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\fluidmechchaptertitle
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\label{chap_five}
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\mecafluboxen

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\section{Motivation}

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	\youtubethumb{BKKXJWgLwJg}{pre-lecture briefing for this chapter (back when it had a different chapter number)}{\oc (\ccby)}
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	In fluid mechanics, only three types of forces apply to fluid particles: forces due to gravity, pressure, and shear. This chapter focuses on shear, and should allow us to answer two questions:
		\begin{itemize}
			\item How is the effect of shear described and quantified?
			\item What are the shear forces generated on walls by simple flows?
		\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Shear forces on walls}

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	%%%%
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	\subsection{Magnitude of the shear force}
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		What is the force which which a fluid shears (i.e.\ “rubs”) against a wall?
		
		When the shear $\tau$ exerted is uniform and the wall is flat, the resulting force $F$ in the direction $i$ is easily calculated:
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			\begin{IEEEeqnarray}{rCcCl}
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					F_{\text{shear, direction } i} & = & \tau_{\text{uniform, direction } i} \ S_\text{flat wall}
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			\end{IEEEeqnarray}

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		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:
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			\begin{IEEEeqnarray}{rCcCl}
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					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}
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			\end{IEEEeqnarray}
			\begin{equationterms}
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				\item for a flat surface, 
				\item where the $S$-integral denotes an integration over the entire surface.
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			\end{equationterms}
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		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
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		\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}
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	\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.
		
		In a general case, the shear on any infinitesimal surface $\diff S$ needs to be expressed as a vector $\vec \tau_n$, where $n$ is the direction perpendicular to the surface. The net force due to shear on the surface is then:
		\begin{IEEEeqnarray}{rCcCl}
				\vec F_\text{shear} & = & \int_S \vec \tau_n \diff S  \label{eq_shear_force_vector}
		\end{IEEEeqnarray}
		

		Much like equation~\ref{eq_pressure_force_vector} in the previous chapter, eq.~\ref{eq_shear_force_vector} is not too hard to implement as a software algorithm to obtain numerically, for example, the force resulting from shear due to fluid flow around a body such as the body of a car. Its computation by hand, however, is far too tedious for us to even attempt.
		
		The position of the shear force is obtained with two moment vector equations, in a manner similar to that described in \S\ref{ch_position_pressure_force} p.\pageref{ch_position_pressure_force} with pressure. This is outside of the scope of this course.
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\section{Shear fields in fluids}
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		We approached the concept of shear in \chapterone with the notion that it represented force parallel to a given flat surface (eq.~\ref{eq_first_def_shear}), for example a flat plate of area $A$:
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			\begin{IEEEeqnarray}{rCl}
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				\tau &\equiv& \frac{F_\parallel}{A} \label{eq_first_def_shear_two}
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			\end{IEEEeqnarray}
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		Like we did with pressure, to appreciate the concept of shear in fluid mechanics, we need to go beyond this equation.
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	\subsection{The direction of shear}
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		Already from the definition in eq.~\ref{eq_first_def_shear_two} we can appreciate that “parallel to a flat plate” can mean a multitude of different directions, and so that we need more than one dimension to represent shear. Furthermore, much in the same way as we did for pressure, we do away with the flat plate and accept that shear is a \vocab{field}, i.e. it is an effort applying not only upon solid objects but also upon and within fluids themselves. We replace eq.~\ref{eq_first_def_shear_two} with a more general definition:
		\begin{IEEEeqnarray}{rCl}
			\vec \tau &\equiv& \lim_{A \to 0} \frac{\vec F_\parallel}{A} \label{eq_def_shear}
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		\end{IEEEeqnarray}
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		\youtubetopthumb{LjWeYPEmCk8}{cloud movements in a time\--lapse video on an interesting day are evidence of a highly\--strained atmosphere: pilots and meteorologists refer to this as \vocab{wind shear}.}{Y:StormsFishingNMore (\styl)}
		Contrary to pressure, shear is not a scalar, i.e. it can (and often does) take different values in different directions. At a given \emph{point} in space we represent it as a vector $\vec \tau = \left(\tau_x, \tau_y, \tau_y\right)$, and in a fluid, there is a shear \vocab{vector field}:
		\begin{IEEEeqnarray}{rCl}
		\vec \tau_{(x, y, z, t)} &\equiv& \left(\begin{array}{c}
								\tau_x\\
								\tau_y\\
								\tau_z
								\end{array}\right)_{(x, y, z, t)}
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		\end{IEEEeqnarray}
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	\subsection{Shear on an infinitesimal volume}
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		Describing the changes in space of the shear vector field requires another mathematical dimension (called \vocab{order}).\\
		Instead of a flat plate, let us consider an infinitesimally small cube within the fluid (\cref{fig_tau_cube}). Because the cube is immersed inside a vector field, the shear vector exerting on each of its six faces may be different.
			\begin{figure}
				\begin{center}
					\includegraphics[width=\textwidth]{particle_shear_tensor}
				\end{center}
				\supercaption{Shear efforts on a cubic fluid particle (with only the efforts on the visible faces 1 to 3 represented). The shear tensor $\vec \tau_{ij}$ has six members of three components each.}{\wcfile{Shear stress infinitesimal volume element.svg}{Figure} \cczero \oc}
				\label{fig_tau_cube}
			\end{figure}
		
