \vec k %
&=& \left(\begin{array}{c}%
\partialderivative{p}{x}\\
\partialderivative{p}{y}\\
\partialderivative{p}{z}\\
\end{array}\right)
\end{IEEEeqnarray}
\subsection{Divergent}
The mathematical operator \vocab{divergent} (first introduced as eq.~\ref{eq_def_divergent} p.\pageref{eq_def_divergent}) is written~$\divergent{}$~ and is defined~as:
\begin{IEEEeqnarray}{rCl}
\divergent{} &\equiv& \partialderivative{}{x} \vec i \cdot \ + \ \partialderivative{}{y} \vec j \cdot \ + \ \partialderivative{}{z} \vec k \cdot
\end{IEEEeqnarray}
When applied on a vector field, it produces a scalar field:
\begin{IEEEeqnarray}{rCl}
\divergent{\vec A} & \equiv & \partialderivative{}{x} \vec i \cdot \vec A \ + \ \partialderivative{}{y} \vec j \cdot \vec A \ + \ \partialderivative{}{z} \vec k \cdot \vec A\\
&=& \partialderivative{A_x}{x} \ + \ \partialderivative{A_y}{y} \ + \ \partialderivative{A_z}{z}
\end{IEEEeqnarray}
When applied on a 2\up{nd} order tensor field, it produces a vector field:
\begin{IEEEeqnarray}{rCcCl}
\divergent{\vec A_{ij}} &\equiv& \left(\begin{array}{c}%
\partialderivative{A_{xx}}{x} + \partialderivative{A_{yx}}{y} + \partialderivative{A_{zx}}{z}\\
\partialderivative{A_{xy}}{x} + \partialderivative{A_{yy}}{y} + \partialderivative{A_{zy}}{z}\\
\partialderivative{A_{xz}}{x} + \partialderivative{A_{yz}}{y} + \partialderivative{A_{zz}}{z}\\
\end{array}\right) &=&%
\left(\begin{array}{c}%
\divergent{\vec A_{ix}}\\
\divergent{\vec A_{iy}}\\
\divergent{\vec A_{iz}}\\
\end{array}\right)
\end{IEEEeqnarray}
For example, the divergent of a velocity field is the scalar field $\divergent{\vec V}$:
\begin{IEEEeqnarray}{rCl}
\divergent{\vec V} &\equiv& \partialderivative{u}{x} \ + \ \partialderivative{v}{y} \ + \ \partialderivative{w}{z}
\end{IEEEeqnarray}
\subsection{Laplacian}
The mathematical operator \vocab{Laplacian} (first introduced as eq.~\ref{eq_def_laplacian} p.\pageref{eq_def_laplacian}) is written~$\laplacian{}$ and defined~as:
\begin{IEEEeqnarray}{rCl}
\laplacian{} &\equiv& \divergent{\gradient{}}\label{eq_def_laplacian}
\end{IEEEeqnarray}
When applied to a scalar field, it is equal to the divergent of the gradient of the field, and produces a scalar field:
\begin{IEEEeqnarray}{rCl}
\laplacian{A} &\equiv& \divergent{\gradient{A}}\\
&=& \secondpartialderivative{A}{x} + \secondpartialderivative{A}{y} + \secondpartialderivative{A}{z}
\end{IEEEeqnarray}
When applied to a vector field, it is equal to the gradient of the divergent of the field, and produces a vector field:
\begin{IEEEeqnarray}{rCl}
\laplacian{\vec A} &\equiv& %
\left(\begin{array}{c}%
\laplacian{A_x}\\
\laplacian{A_y}\\
\laplacian{A_z}\\
\end{array}\right) \ = \
\left(\begin{array}{c}%
\divergent{\gradient{A_x}}\\
\divergent{\gradient{A_y}}\\
\divergent{\gradient{A_z}}\\
\end{array}\right)\\
&=& \left(\begin{array}{c}%
\secondpartialderivative{A_x}{x} + \secondpartialderivative{A_x}{y} + \secondpartialderivative{A_x}{z}\\
\secondpartialderivative{A_y}{x} + \secondpartialderivative{A_y}{y} + \secondpartialderivative{A_y}{z}\\
\secondpartialderivative{A_z}{x} + \secondpartialderivative{A_z}{y} + \secondpartialderivative{A_z}{z}\\
\end{array}\right)
\end{IEEEeqnarray}
For example, the Laplacian of a velocity field is the vector field $\laplacian{\vec V}$:
\begin{IEEEeqnarray}{rCcCl}
\laplacian{\vec V} &\equiv& %
\left(\begin{array}{c}%
\laplacian{u}\\
\laplacian{v}\\
\laplacian{w}\\
\end{array}\right)
&=& \left(\begin{array}{c}%
\secondpartialderivative{u}{x} + \secondpartialderivative{u}{y} + \secondpartialderivative{u}{z}\\
\secondpartialderivative{v}{x} + \secondpartialderivative{v}{y} + \secondpartialderivative{v}{z}\\
\secondpartialderivative{w}{x} + \secondpartialderivative{w}{y} + \secondpartialderivative{w}{z}\\
\end{array}\right)
\end{IEEEeqnarray}
\subsection{Curl}
The mathematical operator \vocab{curl} (sometimes named \vocab{rotational}) is written~$\curl{}$~. It applies to a vector field and produces a vector field. It is defined~as:
\begin{IEEEeqnarray}{rCCCl}
\curl{} &\equiv& \begin{vmatrix}
\vec i &\vec j &\vec k \\
\partialderivative{}{x} &\partialderivative{}{y} &\partialderivative{}{z} \\
& & \end{vmatrix}\\
\curl{\vec A} &\equiv& \begin{vmatrix}
\vec i &\vec j &\vec k \\
\partialderivative{}{x} &\partialderivative{}{y} &\partialderivative{}{z} \\
A_x &A_y &A_z \end{vmatrix} &=& \left(\partialderivative{A_z}{y} - \partialderivative{A_y}{z}\right) \vec i \ \ + \ \ \left(-\partialderivative{A_z}{x} + \partialderivative{A_x}{z}\right) \vec j \ \ + \ \ \left(\partialderivative{A_y}{x} - \partialderivative{A_x}{y}\right) \vec k \nonumber\\\label{eq_def_curl}
\end{IEEEeqnarray}
For example, the curl of velocity is the vector field $\curl{\vec V}$:
\begin{IEEEeqnarray}{rCCCl}
\curl{\vec V} &=& \begin{vmatrix}
\vec i &\vec j &\vec k \\
\partialderivative{}{x} &\partialderivative{}{y} &\partialderivative{}{z} \\
u &v &w \end{vmatrix} &=& \left(\partialderivative{w}{y} - \partialderivative{v}{z}\right) \vec i \ \ + \ \ \left(-\partialderivative{w}{x} + \partialderivative{u}{z}\right) \vec j \ \ + \ \ \left(\partialderivative{v}{x} - \partialderivative{u}{y}\right) \vec k \nonumber\\
\end{IEEEeqnarray}
\clearpage
......
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