chap5.tex 22.1 KB
 Olivier committed Mar 31, 2019 1 \renewcommand{\lastedityear}{2019}  Olivier committed May 10, 2019 2 3  \renewcommand{\lasteditmonth}{05} \renewcommand{\lasteditday}{10}  Olivier committed Mar 18, 2019 4 \renewcommand{\numberofthischapter}{5}  Olivier committed Mar 19, 2019 5 \renewcommand{\titleofthischapter}{\namechapterfive}  Olivier committed Apr 03, 2015 6   Olivier committed Dec 12, 2015 7 \fluidmechchaptertitle  Olivier committed Mar 19, 2019 8 \label{chap_five}  Olivier committed Apr 03, 2015 9 10 11  \mecafluboxen  Olivier committed Mar 17, 2019 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Apr 03, 2015 13 14 \section{Motivation}  Olivier committed Mar 31, 2019 15  \youtubethumb{BKKXJWgLwJg}{pre-lecture briefing for this chapter (back when it had a different chapter number)}{\oc (\ccby)}  Olivier committed Mar 17, 2019 16 17 18 19 20  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}  Olivier committed May 15, 2015 21 22   Olivier committed Mar 17, 2019 23 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Mar 18, 2019 24 25 \section{Shear forces on walls}  Olivier committed May 10, 2019 26  %%%%  Olivier committed Mar 31, 2019 27  \subsection{Magnitude of the shear force}  Olivier committed Mar 18, 2019 28   Olivier committed Mar 31, 2019 29 30 31  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:  Olivier committed Mar 18, 2019 32  \begin{IEEEeqnarray}{rCcCl}  Olivier committed Mar 31, 2019 33  F_{\text{shear, direction } i} & = & \tau_{\text{uniform, direction } i} \ S_\text{flat wall}  Olivier committed Mar 18, 2019 34 35  \end{IEEEeqnarray}  Olivier committed Mar 31, 2019 36  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:  Olivier committed Mar 18, 2019 37  \begin{IEEEeqnarray}{rCcCl}  Olivier committed May 10, 2019 38  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}  Olivier committed Mar 18, 2019 39 40  \end{IEEEeqnarray} \begin{equationterms}  Olivier committed Mar 31, 2019 41 42  \item for a flat surface, \item where the $S$-integral denotes an integration over the entire surface.  Olivier committed Mar 18, 2019 43  \end{equationterms}  Olivier committed Mar 31, 2019 44   Olivier committed May 10, 2019 45  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  Olivier committed Mar 31, 2019 46 47 48 49 50  \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}  Olivier committed May 10, 2019 51 52  %%%%  Olivier committed Mar 31, 2019 53 54 55 56 57 58 59 60 61 62 63 64 65  \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.  Olivier committed Mar 18, 2019 66 67 68 69  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Shear fields in fluids}  Olivier committed May 15, 2015 70   Olivier committed Mar 31, 2019 71  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 $A$:  Olivier committed May 15, 2015 72  \begin{IEEEeqnarray}{rCl}  Olivier committed Mar 17, 2019 73  \tau &\equiv& \frac{F_\parallel}{A} \label{eq_first_def_shear_two}  Olivier committed May 15, 2015 74  \end{IEEEeqnarray}  Olivier committed May 13, 2016 75   Olivier committed Mar 17, 2019 76  Like we did with pressure, to appreciate the concept of shear in fluid mechanics, we need to go beyond this equation.  Olivier committed May 15, 2015 77 78   Olivier committed Mar 17, 2019 79  \subsection{The direction of shear}  Olivier committed Apr 03, 2015 80   Olivier committed Mar 17, 2019 81 82 83  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}  Olivier committed May 10, 2019 84  \end{IEEEeqnarray}  Olivier committed Apr 03, 2015 85   Olivier committed Mar 17, 2019 86 87 88 89 90 91 92 93  \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)}  Olivier committed May 10, 2019 94  \end{IEEEeqnarray}  Olivier committed Apr 03, 2015 95   Olivier committed Mar 17, 2019 96  \subsection{Shear on an infinitesimal volume}  Olivier committed May 17, 2015 97   Olivier committed Mar 17, 2019 98 99 100 101 102 103 104 105 106 107 108 109 110  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}\\  Olivier committed May 10, 2019 111 112 113  &=& \tau_{xx} \vec i + \tau_{xy} \vec j + \tau_{xz} \vec k \end{IEEEeqnarray} \begin{equationterms}  Olivier committed Mar 17, 2019 114  \item where the subscript $xj$ indicates all of the directions ($j = x, y, z$) on a face perpendicular to the $x$-direction.  Olivier committed May 10, 2019 115 116 117  \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.  