		In order to express the efforts on any given face, we express a component of shear with two subscripts, the first indicating the direction normal to the surface of interest, and the second indicating the direction of the effort. For example, $\vec \tau_{xy}$ represents the shear in the $y$-direction on a surface perpendicular to the $x$-direction. On this face, the shear vector would be:
		\begin{IEEEeqnarray}{rCl}
			\vec \tau_{xj} 	&=& \vec \tau_{xx} + \vec \tau_{xy} + \vec \tau_{xz} \label{eq_shear_x}\\
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							&=& \tau_{xx} \vec i + \tau_{xy} \vec j + \tau_{xz} \vec k 
		\end{IEEEeqnarray}
		\begin{equationterms}
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			\item where the subscript $xj$ indicates all of the directions ($j = x, y, z$) on a face perpendicular to the $x$-direction.
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		\end{equationterms}
		
		In eq.~\ref{eq_shear_x}, the reader may be surprised to see the term $\tau_{xx}$ appear — a shear effort perpendicular to the surface of interest. This is because the faces of the infinitesimal cube studied here (shown in \cref{fig_tau_cube}) are not solid. They are permeable, and the local velocity on each one may (in fact, must, if there is to be any flow) include a component of velocity through the face of the cube. Thus, there is no reason for the shear effort, which is three-dimensional, to be aligned along each flat surface. As the fluid travels across any face, it can be sheared (which results in strain) in any arbitrary direction, regardless of the local pressure — and thus shear can and most often does have a component ($\tau_{ii}$) perpendicular to an arbitrary surface inside a fluid.
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		Now, the net shear effect on the cube will have \emph{eighteen} components: one tree-dimensional vector for each of the six faces. Each of those components may take a different value. The net shear could perhaps be represented as en entity —a \vocab{tensor}— containing six vectors $\vec \tau_1, \vec \tau_2, \vec \tau_3\vec \tau_6$. By convention, however, shear is notated using only three vector components: one for each pair of faces. Shear efforts on a volume are thus represented with a \vocab{tensor field} $\vec \tau_{ij}$:
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			\begin{IEEEeqnarray}{rCl}
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				\vec \tau_{ij} 	&\equiv& \left(\begin{array}{c}%
						\vec \tau_{xj} \\
						\vec \tau_{yj} \\
						\vec \tau_{zj} %
						\end{array}\right)%
					\equiv \left(\begin{array}{c}%
							\vec \tau_{xj\ \{1,4\}} \\
							\vec \tau_{yj\ \{2,5\}} \\
							\vec \tau_{zj\ \{3,6\}} %
						\end{array}\right)\nonumber\\
				\vec \tau_{ij} &\equiv& \left(\begin{array}{ccc}%
						\tau_{xx} & \tau_{xy} & \tau_{xz} \\
						\tau_{yx} & \tau_{yy} & \tau_{yz} \\
						\tau_{zx} & \tau_{zy} & \tau_{zz}%
					\end{array}\right)\label{eq_def_shear_tensor}
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			\end{IEEEeqnarray}
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		In this last equation~\ref{eq_def_shear_tensor}, each of the nine components of the tensor acts as the container for two contributions: one for each of the two faces perpendicular to the direction expressed in its first subscript.
		
