Let us start with the unfortunate truth: not only are the methods developed here are incredibly complex, but they are also very ineffective to solve fluid flow problems with a pen and paper. Despite this, this chapter is extraordinarily important, for two reasons:

\begin{itemize}

\item derivative analysis allows us to formally \emph{describe} and \emph{relate} the key parameters that regulate fluid flow, and so is the key to developing an understanding of any fluid phenomenon, even when solutions cannot be derived;

\item it is the backbone for \vocab{computational fluid dynamics} (\textsc{cfd}) in which approximate flow solutions are obtained using numerical procedures.

\item derivative analysis allows us to formally \emph{describe} and \emph{relate} the key parameters that regulate fluid flow, and so, it is the key to developing an understanding of any fluid phenomenon, even when solutions cannot be derived;

\item it is the backbone for \vocab{computational fluid dynamics} (\textsc{cfd}) in which flow solutions are obtained using numerical procedures, in every problem of interest in research and industry today.

\end{itemize}

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@@ -187,7 +187,7 @@

This equation~\ref{eq_continuity_der} is named \vocab{continuity equation} and is of crucial importance in fluid mechanics. It sums up two terms:

\begin{itemize}

\item The first term on the left, $\inlinepartialtimederivative{\rho}$, is the local time-change of density. When the fluid “contracts”, density increases and the term becomes positive. In an incompressible flow, it is always zero.

\item The second term is the divergent of density times velocity, $\divergent{\rho\vec V}=\inlinepartialderivative{\rho u}{x}+\inlinepartialderivative{\rho v}+\inlinepartialderivative{\rho w}{z}$. It sums up the changes in space of the mass fluxes $\rho V_i$. For example, if a particle in a static fluid is heated up suddenly, it will expand and the divergent of $\rho\vec V$ will have positive value.

\item The second term is the divergent of density times velocity, $\divergent{\rho\vec V}=\inlinepartialderivative{\rho u}{x}+\inlinepartialderivative{\rho v}{y}+\inlinepartialderivative{\rho w}{z}$. It sums up the changes in space of the mass fluxes $\rho V_i$. For example, if a particle in a static fluid is heated up suddenly, it will expand and the divergent of $\rho\vec V$ will have positive value.

The Cauchy equation is an expression of Newton’s second law applied to a fluid particle. It expresses the time change of the velocity vector measured at a fixed point (the acceleration field $\text{D}\vec V/\text{D}t$) as a function of gravity, pressure and shear effects. This is quite a breakthrough. Within the chaos of an arbitrary flow, in which fluid particles are shoved, pressurized, squeezed, and distorted, we know precisely what we need to look for in order to quantify the time-change of velocity: gravity, the gradient of pressure, and the divergent of shear.

The Cauchy equation is a formulation of Newton’s second law applied to a fluid particle. It expresses the time change of the velocity vector measured at a fixed point (the acceleration field $\text{D}\vec V/\text{D}t$) as a function of gravity, pressure and shear effects. This is quite a breakthrough. Within the chaos of an arbitrary flow, in which fluid particles are shoved, pressurized, squeezed, and distorted, we know precisely what we need to look for in order to quantify the time-change of velocity: gravity, the gradient of pressure, and the divergent of shear.

Nevertheless, while it is an excellent start, this equation isn’t detailed enough for us. In our search for the velocity field $\vec V$, the changes in time and space of the shear tensor $\vec\tau_{ij}$ and pressure $p$ are unknowns. Ideally, those two terms should be expressed solely as a function of the flow’s other properties. Obtaining such an expression is what \we{Claude-Louis Navier} and \we{Gabriel Stokes} set themselves to in the 19\up{th} century: we follow their footsteps in the next paragraphs.

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@@ -246,14 +246,14 @@

%\subsection{Balance of linear momentum for Newtonian fluids: the Navier-Stokes equation}

\label{ch_navier-stokes}

The Navier-Stokes equation is the Cauchy equation (eq.~\ref{eq_cauchy}) applied to Newtonian fluids. In Newtonian fluids, which we encountered in chapter~2 (\S\ref{ch_newtonian_fluid} p.\pageref{ch_newtonian_fluid}), shear efforts are simply proportional to the rate of strain; thus, the shear component of eq.~\ref{eq_cauchy} can be re-expressed usefully.

The Navier-Stokes equation is the Cauchy equation (eq.~\ref{eq_cauchy}) applied to Newtonian fluids. In Newtonian fluids, which we encountered in \chapterfiveshort (\S\ref{ch_newtonian_fluid} p.\pageref{ch_newtonian_fluid}), shear efforts are simply proportional to the rate of strain; thus, the shear component of eq.~\ref{eq_cauchy} can be re-expressed usefully.

