Commit bf81afb0 by 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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