		So much for the shear \emph{effort} on an element of fluid. What about the net \emph{force} due to shear on the fluid element? Not every element counts:  part of the shear will accelerate (change the velocity vector) the particle, while part of it will merely strain (deform) the particle. Quantifying this force thus requires making a careful selection within the eighteen components of $\vec \tau_{ij}$. We may start with the $x$-direction, which consists of the sum of the component of shear in the $x$-direction on each of the six cube faces:
			\begin{IEEEeqnarray}{rCl}
				\vec F_{\text{shear}\ x} 	&=&  S_3 \vec \tau_{zx\ 3} - S_6 \vec \tau_{zx\ 6}\nonumber\\
											&& + S_2 \vec \tau_{yx\ 2} - S_5 \vec \tau_{yx\ 5}\nonumber\\
											&& + S_1 \vec \tau_{xx\ 1} - S_4 \vec \tau_{xx\ 4}
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			\end{IEEEeqnarray}
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		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:
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			\begin{IEEEeqnarray}{rCl}
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				\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}
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			\end{IEEEeqnarray}
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		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:
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			\begin{IEEEeqnarray}{rCl}
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				\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\\
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											&=&  \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}
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			\end{IEEEeqnarray}
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		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}\\
				\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}
		we can re-write eq.~\ref{eq_shear_force_x_tmp} and see that the net shear force in the $x$-direction is equal to the particle volume times the divergent of the shear in the $x$-direction:
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			\begin{IEEEeqnarray}{rCl}
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				\vec F_{\text{shear}\ x} 	&=&  \diff \vol \ \divergent{\vec \tau_{ix}} \label{eq_fshear_xdir_divergent}
			\end{IEEEeqnarray}
		
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		The $y$- and $z$-direction are taken care of in the same fashion, so that we can gather up our puzzle pieces and express \emph{the force per unit volume due to shear as the divergent of the shear tensor}:
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			\begin{IEEEeqnarray}{rCcCcCl}
				\vec F_{\text{shear}} 	&=&\left(\begin{array}{c}%
						F_{\text{shear}\ x} \\
						F_{\text{shear}\ y} \\
						F_{\text{shear}\ z} %
						\end{array}\right)%
					&=& \diff \vol \left(\begin{array}{c}%
						|\divergent{\vec \tau_{ix}}| \\
						|\divergent{\vec \tau_{iy}}| \\
						|\divergent{\vec \tau_{iz}}| %
						\end{array}\right)%
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					&=& \diff \vol \ \divergent{\vec \tau_{ij}}
			\end{IEEEeqnarray}
			\begin{mdframed}
			\begin{IEEEeqnarray}{rCl}			
				\frac{1}{\diff \vol} \vec F_\text{net, shear} & = & \divergent{\vec \tau_{ij}}\label{eq_shear_force_divergent_shear}
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			\end{IEEEeqnarray}
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			\end{mdframed}
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		This is more than we really need to go through the problems in this chapter, but we will come back to it when we start concerning ourselves with the dynamics of fluid particles in \chaptersix, where the divergent of shear will make part of the glorious \vocab{Cauchy equation}. It suffices for now to sum up our findings as follows:
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			\begin{itemize}
				\item Shear at a point in space has three components — it is a vector field;
				\item The effect of shear on a volume of fluid has eighteen components – it is a second-order tensor field;
				\item The net force due to shear on a volume of fluid, expressed using the divergent of the shear tensor, has three components — it is a vector field.
			\end{itemize}
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	\subsection{The no-slip condition}
	\label{ch_no_slip_condition}
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		We observe that whenever we measure the velocity of a fluid flow along a solid wall, the speed tends to zero as we approach the wall surface. In other words, the fluid “\textit{sticks}” to the surface regardless of the overall faraway flow velocity. This phenomenon is called the \vocab{no-slip condition} and is of paramount importance in fluid mechanics. One consequence of this is that fluid flows near walls are dominated by viscous effects (internal friction) due to the large velocity gradients.
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	\subsection{Viscosity}
	