We had seen with eq.~\ref{eq_shear_velocity_gradient} p.\pageref{eq_shear_velocity_gradient} that the norm $||\vec\tau_{ij}||$ of shear component in direction~$j$ along a surface perpendicular to~$i$ depended on the viscosity and the velocity:

This is a one-dimensional (scalar) equation. Unfortunately, it does not translate easily into three dimensions. The required vector algebra far exceeds our level for this course, and we are interested only in the result (the derivation of equation in Cartesian coordinates is covered in Anderson~\cite{anderson1995} and Versteeg \& Malalasekra \cite{versteegetal2007}, and the vector form can be found in Batchelor~\cite{batchelor1967}). We obtain the heavy-handed result, in the form of a three-dimensional vector:

This is a one-dimensional (scalar) equation. Unfortunately, it does not translate easily into three dimensions. The required vector algebra far exceeds our level for this course, and we are interested only in the result (the derivation of this equation in Cartesian coordinates is covered in Anderson~\cite{anderson1995} and Versteeg \& Malalasekra \cite{versteegetal2007}, and the vector form can be found in Batchelor~\cite{batchelor1967}). We obtain the heavy-handed result, in the form of a (three-dimensional) vector field:

%This equation has several shortcomings, most importantly because the term $\dot q_\text{radiation}$ is not expressed in terms of fluid properties, and because $\mu$ is typically not independent of the temperature $T$.

\item In a description of an ocean flow, a \we{tidal force} may be added;

\item In a description of a large atmospheric flow, a \we{Coriolis force} may be added;

\item In a description of sap flow in a plant, forces related to \we{surface tension} may be added.

\item In a description of sap flow in a tree or plant, forces related to \we{surface tension} may be added.

\end{itemize}

Additionally, modeling some more advanced fluid phenomena requires altogether new equations, for example:

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@@ -377,7 +377,7 @@

To write a balance of mass for incompressible flow, we begin where we left off with the general mass balance (eq.~\ref{eq_continuity_der} p.\pageref{eq_continuity_der}), which we re-write here:

In this equation, we see that if $\rho$ is uniform and constant, the first term will vanish. Once this happens, $\rho$ can be simply dropped from the second term. This leaves us with the (much simpler) \vocab{mass balance equation for incompressible flow}:\pagebreak%handmade

With this new expression, we can come back to the Cauchy equation (eq.~\ref{eq_cauchy}), in which we can replace the shear term with eq.~\ref{shear_tensor_velocity}. This produces the beautifulwe co\vocab{Navier-Stokes equation for incompressible flow}:

With this new expression, we can come back to the Cauchy equation (eq.~\ref{eq_cauchy}), in which we can replace the shear term with eq.~\ref{shear_tensor_velocity}. This produces the beautiful \vocab{Navier-Stokes equation for incompressible flow}:

\begin{mdframed}

\begin{IEEEeqnarray}{rCl}

\rho\totaltimederivative{\vec V}& = &\rho\vec g - \gradient{p} + \mu\laplacian{\vec V}\label{eq_navierstokes}

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@@ -477,7 +477,7 @@

Today indeed, 150 years after it was first written, no general expression has been found for velocity or pressure fields that would solve this vector equation in the general case. Nevertheless, in this course we will use it directly:

\begin{itemize}

\item to understand and quantify the importance of key fluid flow parameters, in \chapterten;

\item to understand and quantify the importance of key fluid flow parameters, in \chapternine;

\item to find analytical solutions to flows in a few selected cases, in the other remaining chapters.

\end{itemize}

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@@ -486,7 +486,7 @@

\subsection{The Bernoulli equation (again)}

We had made clear in \chaptertwo that the Bernoulli equation was very limited in scope, and that it was always safer to approach a problem from the an energy equation instead (\S\ref{ch_bernoulli} p.\pageref{ch_bernoulli}). As a reminder of this fact, and as an illustration of the bridges that can be built between integral and derivative analysis, it can be instructive to derive the Bernoulli equation directly from the Navier-Stokes equation. This derivation is not difficult to follow; it is covered in Appendix~\ref{appendix_benoulli_navier_stokes} p.\pageref{appendix_benoulli_navier_stokes}.

We had made clear in \chaptertwo that the Bernoulli equation was very limited in scope, and that it was always safer to approach a problem from an energy equation instead (\S\ref{ch_bernoulli} p.\pageref{ch_bernoulli}). As a reminder of this fact, and as an illustration of the bridges that can be built between integral and derivative analysis, it can be instructive to derive the Bernoulli equation directly from the Navier-Stokes equation. This derivation is not difficult to follow; it is covered in Appendix~\ref{appendix_benoulli_navier_stokes} p.\pageref{appendix_benoulli_navier_stokes}.

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%\subsection{Principle}

In our analysis of fluid flow from a derivative perspective, our five physical principles from \S\ref{ch_conservation_equations} have been condensed into three equations (often loosely referred together to as the \textit{Navier-Stokes equations}). Out of these, the first two, for conservation of mass (\ref{eq_continuity_der_inc}) and linear momentum (\ref{eq_navierstokes}) in incompressible flows, are often enough to characterize most free flows, and should in principle be enough to find the primary unknown, which is the velocity field $\vec V$:

In our analysis of fluid flow from a derivative perspective, our five physical principles from \S\ref{ch_conservation_equations} have been condensed into three balance equations (often loosely referred together to as \textit{the Navier-Stokes equations}). Out of these, the first two, for conservation of mass (\ref{eq_continuity_der_inc}) and linear momentum (\ref{eq_navierstokes}) in incompressible flows, are often enough to characterize most free flows, and should in principle be enough to find the primary unknown, which is the velocity field $\vec V$:

\begin{IEEEeqnarray*}{rCl}

0 & = &\divergent{\vec V}\\

\rho\totaltimederivative{\vec V}& = &\rho\vec g - \gradient{p} + \mu\laplacian{\vec V}