Olivier committed Mar 17, 2019 118 119  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} $\vec \tau_{ij}$:  Olivier committed May 15, 2015 120  \begin{IEEEeqnarray}{rCl}  Olivier committed Mar 17, 2019 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135  \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}  Olivier committed May 15, 2015 136  \end{IEEEeqnarray}  Olivier committed Mar 17, 2019 137 138 139 140 141 142 143 144  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}  Olivier committed May 15, 2015 145  \end{IEEEeqnarray}  Olivier committed Mar 17, 2019 146  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:  Olivier committed May 15, 2015 147  \begin{IEEEeqnarray}{rCl}  Olivier committed May 10, 2019 148 149 150  \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}  Olivier committed May 15, 2015 151  \end{IEEEeqnarray}  Olivier committed Mar 31, 2019 152  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:  Olivier committed May 15, 2015 153  \begin{IEEEeqnarray}{rCl}  Olivier committed May 10, 2019 154  \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\\  Olivier committed Mar 17, 2019 155  &=& \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}  Olivier committed May 15, 2015 156  \end{IEEEeqnarray}  Olivier committed Mar 17, 2019 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173  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:  Olivier committed May 15, 2015 174  \begin{IEEEeqnarray}{rCl}  Olivier committed Mar 17, 2019 175 176 177  \vec F_{\text{shear}\ x} &=& \diff \vol \ \divergent{\vec \tau_{ix}} \label{eq_fshear_xdir_divergent} \end{IEEEeqnarray}  Olivier committed Mar 31, 2019 178  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}:  Olivier committed Mar 17, 2019 179 180 181 182 183 184 185 186 187 188 189  \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)%  Olivier committed Mar 31, 2019 190 191 192 193 194  &=& \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}  Olivier committed May 15, 2015 195  \end{IEEEeqnarray}  Olivier committed Mar 31, 2019 196  \end{mdframed}  Olivier committed Apr 03, 2015 197   Olivier committed Mar 31, 2019 198  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:  Olivier committed Mar 17, 2019 199 200 201 202 203  \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}  Olivier committed Apr 03, 2015 204   Olivier committed May 15, 2015 205   Olivier committed Apr 03, 2015 206   Olivier committed Mar 31, 2019 207 208  \subsection{The no-slip condition} \label{ch_no_slip_condition}  Olivier committed May 17, 2015 209   Olivier committed Mar 31, 2019 210  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.  Olivier committed Apr 03, 2015 211   Olivier committed Mar 31, 2019 212 213 214 215 216 217  \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*}  Olivier committed Mar 17, 2019 218   Olivier committed Mar 31, 2019 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236  \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}  Olivier committed Mar 17, 2019 237  \begin{equationterms}  Olivier committed Mar 31, 2019 238 239  \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}).  Olivier committed Mar 17, 2019 240  \end{equationterms}  Olivier committed Mar 31, 2019 241 242 243  \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} ($\SI{1}{poise} \equiv \SI{0,1}{\pascal\second}$).  Olivier committed Apr 03, 2015 244   Olivier committed Mar 31, 2019 245 246 247 248 249 250 251 252  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 \nu \equiv \frac{\mu}{\rho} \begin{equationterms} \item where $\nu$ is measured in \si{\metre\squared\per\second}. \end{equationterms}  Olivier committed Apr 03, 2015 253   Olivier committed Mar 31, 2019 254 255 256 257 258  \subsection{Newtonian Fluid} \label{ch_newtonian_fluid} Fluids for which $\mu$ is independent from $\inlinepartialderivative{V_j}{i}$ are called \vocab{Newtonian fluids}.  Olivier committed Apr 03, 2015 259   Olivier committed Mar 31, 2019 260 261 262 263 264 265 266 267 268 269 270 271  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}  Olivier committed Apr 03, 2015 272   Olivier committed May 13, 2016 273   Olivier committed Apr 03, 2015 274   Olivier committed Mar 17, 2019 275 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Olivier committed Mar 31, 2019 276 \section{Special case: shear in simple laminar flows}  Olivier committed May 15, 2015 277   Olivier committed Mar 31, 2019 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313  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.  Olivier committed Apr 03, 2015 314   Olivier committed Apr 02, 2017 315 \atendofchapternotes