		When a solid wall is moved longitudinally within a fluid, the fluid generates an opposing friction force through viscous effects (\cref{fig_shear_velocity_gradient}). We call \vocab{viscosity} (or sometimes “\vocab{dynamic viscosity}”) $\mu$ the ratio between the fluid velocity gradient and the shear effort. For example, the norm of the shear $\vec \tau_{xy}$ on a surface perpendicular to the $x$-direction, in the $y$-direction, can be expressed as:
			\begin{IEEEeqnarray*}{rCcCl}
				||\vec \tau_{xy}|| 	&=& \mu \partialderivative{V_y}{x} &=& \mu \partialderivative{v}{x}
			\end{IEEEeqnarray*}
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		\begin{figure}[ht]
			\begin{center}
				%\vspace{-0.5cm}
				\includegraphics[width=\textwidth]{shear_velocity_gradient}
				%\vspace{-1cm}
			\end{center}
			\supercaption{Any velocity gradient $\partialderivative{V_y}{x} = \partialderivative{v}{x}$ in the flow results in a shear force $F_\parallel$ in the direction~$y$. The ratio between the shear and the velocity gradient is called \vocab{viscosity}.}{\wcfile{Concept of shear in fluid.svg}{Figure} \cczero \oc}\vspace{-0.5cm}%handmade
			\label{fig_shear_velocity_gradient}
		\end{figure}
		
		In the general case, viscosity is defined as the (scalar) ratio between the norm of shear and the corresponding strain rate. The strain rate corresponding to the shear in the $j$-direction is the rate of change in the $i$-direction of the velocity in the $j$-direction ($\inlinepartialderivative{V_j}{i}$):
			\begin{IEEEeqnarray*}{rCl}
				\mu		&\equiv& \frac{||\vec \tau_{ij}||}{\left(\partialderivative{V_j}{i}\right)}
			\end{IEEEeqnarray*}
			\begin{mdframed}
			\begin{IEEEeqnarray}{rCl}
				||\vec \tau_{ij}|| 	&=& \mu \partialderivative{V_j}{i} \label{eq_shear_velocity_gradient}
			\end{IEEEeqnarray}
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			\begin{equationterms}
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				\item in which the subscript $i$ is an arbitrary direction ($x$, $y$ or $z$) and $j$ is the direction following it in order (e.g.\ $j=z$ when $i=y$);
				\item and where $\mu$ is the viscosity (or “dynamic viscosity”) (\si{\pascal\second}).
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			\end{equationterms}
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			\end{mdframed}
		
		Viscosity $\mu$ is measured in \si{\pascal\second}, which is the same as~\si{\newton\second\per\metre\squared} or~\si{\kilogram\per\metre\per\second}. It has historically been measured in \si{poise} ($\SI{1}{poise} \equiv \SI{0,1}{\pascal\second}$).
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		Sometimes, the concept of \vocab{kinematic viscosity} is used. Kinematic viscosity is written $\nu$: the Greek letter \textit{nu}, an unfortunate choice because it is easy to mis-read as the $y$-component of velocity, $v \equiv V_y$. Kinematic viscosity $\nu$ is defined as
		\begin{equation}
			\nu \equiv \frac{\mu}{\rho}
		\end{equation}
		\begin{equationterms}
			\item where $\nu$ is measured in \si{\metre\squared\per\second}.
		\end{equationterms}
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	\subsection{Newtonian Fluid}
	\label{ch_newtonian_fluid}
	
		Fluids for which $\mu$ is independent from $\inlinepartialderivative{V_j}{i}$ are called \vocab{Newtonian fluids}.
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		Most fluids of interest in engineering fluid mechanics (air, water, exhaust gases, pure gases) can be safely modeled as Newtonian fluids. Their viscosity~$\mu$ varies slightly with pressure (a dependency which we ignore) and mildly with temperature (an effect we take into account by reading values in a diagram).
		
		The values of viscosity vary very strongly from one fluid to another: for example, honey is roughly ten thousand times more viscous than water, which is roughly a hundred times more viscous than ambient air. The viscosities of various fluids are quantified in \cref{fig_viscosities_various_fluids} p.\pageref{fig_viscosities_various_fluids}.
		
		Oil-based paint, blood and jelly-based fluids are strongly non-Newtonian; they require more complex viscosity models (\cref{fig_viscosity_characteristics}).
		\begin{figure}
			\begin{center}
				\includegraphics[width=10cm]{viscosity_characteristics}
			\end{center}
			\supercaption{Various possible viscosity characteristics of fluids. Those for which $\mu$ is independent of $\partial V_j/\partial i$ are called \vocab{Newtonian}.}{\wcfile{Fluid viscosity relationships (rheology) 2.svg}{Figure} \cczero \oc}
			\label{fig_viscosity_characteristics}
		\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Special case: shear in simple laminar flows}
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	In any ordinary fluid flow, the velocity field is complex, and it is difficult to express shear and its net effect on particles. Since this requires expressing three values at each point in (three-dimensional) space and time, this would require complex mathematics or large amounts of discrete data.
	
	In simple cases, however, it is possible to express and calculate shear relatively easily. This is especially true in simple, steady, laminar (smooth) flows —typically flows for which the Reynolds number (eq.\ref{eq_def_reynolds_number} p.\pageref{eq_def_reynolds_number}) is low.
	
	In those cases, we can \emph{guess} a reasonably realistic velocity distribution, and then derive an expression for the distribution of shear from it.
	
	One such classical example is the \vocab{Couette flow}, where fluid is prisoner between a static flat surface and another flat surface moving parallel to it, as illustrated in figure~\ref{fig_couette_flow}. In this case, the bottom wall and top wall velocities, as well as the spacing~$H$, are known.
		\begin{figure}[ht]
			\begin{center}
				\includegraphics[width=0.6\textwidth]{couette_flow}
			\end{center}
			\supercaption{A simple flow. The bottom wall is stationary, while the top wall slides from left to right. In between the walls, fluid is strained uniformly.}{\wcfile{Couette flow.svg}{Figure} \ccbysa \wcu{Kulmalukko}}
			\label{fig_couette_flow}
		\end{figure}

	A reasonable guess for the velocity distribution in steady laminar regime is:
	\begin{IEEEeqnarray*}{rCl}
		\left\{ \begin{array}{rcl}
			V_x &=& V_\text{bottom wall} + k y\\
			V_y &=& 0
		\end{array}\right.
	\end{IEEEeqnarray*}
	By applying boundary conditions ($V_\text{bottom wall} = 0$ and $V_{x\ @\ y=H} = V_\text{top wall}$) we can re-write this as:
	\begin{IEEEeqnarray*}{rCl}
		\left\{ \begin{array}{rcl}
			V_x &=& 0 + \frac{V_\text{top wall}}{H} y\\
			V_y &=& 0
		\end{array}\right.
	\end{IEEEeqnarray*}

	And now that the velocity field is known, the shear everywhere in the fluid can be computed. The shear in the $x$-direction is proportional to the derivative in the $y$-direction of the velocity in the $x$-direction:
	\begin{IEEEeqnarray*}{rCl}
		\tau_{yx} 	&=& \mu \derivative{}{y} \left(0 + \frac{V_\text{top wall}}{H} y \right)\\
					&=& \frac{\mu \ V_\text{top wall}}{H}
	\end{IEEEeqnarray*}
	Thus, we see here that the shear applied in the fluid is the same everywhere (it is independent of $y$ and $x$). A few slightly more complex cases are waiting for us in the problem sheet; but to handle more realistic shear distributions, what is needed is a software able to compute the behavior of fluids. The basic but formidable equations to be solved for this are the topic of the upcoming \chaptersix.
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\atendofchapternotes